Astrophysical magnetic ﬁelds and nonlinear dynamo theory4 Dynamos and the ﬂow of energy 43 4.1...
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Astrophysical magnetic fields and nonlinear
Axel Brandenburg a and Kandaswamy Subramanian b
aNORDITA, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark.
email: [email protected]
bInter University Centre for Astronomy and Astrophysics, Post Bag 4, Pune
University Campus, Ganeshkhind, Pune 411 007, India.
email: [email protected]
The current understanding of astrophysical magnetic fields is reviewed with par-ticular emphasis on nonlinear dynamo theory. Analytic and numerical results arediscussed both for small scale dynamos, where helicity is unimportant, and for largescale dynamos, where kinetic helicity is crucial. Large scale dynamos produce smallscale magnetic helicity as a waste product that quenches the large scale dynamoprocess (alpha effect). With this in mind, the microscopic theory of the alpha effectis revisited in full detail and new results for current helicity fluxes are presented.
Key words: magnetohydrodynamics, dynamos, turbulence, mean field theoryPACS: 52.30.-q, 52.65.Kj, 47.11.+j, 47.27.Ak, 47.65.+a, 95.30.Qd
1 Introduction 5
2 Magnetic field observations 7
2.1 Solar and stellar magnetic fields 7
2.2 Magnetic fields in accretion discs 13
2.3 Galactic magnetic fields 14
2.4 Magnetic fields in clusters of galaxies 18
3 The equations of magnetohydrodynamics 21
3.1 Maxwell’s equations 21
Draft available under http://www.nordita.dk/~brandenb/tmp/phys_rep
3.2 Resistive MHD and the induction equation 22
3.3 Stretching, flux freezing and diffusion 26
3.4 Magnetic helicity 29
3.5 The momentum equation 33
3.6 Kinetic helicity evolution 34
3.7 Energy and helicity spectra 34
3.8 Departures from the one-fluid approximation 37
3.9 The Biermann battery 41
4 Dynamos and the flow of energy 43
4.1 Energetics 44
4.2 Kinematic dynamos 45
4.3 Fast dynamos: the stretch-twist-fold picture 50
4.4 Fast ABC-flow dynamos 52
5 Small scale turbulent dynamos 53
5.1 General considerations 54
5.2 Kazantsev theory 54
5.3 Saturation of the small scale dynamo 61
5.4 Simulations 65
5.5 Comments on the Batchelor mechanism 71
5.6 Small and large scale dynamos: a unified treatment 72
6 Large scale turbulent dynamos 76
6.1 Phenomenological considerations 76
6.2 Mean-field electrodynamics 78
6.3 Calculation of turbulent transport coefficients 81
6.4 Transport coefficients from simulations 87
6.5 α2 and αΩ dynamos: simple solutions 90
7 Magnetic helicity conservation and inverse cascade 99
7.1 Inverse cascade in 3-mode interactions 100
7.2 The EDQNM closure model 101
8 Simulations of large scale dynamos 104
8.1 The basic equations 105
8.2 Linear behavior 106
8.3 Nonlinear behavior 107
8.4 Emergence of a large scale field 108
8.5 Importance of magnetic helicity 110
8.6 Alpha effect versus inverse cascade 114
8.7 Nonlinear α effect 114
8.8 Alpha quenching in isotropic box simulations 115
8.9 Dynamo waves in simulations with shear 116
8.10 The magnetic helicity constraint in a closure model 119
9 Magnetic helicity in mean field models 122
9.1 General remarks 122
9.2 A heuristic treatment for the current helicity term 124
9.3 The minimal tau approximation: nonlinear effects 125
9.4 The dynamical quenching model 129
9.5 Saturation behavior of α2 and αΩ dynamos 133
9.6 How close to being resistively limited is the 11 year cycle? 141
9.7 Open boundaries 143
9.8 Mean-field models of magnetic helicity losses 150
10 Revisiting the microscopic theory using MTA 151
10.1 Transport coefficients in weakly inhomogeneous turbulence 152
10.2 Isotropic, helical, nonrotating turbulence 160
10.3 Turbulence with helicity induced by rotation 161
10.4 Nonlinear helicity fluxes using MTA 163
11 Discussion of dynamos in stars and galaxies 169
11.1 General considerations 170
11.2 The solar dynamo problem 171
11.3 Stellar dynamos 175
11.4 Accretion disc dynamos 183
11.5 Galactic dynamos 186
11.6 Cluster magnetic fields 194
12 Where do we stand and what next? 197
A Evolution of the correlation tensor of magnetic fluctuations 199
B Other approaches to calculating α and ηt 204
B.1 The lagrangian approach to the weak diffusion limit 204
B.2 Delta-correlated velocity fields 205
C Calculating the quasi-linear correction EN 207
D Derivation of Eq. (10.28) 210
E Calculation of the second and third terms in (10.38) 211
F Comparison with the paper by Radler, Kleeorin, andRogachevskii 212
G Calculation of Ψ1 and Ψ2 213
Magnetic fields are ubiquitous in the universe. Our most immediate encounterwith magnetic fields is the Earth’s field. This is not only useful for naviga-tion, but it also protects us from hazardous cosmic ray particles. Magneticfields play an important role in various branches of astrophysics. They areparticularly important for angular momentum transport, without which thesun and similar stars would not spin as slowly as they do today . Magneticfields are responsible for the loops and arcades seen in X-ray images of thesun and in heating the coronae of stars with outer convection zones . Theyplay a crucial role in driving turbulence in accretion discs providing perhapsthe crucial viscosity needed for accretion. Large scale fields in these discs arealso thought to be crucial in driving jets. A field permeating a rotating blackhole probably provides one of the most efficient ways of extracting energy topower the jets from active galactic nuclei. Magnetic fields with micro-gaussstrength and coherence scales of order several kiloparsecs are also observedin nearby galaxies and perhaps even in galaxies which have just formed. Themagnetic field strength in galactic spiral arms can be up to 30 microgauss(e.g. in M51). Fields of order several micro-gauss and larger, with even largercoherence scales are seen in clusters of galaxies. To understand the origin ofmagnetic fields in all these astrophysical systems is a problem of great impor-tance.
The universe probably did not begin magnetized. There are various processessuch as battery effects, which can lead to a weak magnetic field, from zero ini-tial fields. Most of these batteries lead to field strengths much weaker than theobserved field. So some way of amplifying the field is required. This is probablyaccomplished by the conversion of kinetic energy into magnetic energy, a pro-cess generally referred to as a dynamo; see Ref.  for a historic account. Somebasic principles of dynamos are well understood from linear theory, but virtu-ally all astrophysical dynamos are in a regime where the field is dynamicallyimportant, and kinematic theory is invalid. In recent years our understandingof nonlinear properties of dynamos has advanced rapidly. This is partly dueto new high resolution numerical simulations which have also triggered fur-ther developments in analytic approaches. An example is the resistively slowsaturation phase of dynamos with helicity that was first seen in numericalsimulations , which then led to the development of a dynamical quenchingmodel [5–8]. The dynamical quenching model was actually developed muchearlier , but it was mostly applied in order to explain chaotic behavior ofthe solar cycle [10–12]. Another example is the so-called small scale dynamowhose theory goes back to the early work of Kazantsev . Again, only in re-cent years, with the advent of fast computers allowing high Reynolds numbersimulations of hydromagnetic turbulence, the small scale dynamo has becomepractical reality. This in turn has triggered further advances in understanding
this problem, especially the nonlinear stages.
Although there have been a number of excellent reviews about dynamo the-ory and comparisons with observations of astrophysical magnetic fields [14–22],there have been many crucial developments just over the past few years involv-ing primarily magnetic helicity. It has now become clear that nonlinearity inlarge scale dynamos is crucially determined by the magnetic helicity evolutionequation. At the same time, magnetic helicity has also become highly topicalin observational solar physics, as is evidenced by a number of recent special-ized meetings on exactly this topic . Magnetic helicity emerges thereforeas a new tool in both observational as well as in theoretical studies of astro-physical magnetohydrodynamics (MHD). This review discusses the details ofwhy this is so, and how magnetic helicity can be used to constrain dynamotheory and to explain the behavior seen in recent simulations of dynamos inthe nonlinear, high magnetic Reynolds number regime.
We also review some basic properties and techniques pertinent to mean field(large scale) dynamos, so that newcomers to the field can gain a broaderview and are able to put new developments into perspective. In particular, wediscuss a simplistic form of the so-called tau approximation that allows thecalculation of mean field turbulent transport coefficients in situations wherethe magnetic fluctuations strongly exceed the magnitude of the mean field.This is when the quasilinear theory (also known as first order smoothing orsecond order correlation approximation) breaks down. We then lead to thenow so intriguing question of what saturates the dynamo and why so muchcan be learned by rather simple considerations in terms of magnetic helicity.
In turbulent fluids, the generation of large scale magnetic fields is genericallyaccompanied by the more rapid growth of small scale fields. The growingLorentz force due to these fields can backreact on the turbulence to modifythe mean field dynamo coefficients. So another topic of great current interestis the nonhelical small scale dynamo, and especially its nonlinear saturation.This could also be relevant for explaining the origin of cluster magnetic fields.These topics are therefore reviewed in the light of recent advances using bothanalytic tools as well as high resolution simulations.
There are obviously many topics that have been left out, because they touchupon nonlinear dynamo theory only remotely. Both hydrodynamic and magne-tohydrodynamic turbulence are only discussed in their applications, but thereare many fundamental aspects that are interesting in their own right; see thetext books by Frisch  and Biskamp  and the work by Goldreich andSridhar ; for a recent review see Ref. . Another broad research areathat has been left out completely is magnetic reconnection and low beta plas-mas. Again, we can here only refer to the text book by Priest and Forbes .More close connections exist with hydrodynamic mean field theory relevant
for explaining differential rotation in stars . Even many of the applicationsof dynamo theory are outlined only rather broadly, but again, we can referto a recent text book by Rudiger and Hollerbach  where many of thoseaspects are addressed.
We begin in the next section with some observational facts that may have achance in finding an explanation in terms of dynamo theory within the not toodistant future. We then summarize some useful facts on basic MHD, and alsodiscuss briefly battery effects to produce seed magnetic fields. Some generalproperties of dynamos are discussed in Section 5 before turning to small scaledynamos in Section 6. The main theme of large scale dynamos is extensivelycovered in Sections 6-10, and in Section 11 we discuss the application of theseideas to various astrophysical systems.
2 Magnetic field observations
In this section we discuss properties of magnetic fields observed in variousastrophysical settings. We focus specifically on aspects that are believed to beimportant for dynamo theory.
2.1 Solar and stellar magnetic fields
The sun has a magnetic field that manifests itself in sunspots through Zee-man splitting of spectral lines . It has long been known that the sunspotnumber varies cyclically with a period between 7 and 17 years. The longi-tudinally averaged component of the radial magnetic field of the sun [32,33]shows a markedly regular spatio-temporal pattern where the radial magneticfield alternates in time over the 11 year cycle and is oppositely oriented on theother hemisphere (Fig. 2.1). One can also see indications of a migration of thefield from mid latitudes toward the equator and the poles. This migration isalso well seen in a sunspot diagram, which is also called a butterfly diagram,because the pattern formed by the positions of sunspots in time and latitudelooks like a sequence of butterflies lined up along the equator (Fig. 2.2).
At the solar surface the azimuthally averaged radial field is only a few gauss(1G = 10−4Tesla). This is rather weak compared with the peak magneticfield in sunspots of about 2 kG. In the bulk of the convection zone, becauseof differential rotation, the magnetic field is believed to point mostly in theazimuthal direction, and it is probably much larger near the bottom of theconvection zone due to an effect known as downward pumping (Section 6.4).
Fig. 2.1. Longitudinally averaged radial component of the observed solar magneticfield as a function of cos(colatitude) and time. Dark shades (blue) denote negativevalues and light shades (yellow) denote positive values. Note the sign changes bothin time and across the equator (courtesy of R. Knaack).
Fig. 2.2. Solar butterfly diagram showing the sunspot number in a space-time dia-gram. Note the migration of sunspot activity from mid-latitudes toward the equator(courtesy of D. N. Hathaway).
2.1.1 Estimates of the field strength in the deeper convection zone
In the bulk of the solar convection zone the thermal energy transport is rea-sonably well described by mixing length theory . This theory yields arough estimate for the turbulent rms velocity which is around urms = 20-50m s−1 at the bottom of the solar convection zone. With a density of aboutρ = 0.2 g cm−3 this corresponds to an equipartition field strength of about3−8 kG. (The equipartition field strength is here defined as Beq =
where µ0 is the magnetic permeability.)
A similar estimate is obtained by considering the total (unsigned) magneticflux that emerges at the surface during one cycle. This argument is dubious,
because there is no compelling reason why all flux tubes in the sun need tohave emerged at the solar surface exactly once. Nevertheless, the notion ofmagnetic flux (and especially unsigned flux) is rather popular in solar physics,because this quantity is readily accessible from solar magnetograms. The totalunsigned magnetic flux is roughly estimated to be 1024Mx. Distributed overa meridional cross-section of about 500Mm in the latitudinal direction andabout 50Mm in radius (i.e. the lower quarter of the convection zone) yieldsa mean field of about 4 kG, which is in fair agreement with the equipartitionestimate above.
