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Magnetic Helicity in fast dynamosAndrew Gilbert aa School of Mathematical Sciences , University of Exeter , Exeter,EX4 4QE, U.K.Published online: 24 Sep 2010.
To cite this article: Andrew Gilbert (2002) Magnetic Helicity in fast dynamos, Geophysical &Astrophysical Fluid Dynamics, 96:2, 135-151, DOI: 10.1080/03091920290027952
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Geophys. Astrophys. Fluid Dynamics, 2002, Vol. 96, No. 2, pp. 135–151
MAGNETIC HELICITY IN FAST DYNAMOS
ANDREW D. GILBERT
School of Mathematical Sciences, University of Exeter, Exeter EX4 4QE, U.K.
(Received 31 July 2001)
The behaviour of magnetic helicity in kinematic dynamos at large magnetic Reynolds number is considered.Hughes, et al. [Phys. Lett. A 223, 167–172 (1996)] observe that the relative helicity tends to zero in the limitof large magnetic Reynolds number. This paper gives upper bounds on the helicity, by relating thehelicity spectrum to the energy spectrum. These bounds are confirmed by numerical simulation and the distri-bution of helicity over scales is considered. Although it is found that the total helicity becomes small in thelimit of high conductivity, there can remain significant, but cancelling, helicity at large and small scales ofthe field. This is illustrated by considering the evolution of helicity in the stretch–twist–fold dynamo picture.
Keywords: Magnetohydrodynamics; Magnetic fields; Dynamo theory; Fast dynamos; Helicity
1. INTRODUCTION
Magnetic fields in the Sun, planets and galaxies are generally believed to be generatedby dynamo processes, governed by the magnetic induction equation
@B
@t¼ J3 ðu3BÞ þ R�1r2B, JEB ¼ 0 ð1:1Þ
(e.g., Moffatt, 1978). Here u is the fluid flow, of plasma or liquid metal, B is the mag-netic field and R is the magnetic Reynolds number, which is large in many astrophysicalenvironments. The equation (1.1) has been non-dimensionalised using the scale andturn-over time of the flow u. In kinematic dynamo theory the flow u is fixed; if itis sufficiently complex and R is large enough, the flow may act as a dynamo, givingexponential growth of magnetic energy. For example, for a steady flow u, we willhave eigenfunctions of the form
B ¼ bðx, y, zÞeði!þpÞt, ð1:2Þ
*E-mail: [email protected].
ISSN 0309-1929 print: ISSN 1029-0419 online � 2002 Taylor & Francis Ltd
DOI: 10.1080/03091920290027952
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where p is the dynamo growth rate. We shall be concerned with the limit of large mag-netic Reynolds number R and in particular ‘fast dynamos’, for which the largest realgrowth rate p remains positive and bounded away from zero in this limit. Our interestin this paper is in understanding the role of magnetic helicity within kinematic fastdynamo theory, although in Section 5 we give some discussion of nonlinear regimes.The general motivation for the study of fast dynamos is that fast processes, such asthe 22-year Solar cycle, are observed in the (nonlinear) Solar dynamo for whichR � 108 (e.g., Childress and Gilbert, 1995).In the idealised situation R ¼ 1, in which the fluid is perfectly conducting, magnetic
energy will still grow in general, but it is known that the magnetic helicity H ¼ hAEBi isconserved (under suitable boundary conditions) as the field evolves, and that this quan-tity has a topological interpretation (Woltjer, 1958; Moffatt, 1969). Here hEi is avolume average over the domain occupied by the fluid, and A is a vector potentialfor the magnetic field. Plainly for R ¼ 1, if a growing eigenfunction of the form(1.2) can be sensibly defined, it must have zero helicity, as pointed out by Moffattand Proctor (1985).This raises the question of what happens to magnetic helicity in the limit when the
magnetic Reynolds number is large, but finite, R! 1. Although the small quantityR�1 multiplies the final, diffusive term in (1.1), this term may not generally be neglected,since the field may adopt small scales, and indeed in the Sun much small scale fieldstructure is evident (e.g., Priest, 1984). The limit R! 1 is highly singular and differentbehaviour may occur for R ¼ 1 and 1 � R < 1. For example, in a kinematicdynamo, a typical flow in the plane amplifies magnetic energy indefinitely forR ¼ 1, but magnetic energy ultimately decays for any finite R, no matter how large(Zeldovich, 1957); the onset of the decay is delayed as R is increased.It is found in MHD, in particular in tokomak and plasma physics, that the ideal
