The α-effect in 2D fast dynamos

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The α-effect in 2D fast dynamosAlice Courvoisier aa Department of Applied Mathematics , University of Leeds , UKPublished online: 19 May 2008.

To cite this article: Alice Courvoisier (2008) The α-effect in 2D fast dynamos, Geophysical &Astrophysical Fluid Dynamics, 102:2, 217-243, DOI: 10.1080/03091920701562095

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Geophysical and Astrophysical Fluid Dynamics,Vol. 102, No. 2, April 2008, 217–243

The a-effect in 2D fast dynamos

ALICE COURVOISIER*

Department of Applied Mathematics, University of Leeds, UK

(Received 5 March 2007; in final form 31 May 2007)

We look at the large-scale dynamo properties of spatially periodic, time dependent, helical 2Dflows of the form u(x, t)¼ (@y (x, y, t), �@x (x, y, t), � (x, y, t). These flows act as kinematicfast dynamos and are able to generate a mean magnetic field uniform and constant in the xy-plane but whose direction varies periodically along z with wavenumber k. Using Mean FieldElectrodynamics, the generation mechanism can be understood in terms of a k-dependent �-effect, which depends on the magnetic Reynolds number, Rm. We calculate this effect fordifferent motions and investigate how its limit as k! 0 depends on Rm and on the properties ofthe flows such as their spatial structure or correlation time. This work generalises earlier studiesbased on 2D steady flows to motions with time dependence.

Keywords: Magnetic fields; Fast dynamos; Alpha-effect

1. Introduction

The Sun possesses a coherent, global magnetic field, which is responsible for the 11-yearsunspot cycle. This field is believed to result from a large-scale dynamo, wherebyinductive motions within the electrically conducting solar plasma are able to generateand sustain a magnetic field on scale orders of magnitude larger than their own. Thedynamo problem is traditionally divided into two complementary steps. First, the studyof the kinematic regime, i.e. that of the amplification of a seed field, is a linear problem,which requires solving the magnetic induction equation for a prescribed velocity field.Then, the study of the dynamical regime, i.e. that of the processes leading to thesaturation of the magnetic field growth and to the maintenance of the field, requires theinclusion of the Lorentz force and to solve the full set of nonlinear MHD equations.

A large-scale dynamo process relevant to the Sun cannot yet be directly simulated oncomputers due to the extreme parameter regimes involved. Since the 1960s, progress hastherefore relied on Mean Field Electrodynamics (MFE). This turbulence closure theoryuses the linearity of the induction equation, valid in the kinematic regime, to derive aclosed equation for the mean magnetic field, in which small-scale effects areparameterised by transport coefficients (Moffatt 1978, Krause and Radler 1980).These are tensor quantities, which need to be determined in terms of the statistical

*Email: [email protected]

Geophysical and Astrophysical Fluid Dynamics

ISSN 0309-1929 print/ISSN 1029-0419 online � 2008 Taylor & Francis

http://www.tandf.co.uk/journals

DOI: 10.1080/03091920701562095

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properties of the velocity fluctuations. Some of these coefficients correspond toturbulent diffusion or advection, effects which are already present in the inductionequation itself, while the so-called �-effect introduces a new source term cruciallyresponsible for the generation of the mean field. In this paper, we are mainly concernedwith the determination of the �-effect in the kinematic regime.

To be more specific, we write here the basic equations of MFE. In the following, h�i isa suitable averaging operation and the velocity field u is incompressible and satisfieshui ¼ 0. Using the mean-field ansatz B¼hBi þ b, the induction equation (in dimension-less form)

@tB ¼ r � u� Bð Þ þ R�1m r2B, ð1Þ

yields evolution equations for both the mean and the fluctuating fields, namely

@thBi ¼ r � E þ R�1m r2hBi, ð2Þ

@tb� r � ðu� b� EÞ � R�1m r2b ¼ r � ðu� hBiÞ, ð3Þ

where E ¼ hu� bi is the mean electromotive force (e.m.f.). Equation (3) is linear in hBi,since u is independent of B in the kinematic regime. Consequently, we expect E to belinearly related to the mean field by an expression of the form

E ¼ a � hBi þ b � rhBi þ � � � , ð4Þ

where a and b are tensors that depend on the properties of the flow and on Rm.Substituting (4) into (2) yields a closed equation for the mean field hBi. The symmetricpart of the a-tensor is responsible for the �-effect and the regeneration of the large-scalefield. It is non-zero only if the velocity field lacks reflexional symmetry, e.g. if it ishelical. This effect is therefore believed to arise naturally in rotating convection, asdescribed originally by Parker (1955).

The mean field equation (2) can be solved in isolation provided the tensors a and b

are known. Analytic progress can be made using the first order smoothingapproximation (FOSA) if there is a short timescale in the system, either due torapid magnetic diffusion, i.e. if the magnetic Reynolds number Rm� 1, or due torapid decorrelations in the flow, i.e. if St� 1, where the Strouhal number St is theratio of the correlation time to the turnover time. Under these rather restrictiveassumptions, the �-effect is simply related to the helicity of the flow, hu �r �ui (seee.g. Moffatt, 1978). In the Sun however, Rm� 1 and St�O(1), which could result in avery different picture. To investigate a wider parameter range, it becomes necessary tounderstand the evolution of the small-scale field. This can be done numerically viatwo complementary methods. One is to perform detailed calculations that retain asmuch as the physics and geometry of the problem as possible; see for exampleOssendrijver et al. (2001), who attempted to simulate part of the solar convectionzone. This approach is however limited by the current computing resources. The otheris to look at simpler flow configurations, which are more amenable to systematicstudies, in order to identify the characteristics that are favourable to the generation ofa large-scale field via an �-effect. This is the approach we pursue here. We show that,even for simple velocity fields, the �-effect displays complex variations with theparameters of the flows and Rm that are not captured by the analytic approximations.

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We insist here that in its basic form, MFE is a kinematic theory, i.e. thebackreaction of the magnetic field on the flow via the Lorentz force is neglected. Thepresent study is also purely kinematic: we solve the induction equation (1) forprescribed velocity fields. Our work follows a line of study started by G.O. Roberts’pioneering paper (Roberts 1970), in which he investigates the possibility of dynamoaction in 2D flows periodic in space and time. Section 2 follows closely his analysisand describes how MFE can be applied to such motions. We also recall resultsobtained for steady flows. Among these, the Roberts flow, introduced by Roberts(1972), has been particularly studied, both numerically (see e.g. Roberts 1972, Plunianand Radler 2002, Radler et al. 2002, Radler and Brandenburg 2003) and analyticallyin the high conductivity (high Rm) regime (Childress 1979, Soward 1987). This flowacts as a slow dynamo: both the growth rate of the magnetic field and the �-effecttend to 0 as Rm!1 (Vainshtein and Zel’dovich 1972). Such field generationmechanisms, relying on diffusion, might not be sufficient to explain magnetic fields inastrophysical situations where Rm is typically huge. Klapper and Young (1995)showed that flows having the fast dynamo property that the maximum growth rateremains strictly positive as Rm!1 must display chaotic regions. To meet thiscriterion without sacrificing the simplicity of dealing with a 2D problem, Gallowayand Proctor (1992) and Otani (1993) used time-periodic motions, which proved to bevery efficient dynamos with an O(1) growth rate as Rm!1. Section 3 generalises thecalculations of the �-effect performed for the Roberts flow to these time dependentvelocity fields. We observe that the �-effect does not vanish in the high Rm limit andalso that it varies with the flow properties. Time-periodic flows correspond, however,to the regime St� 1, while for astrophysical turbulence, it is more likely that St� 1.In section 4 we therefore investigate how the �-effect and the dynamo growth rate aremodified by reducing the time correlations in the flow. Finally, we discuss the mainresults in section 5.