Another type of estimate concerns not the mean field but rather the peakmagnetic field in the strong flux tubes. Such tubes are believed to be ‘stored’either just below or at the bottom of the convection zone. By storage onemeans that the field survives reasonably undisturbed for a good fraction ofthe solar cycle and evolves mostly under the amplifying action of differentialrotation. Once such a flux tube becomes buoyant in one section of the tube itrises, expands and becomes tilted relative to the azimuthal direction owing tothe Coriolis force. Calculations based on the thin flux tube approximation predict field strengths of about 100 kG that are needed in order to producethe observed tilt angle of bipolar sunspots near the surface .
The systematic variation of the global field of the sun is commonly explainedin terms of αΩ dynamo theory (Sections 6.5 and 11.2), but this theory facesa number of problems that will be discussed later. Much of the resolution ofthese problems focuses around magnetic helicity. This has become a very activeresearch field in its own right. Here we discuss the observational evidence.
2.1.2 Magnetic helicity of the solar field
Magnetic helicity studies have become an important observational tool toquantify the complexity of the sun’s magnetic field. Examples of complexmagnetic structures being ejected from the solar surface are shown in Fig. 2.3.For a series of reviews covering the period until 1999 see Ref. . The sig-nificance of magnetic helicity for understanding the nonlinear dynamo hasonly recently been appreciated. Here we briefly review some of the relevantobservational findings.
The only information about the magnetic helicity of the sun available to dateis from surface magnetic fields, and these data are necessarily incomplete.Nevertheless, some systematic trends can be identified.
Vector magnetograms of active regions show negative (positive) current helic-ity in the northern (southern) hemisphere [38–41]. From local measurementsone can only obtain the current helicity density, so nothing can be concludedabout magnetic helicity, which is a volume integral. As we shall show later
Fig. 2.3. The famous prominence of 1946 (left) and a big coronal mass eruptionfrom the LASCO coronograph on board the SOHO satellite (right). Note the com-plexity of the ejected structures, being suggestive of helical nature. Courtesy of theSOHO consortium. SOHO is a project of international cooperation between ESAand NASA.
Fig. 2.4. Net magnetic flux through the solar surface at the northern hemisphere(left hand panel) and magnetic helicity flux for northern and southern hemispheres(right hand panel, lower and upper curves, respectively). Adapted from Berger andRuzmaikin .
(Section 3.7), the spectra of magnetic and current helicity are however simplyrelated by a wavenumber squared factor. This implies that the signs of currentand magnetic helicities agree if they are determined in a sufficiently narrowrange of length scales. We return to this important issue in Section 6.4.
Berger and Ruzmaikin  have estimated the flux of magnetic helicity fromthe solar surface using magnetograms. They discussed α effect and differentialrotation as the main agents facilitating the loss of magnetic helicity. Theirresults indicate that the flux of magnetic helicity due to differential rotationand the observed radial magnetic field component is negative (positive) in thenorthern (southern) hemisphere, and of the order of about 1046Mx2 integratedover the 11 year cycle; see Fig. 2.4.
Chae  estimated the magnetic helicity flux based on counting the crossingsof pairs of flux tubes. Combined with the assumption that two nearly alignedflux tubes are nearly parallel, rather than anti parallel, his results again suggest
Fig. 2.5. X-ray image at 195A showing an N-shaped sigmoid (right-handed writhe)of the active region NOAA AR 8668 at the northern hemisphere (1999 August 21at 18:51 UT). Adapted from Gibson et al. .
that the magnetic helicity is negative (positive) in the northern (southern)hemisphere. The same sign distribution was also found by DeVore  whoconsidered magnetic helicity generation by differential rotation. He finds thatthe magnetic helicity flux integrated over an 11 year cycle is about 1046Mx2
both from active regions and from coronal mass ejections. Thus, the sign agreeswith that of the current helicity obtained using vector magnetograms. Morerecently, Demoulin et al.  showed that oppositely signed twist and writhefrom shear are able to largely cancel, producing a small total magnetic helicity.This idea of a bi-helical field is supported further by studies of sigmoids :an example is Fig. 2.5, which shows a TRACE image of an N-shaped sigmoid(right-handed writhe) with left-handed twisted filaments of the active regionNOAA AR 8668, which is typical of the northern hemisphere. This observationis quite central to our new understanding of nonlinear dynamo theory [47,48]and will be addressed in more detail below.
2.1.3 Magnetic fields of late type stars
Looking at other stars is important for realizing that the solar dynamo isnot unique and just one particular example of a cyclic dynamo. In fact, wenow know that all stars with outer convection zones (usually referred to as‘late-type stars’) have magnetic fields whose strength tends to increase withtheir angular velocity. Some very young stars (e.g. T Tauri stars) have averagefield strengths of about 2 kG . These stars are fully convective and their
Fig. 2.6. Time traces of the relative Calcium H and K line emission, S, for 4 stars(including the sun) with oscillatory activity behavior between the years 1966 and1992 (adapted from Baliunas et al. )
field varies in a more erratic fashion. Cyclic variations are known to existonly for stars with colors B − V in the range 0.57 and 1.37, i.e. for spectraltypes between G0 and K7 . Some examples of the time traces are shownin Fig. 2.6. The sun’s color is 0.66, being close to the upper (bluer) end of themass range where stars show cyclic activity. For the stars in this mass rangethere exists an empirical relation between three important parameters. One isthe inverse Rossby number, Ro−1 ≡ 2Ωτturnover, where τturnover ≈ ℓ/urms is theturnover time of the convection, estimated in terms of the mixing length, ℓ,and the rms velocity of the convection, urms. The second parameter is the ratioof cycle to rotation frequency, ωcyc/Ω, where ωcyc = 2π/Pcyc and Pcyc is thecycle period (≈ 11 years for the sun, but ranging from 7 to 21 years for otherstars). The third parameter is the ratio of the mean chromospheric Calcium Hand K line emission to the bolometric flux, 〈R′
HK〉, which can be regarded as aproxy of the normalized magnetic field strength, with 〈R′
HK〉 ∝ (|〈B〉|/Beq)κ
and κ ≈ 0.47; see Ref. . These three parameters are related to each otherby
ωcyc/Ω = c1Ro−σ, ωcyc/Ω = c2〈R′
HK〉ν 〈R′HK〉 = c3Ro
where c1 = c2cσ3 and σ = µν. It turns out that the slopes σ and ν are positive
for active (A) and inactive (I) stars and that both groups of stars fall on distinctbranches with σA ≈ 0.46 and νA ≈ 0.85 for active stars and σI ≈ 0.48 andνI ≈ 0.72 for inactive stars . In Fig. 2.7 we present scatter plots showingthe mutual correlations between each of the three quantities for all cyclicstars whose parameters have been detected with quality parameters that werelabeled  as ‘good’ and ‘excellent’.
The fact that the cycle frequency depends in a systematic fashion on eitherRo−1 or on 〈R′
HK〉 suggests that for these stars the dynamo has a rather stabledependence on the input parameters. What is not well understood, however,
Fig. 2.7. Mutual correlations between the three quantities ωcyc/Ω, Ro−1, and 〈R′
HK〉.Note the two distinct branches with positive slope in the first two panels. Thenumbers and letters in the plots are abbreviations for specific active and inactivestars, respectively (adapted from Ref. , where also a key with the abbreviationsof all stars is given).
is the slope σ ≈ 0.5 in the relation ωcyc/Ω ∼ Ro−σ, and the fact that there aretwo distinct branches. We note that there is also evidence for a third branchfor even more active (‘superactive’) stars, but there the exponent σ is negative. Standard dynamo theory rather predicts that σ is always negative .We return to a possible interpretation of the exponent σ and the origin of thedifferent branches in Section 11.3.2.
The activity of all stars grows with their rotation rate. Plots similar to thethird panel of Fig. 2.7 have also been produced for other activity proxies .This work shows that there is a relation between activity proxy and inverseRossby number not only for stars with cycles, but for all late type stars withouter convection zones – even when the stars are members of binaries.
2.2 Magnetic fields in accretion discs
Gaseous discs spinning around some central object are frequently found invarious astrophysical settings, for example around young stars, stellar masscompact objects (white dwarfs, neutron stars, or black holes), or in supermas-sive (107 − 109M⊙) black holes that have been found or inferred in virtuallyall galaxies.
Explicit evidence for magnetic fields in discs is sparse: magnetization of me-teorites that were formed in the disc around the young sun  or proxies ofmagnetic activity such as Hβ line emission from discs in binary stars . Adirect search for Zeeman-induced splitting of the maser lines in the accretiondisc of the Seyfert II galaxy NGC 4258 has resulted in upper limits of < 50mG
for the toroidal component of the B field at a distance of about 0.2 pc fromthe central black hole .
On theoretical grounds, however, because of the magnetorotational instability, we strongly expect that all discs are magnetized, or at least that theyhave been magnetized during some stage in their evolution. We also know thatprotostellar discs typically formed in a magnetized environment. Faraday ro-tation measure (RM) maps of the central parsecs of quasars and radio galaxieshosting relativistic jets (cf. ) also reveal that the medium on parsec scalessurrounding AGNs could be significantly magnetized .
Many accretion discs around black holes are quite luminous. For example, theluminosity of active galactic nuclei can be as large as 100 times the luminosityof ordinary galaxies. Here, magnetic fields provide the perhaps only source ofan instability that can drive the turbulence and hence facilitate the conversionof potential energy into thermal energy or radiation. In discs around activegalactic nuclei the magnetic field may either be dragged in from large radiior it may be regenerated locally by dynamo action. The latter possibility isparticularly plausible in the case of discs around stellar mass black holes.Simulations have been carried out to understand this process in detail; seeSection 11.4. Magnetic fields may also be crucial for driving outflows fromdiscs. In many cases these outflows may be collimated by the ambient magneticfield to form the observed narrow jets .
2.3 Galactic magnetic fields
Galaxies are currently the only astrophysical bodies where a large scale mag-netic field can be seen inside the body itself. In the case of stars one onlysees surface manifestations of the field. Here we describe the structure andmagnitude of such fields.
2.3.1 Synchrotron emission from galaxies
Magnetic fields in galaxies are mainly probed using radio observations of theirsynchrotron emission. Excellent accounts of the current observational statuscan be found in the various reviews by Beck [64–67] and references therein. Wesummarize here those aspects which are relevant to our discussion of galacticdynamos. Some earlier reviews of the observations and historical perspectivescan be found in Refs. [15–17,68]. A map of the total synchrotron intensity al-lows one to estimate the total interstellar magnetic field in the plane of the sky(averaged over the volume sampled by the telescope beam). The synchrotronemissivity also depends on the number density of relativistic electrons, andso some assumption has to be made about its density. One generally as-
sumes that the energy densities of field and particles are in equipartition.(Equipartition holds between magnetic fields and relativistic protons so thatthe proton/electron ratio enters as another assumption, with 100 taken asa standard value). In our Galaxy the accuracy of the equipartition assump-tion can be tested, because we have independent measurements of the localcosmic-ray electron energy density from direct measurements and about thecosmic-ray proton distribution from γ-ray data. The combination of thesewith the strength of the radio continuum synchrotron emission gives a localstrength of the total magnetic field of 6± 1µG , which is almost the samevalue as that derived from energy equipartition .
The mean equipartition strength of the total magnetic field for a sample of 74spiral galaxies is 〈Btot〉 = 9µG [64,72]. The total field strength ranges from〈Btot〉 ∼ 4µG, in radio faint galaxies like M31 and M33 to 〈Btot〉 ∼ 15µG ingrand-design spiral galaxies like M51, M83 and NGC 6946 . The strengthof the total field in the inner spiral arms of M51 is about 30µG.
Synchrotron radiation is intrinsically highly linearly polarized – by 70-75% ina completely regular magnetic field. The observable polarization is howeverreduced due to a number of reasons. First the magnetic field usually has atangled component which varies across the telescope beam (geometrical depo-larization); second due to Faraday depolarization in the intervening mediumand third because some part of the radio emission arises due to thermal con-tinuum emission, rather than synchrotron emission. A map of the polarizedintensity and polarization angle then gives the strength and structure of theordered field, say B in the plane of the sky. Note that polarization can alsobe produced by any random field, which is compressed or stretched in onedimension (i.e. an anisotropic field which incoherently reverses its directionfrequently) [70,71]. So, to make out if the field does really have large scaleorder one needs also a map of Faraday rotation measures (RMs), as this willshow large scale coherence only for ordered fields. Such a map also probes thestrength and direction of the average magnetic field along the line of sight.
The large scale regular field in spiral galaxies (observed with a resolution ofa few 100 pc) is ordered over several kpc. The strength of this regular fieldis typically 1...5µG, and up to ∼ 13µG in the interarm region of NGC 6946,which has an exceptionally strong large scale field . In our Galaxy the largescale field inferred from the polarization observations is about 4µG, giving aratio of regular to total field of about 〈B0〉/〈Btot〉 ∼ 0.6−0.7 [74–76]. Howeverthe value inferred from pulsar RM data is [77–79] 〈B0〉 ≈ 1.4±0.2µG, which isless than the above estimate. This may be understood if there is anticorrelationbetween the electron density ne and the total field B .
In the context of dynamo theory it is of great interest to know the ratio of theregular to the random component of the magnetic field in galaxies. This is not
easy to determine, especially because of the systematic biases that can arisein the magnetic field estimates . The ratio of regular to random fields istypically 1 in interarm regions and 0.5 or less in spiral arms (R. Beck, privatecommunication).