helicity invariant tends to be ‘robust’, being approximately conserved in many situa-tions when the dissipation is present but weak (e.g., Taylor, 1974, Berger, 1984,Biskamp, 1993). This suggests that helicity will be approximately conserved in adynamo in the limit R! 1, and so for growing fields the normalised, relative helicityshould be small and tend to zero in this limit. Precisely this behaviour is observed byHughes, et al. (1996), who discuss the behaviour of magnetic fields in fluid flowswith different degrees of flow, or kinetic, helicity, hu EJ 3 ui. In each case consideredthe magnetic helicity falls off with a power law dependence as the magnetic Reynoldsnumber is increased.The aim of the present paper is to give bounds on the level of helicity in growing
magnetic fields, within kinematic dynamo theory, and to discuss its distribution overdifferent scales. We use an exponent defined by Vainshtein and Cattaneo (1992) togive information about the slope of the energy spectrum, and this leads to bounds onthe helicity spectrum, related to results of Berger (1984) (see Sections 2 and 3). InSection 4 we study the magnetic helicity for kinematic dynamo action in a flow firststudied by Otani (1993). This gives information not only about the total helicity,but also about its distribution over scales, and we observe considerable cancellationof helicity as it is summed over a range of scales. In particular we observe helicity atthe very largest scale having opposite sign to that at smaller scales. To understandhow helicity may be distributed over different scales in a fast dynamo, we considerthe idealized ‘stretch–twist–fold’ picture, in Section 5. This makes a link with geometri-cal ideas of writhe and twist (see Moffatt and Ricca, 1992). Finally in Section 6,
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we offer some conclusions and consider the relation of our results to recent discussionof the role of helicity in astrophysical dynamos.
2. ENERGY SPECTRUM AND OHMIC DISSIPATION
In order to discuss the evolution of helicity in dynamos, there are a number of possiblesettings we could consider, three-dimensional steady flows, two- or three-dimensionalunsteady flows, or even random flows. In order to relate our results most cleanly tothe numerical simulations of Section 4, we choose to work with a two-dimensionalincompressible flow field uðx, y, tÞ, which has all three components ðux, uy, uzÞ.We take the flow u to be periodic with period 2� in x and y, and with period T in t.Examples of such flows have been considered by several authors and robust fast dyna-mos have been obtained (Galloway and Proctor, 1992; Otani, 1993). Our analysiscould be adapted to other situations, and the conclusions would be much the same.We therefore consider solutions to the induction equation (1.1) (with J E u ¼ 0) that
are periodic in space. We can define the magnetic energy EðtÞ by
EðtÞ ¼1
2hjBj2i, ð2:1Þ
where h�i represents an average over space. This obeys
@E
@t¼ �hu E J3Bi � �hjJj2i, ð2:2Þ
where J ¼ J3B is the current. Here we have conveniently set � ¼ 1=R, so that � isa dimensionless diffusivity and we are interested in the limit of zero diffusion � ! 0.We suppose that we have integrated the induction equation numerically for the given
flow, and the numerical solution is dominated by the fastest growing eigenmode, whichhere takes the Floquet form
B ¼ eðpþi!Þtþikz bðx, y, tÞ þ c:c: ð2:3Þ
(in the absence of degenerate modes). Here ‘c:c:’ denotes the complex conjugate ofthe previous term. The function b is periodic in t with period T, and p is the realgrowth rate of the mode with wavenumber k 6¼ 0. The above field is periodic in allthree spatial directions, and we take it to lie in a 2�� 2�� 2�=k box.The magnetic energy is given by
EðtÞ ¼ e2pthjbj2i
and grows exponentially in time (superposed on the periodicity inherent in b). Let usfactor out the exponential growth and then time average, to define the magneticenergy of the growing eigenfunction as
E ¼ he�2ptEðtÞit ¼ hhjbj2iit, ð2:5Þ
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where h�it denotes a time average. To avoid cumbersome notation we will drop suchtime-averages from now on, with the understanding that this and similar quadraticquantities are to be time-averaged.It will be useful to look at the magnetic field in Fourier modes, writing
bðx, y, tÞ ¼Xm, n
bmnðtÞeimxþiny: ð2:6Þ
In terms of this we have
E ¼Xm, n
Emn, Emn ¼ jbmnj2, ð2:7Þ
where Emn is the contribution to the energy from the ðm, nÞ-mode. We may normalisethe eigenfunction so that the contribution from the large-scale field is
ELS � E00 ¼ jb00j2 ¼ 1: ð2:8Þ