2. Mean-field electrodynamics for 2D periodic flows

We solve the induction equation (1) together with the solenoidal condition r �B¼ 0 fora prescribed, incompressible velocity field of the form

uðx, tÞ ¼ ð@y ðx, y, tÞ, � @x ðx, y, tÞ, � ðx, y, tÞÞ: ð5Þ

The streamfunction (x, y, t) is initially chosen to be a periodic function of space andtime designed to produce a maximally helical velocity field, hence potentially an�-effect. The analysis below closely follows that in Roberts (1970, 1972), and relatedstudies (see e.g. Radler and Brandenburg (2003) and references therein).

Flows (5) depend only on two spatial coordinates; it is thus convenient to seeksolutions of (1) that are monochromatic in z and read

Bðx, k, tÞ ¼ ReðHðx, y, k, tÞeikzþpðkÞtÞ, ð6Þ

where Re stands for the real part. The z-independent part of the magnetic field,H(x, y, k, t), has the same periodicity as the flow both in space and time. The quantityp(k)¼ �(k)þ i!(k) is the complex growth rate. The functions Hðx, y, k, tÞepðkÞt

� �k2Z

can

The �-effect in 2D fast dynamos 219

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be thought of as the Fourier coefficients of an expansion in z for the magnetic field andthe condition that the magnetic field is a real quantity then implies that

H�ðx, y, k, tÞ ¼ Hðx, y,�k, tÞ, p�ðkÞ ¼ pð�kÞ, ð7Þ

where stars indicate the complex conjugate.Using (6) as an ansatz for B, the induction equation (1) and the solenoidal condition

become

pþ R�1m k2

� �H ¼ �@tHþ r � ðu�HÞ þ ikez � ðu�HÞ þ R�1

m r2H, ð8Þ

@xHx þ @yHy þ ikHz ¼ 0, ð9Þ

where ez is the unit vector in the z-direction. Equations (8) and (9) form a 2D problem,which can be solved for different values of k and Rm.

For future reference we define

H0ðx, y, k, tÞ ¼ Hðx, y, k, tÞei!ðkÞt, ð10Þ

which corresponds to the z-independent part of the magnetic field with theexponentially growing factor taken out.

For periodic flows, it proves useful to define the averaging operators h�i and �, whichcorrespond to averaging over the spatial and the temporal periodicities of the flowrespectively, thus

huðx, y, tÞi ¼1

4�2

Z 2�

0

Z 2�

0

uðxþ �, yþ �, tÞd�d�, uðx, y, tÞ ¼1

2�

Z 2�

0

uðx, y, tþ �Þd�:

ð11Þ

Following Roberts (1970), we then define the mean magnetic field to be

Bðz, k, tÞ ¼ ReðhHieikzþpðkÞtÞ, ð12Þ

which, in this case, corresponds to a filtering in Fourier space: only the componentproportional to eikzþ p(k)t is kept. hHi is a constant and uniform vector that dependsonly on the value of k for a given flow and on Rm. The solenoidal condition (9) impliesthat hHiz ¼ 0. The mean field (12) is thus horizontal, constant and uniform onhorizontal planes with its direction varying along z with periodicity Lz¼ 2�/k.

Using MFE, the growth of the mean field (12) can be understood in term of ak-dependent �-effect, as described below.

We write H as the sum of an average and fluctuating part,Hðx, y, k, tÞ ¼ hHi þ hðx, y, k, tÞ and we consider flows with zero mean. Substitutionfor H into (8) yields evolution equations for both the mean and the fluctuating fields,namely

pþ R�1m k2

� �hHi ¼ ikez � hu� hi, ð13Þ

pþ R�1m k2

� �hþ @th� r � u� hð Þ � ikez � hu� hi � u� h

� �� R�1

m r2h

¼ hHi � r� �

uþ ikez � u� hHi� �

: ð14Þ

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equation (14) determines h if u and hHi are known. The inhomogeneity depends linearlyon hHi and so h is linear in hHi and therefore so is the mean e.m.f. ~E ¼ hu� hi.Furthermore, from equation (13), we see that ~Ez does not appear in the evolution of themean field. Since also hHiz ¼ 0, it is sufficient to study what happens in horizontalplanes and the corresponding component of the e.m.f. can be written as

~EH ¼ ~aðkÞ � hHi, ð15Þ

where ~aðkÞ is a 2� 2 tensor that depends on k, on the statistics of the velocity field andon Rm. Using (15) as an ansatz for the e.m.f. in (13), we can relate the growth rate p(k)to ~aðkÞ. The symmetries of the flow can then be used to simplify expression (15) further(see Appendix A).

To relate the k-dependent �-effect, which describes the growth of a mean field ofarbitrary lengthscale in z to the ‘‘conventional’’ �-effect defined by (4), one needs totake the limit k! 0, which corresponds to a mean field with a lengthscale large in alldirections (Soward 1994). As mentioned in the introduction, a well-studied flow of theform (5) is the Roberts flow defined by (x, y)¼ cos x� cos y. The dynamo growth rate�(k) was first computed by Roberts (1972), and asymptotic studies by Soward (1987)showed that at high Rm it is maximum for kmax� (lnRm/Rm) and goes to zero as�(kmax)� ln(lnRm)/lnRm (see table 1). Here we are concerned with the CP-flow(Galloway and Proctor 1992) and the MWþ flow (Otani 1993), which are efficient fastdynamos, with a growth rate settling down to a strictly positive limit as soon asRm� 100 for kmax¼O(1). The corresponding streamfunctions are listed in table 1 withthe flows’ dynamo properties. Plunian and Radler (2002) calculated the k-dependent�-effect for the Roberts flow and extrapolated its value for k! 0. The symmetries ofthe flow imply that ~aðkÞ is a diagonal matrix with both entries equal to ~�ðkÞ. Inagreement with the asymptotic theory of Childress (1979), they found that ~�ð0Þ � R�1=2

m

in the high conductivity limit. Studies based on other steady flows of the form (5) showthat they are unable to sustain an �-effect at high Rm (Childress and Soward 1989,Plunian et al. 1999). No such calculation has been performed so far for fast dynamosand our aim is to calculate the �-effect in the CP and MWþ flows at high values of Rm.

3. The a-effect in the CP and MWþ flows

In this section we evaluate ~aðkÞ for the CP and MWþ flows as well as the corresponding‘‘conventional’’ �-effect by investigating the limit as k! 0. In section 3.1, we recallsome of the characteristics of the flows and mention in particular the symmetry

Table 1. The flows considered and their dynamo properties. For the CP-flow, the dynamo resultscorrespond to "¼ 1.

Flows (x, y, t) kmax �(kmax)

Roberts flow cosx� cos y O(lnRm/Rm) O(ln(lnRm)/lnRm)

CP-flowffiffi32

qcosðxþ � cos tÞ þ sinðyþ � sin tÞð Þ 0.57 0.3

MWþ flow 2cos2 tcos x� 2sin2 tcos y 0.8 0.4


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properties of ~aðkÞ and of the growing z-dependent mean field. In section 3.2, we describethe numerical method used to determine ~aðkÞ and the results obtained. Finally, insection 3.3, we study the limit k! 0. We explain in more depth how the ‘‘conventional’’�-effect relates to ~aðkÞ and we calculate it using two different methods.

3.1. The properties of the flows and the shape of ~aðkÞ

The CP and MWþ flows are maximally helical, with a constant mean helicityhu � r � ui ¼ �3. They display Lagrangian chaos, as shown by the finite time Lyapunovexponents plotted in figure 1. In the CP-flow, the parameter " controls the size of thechaotic regions; the nearly integrable limit corresponding to "� 1 has been investigatedby Ponty et al. (1995).