2.3.2 Global structure of galactic fields
The global structure of the mean (or regular) magnetic field and the totalfield (mean + random) are also of interest. The random field is almost alwaysstrongest within the spiral arms and thus follows the distribution of cool gasand dust. The regular field is generally weak within spiral arms, except for rarecases like M51 with strong density waves. Thus the total field is also strongestwithin the spiral arms where the random field dominates. The strongest totaland regular fields in M51 are located at the positions of the prominent dustlanes on the inner edges of the optical spiral arms [81,82], as expected if itwere due to compression by density waves, but the regular field also extendsfar into the interarm regions. The regular magnetic field in M31 is nearlyaligned with the spiral arms forming the bright ‘ring’ of emission seen in thisgalaxy . The B vectors of the regular field in several other galaxies (M81,M83, NGC 1566) also follow the optical spiral, though they are generally offset
from the optical arms. A particularly spectacular case is that of the galaxyNGC 6946 [73,84]; here the polarized emission (tracing the regular field) islocated in dominant spiral arms, in between, interlaced and anti-correlatedwith the optical spiral structure. These magnetic spiral arms have widths ofabout 500-1000 pc, and with dominating regular fields of ∼ 13µG. The field inthese arms is also ordered, as inferred from RM observations . In Fig. 2.8we show 6 cm radio observations of the galaxies M51 and NGC 6946, withsuperposed magnetic field vectors. One can clearly see that the magnetic fieldis ordered over large scales.
As we remarked earlier, RM observations are absolutely crucial to distinguishbetween coherent and incoherent fields. Coherency of RM on a large scaleis indeed seen in a number of galaxies (for example, M31 , NGC 6946, NGC 2997 ). The galaxy M31 seems to have a 20 kpc sized torus ofemission, with the regular field nearly aligned with the spiral arms forming anemission ‘ring’, with an average pitch angle of about −15 [83,87,88]. Such afield can probably be produced only by a large scale dynamo.
The structure of the regular field is described in dynamo models by modes ofdifferent azimuthal and vertical symmetry; see Section 6.5.5. Again, Faradayrotation measure observations are crucial for this purpose . The presentdata indicate a singly periodic azimuthal variation of RMs, suggesting a largelyaxisymmetric (ASS, m = 0 symmetry) mean field structure in M31  andIC 342 . There is an indication of a bisymmetric spiral mode (BSS, m = 1
Fig. 2.8. Left: M51 in 6 cm, total intensity with magnetic field vectors. Right:NGC 6946 in 6 cm, polarized intensity with magnetic field vectors. The physicalextent of the images is approximately 28× 34 kpc2 for M51 (distance 9.6Mpc) and22 × 22 kpc2 for NGC 6946 (distance 7Mpc). (VLA and Effelsberg. Courtesy R.Beck.)
symmetry) in M81  and the field in M51 seems to be best described by acombination of ASS and BSS . The magnetic arms in NGC 6946 may bethe result of a superposition of ASS and quadrisymmetric (m = 2) modes .Indeed, in most galaxies the data cannot be described by only a single mode,but require a superposition of several modes which still cannot be resolvedby the existing observations. It has also been noted  that in 4 out of 5galaxies, the radial component of the spiral field could be such that the fieldpoints inward.
The vertical symmetry of the field is much more difficult to determine. Thelocal field in our Galaxy is oriented mainly parallel to the plane (cf. [68,14,78]).This agrees well with the results from several other external edge-on galaxies, where the observed magnetic fields are generally aligned along the discsof the galaxies. Further, in our Galaxy the RMs of pulsars and extragalacticradio sources have the same sign above and below the galactic midplane, forgalactic longitudes between 90 and 270, while toward the galactic centerthere is a claim that RMs change sign cf. [96,97]. This may point to a verti-cally symmetric field in the Galaxy away from its central regions. Note thatthe determination of magnetic field structure in the Galaxy from Faraday ro-tation can be complicated by local perturbations. Taking into account suchcomplications, and carrying out an analysis of the Faraday rotation measuresof extragalactic sources using wavelet transforms, one finds evidence that thehorizontal components of the regular magnetic field have even parity through-out the Galaxy, that is the horizontal components are similarly directed on
both sides of the disc . Note that a vertically symmetric field could arisefrom a quadrupolar meridional structure of the field, while vertically antisym-metric fields could arise from a dipolar structure. A vertically symmetric fieldalso seems to be indicated from RM study of the galaxy M31 .
Although most edge-on galaxies have fields aligned along the disc, severalgalaxies (NGC 4631, NGC 4666 and M82) have also radio halos with a domi-nant vertical field component . Magnetic spurs in these halos are connectedto star forming regions in the disc. The field is probably dragged out by astrong, inhomogeneous galactic wind [98,99].
Ordered magnetic fields with strengths similar to those in grand design spiralshave also been detected in flocculent galaxies (M33 , NGC 3521 and 5055, NGC 4414 ), and even in irregular galaxies (NGC 4449 ). Themean degree of polarization is also similar between grand design and flocculentgalaxies . And a grand design spiral pattern is observed in all the aboveflocculent galaxies; implying that gaseous spiral arms are not an essentialfeature to obtain ordered fields.
There is little direct evidence on the nature of magnetic fields in ellipticalgalaxies, although they may well be ubiquitous in the hot ionized gas seen inthese galaxies (cf. Ref.  for a brief review). This may be due to the paucityof relativistic electrons in these galaxies, which are needed to illuminate themagnetic fields by generating synchrotron emission. Faraday rotation of thepolarized emission from background objects has been observed in a few cases.Particularly intriguing is the case of the gravitationally lensed twin quasar,0957+561, where the two images, have a differential Faraday rotation of ∼ 100rad m−2 . One of the images passes through the central region of a possiblyelliptical galaxy acting as the lens, and so this may indicate that the gas inthis elliptical galaxy has ordered magnetic fields. It is important to search formore direct evidence for magnetic fields in elliptical galaxies.
2.4 Magnetic fields in clusters of galaxies
The most recent area of study of astrophysical magnetic fields is perhaps themagnetic fields of clusters of galaxies. Galaxy clusters are the largest boundsystems in the universe, having masses of ∼ 1014 − 1015M⊙ and typical sizesof several Mpc. Observations of clusters in the X-ray band reveals that theygenerally have an atmosphere of hot gas with temperatures T ∼ 107 to 108K,and extending over Mpc scales. The baryonic mass of clusters is in fact domi-nated by this hot gas component. It has become clear in the last decade or sothat magnetic fields are also ubiquitous in clusters. A succinct review of theobservational data on cluster magnetic fields can be found in Ref. . Here
we gather some important facts that are relevant in trying to understand theorigin of these fields.
Evidence for magnetic fields in clusters again comes from mainly radio obser-vations. Several clusters display relatively smooth low surface brightness radiohalos, attributed to synchrotron emission from the cluster as a whole, ratherthan discrete radio sources. The first such halo to be discovered was that as-sociated with the Coma cluster (called Coma C) . Only recently havemany more been found in systematic searches [107–110]. These radio haloshave typically sizes of ∼ 1 Mpc, steep spectral indices, low surface brightness,low polarizations (< 5% ), and are centered close to the center of the X-rayemission. Total magnetic fields in cluster radio halos, estimated using mini-mum energy arguments  range from 0.1 to 1µG , the value for Comabeing ∼ 0.4µG . (The equipartition field will depend on the assumedproton to electron energy ratio; for a ratio of 100, like in the local ISM, theequipartition field will be larger by ∼ 1002/7 or a factor ∼ 3.7; R. Beck, privatecommunication).
Cluster magnetic fields can also be probed using Faraday rotation studies ofboth cluster radio galaxies and also background radio sources seen throughthe cluster. High resolution RM studies have been performed in several radiogalaxies in both cooling flow clusters and non-cooling flow clusters (althoughthe issue of the existence of cooling flows is at present hotly debated). If oneassumes a uniform field in the cluster, then minimum magnetic fields of 5 to10µG are inferred in cooling flow clusters, whereas it could be a factor of 2lower in non-cooling flow clusters . However the observed RM is patchyindicating magnetic field coherence scales of about 5 to 10 kpc in these clusters.If we use such coherence lengths, then the estimated fields become larger.For example in the cooling flow cluster 3C295 the estimated magnetic fieldstrength is ∼ 12µG in the cluster core , whereas in the non-cooling flowcluster 3C129 the estimated field is about 6µG . In Hydra A, there is anintriguing trend for all the RMs to the north of the nucleus to be positive andto the south to be negative . Naively this would indicate quite a large scalefield (100 kpc) with strength ∼ 7µG , but it is unclear which fraction isdue to a cocoon surrounding the radio source (cf. Ref. ). The more tangledfields in the same cluster were inferred to have strengths of ∼ 30µG andcoherence lengths ∼ 4 kpc . More recently, a novel technique to analyzeFaraday Rotation maps, has been developed, assuming that the magnetic fieldsare statistically isotropic, and applied to several galaxy clusters, [117,118].This analysis yields an estimate of 3µG in Abell 2634, 6µG in Abell 400 and12µG in Hydra A as conservative estimates of the field strengths, and fieldcorrelation lengths of ∼ 4.9 kpc, 3.6 kpc and 0.9 kpc respectively, for these 3clusters.
There is always some doubt whether the RMs in cluster radio sources are
Fig. 2.9. Galaxy-corrected rotation measure plotted as a function of source impactparameter in kiloparsecs for the sample of 16 Abell clusters. The open points rep-resent the cluster sources viewed through the thermal cluster gas while the closedpoints are the control sources at impact parameters beyond the cluster gas. Note theclear increase in the width of the RM distribution toward smaller impact parameter.Adapted from Clarke et al. .
produced due to Faraday rotation intrinsic to the radio source, rather thandue to the intervening intracluster medium. While this is unlikely in mostcases , perhaps more convincing evidence is the fact that studies of RMsof background radio sources seen through clusters, also indicate several µGcluster magnetic fields. A very interesting statistical study in this context is arecent VLA survey , where the RMs in and behind a sample of 16 Abellclusters were determined. The RMs were plotted as a function of distancefrom the cluster center and compared with a control sample of RMs from fieldsources; see Fig. 2.9. This study revealed a significant excess RM for sourceswithin about 0.5Mpc of the cluster center. Using a simple model, where theintracluster medium consists of cells of uniform size and field strength, butrandom field orientations, Clarke et al.  estimate cluster magnetic fieldsof ∼ 5 (l/10 kpc)−1/2 µG, where l is the coherence length of the field.
Cluster magnetic fields can also be probed by comparing the inverse ComptonX-ray emission and the synchrotron emission from the same region. Note thatthis ratio depends on the ratio of the background radiation energy density(which in many cases would be dominated by the Cosmic microwave back-ground) to the magnetic field energy density. The main difficulty is in sepa-rating out the thermal X-ray emission. This separation can also be attemptedusing spatially resolved X-ray data. Indeed, an X-ray excess (compared to
that expected from a thermal atmosphere) was seen at the location of a dif-fuse radio relic source in Abell 85 . This was used in Ref.  to derivea magnetic field of 1.0± 0.1 µG for this source.
Overall it appears that there is considerable evidence that galaxy clusters aremagnetized with fields ranging from a few µG to several tens of µG in somecluster centers, and with coherence scales of order 10 kpc. These fields, if notmaintained by some mechanism, will evolve as decaying MHD turbulence, andperhaps decay on the Alfven crossing time scale of the cluster, which is on theorder of Gyrs, but may be less than the age of the universe. We will have moreto say on the possibility of dynamos in clusters in later sections.
3 The equations of magnetohydrodynamics
In stars and galaxies, and indeed in many other astrophysical settings, thegas is partially or fully ionized and can carry electric currents that, in turn,produce magnetic fields. The associated Lorentz force exerted on the ionizedgas (also called plasma) can in general no longer be neglected in the momen-tum equation for the gas. Magneto-hydrodynamics (MHD) is a study of theinteraction of the magnetic field and the plasma treated as a fluid. In MHDwe combine Maxwell’s equations of electrodynamics with the fluid equations,including also the Lorentz forces due to the electromagnetic fields. We firstdiscuss Maxwell’s equations that characterize the evolution of the magneticfield.
3.1 Maxwell’s equations
In gaussian cgs units, Maxwell’s equations can be written in the form
∂t= −∇×E, ∇ ·B = 0, (3.1)
∂t= ∇×B − 4π
cJ , ∇ ·E = 4πρe, (3.2)
where B is the magnetic flux density (usually referred to as simply the mag-netic field), E is the electric field, J is the current density, c is the speed oflight, and ρe is the charge density.
Although in astrophysics one uses mostly cgs units, in much of the work ondynamos the MHD equations are written in ‘SI’ units, i.e. with magneticpermeability µ0 and without factors like 4π/c. (Nevertheless, the magnetic
field is still quoted often in gauss [G] and cgs units are used for density,lengths, etc.) Maxwell’s equations in SI units are then written as
∂t= −∇×E, ∇ ·B = 0, (3.3)
∂t= ∇×B − µ0J , ∇ ·E = ρe/ǫ0, (3.4)
where ǫ0 = 1/(µ0c2) is the permittivity of free space.
To ensure that ∇ · B = 0 is satisfied at all times it is often convenient toreplace Eq. (3.3) by the ‘uncurled’ equation for the magnetic vector potential,A,
∂t= −E −∇φ, (3.5)
where B = ∇ × A and φ is the scalar potential. Note that magnetic andelectric fields are invariant under the gauge transformation
A′ = A+∇Λ, (3.6)
φ′ = φ− ∂Λ
For numerical purposes it is often convenient to choose the gauge φ′ = 0,which implies that φ = ∂Λ/∂t, so instead of Eq. (3.5) one solves the equation∂A′/∂t = −E.
