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Parametric instability of the helical dynamo
Marine Peyrot1,2, Franck Plunian1,2, Christiane Normand3∗
1Laboratoire de G´ eophysique Interne et Tectonophysique,
CNRS, Universit´ e Joseph Fourier, Maison des G´ eosciences,
B.P. 53, 38041 Grenoble Cedex 9, France
2Laboratoire des Ecoulements G´ eophysiques et Industriels,
CNRS, Universit´ e Joseph Fourier, INPG,
B.P. 53, 38041 Grenoble Cedex 9, France
3Service de Physique Th´ eorique, CEA/DSM/SPhT,
CNRS/URA 2306, CEA/Saclay, 91191 Gif-sur-Yvette Cedex, France
(Dated: February 9, 2008)
We study the dynamo threshold of a helical flow made of a mean (stationary)
plus a fluctuating part. Two flow geometries are studied, either (i) solid body or
(ii) smooth. Two well-known resonant dynamo conditions, elaborated for stationary
helical flows in the limit of large magnetic Reynolds numbers, are tested against lower
magnetic Reynolds numbers and for fluctuating flows (zero mean). For a flow made
of a mean plus a fluctuating part the dynamo threshold depends on the frequency
and the strength of the fluctuation. The resonant dynamo conditions applied on the
fluctuating (resp. mean) part seems to be a good diagnostic to predict the existence
of a dynamo threshold when the fluctuation level is high (resp. low).
PACS numbers: 47.65.+a
arXiv:physics/0703216v1 [physics.flu-dyn] 23 Mar 2007
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I. INTRODUCTION
In the context of recent dynamo experiments [1–3], an important question is to identify the
relevant physical parameters which control the dynamo threshold and eventually minimize
it. In addition to the parameters usually considered, like the geometry of the mean flow
[4, 5] or the magnetic boundary conditions [6, 7], the turbulent fluctuations of the flow seem
to have an important influence on the dynamo threshold [8–11]. Some recent experimental
results [12, 13] suggest that the large spatial scales of these fluctuations could play a decisive
role.
In this paper we consider a flow of large spatial scale, fluctuating periodically in time,
such that its geometry at some given time is helical. Such helical flows have been identified
to produce dynamo action [14, 15]. Their efficiency has been studied in the context of fast
dynamo theory [16–21] and they have led to the realization of several dynamo experiments
[3, 22–24].
The dynamo mechanism of a helical dynamo is of stretch-diffuse type. The radial com-
ponent Br of the magnetic field is stretched to produce a helical field (0, Bθ,Bz), where
(r,θ,z) are the cylindrical coordinates. The magnetic diffusion of the azimuthal component
Bθproduces some radial component Brdue to the cylindrical geometry of the problem [17].
In this paper we shall consider two cases, depending on the type of flow shear necessary for
the Brstretching.
In case (i) the helical flow is solid body for r < 1 and at rest for r > 1 (the same
conductivity is assumed in both domains). The flow shear is then infinite and localized at
∗Electronicaddress:
Marine.Peyrot@ujf-grenoble.fr,Franck.Plunian@ujf-grenoble.fr,
Christiane.Normand@cea.fr
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the discontinuity surface r = 1. Gilbert [17] has shown that this dynamo is fast (positive
growth rate in the limit of large magnetic Reynolds number) and thus very efficient to
generate a helical magnetic field of same pitch as the flow. In case (ii) the helical flow is
continuous, and equal to zero for r ≥ 1. The flow shear is then finite at any point. Gilbert
[17] has shown that such a smooth helical flow is a slow dynamo and that the dynamo action
is localized at a resonant layer r = r0such that 0 < r0< 1. Contrary to case (i), having a
conducting external medium is here not necessary.
In both cases some resonant conditions leading to dynamo action have been derived [16–
18, 20, 21]. Such resonant conditions can be achieved by choosing an appropriate geometry of
the helical flow, like changing its geometrical pitch. They have been derived for a stationary
flow U(r,θ,z) and can be generalized to a time-dependent flow of the form?U(r,θ,z) · f(t)
where f(t) is a periodic function of time. Now taking a flow composed of a mean part U plus
a fluctuating part?U · f(t), we expect the dynamo threshold to depend on the geometry of
each part of the flow accordingly to the resonant condition of each of them and to the ratio
of the intensities |?U|/|U|. However we shall see that in some cases even a small intensity of
the fluctuating part may have a drastic influence. The results also depend on the frequency
of f(t).
The Ponomarenko dynamo (case(i)) fluctuating periodically in time and with a fluctua-
tion of infinitesimal magnitude had already been the object of a perturbative approach [25].
Here we consider a fluctuation of arbitrary magnitude. Comparing our results for a small
fluctuation magnitude with those obtained with the perturbative approach, we found signif-
icant differences. Then we realized that there was an error in the computation of the results
published in [25](though the perturbative development in itself is correct). In Appendix
VIE we give a corrigendum of these results.
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II. MODEL
We consider a dimensionless flow defined in cylindrical coordinates (r,θ,z) by
U = (0,rΩ(r,t),V (r,t)) · h(r) withh(r) =
1, r < 1
0, r > 1
, (1)
corresponding to a helical flow in a cylindrical cavity which is infinite in the z-direction, the
external medium being at rest. Each component, azimuthal and vertical, of the dimensionless
velocity is defined as the sum of a stationary part and of a fluctuating part
Ω(r,t) =
?
Rm+?Rmf(t)
?
ξ(r),V (r,t) =
?
RmΓ +?Rm?Γf(t)
?
ζ(r) (2)
where Rmand Γ (resp.?Rmand?Γ) are the magnetic Reynolds number and a characteristic
pitch of the stationary (resp. fluctuating) part of the flow. In what follows we consider a
fluctuation periodic in time, in the form f(t) = cos(ωft). Depending on the radial profiles
of the functions ξ and ζ we determine two cases (i) solid body and (ii) smooth flow
(i)ξ = ζ = 1 (3)
(ii)ξ = 1 − r, ζ = 1 − r2. (4)
We note here that the magnetic Reynolds numbers are defined with the maximum angular
velocity (either mean or fluctuating part) and the radius of the moving cylinder. Thinking
of an experiment, it would not be sufficient to minimize the magnitude of the azimuthal
flow. In particular if Γ is large (considering a steady flow for simplicity), one would have
to spend too many megawatts in forcing the z-velocity. Therefore the reader interested in
linking our results to experiments should bear in mind that our magnetic Reynolds number
is not totally adequate for it. A better definition of the magnetic Reynolds number could
?
be for exampleˆRm= Rm
1 + Γ
2. For a stationary flow of type (i), the minimum dynamo
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thresholdˆRmis obtained for Γ = 1.3.
