noise-sustained structures due to convective instability...

Physica D 145 (2000) 191–206

Noise-sustained structures due to convectiveinstability in finite domains

M.R.E. Proctora,∗, S.M. Tobiasa, E. Knoblochba Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK

b Department of Physics, University of California, Berkeley, CA 94720, USA

Received 1 March 2000; received in revised form 29 May 2000; accepted 5 June 2000Communicated by F.H. Busse

Abstract

In large aspect ratio systems with broken reflection symmetry the onset of instability is closely related to the thresholdfor absolute instability in the corresponding unbounded system. The upstream boundary (with respect to the group velocity)plays a crucial part in fixing the frequency of the nonlinear wavetrain that results, and hence its stability properties. Incontrast in the convectively unstable regime all perturbations decay, although persistent structures can be maintained by theaddition of small amplitude noise. The upstream boundary, however distant, continues to play an essential role in frequencyselection, with the result that the structures induced by noise are of universal form. A general theory is developed that predictsthe selected frequency and wave number for both primary and secondary convective instabilities, and the results illustratedusing the complex Ginzburg–Landau equation and a mean-field dynamo model of magnetic field generation in the Sun.© 2000 Elsevier Science B.V. All rights reserved.

PACS:05.45; 47.20.K; 47.54; 95.30.Q

Keywords:Noise-sustained structures; Convective instability; Finite domain

1. Introduction

The effects of distant boundaries on the on-set of travelling wave instabilities in systemswith broken reflection symmetry differ remarkablyfrom the corresponding steady-state situation inreflection-symmetric systems. In the latter case theimposition of boundary conditions at the ends of adomain of aspect ratioL leads to a correction tothe instability threshold ofO(L−2) whenL is large.

∗ Corresponding author. Tel.:+44-1223-337913;fax: +44-1223-337918.E-mail address:[email protected] (M.R.E. Proctor).

However, in systems that lack reflection symmetrythe primary instability is always a Hopf bifurca-tion to travelling waves with a preferred direction ofpropagation, and in this case the imposition of sim-ilar boundary conditions results in anO(1) changeto the threshold for instability. Moreover, the initialeigenfunction at the onset is a wall mode rather thanwave-like, and the frequency of this mode differssubstantially from that of the most unstable, spatiallyperiodic solution. With increasing values of the in-stability parameter, hereafter calledµ, a front forms,separating an exponentially small wavetrain near theupstream boundary from a fully developed one down-stream, with a well-defined wave number, frequency

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192 M.R.E. Proctor et al. / Physica D 145 (2000) 191–206

and amplitude, whose location moves further andfurther upstream asµ continues to increase. Both thespatial wave number and the amplitude are controlledby the temporal frequency, which is in turn con-trolled byµ, and in the fully nonlinear regime by theboundary conditions as well. The resulting changesin the frequency may trigger secondary transitions toquasiperiodic and/or chaotic states. These phenomenahave been described in a recent paper by Tobias et al.[24] and the criterion for the presence of a globalunstable mode was found to be closely related tothat for absolute instability of the basic state in anunbounded domain. The secondary transitions arelikewise related to absolute instability of the primarywavetrain. Couairon and Chomaz [7,31] have foundrelated phenomena in a semi-infinite domain.

The results discussed in [24] are based on the as-sumption that no external disturbances are present,apart from a small initial disturbance needed as a seedfor the instability. However, this is not a realistic as-sumption in typical experiments, which are likely tobe affected by small external noise. If the system iscompletely stable, then all such disturbances decay atevery location, and so have little effect. It was noted byDeissler [8], however, that if the system is convectivelyunstable, so that disturbances grow in some movingframe, then even if no global mode is unstable therecan exist structures in the system that are sustainedby the noise. This conjecture has been confirmed bothin experiments [1–3,28] and in various model prob-lems based on the complex Ginzburg–Landau (CGL)equation [4,8–11,15,17,21]. In the present paper weexpand on earlier theoretical work with the aim ofmaking some quantitative prediction of the effects ofnoise. We show by means of numerical and analyt-ical calculations that the addition of noise can leadto the destabilisation not only of the trivial state butalso of the primary wavetrain. Most importantly, weshow that the travelling wave nature of the transientinstability that is sustained by noise injection leadsto a powerful frequency selection effect, which de-termines the spatio-temporal properties of the result-ing noise-sustained structures. For the model problemsanalysed here we are able to give a simple method offinding the selected frequencies from the dispersion