3.2 Resistive MHD and the induction equation
Using the standard Ohm’s law in a fixed frame of reference,
J = σ (E +U ×B) , (3.8)
where σ is the electric conductivity, and introducing the magnetic diffusivityη = (µ0σ)
−1, or η = c2/(4πσ) in cgs units, we can eliminate J from Eq. (3.4),so we have
E = −(
This shows that the time derivative term (also called the Faraday displacementcurrent) can be neglected if the relevant time scale over which the electric fieldvaries, exceeds the Faraday time τFaraday = η/c2. We shall discuss below thatfor ordinary Spitzer resistivity, η is proportional to T−3/2 and varies between10 and 1010 cm2 s−1 for temperatures between T = 108 and T = 102K. Thus,the displacement current can be neglected when the variation time scales arelonger than 3× 10−10 s (for T ≈ 108K) and longer than 0.3 s (for T ≈ 102K).For the applications discussed in this review, this condition is always met,even for neutron stars where the time scales of variation can be of the orderof milliseconds, but the temperatures are very high as well. We can thereforesafely neglect the displacement current and eliminate E, so Eq. (3.4) can bereplaced by the Ampere’s law J = ∇×B/µ0.
It is often convenient to consider J simply as a short hand for ∇ ×B. Thiscan be accomplished by adopting units where µ0 = 1. We shall follow herethis convection and shall therefore simply write
J = ∇×B. (3.10)
Occasionally we also state the full expressions for clarity.
Substituting Ohm’s law into the Faraday’s law of induction, and using Am-pere’s law to eliminate J one can write a single evolution equation for B,which is called the induction equation:
∂t= ∇× (U ×B − ηJ) . (3.11)
We now describe a simple physical picture for the conductivity in a plasma.The force due to an electric field E accelerates electrons relative to the ions;but they cannot move freely due to friction with the ionic fluid, caused byelectron-ion collisions. They acquire a ‘terminal’ relative velocity V, withrespect to the ions, obtained by balancing the Lorentz force with friction.This velocity can also be estimated as follows. Assume that electrons movefreely for about an electron-ion collision time τei, after which their veloc-ity becomes again randomized. Electrons of charge e and mass me in freemotion during the time τei acquire from the action of an electric field E
an ordered speed V ∼ τeieE/me. This corresponds to a current densityJ ∼ eneV ∼ (nee
2τei/m)E and hence leads to σ ∼ nee2τei/me.
The electron-ion collision time scale (which determines σ) can also be esti-mated as follows. For a strong collision between an electron and an ion oneneeds an impact parameter b which satisfies the condition Ze2/b > mev
2. Thisgives a cross section for strong scattering of σt ∼ πb2. Since the Coulomb forceis a long range force, the larger number of random weak scatterings add up togive an extra ‘Coulomb logarithm’ correction to make σt ∼ π(Ze2/mv2)2 ln Λ,where ln Λ is in the range between 5 and 20. The corresponding mean freetime between collisions is
niσtv∼ (kBT )
πZe4ne lnΛ, (3.12)
where we have used the fact that mev2 ∼ kBT and Zni = ne. Hence we get
σ ∼ (kBT )3/2
m1/2e πZe2 ln Λ
where most importantly the dependence on the electron density has canceledout. A more exact calculation can be found, for example, in Landau and Lif-shitz  (Vol 10; Eq. 44.11) and gives an extra factor of 4(2/π)1/2 multiplyingthe above result. The above argument has ignored collisions between electronsthemselves, and treated the plasma as a ‘Lorentzian plasma’. The inclusionof the effect of electron-electron collisions further reduces the conductivity bya factor of about 0.582 for Z = 1 to 1 for Z → ∞; see the book by Spitzer, and Table 5.1 and Eqs. (5)–(37) therein, and leads to a diffusivity, incgs units, of η = c2/(4πσ) given by
η = 104(
As noted above, the resistivity is independent of density, and is also inverselyproportional to the temperature (larger temperatures implying larger meanfree time between collisions, larger conductivity and hence smaller resistivity).
The corresponding expression for the kinematic viscosity ν is quite different.Simple kinetic theory arguments give ν ∼ vtli, where li is the mean free path ofthe particles which dominate the momentum transport and vt is their randomvelocity. For a fully ionized gas the ions dominate the momentum transport,and their mean free path li ∼ (niσi)
−1, with the cross-section σi, is determinedagain by the ion-ion ‘Coulomb’ interaction. From a reasoning very similar tothe above for electron-ion collisions, we have σi ∼ π(Z2e2/kBT )
2 ln Λ, where
Table 3.1Summary of some important parameters in various astrophysical settings. The val-ues given should be understood as rough indications only. In particular, the appli-cability of Eq. (3.17) may be questionable in some cases and has therefore not beenused for protostellar discs (see text). We have assumed lnΛ = 20 in computing Rm
and Pm. CZ means convection zone, CV discs and similar refer to cataclysmic vari-ables and discs around other other compact objects such as black holes and neutronstars. AGNs are active galactic nuclei. Numbers in parenthesis indicate significantuncertainty.
T [K] ρ [ g cm−3] Pm urms [ cm s−1] L [ cm] Rm
Solar CZ (upper part) 104 10−6 10−7 106 108 106
Solar CZ (lower part) 106 10−1 10−4 104 1010 109
Protostellar discs 103 10−10 10−8 105 1012 10
CV discs and similar 106 10−7 102 105 107 107
AGN discs 107 10−5 104 105 109 1011
Galaxy 104 10−24 (1011) 106 1020 (1018)
Galaxy clusters 108 10−26 (1029) 108 1023 (1029)
we have used miv2t ∼ kBT . Substituting for vt and li, this then gives
ν ∼ (kBT )5/2
nim1/2i πZ4e4 ln Λ
More accurate evaluation using the Landau collision integral gives a factor 0.4for a hydrogen plasma, instead of 1/π in the above expression (see the end ofSection 43 in Landau and Lifshitz ). This gives numerically
ν = 6.5× 1022(
)5/2 ( ni
cm2 s−1, (3.16)
so the magnetic Prandtl number is
Pm ≡ ν
η= 1.1× 10−4
0.1 g cm−3
)−1 (ln Λ
Thus, in the sun and other stars (T ∼ 106K, ρ ∼ 0.1 g cm−3) the magneticPrandtl number is much less than unity. Applied to the galaxy, using T =104K and ρ = 10−24 g cm−3, ln Λ ∼ 10, this formula gives Pm = 4× 1011. Thereason Pm is so large in galaxies is mostly because of the very long mean free
path caused by the low density . For galaxy clusters, the temperature ofthe gas is even larger and density smaller, making the medium much moreviscous and having even larger Pm.
In protostellar discs, on the other hand, the gas is mostly neutral with low tem-peratures. In this case, the electrical conductivity is given by σ = nee
2τen/me,where τen is the rate of collisions between electrons and neutral particles. Theassociated resistivity is η = 234x−1
e T 1/2 cm2 s−1, where xe = ne/nn is theionization fraction and nn is the number density of neutral particles .The ionization fraction at the ionization-recombination equilibrium is ap-proximately given by xe = (ζ/βnn)
1/2, where ζ is the ionization rate andβ = 3 × 10−6T−1/2 cm3 s−1 is the dissociative recombination rate [126,127].For a density of ρ = 10−10 g cm−3, and a mean molecular weight 2.33mp ,we have nn = 2.6 × 1013 cm−3. Adopting for ζ the cosmic ray ionizing rateζ ∼ 10−17 s−1, which is not drastically attenuated by the dense gas in thedisk, and a disc temperature T = 103K, we estimate xe ∼ 2 × 10−12, andhence η ∼ 4× 1015 cm2 s−1.
In Table 3.1 we summarize typical values of temperature and density in differ-ent astrophysical settings and calculate the corresponding values of Pm. Herewe also give rough estimates of typical rms velocities, urms, and eddy scales,L, which allow us to calculate the magnetic Reynolds number as
Rm = urms/(ηkf), (3.18)
where kf = 2π/L. This number characterizes the relative importance of mag-netic induction relative to magnetic diffusion. A similar number is the fluidReynolds number, Re = Rm/Pm, which characterizes the relative importanceof inertial forces to viscous forces. (We emphasize that throughout this paperReynolds numbers are defined based on the inverse wavenumber; our valuesmay therefore be 2π times smaller that those by other authors.)
3.3 Stretching, flux freezing and diffusion
The U ×B term in Eq. (3.11) is usually referred to as the induction term. Toclarify its role we rewrite its curl as
∇× (U ×B) = − U ·∇B︸ ︷︷ ︸
+ B ·∇U︸ ︷︷ ︸
+ B∇ ·U︸ ︷︷ ︸
where we have used the fact that ∇ · B = 0. As a simple example, weconsider the effect of a linear shear flow, U = (0, Sx, 0) on the initial field
B = (B0, 0, 0). The solution is B = (1, St, 0)B0, i.e. the field component inthe direction of the flow grows linearly in time.
The net induction term more generally implies that the magnetic flux througha surface moving with the fluid remains constant, in the high-conductivitylimit. Consider a surface S, bounded by a curve C, moving with the fluid asshown in Fig. 3.1. Suppose we define the magnetic flux through this surface,Φ =
S B · dS. Then after a time dt the change in flux is given by
B(t + dt) · dS −∫
B(t) · dS. (3.20)
∇ ·B dV = 0 at time (t+ dt), to the ‘tube’ like volume swept upby the moving surface S, shown in Fig. 3.1, we also have
B(t + dt) · dS =∫
B(t+ dt) · dS −∮
B(t + dt) · (dl ×U), (3.21)
where C is the curve bounding the surface S, and dl is the line element alongC. (In the last term to linear order in dt it does not matter whether we takethe integral over the curve C or C ′). Using the above condition in Eq. (3.20),
[B(t + dt)−B(t)] · dS −∮
B(t+ dt) · (dl ×U). (3.22)
Taking the limit of dt → 0,
∂t· dS +
B · (U × dl) = −∫
(∇× ηJ) · dS. (3.23)
In the second equality we have used∮
C B · (U × dl) =∫
S ∇ × (U ×B) · dStogether with the induction equation (3.11). One can see that, when η → 0,dΦ/dt → 0 and Φ is constant.
Now suppose we consider a small segment of a thin flux tube of length land cross-section S, in a highly conducting fluid. Then, as the fluid movesabout, conservation of flux implies BS is constant, and conservation of massimplies ρSl is constant, where ρ is the local density. So B ∝ ρl. For a nearlyincompressible fluid, or a flow with small changes in ρ, one will obtain B ∝ l.Any shearing motion which increases l will also amplify B; an increase in lleading to a decrease in S (because of incompressibility) and hence an increasein B (due to flux freezing). This effect, also obtained in our discussion ofstretching above, will play a crucial role in all scenarios involving dynamogeneration of magnetic fields.
Fig. 3.1. The surface S enclosed by the curve C is carried by fluid motion to thesurface S′ after a time dt. The flux through this surface Φ is frozen into the fluidfor a perfectly conducting fluid.
The concept of flux freezing can also be derived from the elegant Cauchysolution of the induction equation with zero diffusion. This solution is also ofuse in several contexts and so we describe it briefly below. In case η = 0 inEq. (3.11), the ∇× (U ×B) term can be expanded to give
Dt= B ·∇U −B(∇ ·U), (3.24)
where D/Dt = ∂/∂t+U ·∇ is the Lagrangian derivative. If we eliminate the∇ ·U term using the continuity equation for the fluid,
∂t= −∇ · (ρU), (3.25)
where ρ is the fluid density, then we can write
ρ·∇U . (3.26)
Suppose we describe the evolution of a fluid element by giving its trajectory asx(x0, t), where x0 is its location at an initial time t0. Now suppose we considerthe evolution of two infinitesimally separated fluid elements, A and B, which,at an initial time t = t0, are located at x0 and x0 + δx0, respectively. Thesubsequent location of these fluid elements will be, say, xA = x(x0, t) andxB = x(x0 + δx0, t) and their separation is xB − xA = δx(x0, t). Since the
velocity of the fluid particles will be U(xA) and U(xA) + δx · ∇U , after atime δt, the separation of the two fluid particles will change by δt δx · ∇U .The separation vector therefore evolves as
Dt= δx ·∇U , (3.27)
which is an evolution equation identical to that satisfied byB/ρ. So, if initially,at time t = t0, the fluid particles were on a given magnetic field line with(B/ρ)(x0, t0) = c0δx(t0) = c0δx0, where c0 is a constant, then for all timeswe will have B/ρ = c0δx. In other words, ‘if two infinitesimally close fluidparticles are on the same line of force at any time, then they will alwaysbe on the same line of force, and the value of B/ρ will be proportional tothe distance between the particles’ (Section 65 in Ref. ). Further, sinceδxi(x0, t) = Gijδx0j , where Gij = ∂xi/∂x0j , we can also write
Bi(x, t) = ρc0δxi =Gij(x0, t)
where we have used the fact that ρ(x, t)/ρ(x0, t0) = (detG)−1. We will usethis Cauchy solution in Appendix B.1.
