Page 1
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
Page 2
2
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
Page 3
3
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.
Page 4
4
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
Page 5
5
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
Page 6
6
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.
Page 7
7
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
Page 8
8
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)
Page 9
9
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
Page 10
10
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
Page 11
11
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.
Page 12
12
?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
Page 13
13
-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.
Page 14
14
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.
Page 15
15
?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.
Download full-text