Both cases (i) and (ii) differ in the conductivity of the external medium r > 1. In case
(i) the magnetic generation being in a cylindrical layer in the neighbourhood of r = 1, a
conducting external medium is necessary for dynamo action. For simplicity we choose the
same conductivity as the inner fluid. In the other hand, in case (ii) the magnetic generation
being within the fluid, a conducting external medium is not necessary for dynamo action,
thus we choose an isolating external medium. Though the choice of the conductivity of the
external medium is far from being insignificant for a dynamo experiment [3, 4, 6, 7], we
expect that it does not change the overall meaning of the results given below.
We define the magnitude ratio of the fluctuation to the mean flow by ρ =?Rm/Rm. For
ρ = 0 there is no fluctuation and the dynamo threshold is given by Rm. In the other hand
for ρ ? 1 the fluctuation dominates and the relevant quantity to determine the threshold
is?Rm= ρRm. The perturbative approach of Normand [25] correspond to ρ ? 1.
The magnetic field must satisfy the induction equation
∂B
∂t
= ∇ × (U × B) + ∇2B. (5)
where the dimensionless time t is given in units of the magnetic diffusion time, implying
that the flow frequency ωfis also a dimensionless quantity. As the velocity does not depend
on θ nor z, each magnetic mode in θ and z is independent from the others. Therefore we
can look for a solution in the form
B(r,t) = expi(mθ + kz)b(r,t) (6)
where m and k are the azimuthal and vertical wave numbers of the field. The solenoidality
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of the field ∇ · B = 0 then leads to
br
r+ b?
r+ im
rbθ+ ikbz= 0.(7)
With the new variables b±= br± ibθ, the induction equation can be written in the form
∂b±
∂t
+ [k2+ i(mΩ + kV )h(r)]b±= ±i
2rΩ?h(r)(b++ b−) + L±b±, (8)
with
L±=
∂2
∂r2+1
r
∂
∂r−(m ± 1)2
r2
, (9)
except in case (ii) where in the external domain r > 1, as it is non conducting, the induction
equation takes the form
?L±− k2?b±= 0. (10)
At the interface r = 1, both B and the z-component of the electric field E = ∇×B−U×B
are continuous. The continuity of Brand Bθimply that of b±. The continuity of B and (7)
imply the continuity of b?
rwhich, combined with the continuity of Ezimplies
[Db±]1−
1+±iΩr=1−
2
(b++ b−)r=1= 0 (11)
with D = ∂/∂r and [h]1−
1+= h(r=1−)− h(r=1+). We note that in case (ii) as Ωr=1− = 0, (11)
implies the continuity of Db±at r = 1.
In summary, we calculate for both cases (i) and (ii) the growth rate
γ = γ(m,k,Γ,?Γ,Rm,?Rm,ωf) (12)
of the kinematic dynamo problem and look for the dynamo threshold (either Rmor?Rm)
such that the real part ?γ of γ is zero. In our numerical simulations we shall take m = 1
for it leads to the lowest dynamo threshold.
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A.Case (i): Solid body flow
In case (i) we set
mΩ + kV = Rmµ +?Rm? µf(t),
and (8) changes into
withµ = m + kΓ and
? µ = m + k?Γ, (13)
∂b±
∂t
+ [k2+ i(Rmµ +?Rm? µf(t))h(r)]b±= L±b±. (14)
For mathematical convenience, we take ? µ = 0 . Then the non stationary part of the velocity
does not occur in (14) any more. It occurs only in the expression of the boundary conditions
(11) that can be written in the form
[Db±]1−
1+±i
2(Rm+?Rmf(t))(b++ b−)r=1= 0. (15)
Taking ? µ = 0 corresponds to a pitch of the magnetic field equal to the pitch of the
fluctuating part of the flow −m/k =?Γ. In the other hand it is not necessarily equal to
the pitch of the mean flow (except if Γ =?Γ). In addition we shall consider two situations
depending on whether the mean flow is zero (Rm= 0) or not. The method used to solve
the equations (14) and (15) is given in Appendix VIA.
At this stage we can make two remarks. First, according to boundary layer theory results
[16, 17] and for a stationary flow, in the limit of large Rmthe magnetic field which has the
highest growth rate satisfies µ ≈ 0. This resonant condition means that the pitch of the
magnetic field is roughly equal to the pitch of the flow. We shall see in section IIIA that
this stays true even at the dynamo threshold. Though the case of a fluctuating flow of type
?U·f(t) may be more complex with possibly skin effect, the resonant condition is presumably
analogous, writing ? µ ≈ 0. This means that setting ? µ = 0 implies that if the fluctuations
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are sufficiently large (ρ ? 1), dynamo action is always possible. This is indeed what will
be found in our results. In other words, setting ? µ = 0, we cannot tackle the situation of a
stationary dynamo flow to which a fluctuation acting against the dynamo would be added.
This aspect will be studied with the smooth flow (ii).
Our second remark is about the effect of a phase lag between the azimuthal and vertical
components of the flow fluctuation. Though we did not study the effect of an arbitrary
phase lag we can predict the effect of an out-of-phase lag. This would correspond to take a
negative value of?Γ. Solving numerically the equations (14) and (15) for the stationary flow
and m = 1, we find that dynamo action is possible only if kΓ < 0. For the fluctuating flow
with zero mean, m = 1 and ? µ = 0 necessarily implies that k?Γ = −1. Let us now consider a
flow containing both a stationary and a fluctuating part. Setting?Γ < 0 necessarily implies
that k > 0. Then for Γ > 0, the stationary flow is not a dynamo. Therefore in that case we
expect the dynamo threshold to decrease for increasing ρ. For Γ < 0, together with?Γ < 0
and k > 0, it is equivalent to take?Γ > 0 and Γ > 0 for k < 0 and it is then covered by our
subsequent results.