The symmetries of the flows are detailed in Appendix A. They imply that growingmean fields can belong to two different classes following the relations

ez � hHi ¼ ChHi, r � B ¼ �kB, C ¼ i: ð16Þ

Therefore the mean field B can be a positive or a negative Beltrami wave in z dependingon whether C¼�i or C¼þi respectively. In the MWþ flow, magnetic modes with thesymmetry C¼�i dominate at low k (Courvoisier 2006). In the CP-flow, modes withC¼i have comparable growth rates and dominate in turn depending on the value of k(and "), as shown in figure 2 for "¼ 1. In the range of k investigated, the curves �(k)have two maxima, which correspond to modes with different symmetries. For lowk9 1.3, the fastest growing mode corresponds to the symmetry C¼�i but for larger k(1.39 k9 2), the fastest growing mode has the symmetry C¼ i, as shown, using (A.6),by the timeseries of the horizontal components of hH0i.

As shown in Appendix A, the symmetries of the flows imply that ~aðkÞ is the matrix ofa rotation in the xy-plane and the mean e.m.f. can be written as [see relation (A.13)]

~EHðkÞ ¼ ð�ðkÞ � ik�ðkÞÞhHi þ ððkÞ � ikðkÞÞez � hHi, ð17Þ

where �, , and � are real, even functions of k.Combining (16) and (17) gives

~EHðkÞ ¼ �ðkÞ kðkÞð Þ|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}aðkÞ

i ðkÞ � k�ðkÞð Þ|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}gðkÞ

24

35hHi ¼ ~�ðkÞhHi, ð18Þ

where a(k) and g(k) are real functions of k and ~�ðkÞ can be interpreted as a k-dependent�-effect that describes the evolution of a z-dependent mean field.

Now substituting (18) into (13), we obtain

�ðkÞ þk2

Rm¼ �aðkÞk ¼ ð��ðkÞ � kðkÞÞk, ð19Þ

!ðkÞ ¼ �gðkÞk ¼ ð�ðkÞ k�ðkÞÞk: ð20Þ

Therefore, if �aðkÞ > kR�1m , the real part of the growth rate �(k) is positive and a mean

magnetic field is generated. Moreover, if g(k) 6¼ 0, the growing field takes the form of a

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Figure 1. Finite-time Lyapunov exponents for the MWþ flow and for the CP-flow with different values of ":0.1, 0.25, 0.5, 0.75 and 1 respectively from top to bottom and left to right.


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travelling wave in the z-direction with wavespeed g(k). Relations (19) and (20) provideuseful checks on the numerical results presented below.

3.2. Numerical method and results for ~aðkÞ

We use a pseudo-spectral, 2D-code with a 2nd order timestepping scheme and anintegrating factor method to solve for the z-independent part of the field in (6), namelyfor Hðx, y, k, tÞepðkÞt. The solution rapidly settles to the fastest growing eigenfunctionand the growth rate �(k) is obtained from the timeseries of the magnetic energy using aleast-squares routine. We then take out the exponentially growing factor e�(k)t to obtainH0(x, y, k, t), defined by (10), and finally, based on relation (18), we calculate ~�ðkÞ byevaluating

hu�H0i � hH0i�

hH0i2

ð21Þ

(Radler and Brandenburg 2003). The real and imaginary parts of the cumulative time-average of (21) rapidly converge to a(k) and g(k) respectively. The results are in goodagreement with relations (19) and (20), as shown by the example in figure 3. Thedrawback of the present numerical method however is that we can easily obtainH0(x, y, k, t) but not H(x, y, k, t), which is used in (18) to define ~�ðkÞ. The functions

Figure 2. Top plot: Growth rate versus k for the CP-flow for different values of Rm (for results at highervalues of Rm, see e.g. Hughes et al. 1998). Bottom plots: real and imaginary parts of the horizontalcomponents of hH0i as a function of time for the CP-flow, for Rm¼ 64. Top: ReðhH0ixÞ (dotted line) withImðhH0iyÞ (dashed line). Bottom: ImðhH0ixÞ (dotted line) with ReðhH0iyÞ (dashed line). The left (right) plots arefor k¼ 0.6 (k¼ 1.6).

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H0(x, y, k, t) and H(x, y, k, t) are distinct when !(k) 6¼ 0. This is generally the case for theCP-flow and as a result, the quantity (21) still depends on time and, as described, weneed to perform a further time-average to obtain ~�ðkÞ (for more details, see Courvoisier2006). We note that this method can be preferred to using (19) and (20) since it givesdirect access to a(k) and g(k) and renders the interpolation towards k! 0 somewhateasier.

The results for a(k) and g(k) in the CP-flow for "¼ 1 and different values of Rm

ranging from 8 to 128 are presented in figure 4. Four main remarks can be made.First, there is a discontinuity in the curves for k� 1.3 and a(k) changes sign. The samediscontinuity is present in the curve for �(k) (see figure 2) and corresponds to thecrossing between two modes belonging to the C¼�i and C¼þi symmetry classes forlow and high k respectively. For low k, the mean field is a positive Beltrami wave, ofopposite parity to the flow, and a(k) is positive. For higher k, the mean field is anegative Beltrami wave, with the same parity as the flow, and a(k) is negative.Second, for both functions, the curves do not differ much as Rm is varied, except atsmall values of k; it is indeed expected that for a fast dynamo the dependence on Rm

Figure 3. Results for the CP-flow at Rm¼ 32. The þ signs correspond to ja(k)j (left plot) and g(k) (rightplot) as a function of k, calculated directly from relation (21). The plain line corresponds to their valuescalculated from �(k) and !(k) using relations (19) and (20).

Figure 4. a(k) (left) and g(k) (right) in the CP-flow for different values of Rm and "¼ 1.


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should weaken as Rm is increased. In particular, the curves coincide in the vicinity ofk¼ 0.57 for which the growth rate is maximised and both a(k) and g(k) seem to reachan Rm independent limit (for k¼ 0.6, a(k)� 0.5 and g(k)� 0.15 for all the values ofRm investigated). For comparison, in a slow dynamo like the Roberts flow, a(k) is adecreasing function of Rm as soon as RmO(1) (see figure 3 of Plunian and Radler2002). Third, g(k) 6¼ 0 and the growing solutions are oscillatory. They take the form oftravelling waves along the z-axis. Finally, both ja(k)j and jg(k)j depend sensitively onk and increase sharply as k! 0. As we shall see in the next section, it will be difficultto extrapolate their value at the origin using a linear interpolation.

The second and final remarks also apply to the function a(k) in the MWþ flow,plotted in figure 5. The fastest growing mode for the values of k investigated herebelongs to the C¼�i symmetry class and a(k) remains positive. For the parametersinvestigated here, p(k)2R so g(k)¼ 0.

3.3. Numerical methods and results for ~að0Þ

We now investigate the limit as k! 0 and the ‘‘conventional’’ �-effect. First, weconsider the expansions of a(k) and g(k) to second order in k in order to relate ~�ðkÞ tothe �-effect for k¼ 0 more clearly (Plunian and Radler 2002, Radler and Brandenburg2003). We obtain

aðkÞ ¼ �ð0Þ kð0Þ þOðk2Þ, gðkÞ ¼ ð0Þ � k�ð0Þ þOðk2Þ: ð22Þ

If we assume that the mean field and the mean e.m.f. depend only weakly on z, we get

Bðz, tÞ ¼Zk�1

hHiðkÞeikzþpðkÞtdk, EHðz, tÞ ¼

Zk�1

~EHðkÞeikzþpðkÞtdk: ð23Þ

Combining (17), (22) and (23) we obtain

EH ¼ �ð0ÞB|fflffl{zfflffl}ðiÞ

þ �ð0Þez � ðr �BÞ|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}ðiiÞ

þ ð0Þez � B|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}ðiiiÞ

� ð0Þr �B|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}ðivÞ

, ð24Þ

to 2nd order in k.

Figure 5. a(k) in the MWþ flow for different values of Rm.