3.4 Magnetic helicity
Magnetic helicity plays an important role in dynamo theory. We therefore givehere a brief account of its properties. Magnetic helicity is the volume integral
A ·B dV (3.29)
over a closed or periodic volume V . By a closed volume we mean one in whichthe magnetic field lines are fully contained, so the field has no component nor-mal to the boundary, i.e. B ·n = 0. The volume V could also be an unboundedvolume with the fields falling off sufficiently rapidly at spatial infinity. In theseparticular cases, H is invariant under the gauge transformation (3.6), because
H ′ =∫
A′ ·B′ dV = H +∫
∇Λ ·B dV = H +∮
ΛB · ndS = H (3.30)
where n is the normal pointing out of the closed surface ∂V . Here we havemade use of ∇ ·B = 0.
Fig. 3.2. Two flux tubes with fluxes Φ1 and Φ2 are linked in such a way that theyhave a helicity H = +2Φ1Φ2. Interchanging the direction of the field in one of thetwo rings changes the sign of H.
Magnetic helicity has a simple topological interpretation in terms of the linkageand twist of isolated (non-overlapping) flux tubes. For example consider themagnetic helicity for an interlocked, but untwisted, pair of thin flux tubes asshown in Fig. 3.2, with Φ1 and Φ2 being the fluxes in the tubes around C1 andC2 respectively. For this configuration of flux tubes, B d3x can be replaced byΦ1dl on C1 and Φ2dl on C2. The net helicity is then given by the sum
H = Φ1
A · dl + Φ2
A · dl,= 2Φ1Φ2 (3.31)
where we have used Stokes theorem to transform
A · dl =∫
B · dS ≡ Φ2,∮
A · dl =∫
B · dS ≡ Φ1. (3.32)
For a general pair of thin flux tubes, the helicity is given by H = ±2Φ1Φ2;the sign of H depending on the relative orientation of the two tubes .
The evolution equation for H can be derived from Faraday’s law and its un-curled version for A, Eq. (3.5), so we have
∂t(A ·B)= (−E +∇φ) ·B +A · (−∇×E)
=−2E ·B +∇ · (φB +A×E). (3.33)
Integrating this over the volume V , the magnetic helicity satisfies the evolutionequation
E ·BdV +∮
(A×E + φB) · ndS = −2ηC, (3.34)
where C =∫
V J ·B dV is the current helicity. Here we have used Ohm’s lawE = −U × B + ηJ , in the volume integral and we have assumed that thesurface integral vanishes for closed fields. If the µ0 factor were included, thisequation would read dH/dt = −2ηµ0C.
In the non-resistive case, η = 0, the magnetic helicity is conserved, i.e. dH/dt =0. However, this does not guarantee conservation of H in the limit η → 0, be-cause the current helicity,
J ·B dV , may in principle still become large. Forexample, the Ohmic dissipation rate of magnetic energy QJoule ≡ −η
J2dVcan be finite and balance magnetic energy input by motions, even when η → 0.This is because small enough scales develop in the field (current sheets) wherethe current density increases with decreasing η as ∝ η−1/2 as η → 0, whilstthe rms magnetic field strength, Brms, remains essentially independent of η.Even in this case, however, the rate of magnetic helicity dissipation decreases
with η like ∝ η+1/2 → 0, as η → 0. Thus, under many astrophysical conditionswhere Rm is large (η small), the magnetic helicity H , is almost independent oftime, even when the magnetic energy is dissipated at finite rates. This robustconservation of magnetic helicity is an important constraint on the nonlinearevolution of dynamos and will play a crucial role below in determining howlarge scale turbulent dynamos saturate.
We also note the very fact that the fluid velocity completely drops out from thehelicity evolution equation (3.34), since (U×B)·B = 0. Therefore, any changein the nature of the fluid velocity, for example due to turbulence (turbulentdiffusion), or the Hall effect or ambipolar drift (see below), does not affectmagnetic helicity conservation. We will discuss in more detail the conceptof turbulent diffusion in a later section, and its role in dissipating the meanmagnetic field. However, such turbulent magnetic diffusion does not dissipatethe net magnetic helicity. This property is crucial for understanding why, inspite of the destructive properties of turbulence, large scale spatio-temporalcoherence can emerge if there is helicity in the system.
For open volumes, or volumes with boundaries through which B · n 6= 0, themagnetic helicity H , as defined by Eq. (3.29), is no longer gauge-invariant.One can define a gauge-invariant relative magnetic helicity [130–132]
(A+Aref) · (B −Bref) dV, (3.35)
where Bref = ∇ × Aref is a reference magnetic field that is taken to be thepotential field solution with the boundary condition
n ·Bref = n ·B, (3.36)
i.e. the two fields have the same normal components. The quantity Hrel isgauge-invariant, because in Eq. (3.30) the term n ·B is replaced by n · (B −Bref), which vanishes on the boundaries.
The evolution equation of the relative magnetic helicity is simplified by adopt-ing a specific gauge for Aref , with
∇ ·Aref = 0, Aref · n|∂V = 0. (3.37)
We point out, however, that this restriction can in principle be relaxed. Whenthe gauge (3.37) is used for the reference field, the relative magnetic helicitysatisfies the evolution equation
dt= −2ηC − 2
(E ×Aref) · dS, (3.38)
where dS = n dS is the surface element. The surface integral covers the fullclosed surface around the volume V . In the case of the sun the magnetichelicity fluxes from the northern and southern hemispheres are expected to beabout equally big and of opposite sign, so they would cancel approximatelyto zero. One is therefore usually interested in the magnetic helicity flux out ofthe northern or southern hemispheres, but this means that it is necessary toinclude the contribution of the equator to the surface integral in Eq. (3.38).This contribution can easily be calculated for data from numerical simulations,but in the case of the sun the contribution from the equatorial surface is notobserved.
We have emphasized earlier in this section that no net magnetic helicity canbe produced by any kind of velocity. However, this is not true of the magnetichelicity flux which is affected by the velocity via the electric field. This canbe important if there is differential rotation or shear which can lead to aseparation of magnetic helicity in space. A somewhat related mechanism isthe alpha effect (Section 6) which can lead to a separation of magnetic helicityin wavenumber space. Both processes are important in the sun.
3.5 The momentum equation
Finally we come to the momentum equation, which is just the ordinary Navier-Stokes equation in fluid dynamics supplemented by the Lorentz force, J ×B,i.e.
Dt= −∇p + J ×B + f + Fvisc, (3.39)
where U is the ordinary bulk velocity of the gas, ρ is the density, p is thepressure, Fvisc is the viscous force, and f subsumes all other body forcesacting on the gas, including gravity and, in a rotating system also the Coriolisand centrifugal forces. We use an upper case U , because later on we shall usea lower case u for the fluctuating component of the velocity. Equation (3.39)has to be supplemented by the continuity equation,
∂t= −∇ · (ρU), (3.40)
an equation of state, p = p(ρ, e), an energy equation for the internal energy e,and an evolution equation for the magnetic field.
An important quantity is the adiabatic sound speed, cs, defined by c2s =(∂p/∂ρ)s, evaluated at constant entropy s. For a perfect gas with constantratio γ of specific heats (γ = 5/3 for a monatomic gas) we have c2s = γp/ρ.When the flow speed is much smaller than the sound speed, i.e. when theMach number Ma = 〈U 2/c2s 〉1/2 is much smaller than unity and if, in addition,the density is approximately uniform, i.e. ρ ≈ ρ0 = const, the assumptionof incompressibility can be made. In that case, Eq. (3.25) can be replaced by∇ ·U = 0, and the momentum equation then simplifies to
Dt= − 1
ρ0+ f + ν∇2U , (3.41)
where ν is the kinematic viscosity and f is now an external body force per unitmass. The ratio Pm = ν/η is the magnetic Prandtl number; see Eq. (3.17).
The assumption of incompressibility is a great simplification that is useful formany analytic considerations, but for numerical solutions this restriction isoften not necessary. As long as the Mach number is small, say below 0.3, theweakly compressible case is believed to be equivalent to the incompressiblecase .
3.6 Kinetic helicity evolution
We introduce the vorticity W = ∇ × U , and define the kinetic helicity asF =
W ·U dV . Using Eq. (3.41), and ignoring the magnetic field, F obeysthe evolution equation
W · f dV − 2ν∫
W ·Q dV, (3.42)
where Q = ∇ × W is the curl of the vorticity. Note that in the absence offorcing, f = 0, and without viscosity, ν = 0, the kinetic helicity is conserved,i.e.
dt= 0 (if ν = 0 and f = 0). (3.43)
On the other hand, in the limit ν → 0 the rate of kinetic helicity productionwill diverge, because the rate of energy dissipation, ǫ = ν
W 2dV , is knownto be independent of ν. The magnitude of the vorticity therefore divergeslike |W | ∼ ν−1/2. This implies that the typical wavenumber associated with|W | ≈ k|U | also diverges like k ∼ ν−1/2 and so |Q| ∼ k|W | ∼ ν−1. Thus,
∣∣∣∣∣∼ ν−1/2 → ∞ (for ν → 0). (3.44)
In comparison (as we pointed out earlier), in the magnetic case, the currentdensity also diverges like |J | ∼ η−1/2, but the rate of magnetic helicity pro-duction is only proportional to J ·B, and ηJ ·B ∼ η+1/2 → 0, so
∣∣∣∣∣∼ η+1/2 → 0 (for η → 0). (3.45)
It is worth emphasizing again that it is for this reason that the magnetichelicity is such an important quantity in magnetohydrodynamics.
3.7 Energy and helicity spectra
Magnetic energy and helicity spectra are usually calculated as
2 dΩk, (3.46)
(A∗k·Bk +Ak ·B∗
k) k2 dΩk, (3.47)
where dΩk is the solid angle element in Fourier space and Bk = ik × Ak isthe Fourier transform of the magnetic field and Ak is the Fourier transformof the vectors potential. These spectra are normalized such that
Hk dk = 〈A ·B〉V ≡ H, (3.48)
Mk dk = 〈12B2〉V ≡ M, (3.49)
where H and M are magnetic helicity and magnetic energy, respectively, andangular brackets denote volume averages.
There is a conceptual advantage in working with the real space Fourier filteredmagnetic vector potential and magnetic field,Ak andBk, whereBk = ∇×Ak,and the subscript k (which is now a scalar!) indicates Fourier filtering to keeponly those wavevectors k that lie in the shell
k − δk/2 ≤ |k| < k + δk/2 (k-shell). (3.50)
Magnetic energy and helicity spectra can then be written as
Hk = 〈Ak ·Bk〉V/δk, (3.52)
where angular brackets denote averages over all space. We recall that, for aperiodic domain, H is gauge invariant. Since its spectrum can be written asan integral over all space, see Eq. (3.52), Hk is – like H – also gauge invariant.
It is convenient to decompose the Fourier transformed magnetic vector poten-tial, Ak, into a longitudinal component, h‖, and eigenfunctions h± of the curloperator. Especially in the context of spherical domains these eigenfunctionsare also called Chandrasekhar–Kendall functions , while in cartesian do-mains they are usually referred to as Beltrami waves. This decomposition hasbeen used in studies of turbulence , in magnetohydrodynamics , andin dynamo theory . Using this decomposition we can write the Fouriertransformed magnetic vector potential as
Ak = a+kh+
ik × h±k= ±kh±
k, k = |k|, (3.54)
∗ · h+k〉 = 〈h−
∗ · h−k〉 = 〈h‖
k〉 = 1, (3.55)
where asterisks denote the complex conjugate. The longitudinal part a‖kh
is parallel to k and vanishes after taking the curl to calculate the magneticfield. In the Coulomb gauge, ∇ ·A = 0, the longitudinal component vanishesaltogether.
The (complex) coefficients a±k(t) depend on k and t, while the eigenfunctions
h±k, which form an orthonormal set, depend only on k and are given by
k × (k × e)∓ ik(k × e)
1− (k · e)2/k2, (3.56)
where e is an arbitrary unit vector that is not parallel to k. With these prepa-rations we can write the magnetic helicity and energy spectra in the form
Hk = k(|a+|2 − |a−|2)V, (3.57)
Mk =12k2(|a+|2 + |a−|2)V, (3.58)
where V is the volume of integration. (Here again the factor µ−10 is ignored in
the definition of the magnetic energy). From Eqs. (3.57) and (3.58) one seesimmediately that [129,137]
12k|Hk| ≤ Mk, (3.59)
which is also known as the realizability condition. A fully helical field hastherefore Mk = ±1
For further reference we now define power spectra of those components of thefield that are either right or left handed, i.e.
H±k = ±k|a±|2V, M±
k = 12k2|a±|2V. (3.60)
Thus, we have Hk = H+k +H−
k and Mk = M+k +M−
k . Note that H±k and M±
can be calculated without explicit decomposition into right and left handed
field components using
H±k = 1
2(Hk ± 2k−1Mk), M±
k = 12(Mk ± 1
This method is significantly simpler than invoking explicitly the decompositionin terms of a±
In Section 8.3 plots of M±k will be shown and discussed in connection with
turbulence simulations. Here the turbulence is driven with a helical forcingfunction proportional to h+
k; see Eq. (3.56).
3.8 Departures from the one-fluid approximation
In many astrophysical settings the typical length scales are so large that theusual estimates for the turbulent diffusion of magnetic fields, by far exceed theordinary Spitzer resistivity. Nevertheless, the net magnetic helicity evolution,as we discussed above, is sensitive to the microscopic resistivity and indepen-dent of any turbulent contributions. It is therefore important to discuss indetail the foundations of Spitzer resistivity and to consider more general casessuch as the two-fluid and even three-fluid models. In some cases these gen-eralizations point to important effects of their own, for example the batteryeffect.