relation, and argue that more complex physical sys-tems will exhibit similar behaviour, since the detailedmechanism for instability does not enter into the anal-ysis. We are also able to quantify the effect of thenoise by relating its magnitude to the spatial extentof the noise-sustained structures. Some preliminaryresults can be found in [20].

We shall investigate the two models of travellingwave instability in finite domains studied in [24]. Thefirst example is provided by the CGL equation for acomplex amplitudeA(x, t), cf. [8]:

∂A

∂t= cg∂A

∂x+ µA + a|A|2A + λ∂

2A

∂x2, 0 ≤ x ≤ L

(1)

with λ = 1 + iλI , a = −1 + iaI (see Section 2).The second example, treated in Section 3, derivesfrom a simple theory of the solar magnetic field andtakes the form of a nonlinear mean-fieldα–Ω dynamodescribed by the pair of equations [19]:

∂A

∂t= DB

1 + B2 +∂2A

∂x2− A,

∂B

∂t= ∂A

∂x+ ∂

2B

∂x2− B, (2)

also with 0 ≤ x ≤ L. HereD > 0 is the dynamonumber analogous to the parameterµ of the CGLmodel, andA and B are the poloidal field potentialand the toroidal field itself. Both are real-valued func-tions. This model has the merit of describing a realmagnetic field as opposed to the CGL equation, whichonly describes the evolution of a slowly varying en-velope of a short-wave, high-frequency wavetrain. In[24] these two model systems were solved with homo-geneous boundary conditions to investigate the effectsof distant boundaries. In contrast the problems studiedhere are fundamentally inhomogeneous. We considerthe effects of noise injection at the upstream boundary.While this could be termed ‘inlet noise’ rather than‘additive noise’ it provides, after appropriate scaling,essentially the same effect as when noise is presentthroughout the domain, as noted already by Deissler[8]. This is because the inlet noise is amplified to agreater extent than noise that is added farther down-stream. In the case of the dynamo equations the noise

M.R.E. Proctor et al. / Physica D 145 (2000) 191–206 193

reflects the effects of small scale turbulence on thelarge scale dynamo instability. We choose the param-eters of our models so that the group velocity of anydisturbance is to the left, as in [24]. For our calcula-tions we set the value ofA (for the CGL and for thedynamo model) at the right (upstream) boundary ateach time step to be the value of a random variableof zero mean, uniformly distributed between±�. In afew calculations we added such a random disturbanceat every spatial mesh point, with the purpose of inves-tigating the stochastic properties of the nonlinear re-sponse. Our conclusions are summarised in Section 4.

2. The CGL equation

2.1. Linear theory

We consider first the linearised form of Eq. (1) withA(0, t) = 0 andA(L, t) = �f (t), wheref is a ran-dom variable uniformly distributed on [−1, 1] (whitenoise). In practice, we choose a sample fromf at eachtime step. We can learn about the power spectrum atother values ofx by examining the responsẽA(x) eiωt

to a sinusoidal disturbance� eiωt at x = L. The equa-tion for Ã takes the form

iωÃ = µÃ + cg∂Ã∂x

+ λ∂2Ã

∂x2(3)

and has the solution

Ã = � exp[−cg(x − L)

2λ

]sinhνx

sinhνL, (4)

where

ν2 = 1λ

(iω − µ + c

2g

4λ

). (5)