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The resulting mean field equation obtained by substituting this ansatz for EH in (2)

is then

@tB ¼ r � �ð0ÞBþ ð0Þez � Bð Þ þ ð0Þ þ R�1m

� �r2B� �ð0Þr2ðez �BÞ:

Different effects can be identified from (24):

(i) corresponds to the �-effect, responsible for the regeneration of the mean field.(ii) has been named the ‘‘�-effect’’ or the ‘‘Radler’’-effect (Moffatt 1978) as it was

given particular attention in Radler (1969a,b). This term brings a contribution

to the e.m.f that is perpendicular to the mean current. Owing to this effect, a

toroidal e.m.f. (hence a poloidal field) can be regenerated from a poloidal

current (hence a toroidal field). Similarly to the �-effect, the �-effect can lead

to magnetic field growth.(iii) corresponds to magnetic pumping by the velocity field c¼ (0, 0, (0)), which

advects the magnetic field in the z-direction.(iv) is often interpreted as a turbulent magnetic diffusivity, which in general

enhances the diffusion rate of the field.

We also note that expanding relations (19) and (20) to first order in k implies that

�ð0Þ ¼ ��ð0Þk, !ð0Þ ¼ �ð0Þk: ð25Þ

The coefficients �(0), (0), (0) and �(0) can be calculated by extrapolating the curves

for a(k) and g(k) toward k¼ 0 (Plunian et al. 1999). Results for �(0) and (0)correspond to the stars in figures 6 (CP-flow) and 7 (MWþ flow). The extrapolation

was made using the IDL routine POLYFIT for k2 [0.005, 0.025]. As we recall from

figures 4 and 5, for high values of Rm, ja(k)j and jg(k)j increase sharply as k! 0 and it

becomes necessary to reach very low values of k in order to obtain an accurate

extrapolation. This is numerically expensive as the growth rate approaches zero so that

large integration times are required to isolate the fastest growing mode. Moreover, the

slopes of a(k) and g(k) as k! 0 are very steep and we would need to go to even lower

values of k to obtain accurate results for (0) or �(0). A similar difficulty has been

encountered in the study of the cat’s-eye flow, a symmetry-breaking version of the

Roberts flow defined by ¼A cos x�B cos y, A 6¼B (Courvoisier et al. 2005). The

alternative method described below enables us to overcome this problem (see also

Courvoisier et al. 2006).Here, we follow the asymptotic study of Roberts (1970) and expand the mean field

equations to successive orders in k. We write

p ¼ p0 þ p1 þ � � � , H ¼ H0 þH1 þ � � � , ð26Þ

where pn and Hn are of order kn. At leading order, (13) becomes

p0hH0i ¼ 0: ð27Þ

For a mean field solution p0¼ 0 and (14) becomes

@th0 ¼ r � u� h0ð Þ þ hH0i � r� �

uþ R�1m r2h0: ð28Þ


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This is equivalent to the induction equation for a z-independent field (h0) in the presenceof an imposed mean field in the horizontal plane (hH0i). To first order in k (13) becomes

p1hH0i ¼ r � hu� h0i ¼ r � E0, ð29Þ

where E0 is the e.m.f. to zeroth order in k. Using the expansion of (17) to leading orderwe obtain

E0 ¼ ~að0Þ � hH0i ¼ �ð0ÞhH0i þ ð0Þez � hH0i: ð30Þ

For the MWþ flow, as p02R, ~að0Þ reduces to a diagonal matrix, �(0)I(see Appendix A).

In order to simplify the notations, we now write �(0)¼ �, (0)¼ , h0¼ b, hH0i ¼ B0

and (28) becomes

@tb ¼ B0 � rð Þuþ r � u� bð Þ þ R�1m r2b: ð31Þ

To calculate the mean-field coefficients, we solve (31) using a pseudo-spectral methodwith a 2nd order timestepping scheme. Both field and flow are now z-independent andto ensure that r � b¼ 0, we write the horizontal component of the field in terms of avector potential: b(x, y, t)¼r� (A(x, y, t) ez)þB(x, y, t)ez. To calculate � and , it issufficient to impose B0¼B0ex, evaluate the mean e.m.f. E ¼ hu� bi and use (30) to find

� ¼ ExB�10 , ¼ EyB

�10 : ð32Þ

This second method allows us to explore a wider parameter range and in particular toreach very high values of Rm owing to the 2D nature of the problem considered.

Both a serial and a parallel code were used in this investigation. The parallelcalculations were kindly performed by S.M. Tobias to study the very high Rm regime(Rm>103). We checked that the results of both codes were in agreement at lower valuesof Rm. A version of the same parallel code designed to study the growth rate ofmagnetic modes of the form (6) was used by Cattaneo and Tobias (2005).

Results for � and as a function of Rm correspond to the solid lines in figures 6(CP-flow) and 7 (MWþ flow; recall that in this case, ¼ 0 so only � is shown). The starscorrespond to the values found by extrapolation towards k¼ 0 (described earlier).Both methods give the same results for low Rm, but for Rm 32 the extrapolation tok! 0 clearly underestimates � and in the CP-flow. We believe that this discrepancy isdue to the fact that we did not use sufficiently small values of k.

From figures 6 and 7, we see that � and vary significantly with Rm and values athigh Rm cannot be inferred from those at lower Rm. Moreover, as expected for fastdynamos, � and do not tend to zero as Rm increases – however neither do they seem toreach any well-defined limit, even for values of Rm as high as 2.105 in the CP-flow. Forthe MWþ flow, the �-effect remains positive and seems to keep increasing with Rm. Forthe CP-flow, � and also reach very high values. Most strikingly however, � changessign, even though the helicity of the flow – to which � is often related – remains constantin such kinematic simulations. To check the robustness of this last result, we evaluated �and for different values of " and Rm. Recall from section 3.1 that the parameter "controls the size of the chaotic regions of the flow. The curves for � and versus Rm anddifferent values of " are plotted in figure 8. All show a similar behaviour with variationsin the sign of both effects. We also notice that the curves superpose at low Rm but thatfor higher Rm, the "-dependence becomes very strong.

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At low Rm, the induction equation is accurately described using the FOSA. For theMWþ flow it is possible to evaluate the �-effect analytically in this approximation toobtain � ¼ Rm 1þ 1=½2ð1þ 4R2

mÞ��

(Courvoisier 2006). Results for Rm9 1/2 agree wellwith this formula (see figure 7). For the CP-flow, FOSA predicts that ¼ 0, which isalso observed for Rm9 1/2 but does not apply for any higher values. As Rm increases,the chaotic structure of the flow influences � and , as shown by their growingdependence on �. Furthermore, as already mentioned, � changes sign as Rm varies. Thiscan be intuitively understood by considering Parker’s picture of a cyclonic event (Parker1955, Moffatt 1974). For large Rm, the field lines will be close to frozen so that the

Figure 6. �(0) and (0) versus Rm in the CP-flow for "¼ 1. The stars correspond to the values obtained byinterpolating the data for a(k) and g(k) towards the origin. The solid line joins the values obtained by directlysolving (31) and are more accurate. Data for Rm>103 are courtesy S.M. Tobias.

Figure 7. �(0) versus Rm in Otani’s MWþ flow. The stars correspond to values extrapolated from a(k) ask! 0 and the solid line to values calculated for k¼ 0. The þ joined by a dotted line are obtained using theFOSA approximation and the formula given in the text for the MWþ flow. These match the numerical valuesonly for Rm91=2.