3.8.1 Two-fluid approximation
The simplest generalization of the one-fluid model is to consider the electronsand ions as separate fluids which are interacting with each other throughcollisions. This two-fluid model is also essential for deriving the general formof Ohm’s law and for describing battery effects, that generate fields ab-initiofrom zero initial field. We therefore briefly consider it below.
For simplicity assume that the ions have one charge, and in fact they are justprotons. That is the plasma is purely ionized hydrogen. It is straightforwardto generalize these considerations to several species of ions. The equations ofmotion for the electron and proton fluids may be written as
Dt= − ∇pe
me(E + ue ×B)−∇φg −
(ue − up)
mi(E + ui ×B)−∇φg +
(ue − ui)
Here Djuj/Dt = ∂uj/∂t + uj · ∇uj and we have included the forces due
to the pressure gradient, gravity, electromagnetic fields and electron–protoncollisions. Further, mj , nj , uj, pj are respectively the mass, number density,velocity, and the partial pressure of electrons (j = e) and protons (j = i), φg
is the gravitational potential, and τei is the e–i collision time scale. One canalso write down a similar equation for the neutral component (n) and addingthe e, i and n equations we can recover the standard MHD Euler equation.
More interesting in the present context is the difference between the electronand proton fluid equations. In the limit of me/mi ≪ 1, this gives the general-ized Ohms law; see the book by Spitzer , and Eqs. (2)–(12) therein,
E + ui ×B = −∇peene
J ×B +me
where J = (eniui − eneue) is the current density and
is the electrical conductivity. [If ne 6= ni, additional terms arise on the RHS of(3.64) with J in (3.64) replaced by −eui(ne−ni). These terms can usually beneglected since (ne − ni)/ne ≪ 1. Also negligible are the effects of nonlinearterms ∝ u2
The first term on the RHS of Eq. (3.64), representing the effects of the electronpressure gradient, is the ‘Biermann battery’ term, and provides the source termfor the thermally generated electromagnetic fields [138,139]. If ∇pe/ene can bewritten as the gradient of some scalar function, then only an electrostatic fieldis induced by the pressure gradient. On the other hand, if this term has a curlthen a magnetic field can grow. The next two terms on the RHS of Eq. (3.64)are the usual Ohmic term J/σ and the Hall electric field J ×B/(nee), whicharises due to a non-vanishing Lorentz force. Its ratio to the Ohmic term is oforder ωeτei, where ωe = eB/me is the electron gyrofrequency. The last termon the RHS is the inertial term, which can be neglected if the macroscopictime scales are large compared to the plasma oscillation periods.
Note that the extra component of the electric field introduced by the Hallterm is perpendicular to B and so does not alter E · B in the RHS of thehelicity conservation equation (3.34); so the Hall electric field does not alterthe volume dissipation/generation of helicity. The battery term however can inprinciple contribute to helicity dissipation/generation, but this contribution isgenerally expected to be small. To see this rewrite this contribution to helicity
generation, say (dH/dt)Batt, using pe = nekBTe,
·B dV = −2∫
eB ·∇(kBTe) dV, (3.66)
where kB is the Boltzmann constant, and the integration is assumed to goover a closed or periodic domain, so there are no surface terms. 1 We see fromEq. (3.66) that generation/dissipation of helicity can occur only if there aretemperature gradients parallel to the magnetic field [140–142]. Such parallelgradients are in general very small due to fast electron flow along field lines.We will see below that the battery effect can provide a small but finite seedfield; this can also be accompanied by the generation of a small but finitemagnetic helicity.
From the generalized Ohm’s law one can formally solve for the current compo-nents parallel and perpendicular to B (cf. the book by Mestel ). Definingan ‘equivalent electric field’
one can rewrite the generalized Ohms law as 
J = σE′‖ + σ1E
′⊥ + σ2
1 + (ωeτei)2, σ2 =
1 + (ωeτei)2. (3.69)
The conductivity becomes increasingly anisotropic as ωeτei increases. Puttingin numerical values appropriate to the galactic interstellar medium, say, wehave
ωeτei ≈ 4× 105(
)3/2 ( ne
The Hall effect and the anisotropy in conductivity are therefore importantin the galactic interstellar medium, and in cluster gas with high tempera-tures T ∼ 108K and low densities ne ∼ 10−2 cm−3. Of course, in absolute
1 Note that ne in the above equation can be divided by an arbitrary constant den-sity, say n0 to make the argument of the log term dimensionless, since on integratingby parts,
∫ln(n0)B ·∇(kBTe) dV = 0.
terms, neither the resistivity nor the Hall field are important in these sys-tems, compared to the inductive electric field or turbulent diffusion. For thesolar convection zone with ne ∼ 1018 − 1023 cm−3, ωeτei ≪ 1, even for fairlystrong magnetic fields. On the other hand, in neutron stars, the presence ofstrong magnetic fields B ∼ 1013G, could make the Hall term important, es-pecially in their outer regions, where there are also strong density gradients.The Hall effect in neutron stars can lead to magnetic fields undergoing a tur-bulent cascade  or lead to a nonlinear steepening of field gradients for purely toroidal fields, and hence enhanced magnetic field dissipation. How-ever, even a small poloidal field can slow down this decay considerably .In protostellar discs, the ratio of the Hall term to microscopic diffusion is∼ ωeτen ∼ (8× 1017/nn)
1/2(vA/cs), where vA and cs are the Alfven and soundspeeds respectively [125,127,147]. The Hall effect proves to be important indeciding the nature of the magnetorotational instability in these discs.
A strong magnetic field also suppresses other transport phenomena like theviscosity and thermal conduction, perpendicular to the field. These effects areagain likely to be important in rarefied and hot plasma such as in galaxyclusters.
3.8.2 The effect of ambipolar drift
In a partially ionized medium the magnetic field evolution is governed bythe induction equation (3.11), but with U replaced by the velocity of theionic component of the fluid, ui. The ions experience the Lorentz force due tothe magnetic field. This will cause them to drift with respect to the neutralcomponent of the fluid. If the ion-neutral collisions are sufficiently frequent,one can assume that the Lorentz force on the ions is balanced by their frictionwith the neutrals. Under this approximation, the Euler equation for the ionsreduces to
ρiνin(ui − un) = J ×B (strong coupling approximation), (3.71)
where ρi is the mass density of ions, νin the ion-neutral collision frequencyand un the velocity of the neutral particles. For gas with nearly primordialcomposition and temperature ∼ 104K, one gets the estimate  of ρiνin =niρn〈σv〉eff , with 〈σv〉eff ∼ 4×10−9 cm3 s−1, in cgs units. Here ni is the numberdensity of ions, and ρn the mass density of neutrals.
In a weakly ionized gas, the bulk velocity is dominated by the neutrals, and(3.71) substituted into the induction equation (3.11) then leads to a modifiedinduction equation,
∂t= ∇× [(U + aJ ×B)×B − ηJ ] , (3.72)
where a = (ρiνin)−1. The modification is therefore an addition of an extra
drift velocity, proportional to the Lorentz force. One usually refers to thisdrift velocity as ambipolar drift (and sometimes as ambipolar diffusion) in theastrophysical community (cf. Refs. [143,148,149] for a more detailed discus-sion).
We note that the extra component of the electric field introduced by theambipolar drift is perpendicular to B and so, just like the Hall term, does notalterE·B in the RHS of the helicity conservation equation (3.34); so ambipolardrift, like the Hall effect does not alter the volume dissipation/generation ofhelicity. Due to this feature, ambipolar drift provides a very useful toy modelfor the study of nonlinear evolution. We will study below a closure modelwhich exploits this feature.
Ambipolar drift can also be important in magnetic field evolution in proto-stars, and also in the neutral component of the galactic gas. In the classical(non-turbulent) picture of star formation, ambipolar diffusion regulates a slowinfall of the gas, which was originally magnetically supported (cf.  Chap-ter 11). In the galactic context ambipolar diffusion can lead to the developmentof sharp fronts near nulls of the magnetic field. This can in turn affect the rateof destruction/reconnection of the field [150–152].
3.9 The Biermann battery
Note that B = 0 is a perfectly valid solution of the induction equation (3.11).So there would be no magnetic field generated if one were to start with a zeromagnetic field. The universe probably did not start with an initial magneticfield. One therefore needs some way of violating the induction equation andproduce a cosmic battery effect, to drive currents from a state with initiallyno current. There are a number of such battery mechanisms which have beensuggested [19,138,139,153–158]. Almost all of them lead to only small fields,much smaller than the observed fields. Therefore dynamo action, due to avelocity field acting to exponentiate small seed fields efficiently, is needed toexplain observed field strengths. We briefly comment on one cosmic battery,the Biermann battery.
The basic problem any battery has to address is how to produce finite cur-rents from zero currents? Most astrophysical mechanisms use the fact thatpositively and negatively charged particles in a charge-neutral universe, donot have identical properties. For example if one considered a gas of ionizedhydrogen, then the electrons have a much smaller mass compared to protons.This means that for a given pressure gradient of the gas the electrons tend tobe accelerated much more than the ions. This leads in general to an electric
field, which couples back positive and negative charges. This is exactly thethermally generated field we found in deriving the generalized Ohm’s law.
Taking the curl of Eq. (3.64), using Maxwell’s equations (Faraday’s and Am-pere’s law) and writing pe = nekBT , where kB is the Boltzmann constant, weget
∂t= ∇× (U ×B)−∇× ηJ − ckB
Here we have taken the velocity of the ionic component to be also nearly thebulk velocity in a completely ionized fluid and so put ui = U . And we haveneglected the Hall effect and inertial effects as they are generally very smallfor the fields one generates.
We see that over and above the usual flux freezing and diffusion terms we havea source term for the magnetic field evolution, even if the initial field werezero. This source term is non-zero if and only if the density and temperaturegradients, ∇ne and ∇T , are not parallel to each other. The resulting batteryeffect, known as the Biermann battery, was first proposed as a mechanism forthe thermal generation of stellar magnetic fields [138,139].
In the cosmological context, the Biermann battery can also lead to the ther-mal generation of seed fields in cosmic ionization fronts . These ionizationfronts are produced when the first ultraviolet photon sources, like quasars, turnon to ionize the intergalactic medium (IGM). The temperature gradient in acosmic ionization front is normal to the front. However, a component to thedensity gradient can arise in a different direction, if the ionization front issweeping across arbitrarily laid down density fluctuations. Such density fluc-tuations, associated with protogalaxies/clusters, in general have no correlationto the source of the ionizing photons. Therefore, their density gradients arenot parallel to the temperature gradient associated with the ionization front.The resulting thermally generated electric field has a curl, and magnetic fieldson galactic scales can grow. After compression during galaxy formation, theyturn out to have a strength B ∼ 3 × 10−20G . A similar effect was con-sidered earlier in the context of generating fields in the interstellar mediumin . (This mechanism also has analogues in some laboratory experiments,when Laser generated plasmas interact with their surroundings [161,162]. In-deed our estimate for the generated field is very similar to the estimate of). This field by itself is far short of the observed microgauss strengthfields in galaxies, but it can provide a seed field, coherent on galactic scales,for a dynamo. Indeed the whole of the IGM is seeded with magnetic fields ofsmall strength but coherent on megaparsec scales.
This scenario has in fact been confirmed in detailed numerical simulations
of IGM reionization , where it was found that the breakout of ionizationfronts from protogalaxies and their propagation through the high-density neu-tral filaments that are part of the cosmic web, both generate magnetic fields.The field strengths also increase further due to the gas compression occur-ring as cosmic structures form. The magnetic field at a redshift z ∼ 5 closelytraces the gas density, and is highly ordered on megaparsec scales. They founda mean mass-weighted field strength of B ∼ 10−19 G in their simulation box.
The Biermann battery has also been shown to generate both vorticity andmagnetic fields in oblique cosmological shocks which arise during cosmologicalstructure formation [155,163]. In fact  point out that the well-knownanalogy between the induction equation and the vorticity (without Lorentzforce) extends even to the case where a battery term is present. Suppose weassume that the gas is pure hydrogen, has a constant (in space) ionizationfraction χ, the same temperature for electrons, protons and hydrogen. Thenpe = χp/(1 + χ) and ne = χρ/mp and defining ωB = eB/mp, the inductionequation with the thermal battery term can be written as
∂t= ∇× (U × ωB − η∇× ωB) +
1 + χ. (3.74)
The last term, without the extra factor of −(1 + χ), in fact also correspondsto the baroclinic term in the equation for the vorticity ω = ∇×U .
∂t= ∇× (U × ω − ν∇× ω)− ∇p×∇ρ
So, provided viscosity and magnetic diffusivity were negligible, both ωB(1+χ)and −ω satisfy the same equation, and if they were both zero initially, then forsubsequent times, eB/mp = −ω/(1+χ). Numerically a value of ω ∼ 10−15 s−1
corresponds to a magnetic field of about ∼ 10−19G.
4 Dynamos and the flow of energy
The dynamo mechanism provides a means of converting kinetic energy intomagnetic energy. We shall focus on the astrophysically relevant case of a tur-bulent dynamo, as opposed to a laminar one. Laminar dynamos are easier tounderstand – and we shall discuss some simple examples – while turbulentdynamos have to be tackled via direct numerical simulations or by stochasticmethods. Both will be discussed below.