The power spectrum ofA(x, t) can now easily be con-structed, but this is in general not very illuminating.We prefer to proceed by noting that since (for the cho-senf (t)) the power in each Fourier component is thesame whenx = L, we can find an approximate so-lution at x < L by seeking the valueωmax of ω thatgives the largest response amplitude atx, which wewrite asξL (ξ < 1) to isolate the role ofL. If thereis such a clearly defined maximum, then the solution

in this region will be approximately periodic with fre-quencyωmax. If we suppose thatL � 1, thenÃ(ξL)is given to a good approximation by

Ã ≈ � exp[(cg/2λ − ν) (1 − ξ)L] (6)and so to maximise the amplitude at anyξ , we mustchooseω to maximise the spatial growth rate

κR ≡ Re( cg

2λ− ν

)= cg

2|λ|2 − νR. (7)

Only solutions withκR > 0 are of physical interest.Writing ν = p + iq, we must solve the simultaneousequations

p2 − q2 − 2λIpq = −µ +c2g

4|λ|2 , (8a)

2pq+ λI(p2 − q2) = ω −λIc

2g

4|λ|2 , (8b)

subject to the condition∂p/∂ω = 0. It then followsthatq = −λIp; substituting into the equations we findthe simple relation

ωmax = µλI (9)with the spatial growth rate given by

κR = κmax ≡ 12|λ|2

(cg ±

√c2g − 4µ|λ|2

). (10)

Both solutions satisfy the physical requirementκR >0 but only one is found to agree with simulations[8,9]. This is the solution withp > 0 and it corre-sponds to the lower sign in Eq. (10); moreover, sincep(∂2p/∂ω2) > 0 when∂p/∂ω = 0, this solution cor-responds to aminimumin p and hence to amaximumin κR (see Fig. 1a). The corresponding group velocitydq/dω is downstream[29].

There is a simple argument why this is the correctchoice of the sign ofp. This argument may well beknown but it highlights the important role played bythe downstream boundary in problems of this type.Moreover, the argument generalises to more complexsituations and this generalisation will be useful whenwe consider convective instabilities of finite amplitudewavetrains. We start with the unbounded system andlook for solutions proportional to exp(iωt − κx). The


Fig. 1. Dependence of the rootsκ of the dispersion relation on the frequencyω obtained from Eqs. (7) and (8) withcg = 1, λI = 0.45.(a) κR(ω) and (b)κI (ω), both for µ = 0.95µa, whereµa is the absolute instability threshold. (c, d) The same but forµ = 1.05µa.

resulting dispersion relation has two complex rootsκas shown in Fig. 1. Forµ < µa ≡ c2g/4|λ|2, the thresh-old for absolute instability, the correct solution corre-sponds to choosing theω thatmaximises the real partof the root with the smallest(positive) real part (seeFig. 1a). This procedure differs from the intuitive (butincorrect) idea that the mode growing most rapidly tothe left would be the one observed. This is becauseany solution of the linear problem on a bounded do-main consists of a linear combination of exponentialsof the rootsκ of the dispersion relation. The left (ordownstream) boundary makes the amplitudes of themodes there of the same order; for each value ofωthe mode seen far from either boundary is then theone that decays least rapidly to the right. We can thenselect theω which yields the most rapid leftwardgrowth calculated by this procedure. Observe that thedownstream boundary plays a fundamental role in thisargument even if the precise nature of the boundarycondition there is immaterial. This is because it turnsthe problem into an eigenvalue problem, and this is thereason why a semi-infinite system may behave verydifferently from a finite one however large it may be.

Worledge et al. [30] show that in a semi-infinite systemwith a downstream boundary the intuitive argumentis correct with the consequence that instability sets inat the convective threshold instead of the global insta-bility threshold. Such systems therefore behave likeopen systems. In contrast, semi-infinite systems withan upstream boundary behave like closed systems: inthe convective regime all perturbations are convectedout past any fixed measurement point, and observationof a sustained instability requires that the system beglobally unstable. It is precisely these modes that areselected when the boundary conditions∂A/∂x = 0 or∂2A/∂x2 = 0 are imposed at the downstream bound-ary. However, such boundary conditions do not modelthe system any better thanA = 0, as shown alreadyby Deissler [8]. In particular, the frequency and wavenumber are selected by the minimum/maximum crite-rion just described, and are not those with the fastestspatial growth rate (cf. [2,9,17]).