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magnetic loop formed by the event might well twist more than half a turn and generate

a mean magnetic field whose helicity has the same sign as that of the flow before

diffusion has a chance to operate. In the MWþ flow however the �-effect remains of

constant sign – at least for Rm up to 1000. In this case, the two Beltrami waves making

up the flow give a helical structure to the field lines without letting field loops make a

complete turn (Gilbert et al. 1996). In the CP-flow magnetic modes of different helicities

are generated depending on the values of k and Rm (see section 3.1). The sign change in

� is due to a magnetic mode of negative helicity overtaking the mode of positive helicity

at low values of k. We check this by considering the case "¼ 0.75 for which the sign

change in � occurs for Rm¼O(10) and is thus easier to investigate. The left plot in figure

9 shows the growth rate as a function of k for Rm¼ 64. The main magnetic mode is a

positive Beltrami wave as in the case when "¼ 1 but on the right plot, details for low k

show that another mode is preferred, this time with a negative helicityy. By using the

linear relation (25) between �(k) and �(k) as k! 0 for values of k between 0.001 and

0.003, we obtain �(0)¼�2.80, which is close to �2.94, the value obtained by solving

(31). In this example, we thus need to reach to values of k�O(10�2) to attain the mode

Figure 9. Growth rate versus k in the CP-flow for "¼ 0.75 and Rm¼ 64. The right plot gives details of thebehaviour of �(k) for small values of k.

Figure 8. � and versus Rm in the CP-flow for different values of ". Data for Rm>103 are courtesy S.M.Tobias.

yWe identified the symmetry of these modes using timeseries of the mean field (see figure 2).

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from which we can calculate the ‘‘conventional’’ �-effect by extrapolating to low k. One

might even imagine that many crossings between modes of opposite helicities occur forlow k as Rm is increased. This example shows how the change in the sign of � can beunderstood in the CP-flow. It serves also to underline once more the difficulty ofmatching the results for k! 0 with those calculated directly at k¼ 0.

The main conclusion here is that the �-effect depends crucially on the structure of theflow as soon as Rm is O(1). The velocity fields we use have the same averaged Eulerian

properties, in particular the same helicity, but have different chaotic properties.These cannot be overlooked and the details of the interactions between the small-scalemagnetic and velocity fields must be taken into account. This has importantconsequences for the FOSA, which is valid for low Rm or low St. The latter assumption

clearly does not apply to the flows we have considered so far, but we see that as soon asthe low Rm approximation breaks down, the structure of the flow greatly influences theresult. In particular, even the sign of the �-effect cannot be determined a priori.

4. Results for 2D flows decorrelated in time

In the previous section, we calculated the �-effect for the CP and MWþ flows, whichare periodic in time and therefore have an infinite Eulerian correlation time, whichcorresponds to St� 1. In reality, astrophysical flows are mainly turbulent with

St�O(1). It seems therefore important to modify our model in order to study howreducing the time correlations, i.e. reducing the Strouhal number, influences thedynamo properties of a velocity field.

The new model, based on the CP-flow, is described in section 4.1. The numericalresults for the mean-field coefficients and for the growth rate of the dynamo instabilityare given in sections 4.2 and 4.3 respectively.

4.1. Model

We designed a random version of the CP-flow for which the time correlations can bevaried. The new streamfunction is

�ðx, y, tÞ ¼

ffiffiffi3

2

rðcosðxþ �1ðtÞ cosðtþ �1ðtÞÞ þ �2ðtÞ cosðtþ �2ðtÞÞÞ

þ cosðyþ �1ðtÞ sinðtþ �1ðtÞÞ þ �2ðtÞ cosðtþ �2ðtÞÞÞ: ð33Þ

The �i(t)i¼ 1,2 are piecewise constant functions of time whose values are uniformlydistributed between 0 and 2� and change every 2�c. The random series are obtainedusing the random number generator RAN2 (Numerical Recipes in Fortran). Thefunctions "1(t) and "2(t) are 2�c-periodic and allow us to smooth the phase change; they

are plotted in figure 10 for a typical run. The main results are obtained for �c¼ [�/2, 3.75, 37.5]. For comparison, the dynamical timescales involved in the CP-flow are theturnover time �0 ¼ L=urms ¼ 2�=

ffiffiffi3

p� 3:63 and the temporal periodicity T¼ 2�� 6.28.

The diffusive timescale �d is of order Rm�0.


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With this experimental setup, we can vary three parameters: the magnetic diffusivity,controlled by Rm, the extent of the chaotic regions of the flow, measured by ", and thecorrelation time by changing our new parameter, �c.

To obtain the mean-field coefficients, we solve (31) with an imposed x-directedmean-field as in section 3.3 and calculate

� ¼ limt�!1

ExðtÞB�10 , ExðtÞ ¼

1

t

Z t

0

hu� bixð�Þd�,

¼ limt�!1

EyðtÞB�10 , EyðtÞ ¼

1

t

Z t

0

hu� biyð�Þd�: ð34Þ

Now that the flows have lost their time periodicity, we need to average over very longtimes – beyond the O(Rm) Ohmic decay time – to obtain a converged value for �. Errorbars are calculated using the standard deviation of ExðtÞ. Difficulties in obtainingconverged values for the �-effect have been reported in other numerical studies(Ossendrijver et al. 2001, Cattaneo and Hughes 2006).

To calculate the dynamo growth rate �(k), we use the same method as in section 3.2and solve the induction equation for different values of k.

4.2. Numerical results for the mean field coefficients

The timeseries of Ex(t) for different values of �c are presented in figure 11.For �c� �0�T/2, the timeseries appear completely random with variations increasingin amplitude as �c is increased. When �c goes beyond 2�, the temporal periodicity of theflow becomes apparent. Finally, when �c becomes O(Rm) and greater, time periodicsequences appear in the timeseries interspersed with other events as the phase changes.For �c larger than Rm these sequences become predominant and we shall refer to themas the ‘‘CP-state’’. More precisely, we define the ‘‘CP-state’’ to be the time-periodic statereached by the magnetic field in the CP-flow, i.e. for infinite correlation time. Thevalues of the mean field coefficients for this state and for different values of Rm and "are those computed in the previous section. An important parameter in the presentstudy is the timescale, �s say, needed for the field to reach that state. This is a diffusivetimescale over which the magnetic field rearranges itself and adapts to the structure of

Figure 10. Time series for "1(t) (solid line) and "2(t) (dotted line) for a typical realisation. The functions "1and "2 vary with a tan h profile over a time �c/6, in order to smooth the phase change, and remain constantequal to " or 0 respectively otherwise.

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Figure 11. Time series of Ex(t) for different values of �c. From top to bottom, �c¼ 1.5, 3.75, 7.5, 15, 37.50,75. In all cases, Rm¼ 64 and "¼ 0.75. Note that the sample length is varied for the patterns to emerge moreclearly.


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the flow in which it is embedded. For a steady flow, this timescale is related to the fluxexpulsion time, as investigated by Weiss (1966) and Rhines and Young (1983).

Results for � and as a function of Rm for �c¼�/2, 3.75 and 37.5 are given infigure 12 together with the values for �c¼1 obtained in the previous section. For Rm

less than O(1), the results do not depend on �c. This corresponds to the range of Rm lowenough for FOSA in the low conductivity limit to be applicable and in which theevolution is controlled by the fast diffusive timescale. For larger Rm, however, thebehaviour of the curves depends greatly on �c. For the shortest values of �c investigatedhere, � and seem to settle rapidly to a value independent of Rm and � remains ofconstant sign, opposite to the helicity of the flow. This is particularly clear from thehigh Rm results for �c¼�/2 where � seems to settle down to about 0.55 and remainsclose to zero. We perform further calculations to assess the dependence of the results on�; our findings are presented in table 2 for �c¼ 3.75 and Rm¼ 64. For this value of �c,the mean field coefficients become Rm-independent and their values at Rm¼ 64 is anapproximation of their asymptotic limits. The results in table 2 show that this limit stilldepends on the value of ". For �c¼ 37.5, the curves for � and follow more closelythose for �c¼1. After an initial increase away from zero, � decreases sharply andbecomes negative until it settles down to about �3.25, an Rm-independent limit, forRm� 100. For this value of �c, � again changes sign. For low Rm, remains close tozero, then it decreases to reach a minimum of about �3 for Rm� 100, increases again tovalues close to zero for Rm� 1000 and does not vary much as Rm is increased further.Note that the values of for Rm¼ 20000 and �c¼ 1.57 and 37.5 agree within the errorbars. We believe this agreement to be coincidental.