B. Case (ii): Smooth flow
For the case (ii) we can directly apply the resonant condition made up for a stationary
flow [17, 18], to the case of a fluctuating flow. For given m and k, the magnetic field is
generated in a resonant layer r = r0 where the magnetic field lines are aligned with the
shear and thus minimize the magnetic field diffusion. This surface is determined by the
following relation [17, 18]
mΩ?(r0) + kV?(r0) = 0. (16)
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The resonant condition is satisfied if the resonant surface is embedded within the fluid
0 < r0< 1. (17)
As Ω and V depend on time, this condition may only be satisfied at discrete times. This
implies successive periods of growth and damping, the dynamo threshold corresponding
to a zero mean growth rate. We can also define two distinct resonant surfaces r0and ? r0
corresponding to the mean and fluctuating part of the flow,
mΩ
?(r0) + kV
?(r0) = 0,m?Ω?(? r0(t),t) + k?V?(? r0(t),t) = 0 (18)
with appropriate definition of Ω,V ,?Ω and?V . In addition, if?Ω and?V have the same time
dependency, as in (2), then ? r0becomes time independent. Then we can predict two different
behaviours of the dynamo threshold versus the fluctuation rate ρ =?Rm/Rm. If 0 < r0< 1
and ? r0> 1 then the dynamo threshold will increase with ρ. In this case the fluctuation is
harmful to dynamo action. In the other hand if 0 < ? r0< 1 then the dynamo threshold will
decrease with ρ.
From the definitions (18) and for a flow defined by (1), (2) and (4) we have
r0= −(m/k)/(2Γ)and
? r0= −(m/k)/(2?Γ). (19)
For m = 1 and k < 0, taking?Γ < 0 implies ? r0< 0 and then the impossibility of dynamo
action for the fluctuating part of the flow. Therefore, we expect that the addition of a
fluctuating flow with an out-of-phase lag between its vertical and azimuthal components
will necessarily be harmful to dynamo action. This will be confirmed numerically in section
IIIC.
To solve (8), (10) and (11), we used a Galerkin approximation method in which the
trial and weighting functions are chosen in such a way that the resolution of the induction
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equation is reduced to the conducting domain r ≤ 1 [5]. The method of resolution is given
in Appendix VID. For the time resolution we used a Runge-Kutta scheme of order 4.
III. RESULTS
A. Stationary flow (?Rm= 0)
We solve
?γ(m = 1, k, Γ, 0, Rm, 0, 0) = 0(20)
with k = (µ−1)/Γ for case (i) and k = −1/(2r0Γ) for case (ii). In figure 1 the threshold Rm
and the field frequency ?(γ) are plotted versus µ (resp. r0) for case (i) (resp. (ii)), and for
different values of Γ. Though we do not know how these curves asymptote, and though the
range of µ (resp. r0) for which dynamo action occurs changes with Γ, it is likely that the
resonant condition |µ| < 1 (resp. 0 < r0< 1) is fulfilled for the range of Γ corresponding
to a dynamo experiment (Γ ≈ 1). In case (i) the dispersion relation (32) in Appendix VIA
becomes F0= 0.
B. Periodic flow with zero mean (Rm= 0)
We solve
?γ(m = 1, k, 0,?Γ, 0,
?Rm, ωf) = 0. (21)
In figure 2 the threshold?Rmis plotted versus ωffor both cases (i) and (ii). In both cases we
take ? µ = 0 corresponding to k = −1/?Γ. For the case (ii) it implies from (19) that ? r0= 1/2,
meaning that the resonant surface is embedded in the fluid and then favourable to dynamo
action. In each case (i)?Γ = 1;1.78 and (ii)?Γ = 1;2, we observe two regimes, one at low
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Rm
0
20
40
60
80
100
-1 -0,50 0,51
0.5
0.8
10
4
2
1.3
1
?(p)
0
20
40
60
80
100
-1-0,50 0,51
10
1.3
1
2
4
0.5
0.8
µµ
(i)(i)
Rm
50
70
90
110
130
150
01
0.5
0.8
1
1.3
2
4
10
?(p)
0
20
40
60
80
100
01
r0
r0
(ii) (ii)
FIG. 1: The dynamo threshold Rm(left column) and ?(γ) (right column) versus (i) µ, (ii) r0, for
the stationary case, m = 1 and Γ = 0.5;0.8;1;1.3;2;4;10 .
frequencies for which the threshold does not depend on ωfand the other at high frequencies
for which the threshold behaves like?Rm∝ ω3/4
To understand the existence of these two regimes, we pay attention to the time evolution of
f.
the magnetic field for different frequencies ωf. In figure 3, the time evolution of b−(real and
imaginary parts) for case (ii)?Γ = 1 (case (c) in figure 2) is plotted for several frequencies
ωf.
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?Rm
10
100
1000
10000
0,1110 1001000
(a)
(b)
(c)
(d)
ωf
FIG. 2: Dynamo threshold?Rmversus ωffor case (i) with ? µ = 0 and (a)?Γ = 1.78, (b)?Γ = 1; for
case (ii) with ? r0= 0.5 and (c)?Γ = 2, (d)?Γ = 1.
1.Low frequency regime
For low frequencies (ωf = 1), we observe two time scales : periodic phases of growth
and decrease of the field, with a time scale equal to the period of the flow as expected by
Floquet’s theory. In addition the field has an eigen-frequency much higher than ωf. In fact
the slow phases of growth and decrease seem to occur every half period of the flow. This
can be understood from the following remarks.
First of all the growth (or decrease) of the field does not depend on the sign of the flow.
Indeed, from (8), we show that if b±(m,k) is a solution for (Ω, V ), then its complex conjugate
b±∗(m,k) is a solution for (−Ω, −V ). Therefore we have b±(t + T/2) = b±∗(t) where
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-20
0
20
01
-10
0
10
01
-5
0
5
01
-5
0
5
012
-4
0
4
01234
FIG. 3: Time evolution of ?(b−) (solid lines) and ?(b−) (dotted lines) for several values of ωf
(from top to bottom ωf= 1;2;5;10;100), for case (ii) with?Γ = 1. Time unity corresponds here to
2π/ωf.
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T = 2π/ωf is the period of the flow. Now from Floquet’s theory (see Appendix VIA),
we may write b(r,t) in the form b(r,τ)exp(γt), with b(r,τ) 2π-periodic in τ = ωft. This
implies that changing (Ω, V ) in (−Ω, −V ) changes the sign of ?(γ). This is consistent with
the fact that for given m and k, the direction of propagation of B changes with the direction
of the flow. Therefore changing the sign of the flow changes the sign of propagation of the
field but does not change the magnetic energy, neither the dynamo threshold?Rmwhich are
then identical from one half-period of the flow to another. This means that the dynamo
threshold does not change if we consider f(t) = |cos(ωft)| instead of cos(ωft). It is then
sufficient to concentrate on one half-period of the flow like for example [
π
2ωf,
3π
2ωf] (modulo
π).