(If this mode is identically zero because of a symmetry, some other large-scale modemay be chosen; the choice is not crucial.) We can also define an energy spectrumEð�Þ by
E ¼X1�¼1
Eð�Þ, Eð�Þ ¼X
��1<Kðm, nÞ��
jbmnj2, ð2:9Þ
where Kðm, nÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 þ n2 þ k2
pand � ¼ 1, 2, . . . labels the shells over which the spec-
trum is computed.We now introduce fast dynamo scalings. Moffatt and Proctor (1985) show that the
diffusive cutoff in a fast dynamo is a length-scale of order �1=2 (see also Gallowayand Frisch, 1986), and bearing in mind the normalisation (2.8) above we may postulatean energy spectrum with a power law of the form
Eð�Þ � �2�1, ð19 �9 ��1=2Þ: ð2:10Þ
Here � is taken to mean ‘approximately proportional to’, with similar meaning to theinequality symbols 9 and 0. Such scalings are consistent with numerical simulationsof fast dynamo action (e.g., Childress and Gilbert, 1995, Section 12.2). With this spec-tral slope assumed, the total energy is obtained from integrating (2.10) (and assumingthat Eð�Þ falls rapidly to zero outside the range given),
E � E=ELS �1 < 0,�� > 0 ,
�ð2:11Þ
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and we may identify the exponent with the exponent n of Vainshtein and Cattaneo(1992)1.The global energy balance (2.2) restricts the value of . To see this we first make
the key assumption that in a typical fast dynamo all three terms in (2.2) are of similarmagnitude, and so
E � �J , J � hjjj2i, ð2:12Þ
where the constant of proportionality implicit in our use of ‘�’ depends on the details ofthe fast dynamo but is independent of � as � ! 0. Here j is defined for the current J in asimilar way to b for the magnetic field in (2.3), and in terms of Fourier components(cf. 2.6)
jmn ¼ iK3 bmn, ½K � ðm, n, kÞ�: ð2:13Þ
We may define the spectrum J ð�Þ of the average current squared, analogously to (2.9),with J ¼ hjjj2i ¼
P1
�¼1 J ð�Þ: From (2.13, 2.10),
J ð�Þ � �2þ1, ð19 �9 ��1=2Þ, ð2:14Þ
and so
J �1 < �1,���1 > �1
�ð2:15Þ
(see also Boozer 1992).Clearly (2.11), (2.12) and (2.15) are only compatible if � 0, and we will assume this
holds in what follows. The limiting case ¼ 0 is not ruled out by the above argument,but measurements of in simulations give positive values, as low as 0.25 (Gilbert et al.,1993), but increasing for flows of greater complexity (Cattaneo et al., 1995); see Section4 below. Therefore we shall take >0 and note that for a typical fast dynamo (in thekinematic regime), the total energy must therefore be dominated by small-scale fields(Vainshtein and Cattaneo, 1992). Note that in the two-dimensional case ¼ 1(Vainshtein and Cattaneo, 1992); this is not a dynamo, but we observe ! 1 (frombelow) in the two-dimensional limit k! 0 of large-scale dynamo action, in Section 4below. We therefore expect that in typical fast dynamos 0<<1.It is important to note that there are some examples of fast dynamos which do not
fall into our framework. Models such as baker’s maps with uniform stretching(Finn and Ott, 1988), and that of Arnold et al. (1981), amplify magnetic fields withoutgenerating fine structure. There is no cascade of magnetic energy to small scales andso (2.10) does not hold; furthermore magnetic dissipation is negligible in (2.2).However these models cannot straightforwardly be embedded as smooth flows inthree-dimensional space. For realistic flows, as have been simulated numerically, it is
1Note that for simplicity and brevity we shall here and elsewhere ignore special cases such as ¼ 0 in (2.11)which give rise to the logarithmic scaling E � log �. Such cases are probably only of academic interest and inany case our scaling assumption (2.10) is too crude to make deductions about any variation as weak as log �.
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clear that there is a cascade of magnetic energy to small scales (see Galloway andProctor, 1992; Otani, 1993, Section 4 below), and our framework is correct.
3. HELICITY SPECTRUM AND DISSIPATION
We now consider the helicity per unit volume given by
HðtÞ ¼ hA EBi, ð3:1Þ
where A is the vector potential defined by B ¼ J3A, J EA ¼ 0 and hAi ¼ 0. We takethere to be no mean field, hBi ¼ 0, this being a dynamo calculation, in which case A
is periodic in space with the same periodicity as B, and may be taken to haveFourier components amn (cf. 2.6). The evolution of helicity is governed by
@H
@t¼ �2�hBEJi: ð3:2Þ
If the diffusivity � is strictly zero, then H is a constant, in keeping with its topologicalinterpretation in terms of the linkage of magnetic field lines (Moffatt, 1969).Now suppose we have 0 < � � 1 and a growing magnetic field eigenmode (2.3).