Some interesting properties of the result (10)should be noted. The quantityκmax is only defined forµ < µa, i.e., for µ below the threshold for absoluteinstability. The reason is made clear in Fig. 1, which


shows the behaviour of the two roots of the dispersionrelation for values ofµ both below and above thisthreshold. Below the threshold the two branches ofthe roots of the dispersion relation have disjoint realparts. The frequency can then be calculated by deter-mining the maximum of the root with thesmallerrealpart as discussed above (see Figs. 1a and b). Beyondthe absolute instability threshold, characterised by thepresence of a double root of the dispersion relation,the real parts of the roots cross and a turning pointcan no longer be found (see Fig. 1c). In addition,κmax < 0 for µ < 0, i.e., forµ below the convectivethreshold. In this case the driven response has itsgreatest amplitude at the right boundary and no finiteamplitude solutions will be seen when the amplitudeof the injected noise is small. This physically reason-able requirement constitutes an independent reasonfor selecting the root withp > 0.

2.2. Nonlinear structures

From the linear theory just described we may pre-dict that the amplitude of any disturbance driven bynoise atx = L will have an r.m.s. amplitude atx < Lgiven by� exp[κmax(L − x)]. Linear theory thereforebreaks down whenx ≈ xfront, wherexfront ≡ L − κ−1max| ln �|. (11)For x < xfront and for O(1) values ofµ not toolarge, the solution of the nonlinear problem (1) takesthe form of a finite amplitude Stokes wave with fre-quency ωmax, i.e., A = A0 exp[iΩt + ikx], whereΩ = ωmax = µλI and k is now real. Thus xfrontmeasures the extent of the nonlinear noise-sustainedstructure; the result (11) shows that at fixedµ thisquantity depends logarithmically on the noise am-plitude. Since in the convectively unstable regimeµ < µa, the equations fork andA0 (cf. Eqs. (16)–(18)of [24]) have theuniquesolution

k = 12(aI + λI)

(cg −

√c2g − 4µ(λ2I − a2I )

),

|A0|2 = 12(aI + λI)2

(4µλI(aI + λI) − c2g

+cg√

c2g − 4µ(λ2I − a2I ))

. (12)

Although this wavetrain exists for allµ in the convec-tive range, it may be unstable. We can investigate itsstability by linearising about the Stokes wave [16,22].The stability criteria are quite involved for generalvalues ofaI , λI , but if we restrict attention to the rangeof λI given by 1− λIaI > 0, λ2I < 3, the stabilityboundary (for a general Stokes wave) is given by

µ(1 − λIaI) < k2(1 − λIaI + 2|a|2). (13)The reverse condition defines the region ofconvectiveinstability. Consequently, on the convective instabilityboundary,

k = cg(1 − λIaI)2(2λI − λ2I aI + a2I λI − aI − a3I )

, (14)

and the condition for stability becomes

µ <(1 − λIaI)(2a2I − aIλI + 3)c2g

4(2λI − λ2I aI + a2I λI − aI − a3I )2. (15)

In Fig. 2 we show the resulting stability region in the(µ, λI) plane for various values ofaI (note that wecan takeaI ≥ 0 without loss of generality).