The second remark uses the fact that the flow geometry that we consider does not change in
time (only the flow magnitude changes). For such a geometry we can calculate the dynamo
threshold Rmcorresponding to the stationary case. Then coming back to the fluctuating
flow, we understand that?Rm|f(t)| > Rm(resp.?Rm|f(t)| < Rm) corresponds to a growing
(resp. decreasing) phase of the field. Assuming that the dynamo threshold?Rmis given by
the time average < . > of the flow magnitude leads to the following estimation for?Rm
?Rm≈π
as < |cos(ωft)| >= 2/π. For the three cases (a), (b) and (c) in figure 2 we give in table
IIIB1 the ratio 2?Rm/πRmwhich is found to be always close to unity. In this interpretation
of the results the frequency ωfdoes not appear, provided that it is sufficiently weak in order
2Rm
(22)
that the successive phases of growth and decrease have sufficient time to occur. This can
explain why for low frequencies in figure 2 the dynamo threshold?Rmdoes not depend on
ωf.
Finally the frequencies ω of the stationary case for Γ =?Γ are also reported in table IIIB1.
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?RmRm
2e Rm
πRm
ω
(a) 2113 1.03 4.4
(b) 33211 3.1
(c) 143 84 1.08 28.8
(d) 170 100 1.08 33
TABLE I: see in the text.
For a geometry identical to case (c), we find, in the stationary case, ω = 33 which indeed
corresponds to the eigen-frequency of the field occurring in figure 3 for ωf= 1. The previous
remarks assume that the flow frequency is sufficiently small compared to the eigen-frequency
of the field, in order to have successive phases of growth and decrease of the field. We can
check that the values of ω given in table IIIB1 are indeed reasonable estimations of the
transition frequencies between the low and high frequency regimes in figure 2.
2. High frequency regime
In case (ii) and for high frequencies (figure 3, ωf= 100), the signal is made of harmonics
without growing nor decreasing phases. We note that the eigen-frequency of the real and
imaginary parts of b−are different, the one being twice the other.
In case (i), relying on the resolution of equations (14) and (15) given in Appendix VIA, we
can show that?Rm∝ ω3/4
the case (ii) can emerge from an approximate 3 × 3 matrix system. As these developments
f. We also find that some double frequency as found in figure 3 for
necessitate the notations introduced in Appendix VIA, they are postponed in Appendix
VIB.
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3.Further comments about the ability for fluctuating flows to sustain dynamo action
We found and explained how a fluctuating flow (zero mean) can act as a dynamo. We
also understood why the dynamo threshold for a fluctuating flow is higher than that for
a stationary flow with the same geometry. It is because the time-average of the velocity
norm of the fluctuating flow is on the mean lower than that of the stationary flow. This can
be compensated with other definitions of the magnetic Reynolds number. Our definition is
based on maxt|Ω(r,t)|. An other definition based on < |Ω(r,t)| >twould exactly compensate
the difference.
Recently a controversy appeared about the difficulty for a fluctuating flow (zero mean) to
sustain dynamo action at low Pm[26], whereas a mean flow (non-zero time average) exhibits
a finite threshold at low Pm[10, 27] (the magnetic Prandtl number, Pm, being defined as
the ratio of the viscosity to the diffusivity of the fluid). This issue is important not only
for dynamo experiments but also for natural objects like the Earth’s inner-core or the solar
convective zone in which the electro-conducting fluid is characterized by a low Pm. Though
we did not study this problem, our results suggest that the dynamo threshold should not
be much different between fluctuating and mean flows, provided an appropriate definition
of the magnetic Reynolds number is taken. In that case why does it seem so difficult to
sustain dynamo action at low Pmfor a fluctuating flow [26] whereas it seems much easier
for a mean flow [10, 27] ?
In the simulations with a mean flow [10, 27], two dynamo regimes have been found,
the one with a threshold much lower than the other. In the lowest threshold regime, the
magnetic field is generated at some infinite scale in two directions [27]. There is then an
infinite scale separation between the magnetic and the velocity field, and the dynamo action
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is probably of mean-field type and might be understood in terms of α-effect, β-effect, etc.
In that case, removing the periodic boundary conditions would cancel the scale separation
and imply the loss of the dynamo action. In the highest threshold regime, the magnetic
field is generated at a scale similar to the flow scale, the periodic boundary conditions are
forgotten and the dynamo action can not be understood in terms of an α-effect anymore.
In order to compare the mean flow results [27] with those for a fluctuating flow [26] we have
to consider only the highest threshold regime in [27], the lowest one relying on mean-field
dynamo processes due to the periodic boundary conditions and which are absent in the
fluctuating flow calculations [26].
Now when comparing the threshold of the highest threshold regime for a mean flow with
the threshold obtained for a fluctuating flow and with appropriate definitions of Rm, a strong
difference remains at low Pm[28]. A speculation made by Schekochihin et al. [29] is that
the highest threshold regime obtained for the mean flow at low Pm[27] would correspond in
fact to the large Pmresults for the fluctuating flow in [28]. Their arguments rely on the fact
that the mean flow in [27] is peaked at large scale and so spatially smooth for the generated
magnetic field. It would then belongs to the same class as the large-Pmfluctuation dynamo.
Both dynamo thresholds are found to be similar indeed, and thus the discrepancy vanishes.
Though the helical flow that we consider here is noticeably different (no chaotic trajec-
tories) it may have some consistency with the simulations at large Pmmentioned above and
at least supports the speculation by Schekochihin et al. [29].
C. Periodic flow with non zero mean
We are now interested in the case where both?Rm?= 0 and Rm?= 0. The flow is then the
sum of a non zero mean part and a fluctuating part. We have considered two approaches
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depending on which part of the flow geometry is fixed, either the mean or the fluctuating
part.
1. Γ = 1
Here we fix Γ = 1, m = 1 and k = −1 and vary?Γ, ρ and ωf for the case (ii). Then we
solve the equation
?γ(m = 1, k = −1, Γ = 1,?Γ = 1/2? r0, Rm,
to plot Rmas a function of ρ in figure 4 for values of ? r0and ωf. From (19) we have r0= 1/2
which corresponds to a mean flow geometry with a dynamo threshold about 100. The curves
?Rm= ρRm, ωf) = 0(23)
are plotted for several values of?Γ leading to values of ? r0not necessarily between 0 and 1.