By analogy with the discussion in Section 2 above, we may define the helicity of theeigenmode H ¼ haEb� þ a�Ebi, its spectrum Hð�Þ, and the contributions Hmn fromwavenumber ðm, nÞ in the horizontal. We have Hmn ¼ amnEb
�mn þ a�mn E bmn with
amn ¼ K�2iK3 bmn. There follows the inequality jamnj � K
�1jbmnj and so
jHmnj �2
Kðm, nÞEmn, jHð�Þj9
2
�Eð�Þ: ð3:3a, bÞ
The first of these is exact, but the second is approximate, because of the bundling ofmodes into shells used to define the spectra.We wish to bound the helicity in a fast dynamo. The most straightforward inequality
comes from taking the minimum value, k, of Kðm, nÞ in (3.3a) and summing over allmodes, to obtain jHj � 2E=k. This is rather crude, though, as it does not take intoaccount the fact that the energy is predominantly at small scales, where K is muchlarger than k. To use this fact we substitute (2.10) into (3.3b) and integrate to obtain
jHj91 < 1=2,�1=2� > 1/2.
�ð3:4Þ
However a tighter bound still may be derived by considering dissipation of heli-city according to (3.2) above. We define the current helicity of the eigenmodeby D ¼ hbEj� þ b�Eji, with contributions Dmn ¼ bmnEj
�mn þ b�mnEjmn. Using jjmnj �
K jbmnj we have that
jDmnj � 2Kðm, nÞEmn, jDð�Þj9 2�Eð�Þ ð3:5Þ
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and so
jDj9 ��1=2�, ð > 0Þ: ð3:6Þ
We then have from (3.2) for the total helicity
jHj9 �1=2�, ð > 0Þ, ð3:7Þ
which is tighter than the inequality in (3.4) (for 0 < < 1=2). This is our main result forthe helicity. If the magnetic field is normalised so that the large-scale flux is of orderunity, the total energy increases as � ! 0, but the helicity decreases for < 1=2 or,for > 1=2, increases more slowly than energy, as no more than �1=2E.We note that this result may also be obtained by an argument of Berger (1984).
Assuming (2.12) holds, we have by the Cauchy–Schwartz inequality and (3.2)
@H
@t
� �2
� 4�2hjBj2i hjJj2i9�E @tE � �@tE2 ð3:8Þ
and so (since growth rates are of order unity),
jHj9 �1=2E, ð3:9Þ
in agreement with (3.7) above.Another measure of the helicity in the magnetic field is the relative helicity
HR ¼ H=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihjaj2ihjbj2i
pof the field, which satisfies jHRj � 1, and is used by Hughes
et al. (1996). Here, using now familiar arguments,
A � hjaj2i �1 < 1,�1� > 1,
�ð3:10Þ
and so putting this together with (2.11) and (3.7) we have
jHRj9�ð1�Þ=2 0 < < 1,1 > 1:
�ð3:11Þ
Assuming that 0 < < 1 in a typical fast dynamo,HR tends to zero in the high-conduc-tivity limit � ! 0.Hughes et al. (1996) also measure the two lengthscales given by
lA ¼ ðhjaj2i=hjbj2iÞ1=2 ��=2 0 < < 1,�1=2 > 1,
�ð3:12Þ
and
lB ¼ ðhjbj2i=hjjj2iÞ1=2 � �1=2, ð > 0Þ: ð3:13Þ
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The same results are obtained by Hughes et al. (1996), who give discussion similar toours. With 0 < < 1 we may identify the scaling exponents of these authors givenby HR � ��1 , lB � ��2 , lA � ��3 with �1 � ð1� Þ=2, �2 ¼ 1=2 and �3 ¼ =2. We alsoobtain the inequality �1 þ �3 � 1=2, which is satisfied by their numerical results forthese flows. Finally note that relative helicity of order unity in a fast dynamo is onlypossible if � 1, in which case lA, lB � �1=2. This is in agreement with the discussionof Moffatt and Proctor (1985), although we know of no numerical examples whichshow � 1.
4. NUMERICAL RESULTS
We now present some numerical simulations. Our aims are first to confirm theabove upper bounds on helicity as � ! 0, and to see how tight these bounds are.Secondly we wish to look in more detail at the distribution of helicity over scales,given by its spectrum Hð�Þ. We study a specific example, the MWþ flow of Otani(1993) given by
uðx, y, tÞ ¼ 2 cos2 t ð0, sin x, cos xÞ þ 2 sin2 t ðsin y, 0, � cos yÞ, ð4:1Þ
based on modulating two Beltrami waves of positive helicity. Related models involvingpulsed Beltrami waves were studied by Bayly and Childress (1988) and related smoothflows by Galloway and Proctor (1992) and Hughes et al. (1996). These flows are readilysimulated numerically, and we used resolutions of up to 5122 Fourier modes.