It should be noted from this figure that for allvalues of aI , λI such that 1− λIaI > 0 (i.e., inthe Benjamin–Feir stable regime), the noise-inducedwavetrain is stable at onset (µ ≈ 0) but is alwaysconvectively unstable atµa = c2g/4|λ|2 regardless ofthe value ofλI as soon asaI ≥ 1.23. This meansthat in sufficiently long domains it may be possibleto observe, forµ < µa, the production ofsecondarystructures that, like the primary one, owe their ex-istence entirely to the noise. This possibility in theBenjamin–Feir unstable regime is considered byDeissler [9], although his argument that his wave-train can only be convectively unstable is, like that ofOuyang and Flesselles [18], incorrect (see [25]). Infact, the presence of the ‘fully developed region’ inthe simulations could be due toeither noise ampli-fication by the convective instability of the primarywavetrain (as suggested by Deissler), or the manifes-tation of an absolute instability of the (noise-induced)primary wavetrain, with the position of the secondaryfront controlled by the upstream boundary. The lattersituation arises in the absence of noise when the pri-mary wavetrain becomes unstable to a global mode


Fig. 2. Convective stability boundaries (thin lines) of the noise-sustained Stokes solution of the CGL equation in the(µ/c2g, λI ) plane forthe following values ofaI : (a) 0.0, (b) 1.05, (c) 1.25, and (d) 2.0. Unstable regions are shaded. The heavy line shows the upper limit for theexistence of such noise-sustained solutions; in the absence of noise this boundary corresponds to the absolute instability of theA = 0 state.

(see [24]), but in this case neither the primary nor thesecondary structure would be sensitive to the additionof noise. An intermediate situation arises when theprimary wavetrain (a global mode of the system) isonly convectively unstable. In this case the primarystructure is noise-insensitive but any secondary struc-ture will be noise-induced. We discuss the nature ofthese states in the next subsection.

2.3. Noise-sustained secondary instability

In finite but large domains the global instabilitythreshold is given byµf = µa + O(L−2). As shownin [24], whenµ − µf = O(L−2) or larger, the finiteamplitude solution (in the absence of noise) takes theform of a periodic Stokes wave, except near the bound-aries, with a frequency that is determined by solvinga nonlinear eigenvalue problem. This solution may beconvectively unstable even at onset, i.e., forµ ≈ µf , ascan be seen from Fig. 2. As noted by Deissler [8–10],the action of additive noise on such flows will leadto secondary structures analogous in every way to theprimary solutions discussed above. It is known from

[24] that at larger values ofµ these secondary struc-tures can arise spontaneously due to absolute instabil-ity, and that they also take the form of a front separat-ing an exponentially small wave with a definite sec-ondary frequency and spatial growth rate from a fullydeveloped secondary wavetrain. This wavetrain maybe laminar or spatio-temporally chaotic depending onthe wave number that is selected by the secondaryfrequency. If the group velocity associated with thesecondary instability is leftward the front separatingthe primary and secondary wavetrains first appears atx ≈ 0 and moves, with increasingµ, to larger val-ues ofx. One might surmise, therefore, that any sec-ondary noise-generated structures might exhibit simi-lar behaviour. Fig. 3 shows that this is in fact the case.The figure shows a noise-sustained spatio-temporallychaotic secondary structure forµ = 1, � = 10−4,λI = 0.45 andaI = 2. For these parameters the sec-ondary instability boundary in the absence of noise isat µ ≈ 1.5, i.e., the chaotic structure present near theleft boundary in Fig. 3 decays away when� is set tozero. Guided by the results of [24] we expect that in itslinear phase, the secondary mode has a well-defined


Fig. 3. Noise-sustained secondary structures in the CGL equation. Space–time plot of Re(A) for µ = 1.0, � = 10−4, λI = 0.45 andaI = 2.0 with x increasing to the right and time increasing upwards. The irregular phase at theleft of the picture is sustained by noiseinput at theright boundary, and decays away in its absence.