Important insight can be gained by considering the magnetic energy equation.By taking the dot product of Eq. (3.11) with B/(2µ0) and integrating overthe volume V we obtain
dV = −∫
U · (J ×B) dV −∫
This equation shows that the magnetic energy can be increased by doing workagainst the Lorentz force, provided this terms exceeds resistive losses (secondterm) or losses through the surface (Poynting flux). Likewise, by taking thedot product of Eq. (3.39) with ρU and integrating one arrives at the kineticenergy equation
12ρU 2 dV = +
p∇ ·U dV +∫
U · (J ×B) dV
ρU · g dV −∫
2νρS2 dV, (4.2)
where we have assumed stress-free boundary conditions, so there are no surfaceterms and no kinetic energy is lost through the boundaries. Equations (4.1)and (4.2) show that the generation of magnetic energy goes at the expense ofkinetic energy, without loss of net energy.
In many astrophysical settings one can distinguish four different energy reser-voirs that are involved in the dynamo process: magnetic, kinetic, thermal, andpotential energy. In accretion discs the magnetic energy comes ultimately frompotential energy which is first converted into kinetic energy through resistiveprocesses. This requires turbulence to produce small enough length scales sothat enough kinetic energy can be dissipated on a dynamical time scale. Thisturbulence is most likely driven by the Balbus-Hawley (or magneto-rotational)instability; see Ref.  for a review. In Fig. 4.1 we show the energy diagramfrom a local simulation of the Balbus-Hawley instability where a dynamo pro-cess is able to replenish magnetic energy lost through Joule heating .
In the case of solar convection the energy for the dynamo comes ultimatelyfrom the nuclear reactions in the center of the star. This acts as a source ofthermal energy which gets converted into kinetic energy via the convection in-stability. The corresponding energy diagram for this case is shown in Fig. 4.2.Potential energy does not contribute directly: it only contributes through re-arranging the mean density stratification .
Fig. 4.1. Energy budget in a local accretion disc simulation where the turbulence ismaintained by the Balbus-Hawley instability. The numbers on the arrows indicatethe energy conversion rates in units of ΩEM, where Ω is the angular velocity andEM is the steady state value of the magnetic energy (adapted from Ref. ).
Fig. 4.2. Energy budget in a local convection simulation. The dynamo is convectivelydriven by the luminosity from below, giving rise to convection via work done byadiabatic compression, Wc ≡
∫p∇ · udV , and through work done against the
Lorentz force, WL =∫u · (J ×B) dV . Energy is being fed back from magnetic and
kinetic energy to thermal energy via Joule and viscous heating, QJ and Qv. Someof the kinetic energy is constantly being exchanged with potential energy EP viaWb =
∫ρg · udV (adapted from Ref. ).
4.2 Kinematic dynamos
The onset of dynamo action can be studied in the linear approximation, i.e.the velocity field is assumed to be given (kinematic problem). There is ingeneral a critical value of the magnetic Reynolds number above which themagnetic field grows exponentially. A lot of work has been devoted to the
question of whether the growth rate can remain finite in the limit Rm → ∞(the so-called fast dynamo); see Refs. [166–168] for reviews. Fast dynamos arephysically meaningful only until nonlinear effects begin to modify the flow tolimit further growth of the field.
In the following we consider two simple examples of a dynamo. Both are slowdynamos, i.e. magnetic diffusion is crucial for the operation of the dynamo. Wealso discuss the stretch-twist-fold dynamo as a qualitative example of what ispossibly a fast dynamo.
4.2.1 The Herzenberg dynamo
In the wake of Cowling’s antidynamo theorem  the Herzenberg dynamo played an important role as an early example of a dynamo where theexistence of excited solutions could be proven rigorously. The Herzenberg dy-namo does not attempt to model an astrophysical dynamo. Instead, it wascomplementary to some of the less mathematical and more phenomenologicalmodels at the time, such as the Parker migratory dynamo  as well asthe observational model of Babcock , and the semi-observational modelof Leighton , all of which were specifically designed to describe the solarcycle.
The Herzenberg dynamo is based on the mutual interaction of the magneticfields produced by two spinning spheres in a conducting medium. In its sim-plest variant, the axes of the two spheres lie in planes that are parallel to eachother, but the axes have an angle ϕ to each other; see Fig. 4.3, which showsthe field vectors from a numerical simulation of the Herzenberg dynamo .
Dynamo action is possible unless the angle ϕ is exactly 0, 90, or 180. For90 < ϕ < 180 nonoscillatory dynamo action is possible. In the limit wherethe radius of the spheres, a, is small compared with their separation d, onecan expand the field locally in terms of multipoles to lowest order. The criticalmagnetic Reynolds number for dynamo action is Rcrit, given by 
R−2crit = − 1
sin2ϕ cosϕ, (4.3)
which shows that the smallest value of Rcrit is reached for ϕ ≈ 125. Thedynamo works on the principle that each sphere winds up its ambient field,creates thereby a strong toroidal field around itself, but because there is anangle between the two spheres the toroidal field of one sphere acts as a poloidalfield for the other sphere. For the toroidal field of each sphere to propagate tothe other sphere, a non-zero diffusion is necessary, hence making this dynamoa slow dynamo.
Fig. 4.3. Three-dimensional visualization of the magnetic field geometry of theHerzenberg dynamo. B-vectors are shown when their length exceeds about 25%of the maximum value (adapted from Ref. ).
Already back in the sixties, the idea of the Herzenberg dynamo has beenverified experimentally [175,176] using two conducting cylinders embedded ina solid block of the same material. The cylinders were in electric contact withthe block through a thin lubricating film of mercury.
The asymptotic theory of Herzenberg  assumed that a/d ≪ 1; for excel-lent reviews of the Herzenberg dynamo see Refs. [129,177]. Using numericalsimulations , it has been shown that Eq. (4.3) remains reasonably ac-curate even when a/d ≈ 1. These simulations also show that in the range0 < ϕ < 90 dynamo action is still possible, but the solutions are no longersteady but oscillatory; see Ref.  for an asymptotic treatment. In the earlypapers, only steady solutions were sought, which is the reason why no solutionswere originally found for 0 < ϕ < 90.
4.2.2 The Roberts flow dynamo
In the early years of dynamo theory most examples were constructed andmotivated based on what seems physically possible and plausible. An impor-tant element of astrophysical dynamos is that the flow is bounded in spaceand that the magnetic field extends to infinity. Later, and especially in recent
Fig. 4.4. Roberts flow pattern with periodicity 2a, corresponding to Eq. (4.4)(adapted from Ref. ).
years, these restrictions were relaxed in may approaches. One of the first ex-amples is the G. O. Roberts dynamo [178,179]. The flow depends on only twocoordinates, U = U(x, y), and can be written in the form
U(x, y) =√2∇× (ϕz) +∇×∇× (ϕz), (4.4)
with the stream function ϕ = (U0/kf) sin kxx sin kyy, where kx = ky = π/a;see Fig. 4.4. This flow is fully helical with W = kfU , were k2
f = k2x + k2
y .Since the flow is only two-dimensional (in the sense that U is a function onlyof x and y) for growing solutions (dynamo effect) the magnetic field must bethree-dimensional and must therefore also depend on z.
The governing equations are homogeneous with coefficients that are indepen-dent of z and t. The solutions of the kinetic problem can therefore be writtenin the form
B(x, y, z, t) = Re[
Bkz(x, y) exp(ikzz + λt)]
where Bkz is the eigenfunction, which is obtained by solving the eigenvalueproblem
λAkz = U × Bkz + η(
∇2 − k2z
Akz , (4.6)
Fig. 4.5. Critical magnetic Reynolds number, Rcrit = U0/(ηkf )crit, for the Robertsflow as a function of kz/kf , where k2f = k2x + k2y. The critical magnetic Reynoldsnumber based on kz, Rcrit,kz = U0/(ηkz)crit, has a minimum at kz ≈ 0.34kf ≈ 0.48kx.The case of a squared domain with kx = ky = kz, i.e. kz/kf = 1/
√2, is indicated by
the vertical dash-dotted line.
where Bkz = ∇ × Akz + ikz × Akz is expressed in terms of Akz , which is amixed representation of the vector potential; in real space the vector potentialwould be Re
Akz(x, y) exp(ikzz + λt)]
. In Fig. 4.5 we present critical valuesof the magnetic Reynolds number as a function of kz, obtained by solvingEq. (4.6) numerically as described in Refs. [180,181]. For kz = kx = ky ≈0.71kf , the marginal state (λ = 0) is reached when Rcrit ≡ U0/(ηkf)crit ≈ 3.90.The larger the domain in the z-direction, the lower is the critical magneticReynolds number. However, the critical magnetic Reynolds number based onkz, Rcrit,kz = U0/(ηkz)crit, has a minimum at kz ≈ 0.34kf ≈ 0.48kx withRcrit,kz ≈ 3.49; cf. Fig. 4.5.
The horizontally averaged eigenfunction is Bkz = (i, 1, 0), corresponding to aBeltrami wave (see Section 3.7), which has maximum magnetic helicity witha sign that is opposite to that of the flow. In the present case, the kinetichelicity of the flow is positive, so the magnetic and current helicities of themean field are negative.
The significance of this solution is two-fold. On the one hand, this dynamo isthe prototype of any fully helical dynamo capable of generating a large scalefield. On the other hand, it is a simple model of the Karlsruhe dynamo exper-iment where a similar flow of liquid sodium is generated by an arrangementof pipes with internal ‘spin generators’ making the flow helical. It is also anexample of a flow where the generation of the magnetic field can be describedin terms of mean field electrodynamics.
Unlike the original Roberts flow dynamo, the flow in the Karlsruhe dynamoexperiment is bounded and embedded in free space. Within the dynamo do-
main, the mean field, Bkz = (i, 1, 0), has only (x, y)-components. The fieldlines must close outside the dynamo domain, giving therefore rise to a dipolelying in the (x, y)-plane. Similar fields have long been predicted for rapidlyrotating stars . This will be discussed in more detail in Section 11.3.3.
4.3 Fast dynamos: the stretch-twist-fold picture
An elegant heuristic dynamo model illustrating the possibility of fast dynamosis what is often referred to as the Zeldovich ‘stretch-twist-fold’ (STF) dynamo(see Fig. 4.6). This is now discussed in many books ([168,184]) and we brieflyoutline it here, as it illustrates nicely several features of more realistic dy-namos.
The dynamo algorithm starts with first stretching a closed flux rope to twice itslength preserving its volume, as in an incompressible flow (A→B in Fig. 4.6).The rope cross-section then decreases by factor two, and because of flux freez-ing the magnetic field doubles. In the next step, the rope is twisted into afigure of eight (B→C in Fig. 4.6) and then folded (C→D in Fig. 4.6) so thatnow there are two loops, whose fields now point in the same direction andtogether occupy a similar volume as the original flux loop. The flux throughthis volume has now doubled. The last important step consists of mergingthe two loops into one (D→A in Fig. 4.6), through small diffusive effects.This is important in order that the new arrangement cannot easily undo it-self and the whole process becomes irreversible. The newly merged loops nowbecome topologically the same as the original single loop, but now with thefield strength scaled up by factor 2.
Repeating the algorithm n times, leads to the field in the flux loop growingby factor 2n, or at a growth rate ∼ ln 2/T where T is the time for the STFsteps. This makes the dynamo potentially a fast dynamo, whose growth ratedoes not decrease with decreasing resistivity. Also note that the flux througha fixed ‘Eulerian surface’ grows exponentially, although the flux through anyLagrangian surface is nearly frozen; as it should be for small diffusivities.
The STF picture illustrates several other features: first we see that shear isneeded to amplify the field at step A→B; but without the twist part of thecycle, the field in the folded loop will cancel rather than add coherently. Totwist the loop the motions need to leave the plane and go into the thirddimension; this also means that field components perpendicular to the loopare generated, albeit being strong only temporarily during the twist part ofthe cycle. The source for the magnetic energy is the kinetic energy involved
Fig. 4.6. A schematic illustration of the stretch-twist-fold-merge dynamo.
in the STF motions.
Most discussions of the STF dynamo assume implicitly that the last step ofmerging the twisted loops can be done at any time. And the dynamo growthrate is not limited by this last step. This may well be true when the fieldsin the flux rope are not strong enough to affect the motions, that is in thekinematic regime. However as the field becomes stronger, and if the mergingprocess is slow then the Lorentz forces due to the small scale kinks and twistswill gain in importance before the forces associated with the loop as a whole,and limit the efficiency of the dynamo.
Indeed, the growing complexity of the field, in the repeated application of theSTF process, without the merging, can be characterized particularly by theevolution of the magnetic helicity spectrum. This is discussed for example inRef. , where it is pointed out that the repeated application of the STFcycle (with the same sense for the twist part of the cycle), under flux freez-ing, leads to both a large scale writhe helicity associated with the repeatedcrossings of the flux tube, and oppositely signed twist helicity at much smallerscales. If one does not destroy this small scale structure by diffusion, then theLorentz forces associated with these structures will interfere with the STFmotions. A somewhat similar situation holds for the mean field turbulent dy-namos, where we will find oppositely signed (almost equal strength) helicitiesbeing generated by the motions on different scales. In that case we also findthe dynamo to be eventually resistively limited, when there is strict helicityconservation!
In this context one more feature deserves mention: if in the STF cycle one
twists clockwise and folds, or twists anticlockwise and folds one will still in-crease the field in the flux rope coherently. But one would introduce oppositesense of writhe in these two cases, and so opposite internal twists. So althoughthe twist part of the cycle is important for the mechanism discussed here, thesense of twist can be random and does not require net helicity. This is anal-ogous to a case when there is really only a small scale dynamo, but one thatrequires pointwise helicity. We should point out, however, that numerical sim-ulations  have shown that dynamos work and are potentially independentof magnetic Reynolds number even if the flow has pointwise zero helicity.