From the results beyond the low Rm-FOSA stage, it seems that when �c is small,the value of � settles rapidly to a positive Rm-independent limit, whereas, when �c islarger, it remains Rm-dependent until higher values of Rm are reached and it can stillchange sign. We conjecture that for any finite correlation time, �() reaches anRm-independent limit �*(*) for Rm > R�

m, with R�m an increasing function of �c. If

�c is infinite, � and might never settle downy. Indeed, as Rm increases, the

Figure 12. � (left plot) and (right plot) as a function of Rm for �c¼ 1.57 (dotted line), �c¼ 3.75 (dashedline), �c¼ 37.5 (dashed-dotted line), and �c¼1 (solid line); "¼ 0.75. Data for Rm>103 are courtesy S.M.Tobias.

yThis might however depend on the value of ". As "! 0, the chaotic regions shrink and the �-effect might beaffected. Indeed for "¼ 0, we recover the Roberts flow for which �! 0 as Rm!1. The results presentedhere are however limited to "¼ 0.5, 0.75 and 1.0 for which the chaotic regions are substantial (see figure 1).

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timescale necessary to reach the CP-state, �s, also increases. When �c> �s, the fieldhas the time to rearrange itself via diffusive processes and the results are Rm-dependent. As �s increases beyond �c however, diffusive effects are less influentialand the mean field coefficients become Rm-independent.

4.3. Results for the growth rate of a z-dependent magnetic field

We saw in section 3.3 that the sign of the �-effect is linked to the symmetry of themagnetic mode generated at low k. For short �c, it seems that � remains positive and inthis section, we study how the growth rate of the dynamo instability is affected byreducing the time correlations in the flow.

Figure 13 shows how the growth rate varies with k at fixed Rm, for different values of�c. We see that the growth rate for longer �c (dashed line) is very close to that for�c¼1. For the values of k investigated, the curve has two maxima. Figure 14 presentsthe timeseries of the real and imaginary part of the x and y-components of hH0i whereH0(x, y, k, t) is defined by (10), for �c¼ 37.5 and for values of k corresponding to the twomaxima in figure 13. Comparison with figure 2 and relation (A.6) shows that thegrowing field for long �c has the same symmetry properties as in the case of infinite �c.If we return to figure 13, we see that the curve for short �c (dotted line) has a singlemaximum. The analysis of the timeseries for the components of hH0i plotted in figure 15show that the maximum at low k again has the same symmetry as in the CP-flow. Fork¼ 1.6 however, the timeseries do not give any clear indication of the symmetry of thegrowing mode. From figure 13, we also note that even if the fastest growth rate occursfor the shorter correlation time, it is also substantially lower for shorter k, hence longerwavelength in z. This hints at the fact that it is more difficult to generate a field on largescales in a very random flow.

It seems therefore that reducing the correlation time does not affect the efficiency ofthe dynamo in the sense that the maximum growth rate over all values of k is at least ashigh as for the CP-flow. The geometry of the growing mode is however affected and, fora short correlation time, there is a clear preference for the positive Beltrami wave, ofparity opposite to that of the flow. A similar phenomenon was also observed in therandom model of Gilbert and Bayly (1992).

5. Discussion

In the present paper, we studied how Mean-Field Electrodynamics is used to describedynamo action in a family of time-dependent 2D-flows of the form (5), with varying

Table 2. � and for different values of " at Rm¼ 64.

" �

0.5 0.4564 0.0044 �0.0690 0.0060.75 0.824 0.018 �0.185 0.021.0 1.134 0.011 0.197 0.015


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chaotic properties and time correlations. Suchmotions are able to generate a helical meanfield, of infinite horizontal extent but with an O(k� 1) vertical lengthscale in thez-direction. The growth mechanism can be described in terms of a k-dependent �-effect,~�ðkÞ, which we determined for the Galloway-Proctor CP-flow and Otani’s MWþ flow.Particular attention was paid to the ‘‘conventional’’ �-effect, �, that corresponds to a

Figure 13. Growth rate of the magnetic energy versus k at Rm¼ 64, for different values of the correlationtime �c¼ 3.75 (dotted line), �c¼ 37.5 (dashed line), �c¼1 (solid line).

Figure 14. Plots of the real and imaginary parts of the components of hH0i as a function of time for�c¼ 37.5. Top ReðhH0ixÞ (solid line) with ImðhH0iyÞ (dashed line). Bottom ImðhH0ixÞ (solid line) with ReðhH0iyÞ

(dashed line). The left (right) plot is for k¼ 0.6 (k¼ 1.6). Compare with figure 2.

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uniform mean field and to the limit k! 0. We evaluated it both by extrapolating ~�ðkÞtowards the origin and, more successfully, by solving directly the induction equation fork¼ 0. We studied systematically how � varies with Rm and with the flow properties. Thiswork extends to fast dynamos earlier investigations performed for slow dynamos, such asthe steady Roberts flow.

The calculation of � via the resolution of the induction equation for k¼ 0 in thepresence of an imposed mean-field is a self-consistent 2D-problem. For slow dynamos,earlier studies showed that � tends to 0 as Rm tends to infinity. In contrast, the presentresults show that for fast dynamos, the value of the �-effect at high Rm depends on thechaotic properties and correlation time of the motions and on Rm itself; only in the caseof finite time-correlations and for sufficiently high Rm did � reach an Rm-independentlimit. This emphasizes that even in the kinematic regime, even for flows with a simpleanalytical formulation, the determination of the �-effect remains a delicate issue.Indeed, with exception of the narrow parameter range for which the First OrderSmoothing Approximation is valid (Rm� 1 or St� 1), � has an unpredictiblebehaviour, strongly dependent on the exact details of the underlying motion. Inastrophysical applications, for which Rm� 1 and St� 1, it is therefore probablymisleading and certainly not justified to use a simple parameterisation of the �-effect(traditionally, �� flow helicity) in the derivation of mean field models. Along theselines, the most striking feature of our findings is perhaps that, in the CP-flow, � canchange sign, even in kinematic simulations where the flow helicity remains constant.

Figure 15. Plots of the real and imaginary parts of the components of hH0i as a function of time for�c¼ 3.75. Top ReðhH0ixÞ (solid line) with ImðhH0iyÞ (dashed line). Bottom ImðhH0ixÞ (solid line) with ReðhH0iyÞ

(dashed line). The left (right) plot is for k¼ 0.6 (k¼ 1.6). Compare with figure 2.