Fig. 4. Real and imaginary parts ofq for perturbations of the Stokes solution of the CGL equation withµ = 1.2, λI = 0.45 andaI = 2.0as a function of the perturbation frequencyδ. The predicted value ofqR, qR ≈ −0.35, is given by the minimum value of the real partalong the second curve from the top nearδ = 2.


spatial growth rate, wave number and temporal fre-quency, and that these are determined by a criterionsimilar to that for the primary noise-sustained mode.This expectation is borne out by an investigation of thelinear instability of the primary wavetrain. If we lookfor disturbances proportional to exp(iδt + qx) (and itscomplex conjugate), whereδ is real butq complex,we find the complicated dispersion relation given inEqs. (29) and (30) of [24]. In that paper we lookedfor a double root of this dispersion relation in order todetermine the onset of absolute secondary instability.Typical behaviour of the roots of the relation belowthis threshold is shown in Fig. 4 as a function of thefrequencyδ.

As discussed in Section 2.1 the minimum/maximumcriterion for frequency selection applies here as well,even though we are now dealing with a secondary

Fig. 5. Comparison for various values ofµ of the theoretical pre-diction (+) of the spatial growth rate−qR for the noise-sustainedsecondary structure with the results of direct numerical simulation(1). HereλI = 0.45, aI = 2.0.

instability. This is confirmed in Fig. 5 which com-pares the resulting prediction of the growth rate−qRwith the results of a direct simulation of the equa-tions. These results lead to a general picture of the roleplayed by small amplitude noise in the CGL equation(1) that is summarised in Fig. 6. In the following sec-tion we present evidence that this picture extends tothe dynamo equations (2) as well, and use this fact tosuggest that it has in fact broad applicability.

3. Noise-sustained structures in the dynamoequations

Although the dynamo equations (2) are not as sus-ceptible to analysis as the CGL equation having nosimple rotating wave solutions, they have been ex-tensively investigated numerically (see [24] as wellas [20,23,30]). In the following we use the boundaryconditionsBx(0, t) = A(0, t) = 0, B(L, t) = 0 withA(L, t) = 0 replaced byA(L, t) = �f (t), as in theCGL case. As in that case, the precise boundary con-ditions atx = 0 have a negligible effect on the so-lution in the interior. It should be noted that Eqs. (2)contain no free parameters apart from the dynamonumberD which plays the role ofµ for this system,although the form of the nonlinearity adopted here,albeit conventional, is essentially arbitrary. It is alsoimportant to emphasise that the role of noise in thisformulation of the dynamo problem is fundamentallydifferent from previous studies of stochastic fluctua-tions in mean-field dynamos (e.g., [14]) which rely onstochastic perturbations at a marginal dynamo num-ber to produce random modulation of thekinematicsolutions.

The investigation of the linearised theory proceedsin a similar fashion to that in Section 2. For modesproportional to exp(iωt−κx) the dispersion relation is(iω+1−κ2)2+κD = 0. From this relation it followsthat the critical value ofD for convective instability isDc = 32/3

√3 corresponding toκ = κc = −i/

√3 and

ω = ωc = 43. For absolute instability the critical valueDa =

√2Dc with corresponding frequencyωa =

√3.

When Dc < D < Da we may therefore seek theselected frequency by the same procedure as for the


Fig. 6. Schematic diagram of the possible roles of noise in the CGL equation. The two scenarios presented are for (a) Benjamin–Feirstable parameter values and (b) Benjamin–Feir unstable parameter values. In (a) the role of the noise depends critically on the value ofthe bifurcation parameter, while in (b) the noise-induced primary wavetrain is always susceptible to noise-induced disruption as discussedby Deissler [9].

CGL equation. As in Section 2.1 we writeκ = p +iq and solve the dispersion relation together with thecondition∂p/∂ω = 0. In terms of∂q/∂ω ≡ 2/β theselected frequency is then given as a function ofD bythe parametric relations

q2 = 132(16− β2), ω2 = 4q2(1 + q2),

p = 16

(β +

√6 − 18β2

),

D = −β2p + 8β(p2 − q2) − 16p(p2 − 3q2). (16)


Fig. 7. Noise-sustained dynamo waves: selected frequencyω as a function of the dynamo numberD. The solid line is the boundary ofthe region in which wavetrains occur. The dashed line shows the theoretical prediction from Eq. (16). The symbols show the results ofnumerical simulations with different values ofD and noise level�. Dotted lines show thresholds for convective and absolute instabilityof the trivial state.