We consider two fluctuation frequencies ωf = 1 and ωf = 50. We find that the dynamo
threshold Rmincreases asymptotically with ρ unless the resonant condition 0 < ? r0< 1 is
satisfied, here the curves (a), (b) and (e). For these three curves we checked that in the
limit of large ρ, Rm= O(ρ−1). For ? r0= 1/4 (curve (e)) and for ρ ≈ 1 we do not know if a
dynamo threshold exists.
2.
?Γ = 1
Here we fix?Γ = 1, m = 1 and k = −1 and vary Γ, ρ and ωf. We then solve the equation
?γ(m = 1, k = −1, Γ,?Γ, Rm,
with Γ = 1 − µ in case (i) and Γ = 1/2r0in case (ii). In figure 5, Rmis plotted versus ρ
for values of µ (resp. r0) in case (i) (resp. (ii)) and ωf. Taking?Γ = 1, m = 1 and k = −1
?Rm= ρRm, ωf) = 0(24)
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Rm
10
100
1000
0,11 10
(a)
(b)
(c)
(d)
(e)
(e)
(f)(g)
Rm
10
100
1000
0,11 10
(b)
(c)
(d)
(e)
(a)
(e)
(f)
(g)
ρρ
(ii) ωf = 50 (ii) ωf = 1
FIG. 4: Dynamo threshold Rmversus ρ for case (ii), for two frequencies ωf= 50 and ωf= 1 and
r0= 1/2 (Γ = 1, m = 1, k = −1). The different curves correspond to ? r0= (a) 1/2; (b) 2/3; (c) 1;
(d) ∞; (e) 1/4; (f) -1; (g) -1/2 (?Γ = (a) 1; (b) 0.75; (c) 0.5; (d) 0; (e) 2; (f) -0.5; (g) -1).
implies ? µ = 0 in case (i) and ? r0= 0.5 in case (ii). In both cases (i) and (ii) the fluctuating
part of the flow satisfies the resonant condition for which dynamo action is possible. This
implies that Rmshould scale as O(ρ−1) provided that ρ is sufficiently large. In each case
we consider two flow frequencies, ωf = 1 and ωf = 10 for case (i), ωf = 1 and ωf = 50
for case (ii). The curves are plotted for different values of Γ corresponding to |µ| < 1 for
case (i) and 0 < r0< 1 for case (ii). For large ρ we checked that Rm= O(ρ−1). The main
differences between the curves is that Rmversus ρ may decrease monotonically or not. In
particular in case (i) for µ = 0.4, Rmdecreases by 40% when ρ goes from 0 to 1 showing
that even a small fluctuation can strongly decrease the dynamo threshold. In most of the
curves there is a bump for ρ around unity showing a strong increase of the threshold before
the final decrease at larger ρ.
Page 20
20
Rm
0
10
20
30
40
50
60
0,11 10100
0.4
-0.3
0.2
0
Rm
0
10
20
30
40
50
60
70
80
0,11 10100
0.4
-0.3
0.2
0
ρρ
(i)ωf = 1(i)ωf = 10
Rm
0
200
400
600
800
1000
1200
0,1110 100
0.94
0.28
0.33
0.83
0.5
0.625
Rm
0
200
400
600
800
1000
0,1110100
0.94
0.28
0.33
0.83
0.5
0.625
ρρ
(ii)ωf = 1(ii)ωf = 50
FIG. 5: The dynamo threshold Rmversus ρ for k = −1, m = 1 and?Γ = 1 (? µ = 0 in case (i) and
? r0= 0.5 in case (ii)) and ωf= 1,10 or 50. The labels correspond to µ in case (i) and r0in case
(ii).
3. Γ =?Γ = 1
Here we fix Γ =?Γ = 1, m = 1 and k = −1 for the case (ii) and vary ωfand ρ. We then
solve the equation
?γ(m = 1, k = −1, Γ = 1,?Γ = 1, Rm,
?Rm= ρRm, ωf) = 0(25)
Page 21
21
Rm
0
20
40
60
80
100
120
140
0,1110 100
1
16.8
25
33.52
40
50
67
100
ρ
FIG. 6: Dynamo threshold Rmversus ρ for r0= ? r0= 0.5 (Γ =?Γ = 1). The labels correspond to
different values of ωf. The eigen-frequency for ρ = 0 is ω = 33.
to plot Rmversus ρ in figure 6 for various frequencies ωf. Taking Γ =?Γ = 1, m = 1 and
k = −1 implies r0= ? r0= 0.5 For ρ larger than 1, Rmdecreases as O(ρ−1) as mentioned
earlier. For ρ smaller than unity, Rmdecreases versus ρ monotonically only if ωf is large
enough. In fact the transition value of ωfabove which Rmdecreases monotonically is exactly
the field frequency ω (here ω = 33) corresponding to ρ = 0 . This shows that a fluctuation
of small intensity (ρ ≤ 1) helps the dynamo action only if its frequency is sufficiently high.
This is shown in Appendix VIC for the case (i). Though, the frequency above which a small
fluctuation intensity helps the dynamo may be much larger than ω. For example in case
(i) for µ = 0.4 and ? µ = 0 represented in figure 5, we have ω = 0.51. For ωf = 1 small
fluctuation helps, for ωf= 10 they do not help, and for higher frequencies they help again.
Page 22
22
IV.DISCUSSION
In this paper we studied the modification of the dynamo threshold of a stationary helical
flow by the addition of a large scale helical fluctuation. We extended a previous asymptotic
study [25] to the case of a fluctuation of arbitrary intensity (controlled by the parameter
ρ). We knew from previous studies [17, 18] that the dynamo efficiency of a helical flow is
characterized by some resonant condition at large Rm. First we verified numerically that
such resonant condition holds at lower Rmcorresponding to the dynamo threshold, for both
a stationary and a fluctuating (no mean) helical flow. Then for a helical flow made of a mean
part plus a fluctuating part we showed that, in the asymptotic cases ρ ? 1 (dominating
mean) and ρ ? 1 (dominating fluctuation), it is naturally the resonant condition of the
mean (for the first case) or the fluctuating (for the second case) part of the flow which
governs the dynamo efficiency and then the dynamo threshold. In between, for ρ of order
unity and if the resonant condition of each flow part (mean and fluctuating) is satisfied, the
threshold first increases with ρ before reaching an asymptotic behaviour in O(ρ−1). However
there is no systematic behaviour as depicted in figure 5 (i) for ωf= 1 and µ = 0.4, in which
a threshold decrease of 40% is obtained between ρ = 0 and ρ = 1. If the fluctuation part
of the flow does not satisfy the resonant condition, then the dynamo threshold increases
drastically with ρ.