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The sign change just mentioned can be explained by considering the dominantgrowing magnetic mode for sufficiently low k. Indeed, the study of the growth rateof the dynamo instability �(k) and that of the symmetries of the growing magneticfield reveal that in the CP-flow, magnetic modes of polarities identical or oppositeto that of the motion can be generated; which polarity dominates at low k anddictates the sign of � depends on the values of Rm, " and �c. This phenomenon isreminiscent of the discussion of Parker’s scenario (Parker 1955) by Moffatt (1974),already mentioned in section 3.3 and summarised below. Figure 16 depicts twopossible outcomes of the action of a cyclonic event on a field line. An upwardhelical flow creates a loop of magnetic field. In case (a), the loop has turned lessthat 180o and a current anti-parallel to the original field is created, while in case(b), the loop has turned more that 180o (and less that 360o) and the inducedcurrent is now parallel to the original field. Case (a) produces a helical field withopposite polarity to that of the flow. It is likely to occur if there is a shorttimescale in the system, i.e. if Rm� 1, when the magnetic field diffuses rapidly, or ifSt � 1, when the cyclonic event is short-lived. In these circumstance, the loop ofmagnetic field will not perform more than half a turn. This scenario seems tohappen in the random CP-flow for short correlation time: � remains of oppositesign to the flow helicity and growing magnetic modes of opposite polarity to thatof the flow are clearly preferred. Now if there is no such short timescale, then case(b) can occur and therefore, changes even in the sign of the �-effect are possible.This indeed happens for the 2�-periodic CP-flow and also for the longer value of �cselected, when � settles to a limit of the same sign as the flow helicity. We notehowever that although this intuitive picture of the twisting of field loops provides auseful framework to the interpretation of the present results, one needs to bear inmind that the complicated Lagrangian dynamics of the flows and the resultingmovements of the field lines are presumably more complex and not so easy tovisualise. Nonetheless, the above discussion emphasises yet again that at high Rm,the flow details – as opposed to averaged Eulerian quantities alone – are importantfor determining the mean field coefficients and it seems that the difficulty ofobtaining meaningful high Rm results for the motions considered could have beenforeseen.

Figure 16. Possible outcomes of the action of a cyclonic event on a magnetic field line (after Moffatt 1974).An upward helical flow creates and twists a loop of magnetic field. In (a), the loop has turned 90o and acurrent anti-parallel to the original field is created. In (b), the loop has turned 270o and the induced current isnow parallel to the original field. Situations (a) and (b) yield �-effects of opposite signs.

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The above paragraph highlights the link between � and the dominant magnetic modeat low k. For modes with scales large in the z-direction, the k-dependent �-effect ~�ðkÞcan be approximated using its Taylor series expansion

~�ðkÞ ¼ ~�ð0Þ þ k@k ~�ð0Þ þ � � � : ð35Þ

Different effects can then be identified, such as the ‘‘conventional’’ �-effect and thepumping velocity, included in the leading order term, or the magnetic diffusivity,included in the first order term in k (see section 3.3). As emphasized by Soward (1994)however, the ‘‘conventional’’ �-effect is a useful concept only if the series (35) is rapidlyconvergent. This is clearly not the case for the present examples: as the results of section3 show, a(k) and g(k), the real and imaginary parts of ~�ðkÞ cannot be approximated by alinear function of k (i.e. by truncating the series (35) to first order) unless k� 1.Therefore a description of dynamo action based on a mean field model with ak-independent �-effect has a very limited range of validity for the motions consideredhere. As a result, it fails to account for the main dynamo mechanisms in the flowsstudied, for which the field growth is more efficient for k�O(1) (see also Courvoisieret al. 2005). Our findings also show that the low-k modes described by the‘‘conventional’’ �-effect are more sensitive to variations in Rm than the dominantk�O(1) mode, which retains an O(10�1) growth rate and the same symmetry property.

Finally, the present model, which uses velocity fields dependent on two spatialcoordinates only, enabled us to conduct a systematic survey of various parameter regimesand in particular to determine the mean field coefficients for values of Rm as high as 105,owing to the two-dimensional nature of the problem considered. This numericalsimplicity is balanced by the fact that we are considering the large-scale dynamoproperties (i.e. �(k¼ 0)) of what are essentially small-scale dynamos in the sense thatk� 1 corresponds to the fastest growing mode. To remedy this, further studies should bebased on more realistic velocity fields. In particular it seems important to contrast thesefindings with proper 3D calculations, for which the flow will have a well-definedlengthscale in all three directions.

Acknowledgements

The author is deeply grateful to D.W. Hughes and S.M. Tobias for the supportprovided whilst carrying out this research and for numerous useful comments on themanuscript, to A.D. Gilbert, F. Plunian and A.M. Rucklidge for fruitful discussionsand to the anonymous referees for helpful comments. This project was carried out withthe financial support of the EPSRC and the University of Leeds and completed underPPARC grant PP/D00179X/1.

References

Cattaneo, F. and Hughes, D.W., Dynamo action in a convective rotating layer. J. Fluid Mech., 2006, 553,401–418.

Cattaneo, F. and Tobias, S.M., Interactions between dynamos at different scales. Phys. Fluids, 2005, 17,127105.


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Childress, S., Alpha-effect in flux ropes and sheets. Phys. Earth Planet Int., 1979, 20, 172–180.Childress, S. and Gilbert, A.D., Stretch, twist, fold, the fast dynamo, 1995 (Springer: Berlin, Germany).Childress, S. and Soward, A.M., Scalar transport and alpha-effect for a family of cat’s-eye flows. J. Fluid

Mech., 1989, 205, 99–133.Courvoisier, A., Dynamo action and the generation of large-scale magnetic fields in Astrophysics. PhD thesis,

2006 (University of Leeds: Leeds, England).Courvoisier, A., Gilbert, A.D. and Ponty, Y., Dynamo action in flows with cat’s-eyes. Geophys. Astrophys.

Fluid Dyn., 2005, 99, 413–429.Courvoisier, A., Hughes, D.W. and Tobias, S.M., �� effect in a family of chaotic flows. Phys. Rev. Lett.,

2006, 96, 034503.Galloway, D.J. and Proctor, M.R.E., Numerical calculations of fast dynamos for smooth velocity fields with

realistic diffusion. Nature, 1992, 356, 691–693.Gilbert, A.D. and Bayly, B.J., Magnetic field intermittency and fast dynamo action in random helical flows.

Phil. Trans. R. Soc. Lon. A, 1992, 241, 199–214.Gilbert, A.D., Soward, A.M. and Childress, S., A fast dynamo of �!-type. Geophys. Astrophys. Fluid Dyn.,

1996, 85, 279–314.Hughes, D.W., Cattaneo, F. and Kim, E.-J., Nonlinear multi-cellular fast dynamos. Studia geoph. et. geod.,

1998, 42, 328–334.Klapper, I. and Young, L.S., Bounds on the fast dynamo growth rate involving topological entropy.

Comm. Math. Phys., 1995, 173, 623–646.Krause, F. and Radler, K.-H., Mean field magnetohydrodynamics and dynamo theory, 1980 (Pergamon Press:

Oxford, England).Moffatt, H.K., The mean electromotive force generated by turbulence in the limit of perfect conductivity.

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Cambridge, England).Ossendrijver, M., Stix, M. and Brandenburg, A., Magnetoconvection and dynamo coefficients I: Dependence

of the �-effect on rotation and magnetic field. Astron. Astrophys., 2001, 376, 713–726.Otani, N.F., A fast kinematic dynamo in time-dependent two-dimensional flows. J. Fluid Mech., 1993, 253,

327–340.Parker, E.N., Hydromagnetic dynamo models. Astrophys. J., 1955, 122, 293–314.Plunian, F., Marty, G. and Alemani, A., Kinematic dynamo action in a network of screw motions:

application to the core of a fast breeder reactor. J. Fluid Mech., 1999, 382, 137–154.Plunian, F. and Radler, K.-H., Harmonic and subharmonic solutions of the Roberts dynamo problem.

Application to the Karlsruhe experiment. Magnetohydrodynamics, 2002, 38, 92–103.Ponty, Y., Pouquet, A. and Sulem, P.-L., Dynamos in weakly chaotic two-dimensional flows. Geophys.