Note that, for reasons discussed in Section 2.1, wehave chosen this time the positive sign of the squareroot in the expression forp.

The selected frequencyω(D) is shown in Fig. 7 to-gether with the curve delimiting the region in whichuniform wavetrains occur. This curve joins the points(Dc, ωc) and(Da, ωa). Also plotted are numerical datafrom a direct simulation of Eqs. (2) withL = 300;these follow closely the theoretical prediction. Notethat the noise level does not affect the realised valueof ω, since the frequency selection mechanism is lin-ear. Fig. 8 compares instead the spatial growth ratefrom the simulations with the predictionp obtainedfrom Eqs. (16). As expected, this rate drops to zero atD = Dc, and has an infinite gradient atD = Da,where the dispersion relation has a double root.

The effect of the noise can be seen more clearlyin Figs. 9–11, which show snapshots of bothB andln B as functions ofx for several values ofD and

noise levels�. For these three figures only, the dy-namo numberD is not constant, but falls rapidly tozero atL ≈ 270. Thus theeffectivevalues of� shownin the figure areO(10−9) lower than the values given.As predicted, the linear growth region is characterisedby a slope depending only onD, and not on�. Fur-thermore, the nonlinear part of the structure has awell-defined amplitude and (spatial) period that is re-lated to its temporal frequency in a way analogousto that for the CGL equation, although no analyti-cal form of the relation is available. While there aresome long-timescale fluctuations in the amplitude, thisis only to be expected given the stochastic nature ofthe forcing and the (inferred) convective instabilityof the nonlinear wavetrain. It will be noticed that forthe largest value ofD shown, namely 8.70, the ef-fect of the noise on the spatial extent of the struc-ture is reduced. This is because the system is al-most at the global linear stability boundary, beyond


Fig. 8. Noise-sustained dynamo waves. As for Fig. 7 but showing the comparison as a function ofD between the spatial growth rateobtained from simulations with several different noise levels� and its theoretical valuep.

which the noise would in any event have little ef-fect.

For supercritical values ofD, secondary struc-tures induced by noise can also be found. No theoryis available, however, since the nonlinear primarywavetrain is not a simple sinusoidal function of time.Nonetheless, Fig. 12 indicates that these secondarystructures do indeed have a well-defined secondaryfrequency, wave number and amplitude, making themdirect analogues of the CGL results, with the addedfeature that the secondary structures do not breakdown into spatio-temporal chaos nearx = 0. Onceagain, it is hard to distinguish these solutions withnoise-sustained secondary modes from those that ap-pear when the primary mode is globally unstable,as described in [24]. Note, however, that the extentof the secondary noise-sustained structures is muchmore sensitive to the noise amplitude�. We surmisethat this is because for these structures the noise en-ters in effect parametrically: in addition to its rolein sustaining the secondary structure, the noise nowalso alters the basic state which is itself noisy, as

well as shifting the threshold for secondary instabilityby changing the selected wave number of the basicwavetrain through changes to the selected frequency.All these effects make the secondary structures muchmore noise-sensitive than the primary ones.

4. Discussion

In this paper we have investigated the effects of ad-ditive (inlet) noise on two particular instability prob-lems, which may be taken as prototypes of a widevariety of systems exhibiting instabilities to travellingwaves with a preferred direction of propagation. Insuch systems there is a distinction between the abil-ity of the system to support transiently growing wavepackets (convective instability) and an absolute insta-bility that produces a persistent finite amplitude stateat every point in the laboratory frame. As discussedin detail in [24] this distinction carries over to finitebut extended systems. In the present paper we haveshown that the behaviour of such extended systems


Fig. 9. Noise-sustained dynamo waves. Left column: instantaneous plots ofB as a function ofx for D = 7.20 and various noise levels�.Right column: the same but for ln|B|. Note that the slope in the latter is independent of� at least for sufficiently small�.