Contrary to the case of a cellular flow [11], there is no systematic effect of the phase lag
between the different components of the helical flow. For the helical flow geometry it may
imply an increase or a decrease of the dynamo threshold, depending how it changes the
resonant condition mentioned above.
There is some similarity between our results and those obtained for a noisy (instead of
Page 23
23
periodic) fluctuation [8]. In particular in [8] it was found that increasing the noise level
the threshold first increases due to geometrical effects of the magnetic field lines and then
decreases at larger noise. This could explain why at ρ ≈ 1 we generally obtain a maximum
of the dynamo threshold.
Finally these results show that the optimization of a dynamo experiment depends not only
on the mean part of the flow but also on its non stationary large scale part. If the fluctuation
is not optimized then the threshold may increase drastically with (even small) ρ, ruling out
any hope of producing dynamo action. In addition, even if the fluctuation is optimized
(resonant condition satisfied by the fluctuation), our results suggest that there is generally
some increase of the dynamo threshold with ρ when ρ ≤ 1. If the geometry of the fluctuation
is identical to that of the mean part of the flow, there can be some slight decrease of the
threshold at high frequencies but this decrease is rather small. When ρ > 1 the dynamo
threshold decreases as O(ρ−1) which at first sight seems interesting. However we have to
keep in mind that as soon as ρ > 1 the driving power spent to maintain the fluctuation
is larger than that to maintain the mean flow. Then the relevant dynamo threshold is not
Rmany more, but?Rm= ρRminstead. In addition monitoring large scale fluctuations in an
experiment may not always be possible, especially if they occur from flow destabilisation.
In that case it is better to try cancelling them as was done in the VKS experiment in which
an azimuthal belt has been added [30].
V. ACKNOWLEDGMENTS
We acknowledge B. Dubrulle, F. P´ etr´ elis, R. Stepanov and A. Gilbert for fruitful discus-
sions.
Page 24
24
VI.APPENDIX
A. Resolution of equations (14) and (15) for the case (i): solid body flow
As f(t) is time-periodic of period 2π/ωf, we look for b(r,t) in the form b(r,τ)exp(γt)
with b(r,τ) being 2π-periodic in τ = ωft. Thus we look for the functions b±(r,τ) in the
form
b±(r,τ) =
?
b±
n(r)exp(inτ) (26)
where, from (14) and for ? µ = 0, the Fourier coefficients b±
[γ + k2+ i(Rmµh(r) + nωf)]b±
n(r) must satisfy
n= L±b±
n.(27)
In addition, the boundary condition (15) with f(t) = cos(τ) implies
[Db±
n]1−
1+±i
2Rm(b+
n+ b−
n)r=1±i
4
?Rm(b+
n−1+ b−
n−1+ b+
n+1+ b−
n+1)r=1= 0. (28)
The solutions of (27) which are continuous at r = 1 can be written in the form
b±
n= C±
nψ±
n, withψ±
n=
I±(qnr)/I±(qn), r < 1
K±(snr)/K±(sn), r > 1
,(29)
with
q2
n= k2+ γ + i(Rmµ + nωf),s2
n= k2+ γ + inωf. (30)
Substituting (29) in (28), we obtain the following system
C±
nR±
n± iRm
2(C+
n+ C−
n) ± i
?Rm
4(C+
n−1+ C−
n−1+ C+
n+1+ C−
n+1)) = 0(31)
with R±
n= qnI±?
n/I±
n− snK
?±
n/K±
nand where I±
n= Im±1(qn) and K±
n= Km±1(sn) are
modified Bessel functions of first and second kind.
Page 25
25
The system (31) implies the following matrix dispersion relation
FnCn− i
?Rm
4(R+
n− R−
n)(Cn−1+ Cn+1) = 0 (32)
with Cj= C+
j+ C−
jand
Fn= R+
nR−
n− i(Rm/2)(R+
n− R−
n). (33)
Solving the system (32) is equivalent to setting to zero the determinant of the matrix A
defined by
Ann= Fn
An n−1= An n+1= −i
?Rm
4(R+
n− R−
n) (34)
and with all other coefficients being set to zero.
B.High frequency regime for the periodic flow (i) with zero mean
Following the notation of section VIA, and considering a periodic flow (i) with zero mean,
we have µ = 0. From (30) this implies that qn= sn. Using the identity
I±?
nK±
n− K
?±
nI±
n=
1
sn
(35)
we obtain R±
n= (I±
nK±
n)−1. As Rm= 0, the equation (33) becomes Fn= R+
nR−
n. Then we
can rewrite the system (32) in the form
Cn+ i
?Rm
4(I+
nK+
n− I−
nK−
n)(Cn−1+ Cn+1) = 0.(36)
From the asymptotic behaviour of the Bessel functions for high arguments, we have
αn≡ I−
nK−
n− I+
nK+
n≈ 1/s3
n.(37)
For the high values of n these terms are negligible and in first approximation we keep in the
system (36) only the terms corresponding to n = 0,±1. This leads to a 3×3 matrix system
Page 26
26
whose determinant is
1 +(?Rm)2
16
α0(α−1+ α1) = 0.(38)
At high forcing frequencies ωfwe have s±1≈√ωf. Together with (38), it implies
?Rm≈ ω3/4
f. (39)
In addition, from the approximate 3 × 3 matrix system, the double-frequency 2ωf emerges
for n = ±1.
C. High frequency regime and small modulation amplitude for the periodic flow (i)
with non zero mean
For small amplitude modulation ρ << 1 the system (31) is truncated so as to keep the
first Fourier modes n = 0 and n = ±1. The dispersion relation :
?Rm
F0+ ρ2
4
?2
(R+
0− R−
0)
?R+
−1− R−
F−1
−1
+R+
+1− R−
F+1
+1
?