4.1. Fields of Moderate Scale
At high R ¼ 1=� the most unstable magnetic field mode has k ’ 0:8 and shows struc-ture over a range of scales; its growth rate is p ’ 0:393 for small �. We first investigatethe behaviour of helicity and energy in this case.Figure 1 shows the scaling of E (triangles), J (pluses) and jHj (crosses) as functions
of 1=� on a log–log plot. The scaling of E is very clear: the straight line observed givesthe value ’ 0:25 as obtained by Gilbert et al. (1993). The scaling for the current J isless clear; the best estimate of the slope for the smallest values of � would give ’ 0:18,but the graph has some upward curvature. In measuring the current the dissipationscale is more heavily weighted than for the energy, leading to slower convergence tothe scaling law as � ! 0. For a given � higher numerical resolutions are also requiredto obtain the data point on the J curve accurately, than for E. We therefore use theE scaling to estimate ’ 0:25.This implies from (3.7) that the total helicity H must fall off at least as fast as �0:25.
This slope is shown by the dashed line in Fig. 1 and it may be seen that the total helicityfalls off faster than this upper bound requires, more as �0:5, with some downwardcurvature. To confirm the connection with the energy spectrum, Fig. 2(a) shows Eð�Þplotted for several values of �. The spectral slope shows reasonable agreement withEð�Þ � ��0:50 (shown dashed) as suggested theoretically, but is rather noisy, despitetime averaging over one period.Although the total helicity goes to zero with �, examination of Hð�Þ shows that this
arises from significant cancellation of positive and negative helicity at different scales.
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Figure 2(b) shows the logarithm of jHð�Þj. In order to show its sign and yet use alog–log graph we in fact plot f ½Hð�Þ� against log10 �, where
f ðxÞ ¼ signðxÞ log10½maxð1, 106xÞ�: ð4:2Þ
Thus positive values of helicity appear on the graph on the range 0 to 6, and negativevalues on the range �6 to 0. Also shown by dashed lines are the bounds (for 1=� ¼ 105)
FIGURE 1 log10 E (triangles), log10 J (pluses) and log10 jHj (crosses) are plotted against log10 1=�,for Otani’s flow with k ¼ 0:8. The dashed line indicates a power law dependence of �0:25.
FIGURE 2 (a) log10 Eð�Þ (b) f ðHð�ÞÞ, plotted against log10 � for 1=� ¼ 10, 20, 50, . . . 105. In (a) the dashedline indicates a power law dependence ��0:5. In (b) the dashed lines show the upper and lower bounds obtainedfrom (3.3a).
(a)
(b)
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obtained from summing (3.3a) over all the modes in the calculation. At the very largestscale, when � ¼ 1 and only the large-scale b00 mode is included in the spectral sum, thehelicity is significant, and negative with H00 ’ �1:5 for small �. We see that althoughthe total helicity goes to zero even faster than our bounds indicate, this is not much ofa constraint on the magnetic field at large scales. The large scale modes generated in thegrowing field can have significant helicity, here negative, while helicity of the oppositesign is spread out along the cascade to small scales, the total helicity being negligiblysmall as � ! 0.
4.2. Large-scale Fields
In the above example we took k ¼ 0:8, giving the fastest growing mode in which thescale of the magnetic field is comparable with that of the fluid flow. Also of interestis the behaviour of large-scale fields with small values of k. Although these have smallergrowth rates, tending to zero as k! 0, study of these modes gives insight into themean-field dynamo regime. Therefore, as a further example, we consider Otani’sMWþ flow (4.1) with k ¼ 0:05 and k ¼ 0:01. Growth rates are shown in Fig. 3(a),
FIGURE 3 (a) Growth rate p plotted against log10 1=� for Otani’s flow with k ¼ 0:05 (upper curve),k ¼ 0:01 (lower curve). (b,c) log10 E (triangles) and log10 jHj (crosses) are plotted against log10 1=�, forOtani’s flow with (b) k ¼ 0:05 and (c) k ¼ 0:01. The dashed lines indicate a power law dependence of(b) ��0:10 and (c) ��0:22.