in the regime between the convective and absolute in-stability thresholds depends sensitively on the amountof external noise that may be present. For very longdomains for which the difference between the con-vective and global instability criteria is most marked,a small disturbance level in the convectively unstableregime can lead to a finite amplitude, almost periodicresponse, which greatly resembles the global modes inthe absence of noise. This strong frequency selectionmay have important consequences for understandingthe phase stability of the solar magnetic cycle. Thisselection mechanism is, however, very different fromthat operating in the globally unstable regime. In the

latter case the selected frequency solves a nonlineareigenvalue problem. In contrast, the frequency char-acterising a noise-sustained structure is selected by alinear mechanism and maximises the minimum spa-tial growth rate. As a consequence the selected fre-quency is independent of the noise level�, as is thespatial growth rate. Both depend only on the distancefrom the convective instability threshold. In contrast,the spatial extent of the noise-sustained structure de-pends linearly on| ln �|. In addition, we have shownthat the wavetrains present in the absolutely unstableregime may themselves exhibit convective instabilitythat generates secondary structures which are likewise


Fig. 10. Noise-sustained dynamo waves. As for Fig. 9 but withD = 8.0.

sustained by the noise. These also resemble the globalmodes that set in at the threshold for secondary abso-lute instability but require noise for their presence be-low this threshold. An example of such a structure inthe context of spiral wave breakdown is described byTobias and Knobloch [25]. While the qualitative as-pects of these noise-driven responses were discussedalready by Deissler [8–10], we have given quantitativepredictions concerning onset criteria, spatial and tem-poral periods, and the spatial extent of the responseas functions of both the parameters and of the noiselevel.

When noise is added instead at every (spatial) meshpoint, the same frequency selection process remains

in evidence, but with significantly larger fluctuationsin the amplitude. In this case the determination of theultimate statistically steady state requires substantiallylonger computations, the results of which will be re-ported elsewhere.

In some ways the problems we discuss bear a con-siderable similarity to ‘non-normal’ bifurcation prob-lems [26,27], which are characterised by long transientgrowth phases even when there is ultimate exponen-tial decay. Trefethen and coworkers have shown howsome nonlinear dynamical systems, with non-normallinear dynamics, can be very sensitive to noise. Indeed,the non-normality of the linear dynamo problem wasdiscussed extensively by Farrell and Ioannou [12,13]


Fig. 11. Noise-sustained dynamo waves. As for Fig. 9 but withD = 8.70. SinceD is very close toDa the noise has a very small effecton the structure.

who demonstrated that noise can lead to very largelinear responses even when the trivial state is stableaccording to normal mode analysis. Thus finite thoughsmall perturbations can be amplified substantially,and may as a result be attracted to a finite amplitudesubcritical solution if such a solution is present. Weemphasise here that the states we find, although fullynonlinear, are not subcritical; they do not correspondto any nonlinear noise-free solution and owe theirexistence solely to the presence of noise. We havenot addressed the interesting question of what hap-pens when the initial instability is indeed subcritical(see [5,6]). Moreover the convective instability gives

a spatial dependence to the role of the noise that islargely absent from the work on non-normal matrices.

The results of this paper, although based on twomodel equations, suggest that the phenomena we havedescribed are relevant to a large class of problemsinvolving extended domains and instability to unidi-rectional travelling waves. Although the experimentsreferred to in Section 1 have identified unambigu-ously primary noise-sustained structures, there appearat present to be no observations of noise-inducedsecondary structures of the type described here.An experimental observation of such structures anda verification of the frequency selection criterion


Fig. 12. Noise-sustained secondary instabilities for dynamo waves. As for Fig. 9 but withD = 11.0.


established here would therefore be of considerableinterest.

Acknowledgements

The work of EK was supported in part bythe National Science Foundation under GrantDMS-9703684. SMT is grateful to Trinity College,Cambridge, for a Research Fellowship.

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