= 0 (40)
is then solved perturbatively setting Rm = R0+ δR and ω = ω0+ δω and expanding
F0(Rm,ω) to first order in δR and δω given that F0(R0,ω0) = 0 and with the constants
C∓
0= ±R±
0. The dispersion relation (40) becomes
δR∂F0
∂Rm
+ δω∂F0
∂ω
= −ρ2
?R0
4
?2
C0
?β−1
F−1
+β+1
F+1
?
(41)
with βn= R+
n− R−
n. The threshold and frequency shifts which behave like ρ2are written
δR = ρ2R2and δω = ρ2ω2. In the left-hand-side of (41) the partial derivatives are given by
∂F0
∂ω
∂F0
∂Rm
= −1
= −1
C0
?
?
(C+
0)2∂R+
0
∂ω
− (C−
0)2∂R−
0
∂ω
?
?
(42)
C0
(C+
0)2∂R+
∂Rm
0
− (C−
0)2∂R−
∂Rm
0
−i
2C0
(43)
Page 27
27
One can show that the partial derivatives of R±
0are related to integrals calculated in [25]
through the relations
∂R±
∂ω
0
≡ i
?∞
0
(Ψ±
0)2rdr
∂R±
∂Rm
0
≡ iµ
?1
0
(Ψ±
0)2rdr (44)
In the following we shall focus on the case µ = 0 and we introduce the notations
∂F0
∂Rm
= −iC0(f1+ if2),
∂F0
∂ω
= −iC0(g1+ ig2),
β−1
F−1
+β+1
F+1
= X + iY(45)
Solutions of (41) are
R2=
?R0
4
?2Xg1+ Y g2
f1g2− f2g1
ω2=
?R0
4
?2Xf1+ Y f2
f1g2− f2g1
(46)
recovering results similar to those obtained in [25] using a different approach.
We have in mind that for some values of ωfresonance can occur. An oscillating system
forced at a resonant frequency is prone to instability and a large negative threshold shift
is expected. However, inspection of (46) reveals no clear relation between the sign of R2
and the forcing frequency which appears in the quantities X and Y . We only know that
f1g2− f2g1< 0, since near the critical point (δR = Rm− R0) the denominator in (46) is
proportional to the growth rate of the dynamo driven by a steady flow. When ρ = 0, we
shall consider Eq. (41) for an imposed δR and complex values of δω = ω1+ iσ1where ω1is
the frequency shift and ?(γ) = −σ1is the growth rate, given by
σ1= δRf1g2− f2g1
g2
1+ g2
2
(47)
Above the dynamo threshold, (δR > 0) the field is amplified (?(γ) > 0) thus σ1< 0 and
f1g2− f2g1< 0.
In the high frequency limit (ωf>> ω0) expressions for X and Y can be derived explicitly
using the asymptotic behaviour of the Bessel functions for large arguments. For µ = 0 with
Page 28
28
q±1= s±1≈ (ωf±ω0)1/2(1±i)/√2 and using the asymptotic behaviour : β±1/F±1→ (s±1)−3
one gets
X + iY = −√2ω−3/2
f
(1 − i3ω0
2ωf) (48)
When µ = 0, we have also f1= 1/2 and f2= 0, leading to the expression for R2:
R2≈ −R2
0
4√2ω−3/2
f
?g1
g2
−3ω0
2ωf
?
(49)
For the wave numbers m = −k = 1, numerical calculations of g1and g2which only depend
on the critical parameters R0and ω0give g1/g2= 1.626, and thus R2< 0 when ωf→ ∞.
When µ = 0, there are several reasons to consider the particular value of the forcing : ωf=
2ω0. One of them is that for Hill or Mathieu equations it is a resonant frequency. Moreover, in
the present problem it leads to simplified calculations. In particular the asymptotic behavior
of βn/Fncan still be used for n = +1 since ωf+ ω0is large, while the approximation is no
longer valid for n = −1. Nevertheless, the mode n = −1 is remarkable since it corresponds
to s−1= s∗
0from which it follows that β−1= β∗
0and F−1= −iR0β∗
0. Finally one gets the
exact result : β−1/F−1= i/R0, which leads to
X + iY ≈
i
R0
+1
s3
1
withs3
1≈ −2
?3ω0
2
?3/2
(1 − i)(50)
For the values R0= 20.82 and ω0= 4.35 corresponding to Fig. 8 (f) one gets X = −1.5×10−2
and Y = 3.3 × 10−2. The threshold shift is
R2≈R2
0
8(1.62X + Y ) = 0.48(51)
showing that the sign of R2changes when ωf decreases from infinity to 2ω0. This result
is in qualitative agreement with the exact results reported in Fig. 8 (f) where R2= 0 for
ωf = 8.3 ? 2ω0. When the forcing frequency is exactly twice the eigen-frequency ω0we
had rather expected a large negative value of R2 on the basis it is a resonant condition
Page 29
29
for ordinary differential system under temporal modulation. In Fig. 8 (f) the maximum
negative value of R2occurs for ωf? 4ω0which cannot be explained by simple arguments.
For µ ?= 0, we have not been able to find resonant conditions like : nωf+ mω0 = 0
(n, m integers) between ωf and ω0such that ωf would be associated to a special behavior
of the threshold shift. Contrary to the Hill equation, the induction equation is a partial
differential equation with the consequence that the spatial and temporal properties of the
dynamo are not independant. The wave numbers k and m are linked to the frequencies ω0
and ωfthrough q±nand s±nwhich appears as arguments of Bessel functions having rules of
composition less trivial than trigonometric functions. Exhibiting resonant conditions implies
to find relationship between q±n, s±nand q0, s0for specific values of ωf. We have shown
above for µ = 0 that a relation of complex conjugaison exists for n = −1 when ωf = 2ω0
but we have not yet found how to generalize to other values of µ and have left this part for
a future work.
D.Resolution of case (ii): smooth flow
We define the trial functions ψ±
j= Km±1(k)Jm±1(αjr)/Jm±1(αj) where the αj are the
roots of the equation
αj
?Km+1(k)
Jm+1(αj)−Km−1(k)
Jm−1(αj)
?
+ 2kKm(k)
Jm(αj)= 0, (52)
and where J and K are respectively the Bessel functions of first kind and the modified Bessel
functions of second kind. For r ≤ 1, we look for solutions in the form
b±=
N
?
j=1
bj(t)ψ±
j(αjr) (53)
Page 30
30
where N defines the degree of truncature. For r ≥ 1 the solutions of (10) are thus of the
form
b±= Km±1(kr)
N
?
j=1
bj(t)(54)
and, from (52) these solutions satisfy the conditions (11) at the interface r = 1. To determine
the functions bj(t) it is sufficient to solve the induction equation (8) for r ≤ 1. For that
we replace the expression of b±given by (53) into the induction equation (8), in order to
determine the residual
R±=
N
?
j=1
?˙bj+ [k2+ α2
j+ i(mΩ + kV )]bj
?