(a)
(b)
(c)
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and it may be seen that these saturate rather slowly as � ! 0. Figures 3(b, c) are similarto Fig. 1 and show E (triangles) and jHj (crosses) plotted against 1=� in a log–log plot(J is not shown); in Fig. 3(b) k ¼ 0:05 while in 3(c) k ¼ 0:01. Here the power law forenergy is steeper and takes longer to become established as � ! 0. Measurements of theexponent give ’ 0:60 for k ¼ 0:05 (using the range 500 � 1=� � 104) and ’ 0:72for k ¼ 0:01 (using 103 � 1=� � 104). The upper bounds on the helicity are ��0:10
and ��0:22 respectively, shown dashed, and the actual helicity lies well within thesebounds, decaying rather than growing.We note that the value of appears to increase as k is reduced, corresponding to
more energy in the small scales. With a field of larger scale in the z-direction, weapproach the situation where the field is independent of z, and the arguments ofVainshtein and Cattaneo (1992) come into play, which predict ¼ 1 for the non-dynamo case k ¼ 0, from consideration of the vector potential for the field. Theenergy spectrum is flatter, and the helicity spectrum shows greater fluctuationsthan for k ¼ 0:8 (we do not plot either of these here). There is strong negative helicityat the largest scale and positive helicity further down the spectrum.
5. HELICITY IN THE STRETCH–TWIST–FOLD PICTURE
We have seen that in Otani’s flow there is significant helicity at the largest scale,balanced by opposite helicity at smaller scales, to give a total helicity that is vanishinglysmall in the fast dynamo limit � ! 0. In this section we consider the behaviour of heli-city and its distribution in the original stretch–twist–fold (STF) fast dynamo picture ofVainshtein and Zeldovich (1972). Although the STF dynamo is only a picture, raisesmany questions, and indeed is difficult to realise numerically (see Vainshtein et al.,1996, 1997), it remains an important guide to intuition. Below we shall see that it mir-rors closely the evolution of helicity seen above in Otani’s flow. Our aims here are tounderstand how helicity is distributed over scales in a fast dynamo with a helicalflow field, and to give indications of how this is modified in a dynamical regime inwhich the Lorentz force becomes important.Figure 4 shows the basic STF process on a tube of flux (Vainshtein and Zeldovich,
1972). We note that the flow field as depicted by the bold arrows itself has a positivesense of kinetic helicity, measured by uEJ3 u, as in a basic ‘cyclonic event’ of Parker(1955). We take the helicity of the magnetic field in Fig. 4(a) to be H ¼ 0, for exampleby taking the field lines to be circular and unlinked. We also take the flux of the tube tobe � ¼ 1 and the fluid to be perfectly conducting, � ¼ 0, so that the magnetic field linetopology is preserved subsequently. Thus in Fig. 4(d) the helicity remains zero, but wemay identify two, cancelling contributions as follows. To do this we use Moffatt andRicca (1992) in the rest of this section; this paper should be consulted for further dis-cussion, references and historical perspective.The first contribution to the helicity comes from considering the centre line of the
tube; initially this is just a circle C0, but is depicted after the stretch–twist operationsin Fig. 5(a), and as the curve C1 after after one complete STF operation in Fig. 5(b).When deformed to lie almost in a plane, C1 (nearly) crosses itself just once. If we associ-ate an integer �1 with the crossing according to its sense as also shown in Fig. 5(a) (onthe right), we obtain that the ‘writhe’ of C1 is �1 and this gives a ‘writhe contribution’
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to the helicity of Hwr ¼ �1. More generally the writhe of any curve C is given by theintegral
WðCÞ ¼1
4�
IC
IC
dr � dr0 � ðr � r0Þ
jr � r0j3: ð5:1Þ
There is a compensating ‘twist’ of field lines inside the tube : relative to the tube’s centreline (or more precisely a Serret–Frenet framing of the curve C1) the field lines rotatethrough an angle 2� when the centre line is traversed once (Moffatt and Proctor,1985). This gives a ‘twist’ of þ1, and the ‘twist contribution’ to the helicity isHtw ¼ þ1. The total helicity after one STF step is H ¼ Hwr þHtw ¼ 0.Although the two forms of helicity sum to zero, they are associated with different
scales of the field. The positive twist helicity is associated with structure internalto the tube, whereas the negative writhe helicity is related to the twisting of the pairof tubes about each other. Thus the STF process has generated positive helicityat the smallest scale, and negative helicity at a slightly larger scale. Although idealised,the STF picture captures nicely the well-known transfer of negative magnetic helicityto larger scale, and positive to smaller scale, in a flow field with positive kinetic helicity(see, e.g., Moffatt, 1978; Pouquet et al., 1976; Brandenburg, 2001).Now consider applying the stretch–twist [Fig. 5(c)] and fold [Fig. 5(d)] operations a
second time; the centre line C2 of the tube is shown. This time the stretch–twist of apair of flux tubes gives a writhe of �4 as pairs of 2 tubes cross each other. Howeverat the same time two individual tubes wrap around each other and this givesanother contribution to the writhe of þ 2 as shown. Again we may consider the
FIGURE 4 The stretch–twist–fold process: an initial flux tube (a), is stretched (b), twisted (c) and folded(d), to obtain a doubled flux tube.
