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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
0167-2789/00/$ – see front matter © 2000 Elsevier Science B.V.
All rights reserved.PII: S0167-2789(00)00127-5
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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
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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
iωà = µÃ + cg∂Ã∂x
+ λ∂2Ã
∂x2(3)
and has the solution
à = � 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
à ≈ � 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
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194 M.R.E. Proctor et al. / Physica D 145 (2000) 191–206
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
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M.R.E. Proctor et al. / Physica D 145 (2000) 191–206 195
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
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196 M.R.E. Proctor et al. / Physica D 145 (2000) 191–206
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
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M.R.E. Proctor et al. / Physica D 145 (2000) 191–206 197
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.
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198 M.R.E. Proctor et al. / Physica D 145 (2000) 191–206
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
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M.R.E. Proctor et al. / Physica D 145 (2000) 191–206 199
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)
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200 M.R.E. Proctor et al. / Physica D 145 (2000) 191–206
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
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M.R.E. Proctor et al. / Physica D 145 (2000) 191–206 201
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
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202 M.R.E. Proctor et al. / Physica D 145 (2000) 191–206
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
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M.R.E. Proctor et al. / Physica D 145 (2000) 191–206 203
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]
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204 M.R.E. Proctor et al. / Physica D 145 (2000) 191–206
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
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M.R.E. Proctor et al. / Physica D 145 (2000) 191–206 205
Fig. 12. Noise-sustained secondary instabilities for dynamo
waves. As for Fig. 9 but withD = 11.0.
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206 M.R.E. Proctor et al. / Physica D 145 (2000) 191–206
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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