ψ±
j∓i
2rΩ?
N
?
j=1
bj(ψ+
j+ ψ−
j).(55)
Then we solve the following system
?1
0
R+φ+
irdr +
?1
0
R−φ−
irdr = 0,i = 1,··· ,N(56)
where the weighting functions are defined by φ±
j= Jm±1(αjr)/Jm±1(αj). Using the orthog-
onality relation
?1
0
φ+
iψ+
jrdr +
?1
0
φ−
iψ−
jrdr = δijGij
(57)
with
Gii=Km+1(k)
J2
m+1(αi)
?1
0
J2
m+1(αir)rdr +Km−1(k)
J2
m−1(αi)
?1
0
J2
m−1(αir)rdr,(58)
we write the system (56) in the following matrix form
˙X = MX withX = (b1,··· ,bN)(59)
with
Mij= δij(k2+ α2
j) +
i
Gii
?1
?1
0
(φ+
iψ+
j+ φ−
iψ−
j)(mΩ + kV )rdr (60)
−
i
2Gii
0
(φ+
i− φ−
i)(ψ+
j+ ψ−
j)Ω?r2dr. (61)
The numerical resolution of this system is done with a fourth order Runge-Kutta time-step
scheme. We took a white noise as an initial condition for the bj.
Page 31
31
E.Corrigendum of Normand (2003) results [25]
For very small values of the fluctuation rate ρ and for an infinite shear (case (i)), a
comparison can be made between the results obtained for ? µ = 0 by the method based on
Floquet theory (see Appendix VIA) and those obtained by a perturbative approach [25]
which consists in expanding Rmand the frequency ?(p) in powers of ρ according to
?(p) = ω0+ ρω1+ ρ2ω2+ ....(62)
Rm = R0+ ρR1+ ρ2R2+ .... (63)
where R0and ω0are respectively the critical values of the Reynolds number and the fre-
quency in the case of a stationary flow. At the leading order it appears that : ω1= R1= 0.
At the next order, the expressions of R2and ω2are given in [25], however their numerical
values are not correct due to an error in their computation.
After correction, the new values of R2and ω2are given in Figure 7 for the set of parameters
considered in [25]: m = 1, k = −0.56 and Γ = 1 (µ = 0.44). The different curves correspond
to values of?Γ which are not necessarily the same than those taken in [25]. For |? µ| sufficiently
small (curves e and f), R2changes its sign twice versus the forcing frequency. We find that
R2is negative for low and high frequencies implying a dynamo threshold smaller than the
one for the stationary flow. At intermediate frequencies R2 is positive with a maximum
value, implying a dynamo threshold larger than the one for the stationary flow. For larger
values of |? µ| (curves a, b, c and d) R2is positive for all forcing frequencies with a maximum
value at a low frequency which increases with |? µ|. This implies that for |? µ| sufficiently large
the dynamo threshold is larger than the one obtained for the stationary flow as was already
mentioned in section IIIA. For?Γ = 1.78 we have ? µ = 0. In this case, we have checked that
the values of R2and ω2are in good agreement with the values of Rmand ω obtained by the
Page 32
32
R2
-10
0
10
20
30
40
50
60
70
80
0,11 10 100
(a)
(g)
(c)
(e)
(f)
(d)
(b)
ω2
-1
0
1
2
3
4
5
0,1110 100
(c)
(a)
(d)
(g)
(e)
(f)
(b)
ωf
ωf
FIG. 7: Results obtained by the perturbative approach for m = 1, k = −0.56 and µ = 0.44 (Γ = 1).
The dynamo threshold R2is plotted versus ωffor several values of ? µ = (a) 1.56; (b) 1.28; (c) 1;
(d) 0.72; (e) 0.44; (f) 0.16 and (g) 0.
R2
-10
0
10
20
30
40
50
60
70
80
90
100
0,11 10100
(b)
(a)
(c)
(d)
(e)
(f)
ω2
-5
0
5
10
15
20
0,1110 100
(b)
(a)
(c)
(d)
(e)
(f)
ωf
ωf
FIG. 8: Same as Figure 7 but for m = 1, k = −1 and µ = 0 (Γ = 1). The labels correspond to ? µ =
(a) 1.5; (b) 1.25; (c) 1; (d) 0.5; (e) -0.5 and (f) 0.
method of Appendix VIA, provided ρ ≤ 0.1.
For completeness we have also calculated the values of R2and ω2for m = 1, k = −1 and
Γ = 1, (µ = 0) as considered in the body of the paper. The results are plotted in Figure
Page 33
33
8. Qualitatively the results are in good agreement with those of Figure 7. For ? µ = 0 again,
we have checked that the values of R2and ω2are in good agreement with the values of Rm
and ω obtained by the method of Appendix VIA, provided ρ ≤ 0.1. For higher values of
the modulation amplitude ρ the relative difference between the results obtained by the two
methods can reach 10% on R2for ρ = 0.4.
Finally it must be noticed that our parameters?Γ and ωfare strictly equivalent to respec-
tively ε1and σ in [25].
[1] P. Cardin, D., D. Jault, H.-C. Nataf and J.-P. Masson, “Towards a rapidly rotating liquid
sodium dynamo experiment”, Magnetohydrodynamics 38, 177-189 (2002)
[2] M. Bourgoin, L. Mari´ e, F. P´ etr´ elis, C. Gasquet, A. Guigon, J.-B. Luciani, M. Moulin, F.
Namer, J. Burgete, A. Chiffaudel, F. Daviaud, S. Fauve, P. Odier and J.-F. Pinton, “Mag-
netohydrodynamics measurements in the von Karman sodium experiment”, Phys. Fluids 14,
3046-3058 (2002)
[3] P. Frick, V. Noskov, S. Denisov, S. Khripchenko, D. Sokoloff, R. Stepanov, A. Sukhanovsky,
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?RmRm
2e Rm
πRm
ω
(a) 2113 1.03 4.4
(b) 33 211 3.1
(c) 143 84 1.08 28.8
(d) 170 100 1.08 33
TABLE II: see in the text.
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