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writhe as distributed over different scales. On the largest scale involving 4 ¼ 22 tubes,there is a writhe of Hwr2 ¼ �4. On the next scale, involving 2 ¼ 21 tubes, there is awrithe of Hwr1 ¼ 2� 1 ¼ 1. The total writhe is then Hwr ¼ Hwr1 þHwr2 ¼ �3 and sothe corresponding twist is Htw ¼ 3 to compensate.The general picture now becomes clear. After the step n� 1 of the STF process
we have 2n�1 tubes of field with centre line Cn�1. We then do the stretch–twist–foldoperation. This generates a bundle of 2n tubes of flux with centre line Cn. Associatedwith the fold operation is positive and negative writhe. The negative writhe is on thelargest scale, involving all 2n tubes, and corresponds to 2n�1 tubes crossing over 2n�1
tubes and is �2n�1 � 2n�1 ¼ �22n�2 [Fig. 5(e)]. The positive writhe is on the nextscale down, involving 2n�1 tubes being twisted through 2�, and gives a writhe of2n�1ð2n�1 � 1Þ [Fig. 5(f)]. If after n steps we let Hwr j denote the writhe that involves2j tubes we have for the largest scale
Hwr n ¼ �22n�2, ð j ¼ nÞ, ð5:2Þ
while for smaller scales,
Hwr j ¼ �22j�2 þ 2jð2j � 1Þ, ð j ¼ 1, 2, . . . , n� 1Þ: ð5:3Þ
FIGURE 5 The centre line of the flux tube after (a,b) one and (c,d) two iterations of the stretch–twist–foldprocess. Contributions to the total writhe of the curve (with sign convention shown in (a)) are marked neareach set of crossings. The writhe generated at the nth iteration is shown as (e) the negative contribution, (f) thepositive contribution.
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In (5.3) the negative writhe is generated at step j, and the positive writhe at step j þ 1.The total writhe involving 2j tubes is
Hwr j ¼ 2j�1ð3� 2j�1 � 2Þ, ð j ¼ 1, 2, . . . , n� 1Þ: ð5:4Þ
The total twist helicity is Htw ¼ 2n � 1, given by the field structure internal to the tubes.Although the STF picture is very idealised, it does throw light on the numerical
results and their relation with the analysis of Sections 2 and 3. First, we observe thesame structure to the helicity spectrum, with negative helicity at the largest scale, andpositive helicity at all smaller scales. Second, we note that the net positive writhe at agiven intermediate scale comes from a positive and a negative contribution, which wehave been careful to isolate above in (5.3). Kinematically there is no reason why thepositive and negative contributions, for example the þ 2 and �1 shown in Fig. 5(d),should cancel other than fortuitously. This agrees with the numerical results thatindicate that the inequality jHð�Þj92Eð�Þ=� is not well saturated in simulations [seeFig. 2(b)]; there is more energy at a given scale than is required by the total helicityat that scale. Finally if weak diffusion � > 0 is now introduced, obviously the tubesof field will start to reconnect when they reach scales of order �1=2. The twist helicitywill be destroyed first, and then writhe Hwr j for increasing j. The total helicity willbe negative and small.The above picture, although crude, also gives some indications about what may
occur initially when Lorentz forces become important. Let us consider the case whenthe fluid Reynolds number is of a similar size to the magnetic Reynolds number for sim-plicity. It is clear, looking at Fig. 5(d), that it is energetically favourable for the þ 2writhe and the �1 writhe now to cancel by the interchange of sections of field lines.This would immediately reduce the energy while preserving the field line topology atthis scale. Similarly the plus and minus contributions (5.3) to the writhe at a givenscale may be partially cancelled by interchange motions. Thus we may expect that atthe onset of dynamical effects the total number of crossings of the magnetic field willbe reduced by dynamical, magnetic relaxation effects (Moffatt, 1985). The ‘crossingnumber’ of a continuous vector field has been defined by Freedman and He (1991)(see also Berger, 1993) as
CðBÞ ¼1
4�
Z ZjBðrÞ3Bðr0ÞEðr � r0Þj
jr � r0j3d3r d3r0: ð5:5Þ
This definition is analogous to (5.1) for a continuous field, but with the absolute valueof the integrand taken. We may expect that this quantity decreases significantly at theonset of saturation, while plus and minus writhe cancel through interchange motions.This also suggests that the upper bound jHð�Þj92Eð�Þ=� will be quite closely attainedafter equilibration, and this is in fact seen in numerical simulations (Brandenburg, 2001).
6. DISCUSSION
We have discussed the role of helicity in kinematic dynamos for the astrophysicallyimportant limit of large magnetic Reynolds number R. We have focussed on fast dyna-mos, in which the growth rates remain of order the turn-over time-scale in this limit.