Astrophys. Fluid Dyn., 1995, 79, 239–257.Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P., Numerical Recipes in Fortran 77, Second

Edition, 1992 (Cambridge University Press: Cambridge, England).Radler, K.-H., A new turbulent dynamo [English translation by Roberts and Stix (1971)]. Monats Dt Akad

Wiss, 1969a, 11, 272–279.Radler, K.-H., On the electrodynamics of turbulent fluids under the influence of Coriolis forces [English

translation by Roberts and Stix (1971)]. Monats Dt. Akad. Wiss., 1969b, 11, 194–201.Radler, K.-H. and Brandenburg, A., Contributions to the theory of a two-scale homogeneous dynamo

experiment. Phys. Rev. E., 2003, 67, 2–026401.Radler, K.-H., Rheinhardt, M., Apstein, E. and Fuchs, H., On the mean-field theory of the Karlsruhe

dynamo experiment. Nonlinear Processes in Geophysics, 2002, 9, 171–187.Rhines, P.B. and Young, W.R., How rapidly is a passive scalar mixed within closed streamlines. J. Fluid

Mech., 1983, 133, 133–145.Roberts, G.O., Spatially periodic dynamos. Phil. Trans. R. Soc. Lon. A, 1970, 266, 535–558.Roberts, G.O., Dynamo action of fluid motions with two-dimensional periodicity. Phil. Trans. R. Soc.

Lon. A, 1972, 271, 411–454.Roberts P.H. and Stix M. 1971. The turbulent dynamo: a translation of a series of papers by Krause F.,

Radler K.-H., Steenbeck M. Tech. Note 60, NCAR, Boulder, Colorado.Soward, A.M., Fast dynamo action in a steady flow. J. Fluid Mech., 1987, 180, 267–295.Soward, A.M., Fast dynamos. In Lectures on Solar and Planetary Dynamos, edited by M.R.E. Proctor and

A.D. Gilbert, pp. 181–217, 1994 (Cambridge University Press: Cambridge, England).Vainshtein, S.I. and Zel’dovich, Y.B., Origin of magnetic fields in astrophysics. Sov. Phys. Usp., 1972, 15,

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Appendix A: Symmetries of the flows and simplifications to the dynamo problem

A1. Symmetry properties of the flows

Starting with Roberts (1972), the symmetry properties of different 2D cellular flows andhow they contribute to simplifying the dynamo problem has been detailed in variouspapers, e.g. Soward (1987), Childress and Soward (1989), Childress and Gilbert (1995),Plunian et al. (1999), Plunian and Radler (2002). The derivation below adapts to thetime dependent flows the derivations made for the Roberts flow or other steady flows.

The symmetries M that leave the CP and MWþ flows invariant are listed in table A1.They correspond to a rotation of R3 defined by a matrix A accompanied by a space-time translation defined by a vector (d, t). It can be verified that for thesetransformations, the velocity field satisfies

uðx, tÞ ¼ Muðx, tÞ ¼ Au A�1ðx� dÞ, t� �

� �: ðA:1Þ

The maps Ri – where the subscript i indicates the flow: 1 for the CP-flow and 2 for theMWþ flow – are rotations of �/2 around the z-axis with flow-dependent shifts in spaceand time. The flows are also invariant under the successive application of Ri and sinceR4

i ¼ I, the identity transformation, they are invariant under the transformations of thecyclic group of order 4. The maps Si are rotations of � around the y-axis with flow-dependent shifts in space and time. As S2

i ¼ I, we only obtain new transformations bycombining Si with Ri. We note that for the CP-flow, the maps involving S1 reverse time,and we disregard these symmetries.

A2. Symmetry properties of the mean field

Knowing the symmetry properties of the flow it is possible to simplify the dynamoproblem by determining the symmetries of the corresponding growing magnetic field, asexplained below.

Table A1. Symmetry properties of the fast dynamos considered (see Childress and Gilbert 1995).

Flows Symmetries

CP-flow I, R1, R21, R

31, S1, R1S1, R

21S1, R

31S1

R1:

x0 ¼ �=2� y

y0 ¼ �=2þ x

z0 ¼ z

t0 ¼ tþ �=2!0

0BBB@

1CCCA, S1:

x0 ¼ �� x

y0 ¼ �þ y

z0 ¼ �z

t0 ¼ �tþ �=!0

0BBB@

1CCCA

MWþ flow I, R2, R22, R

32, S2, R2S2, R

22S2, R

32S2

R2

x0 ¼ �� y

y0 ¼ �þ x

z0 ¼ z

t0 ¼ tþ �=2

0BBB@

1CCCA, S2:

x0 ¼ �� x

y0 ¼ �þ y

z0 ¼ �z

t0 ¼ t

0BBB@

1CCCA


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The maps M (Ri and Si) have the properties that

Mð@tvÞ ¼ @t Mvð Þ,

M r � vð Þ ¼ detðAÞ r �Mvð Þ,

M r2v� �

¼ r2 Mvð Þ,

M v1 � v2ð Þ ¼ detðAÞ Mv1 �Mv2ð Þ: ðA:2Þ

The induction equation (8) is an eigenvalue problem for the reduced growth rate definedby pr(k)¼ p(k)þ k2/Rm. Since u(x, t) and k¼ kez are invariant under Ri, the properties(A.2) imply that H and RiH are solutions to the same induction equation. If thecorresponding eigenvalue pr(k) is non-degenerate, they are constant multiple of oneanother and therefore they are eigenvectors of the transformation Ri and satisfy

RiH ¼ CH, C4 ¼ 1: ðA:3Þ

This property has important consequences for the mean field (12). Indeed, we have

RihHi ¼ ez � hHi ¼ ChHi, ðA:4Þ

which implies hHix ¼ ChHiy ¼ �C2hHix. Therefore, solutions with a non-zero meanfield must have C2

¼�1. In this case

ez � hHi ¼ ihHi ðA:5Þ

and therefore

ez �ReðhHiÞ ¼ �ImðhHiÞ, C ¼ i: ðA:6Þ

Finally, if we consider the mean field defined by (12), using relation (A.6) we can showthat

r � B ¼ �kB, C ¼ i: ðA:7Þ

Therefore, the mean field can take the form of a positive or negative Beltrami wave inthe z-direction, depending on the sign of C.

For the MWþ flow, the map S2 transforms ez into �ez so that if pr(k)2R, H andðS2HÞ

�, the complex conjugate of S2H, are solutions to the same induction equation. Tokeep the discussion general, we do not consider the simplifications introduced by thissymmetry, except when applied to the �-effect, as explained in the next section.

A3. Symmetry properties of ~aðkÞ

The a-tensor can also be simplified. We already know that H and RiH satisfy the sameinduction equation. As ~aðkÞ defined above depends only on the flow if k and Rm arefixed, then

hu� hi ¼ ~aðkÞ � hHi ðA:8Þ

and

hu� Rihi ¼ ~aðkÞ � hRiHi: ðA:9Þ

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Now, as Riu¼ u and using the properties (A.2), we obtain

hRiðu� hÞi ¼ ~aðkÞ � hRiHi, ðA:10Þ

which yields, using (A.1),

hu� hi ¼ A�1 ~aðkÞA � hHi, ðA:11Þ

since the average is taken over the space and time periodicities of the flows.Comparison with (A.8) gives

~aðkÞ ¼ A�1 ~aðkÞA: ðA:12Þ

In the present case, this last relation implies that ~aðkÞ is the matrix of a rotation in the

xy-plane. Furthermore, analogous to relation (7), we have ~a�ðkÞ ¼ ~að�kÞ so that

Reð~aðkÞÞ is even while Imð~aðkÞÞ is odd in k. We can therefore write

~EHðkÞ ¼ ð�ðkÞ � ik�ðkÞÞhHi þ ððkÞ � ikðkÞÞez � hHi, ðA:13Þ

where �, , and � are real and even functions of k.For the MWþ flow, it is possible to simplify ~aðkÞ further for solutions with a real

growth rate. In that case, S2H obeys the same induction equation asH*. By a derivation

similar to that leading to (A.12), we find that

~aðkÞ ¼ A�1 ~a�ðkÞA ¼ A�1 ~að�kÞA, ðA:14Þ

which implies that in this case, (k)¼ �(k)¼ 0.


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The α-effect in 2D fast dynamos

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Transcript of The α-effect in 2D fast dynamos