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Hughes et al. (1996) showed that the relative magnetic helicity goes to zero as R! 1
in a growing field, and we have derived an upper bound on the helicity, relating it tothe slope of the energy spectrum. Studying the distribution of helicity over differentscales revealed however that there is a great deal of cancellation: in the exampleswe considered there was strong negative helicity at the largest scale, almost entirelycancelled by positive helicity at moderate and small scales. A similar behaviour wasseen in the idealised STF picture of a fast dynamo, and some predictions were madeof behaviour at the onset of saturation, when Lorentz tension forces come into play.The flow fields we have used have maximal kinetic helicity uEJ3 u. Hughes et al.
(1996) study a number of dynamo flows which have zero kinetic helicity and itwould be interesting to consider the role of magnetic helicity in these dynamos inmore detail. Obviously the inequalities we have derived remain correct as they useno information about the flow field. However the distribution of magnetic helicityover the different scales will probably be less coherent in fast dynamos with zero kinetichelicity.Our study may be summarised by saying that the exact conservation of helicity in
ideal MHD is replaced by very low levels of total helicity at large R in a kinematicfast dynamo. However this low total helicity does not seem to act as a constrainton field growth nor on the helicity of the large-scale field. In our example we haveseen a strongly helical field on the large scale, with smaller scales containing cancellinghelicity of the opposite sign.This may be contrasted with recent results on nonlinear dynamos, in which helicity
appears to exert a controlling influence on the growth of large-scale fields(Brandenburg, 2001; see also Vishniac and Cho, 2001). Here the loss of helicity atsmall scales by diffusion, or through open boundaries of the system (Blackman andField, 2000; Brandenburg and Dobler, 2001) appears to limit the total helicity in thesystem, and so the magnitude of the large-scale field. It is also known that non-lineardynamos with helical and non-helical forcing of the velocity field show very differentbehaviour of the large-scale fields (e.g., Pouquet et al., 1976; Maksymczuk andGilbert, 1999), whereas in kinematic fast dynamo theory both helical and nonhelicalflows allow seemingly similar fast dynamo action (Otani, 1993; Hughes et al., 1996).We have to ask why the linear and nonlinear regimes are so different when the
flow or forcing is helical. In the linear models we have studied the slope of theenergy spectrum is quite shallow and this allows cancellation of helicity over a rangeof scales without violating the constraint (3.3). However in nonlinear MHD the fieldand flow usually adopt steeper spectral slopes through nonlinear interactions, aroundk�5=3, and this means that cancellation of helicity cannot occur over a wide range ofscales. There is still the possibility that there could be cancelling helicity in the modesat the largest scales; however in simulations (e.g., Brandenburg, 2001) it is apparentthat there is a strong tendency for the largest scales to have a single sign of helicity.It appears that this dominant helicity can then only change through dissipation atthe small scales or loss through boundaries.On this note, the following numerical simulation might be of interest to determine to
what extent dissipation of helicity controls a large-scale field. A nonlinear run (1) couldbe started using a velocity forcing with positive helicity and run up to time t ¼ T . Inanother run (2) the forcing would have positive helicity for 0 � t � T=2 but negativefor T=2 � t � T . There are then two possible outcomes for the second run. First,the field may take a significant time to adjust after t ¼ T=2 and to reverse its magnetic
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helicity at large scales. During this adjustment period we would expect to see a decayof magnetic energy and halting of any inverse cascade (see Pouquet et al., 1976,Gilbert and Sulem, 1990). If this long adjustment occurs (and its length increasesfor R! 1) before the evolution resumes as the mirror image of run (1), we mayconclude that the large-scale field is controlled by dissipation of helicity. The secondpossibility is that the large-scale fields adjust rapidly at t ¼ T=2 and change sign ofhelicity (in a fashion independent of R for large R), this change being taken up bythe helicity at lower scales. During the transition the magnetic energy would notbe diminished significantly, and the inverse cascade would be only interrupted briefly.If this is the case then it will be clear that helicity is playing a minor role in drivingthe nonlinear dynamo.
Acknowledgements
I am grateful to Axel Brandenburg, Mitch Berger, Steve Childress, Keith Moffatt,Mike Proctor, Renzo Ricca and Andrew Soward for useful discussions and helpfulreferences. This work was undertaken at the Geometry and Topology of Fluid Flowprogramme at the Isaac Newton Institute of Mathematical Sciences in Cambridge.I would like to thank the organisers and staff for a very fruitful programme. Largenumerical runs were performed on the University of Exeter IBM SP/3 supercomputer,suported by the HEFCE JREI.
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