If the twisted loops can be made to merge efficiently, the saturation of the STFdynamo would probably proceed differently. For example, the field in the loopmay become too strong to be stretched and twisted, due to magnetic curva-ture forces. Another interesting way of saturation is that the incompressibilityassumed for the motions may break down; as one stretches the flux loop thefield pressure resists the decrease in the loop cross-section, and so the fluiddensity in the loop tends to decrease as one attempts to make the loop longer.(Note that it is B/ρ which has to increase during stretching.) The STF pic-ture has inspired considerable work on various mathematical features of fastdynamos and some of this work can be found in the book by Childress andGilbert  which in fact has STF in its title!
4.4 Fast ABC-flow dynamos
ABC flows are solenoidal and fully helical with a velocity field given by
C sin kz +B cos kyA sin kx+ C cos kzB sin ky + A cos kx
When A, B, and C are all different from zero, the flow is no longer integrableand has chaotic streamlines. There is numerical evidence that such flows actas fast dynamos . The magnetic field has very small net magnetic he-licity [137,185]. This is a general property of any dynamo in the kinematicregime and follows from magnetic helicity conservation, as will be discussedlater. Even in a nonlinear formulation of the ABC flow dynamo problem,where the flow is driven by a forcing function similar to Eq. (4.7) the netmagnetic helicity remains unimportant [188,189]. This is however not sur-prising, because the development of net magnetic helicity requires sufficientscale separation, i.e. the wavenumber of the flow must be large compared withthe smallest wavenumber in the box (k = k1). If this is not the case, helicalMHD turbulence behaves similarly to nonhelical turbulence . A signifi-cant scale separation also weakens the symmetries associated with the flow
and the field, and leads to a larger kinematic growth rate, more compatiblewith the turnover time scale . These authors also find that the cigar-likemagnetic field structures which develop in canonical A = B = C flows withstagnation points, are replaced by more ribbon like structures in flows withoutsuch stagnation points.
Most of the recent work on nonlinear ABC flow dynamos has focused on thecase with small scale separation and, in particular, on the initial growth andpossible saturation mechanisms . In the kinematic regime, these authorsfind a near balance between Lorentz work and Joule dissipation. The balanceoriginates primarily from small volumes where the strong magnetic flux struc-tures are concentrated. The net growth of the magnetic energy comes aboutthrough stretching and folding of relatively weak field which occupies mostof the volume. The mechanism for saturation could involve achieving a localpressure balance in these strong field regions .
5 Small scale turbulent dynamos
Dynamos are often divided into small scale and large scale dynamos. Largescale dynamos are those responsible for the solar cycle, for example. Theyshow large scale spatial coherence and, in the case of the sun, they also showlong-term temporal order in the sense of the 11 year cycle, i.e. much longerthan the time scale of the turbulent motions. Small scale dynamos producemagnetic fields that are correlated on scales of the order of or smaller thanthe energy carrying scale of the velocity. In the literature such dynamos aresometimes also referred to as ‘fluctuation dynamos’. Even nonhelical turbulentflows can act as small scale dynamos, while flows with significant amounts ofkinetic helicity act as large scale dynamos. Inhomogeneous and anisotropicflows are potential candidates for producing large scale dynamo action.
Small scale dynamos are potentially important for several reasons. Firstly, theytypically have larger growth rates than large scale dynamos. The question nowarises as to what effect this rapidly generated magnetic noise has on the largescale dynamo action. Further there could be physical settings where large scaledynamos do not work, like in clusters of galaxies or elliptical galaxies whererotation effects are negligible and hence any turbulent flow lacks helicity. Insuch systems, turbulence may still lead to small scale dynamo action andgenerate magnetic fields. Whether such fields are coherent enough to lead tothe observed cluster rotation measures for example, is an important questionto settle.
5.1 General considerations
In a turbulent flow fluid particles random walk away from each other in time.A magnetic field line frozen into the fluid (assuming large Rm) will then alsolengthen by this random stretching. This leads to an increase in B/ρ and forflows with ρ ≈ const, the magnetic field will be amplified. The lengtheningof the field line in a given direction also leads to a decrease in its scale inthe directions perpendicular to the stretching. As the field strength increases,the scale of individual field structures decreases and the Ohmic dissipationincreases, until it roughly balances the growth due to random stretching. Whathappens after this? This question of the long term behavior of the magneticfield in a turbulent flow was first raised by Batchelor . He argued on thebasis of the analogy of the induction equation to the equation for vorticity,that the field will grow exponentially if the magnetic Prandtl number is largerthan unity. This argument is dubious and the possibility of dynamo action inturbulent flows that lack helicity was first elucidated in a decisive manner byKazantsev , for a special kind of flow. Numerical simulations of turbulentflows also show invariably that dynamo action can occur for forced turbulence.We first discuss the Kazantsev dynamo model and then present some resultsfrom simulations of turbulent flows.
5.2 Kazantsev theory
Kazantsev considered a velocity field, v, which is an isotropic, homogeneous,gaussian random field with zero mean and also more importantly δ-correlatedin time. [We use the symbol v instead of U to emphasize that v is not thesolution of the momentum equation (3.41).] The two point spatial correlationfunction of the velocity field can be written as 〈vi(x, t)vj(y, s)〉 = Tij(r)δ(t−s),where
Here 〈·〉 denotes averaging over an ensemble of the stochastic velocity fieldv, r = |x − y|, ri = xi − yi and we have written the correlation functionin the form appropriate for a statistically isotropic and homogeneous tensor(cf. Section 34 of Ref. ). Note that homogeneity implies the two pointcorrelation function depends only on x − y. Together with isotropy this alsoimplies that the correlation tensor can only contain terms proportional to δij ,rirj and ǫijk and the functions multiplying these tensors depend only on r.TL(r) and TN(r) are the longitudinal and transverse correlation functions forthe velocity field. (The helical part of the velocity correlations is assumed to be
zero in this section; see however Section 5.6.) If v is assumed to be divergencefree, then TN is related to TL via
〈v(t) · v(t′)〉 dt′. (5.3)
We will see in the next section that TL(0) is actually the turbulent diffusioncoefficient for the mean field.
5.2.1 Kazantsev equation in configuration space
The stochastic induction equation can now be converted into equations for thevarious moments of the magnetic field. Assume that there is no mean field orfirst moment, and that the magnetic correlation has the same symmetries asthe flow; i.e it is isotropic and homogeneous. Then its equal time, two pointcorrelation is given by 〈Bi(x, t)Bj(y, t)〉 = Mij(r, t), where
MN(r, t) +rirjr2
ML(r, t), (5.4)
and ML(r, t) and MN(r, t) are, respectively, the longitudinal and transversalcorrelation functions of the magnetic field. Since ∇ ·B = 0,
Kazantsev derived an equation for ML(r, t) by deriving the equation for thek-space magnetic spectrum using diagram techniques, and transforming theresulting integro-differential equation in k-space into a differential equation inr-space. This k-space equation was also derived by Kraichnan and Nagarajan. Subsequently, Molchanov et al. [196,197] derived this equation directlyin r-space by the Wiener path integral method. We present a simple deriva-tion of the Kazantsev equation in the more general case of helical turbulencein Appendix A, following the method outlined in Subramanian . For anonhelical random flow we have
where ηT(r) = η + ηt(r) is the sum of the microscopic diffusivity, η, andan effective scale-dependent turbulent magnetic diffusivity ηt(r) = TL(0) −TL(r). The term G = −2(T ′′
L + 4T ′L/r), where primes denote r derivatives,
describes the rapid generation of magnetic fluctuations by velocity shear andthe potential existence of a small scale dynamo (SSD) independent of any largescale field; see the book by Zeldovich et al.  and references therein.
One can look for eigenmode solutions to Eq. (5.6) of the form Ψ(r) exp(2Γt) =r2√ηTML. This transforms Eq. (5.6) for ML(r, t), into a time independent,
Schrodinger-type equation, but with a variable (and positive) mass,
−ΓΨ = −ηTd2Ψ
dr2+ U0(r)Ψ. (5.7)
The ‘potential’ is
U0(r) = T ′′L +
rT ′L +
2η′′T − (η′T)
for a divergence free velocity field. The boundary condition is Ψ → 0 forr → 0,∞. Note that U0 → 2η/r2 as r → 0, and since TL(r) → 0 as r → ∞, itfollows that U0 → 2[η+TL(0)]/r
2 as r → ∞. The possibility of growing modeswith Γ > 0 is obtained, if one can have a potential well with U0 sufficientlynegative in some range of r. This allows for the existence of bound states
with ‘energy’ E = −Γ < 0. The solutions to the Kazantsev equation forvarious forms of TL(r) have been studied quite extensively by several authors[13,184,198–204].
Suppose we have random motions on a single scale L, with a velocity scale V .Define the magnetic Reynolds number Rm = V L/η. Such a random flow mayarise if the fluid is highly viscous, or will also be relevant for the viscous cut-offscale eddies in Kolmogorov turbulence. Then one finds that there is a criticalRm = Rcrit, so that for Rm > Rcrit, the potential U0 allows for the existence ofbound states. For Rm = Rcrit, Γ = 0, and this marginal stationary state is the‘zero’ energy eigenstate in the potential U0. The value of Rcrit one gets rangesbetween 30−60 depending on assumed form of TL(r) [198–200]. ForRm > Rcrit,Γ > 0 modes of the SSD can be excited, and the fluctuating field correlatedon a scale L, grows exponentially, on the corresponding ‘eddy’ turnover timescale. For example, suppose one adopts TL(r) = (V L/3)(1− r2/L2) for r < L,and zero otherwise, as appropriate for a single scale flow (or the flow belowthe viscous cut-off). Then a WKBJ analysis , gives the growth rate forthe fastest growing mode as Γ = 5
To examine the spatial structure for various eigenmodes of the small scaledynamo, it is more instructive to consider the function
w(r, t) = 〈B(x, t) ·B(y, t)〉 = 1
which measures the ensemble average of the dot product of the fluctuating fieldat two locations, with w(0, t) = 〈B2〉. We have
∫∞0 w(r)r2dr =
′ = 0,since ML is regular at the origin and vanishes faster than r−3 as r → ∞.Therefore the curve r2w(r) should have zero area under it. Since w(0, t) =〈B2〉, w is positive near the origin. Therefore, B points in the same directionfor small separation. As one goes to larger values of r, there must then bevalues of r, say r ∼ d, where w(r) becomes negative. For such values of r,the field at the origin and at a separation d are, on the average, pointing inopposite directions. This can be interpreted as indicating that the field lines,on the average, are curved on the scale d.
For the growing modes of the small scale dynamo, one finds [198–202] thatw(r) is strongly peaked within a region r = rd ≈ LR−1/2
m about the origin, forall the modes, and for the fastest growing mode, changes sign across r ∼ Land rapidly decays with increasing r/L. The scale rd is in fact the diffusivescale determined by the balance of rate of Ohmic decay η/r2d, and the growthrate V/L due to random shearing. (Note for the single scale flow one has linearshear at small scales, and hence a scale independent shearing rate). Detailedasymptotic solutions for w(r) have been given by  and a WKBJ treat-ment can also be found in . A physical interpretation of this correlationfunction [184,201,200] is that the small scale field in the kinematic regimeis concentrated in structures with thickness rd and curved on a scale up to∼ L. How far such a picture holds in the nonlinear regime is still matter ofinvestigation (see below).
The small scale dynamo in Kolmogorov turbulence can be modeled [198,202],by adopting TL(r) = 1
3V L[1 − (r/L)4/3] in the inertial range ld < r < L,
a form suggested in Ref. . Here L is the outer scale and ld ≈ LRe−3/4
is the viscous cut-off scale of the turbulence, where Re = V L/ν is the fluidReynolds number. For Kolmogorov turbulence, the eddy velocity at any scalel, is vl ∝ l1/3, in the inertial range. So the scale dependent diffusion coefficientscales as vll ∝ l4/3. This scaling, also referred to as Richardson’s law, is themotivation for the above form TL(r). Also, one should ensure T ′
L(0) = 0 andso TL is continued from its value at r = ld to zero, satisfying this constraintand was taken to be zero for r > L. (The exact form of the continuation haslittle effect on the conclusions). In the inertial range the potential then has
the scale invariant form,
3/Rm(l) + (r/l)4/3+
where Rm(l) = vll/η = Rm(l/L)3/4 is the magnetic Reynolds number asso-
ciated with a scale l. Note that U has the same form at any scale l, withRm(l) appropriate to that scale. This suggests that a number of conclusionsabout the generation of small scale fields can be scaled to apply to an arbi-trary scale, l, provided we use the corresponding velocity scale vl and Reynoldsnumber Rm(l) appropriate to the scale l. For example, the condition for ex-citation of small scale dynamo modes which are concentrated at a scale l, isalso Rm(l) = Rcrit ≫ 1.
Note that Rm(l) decreases as one goes to smaller scales, and so the smallscale dynamo will be first excited when the magnetic Reynolds number atthe outer scale satisfies Rm(L) > Rcrit. For Kolmogorov turbulence describedby the above TL(r)  estimated from a WKBJ analysis Rcrit ∼ 60. Also,the marginal mode which has zero growth rate, in this case, has w(r) peakedwithin r ∼ L/R3/4
m . This different scaling can be understood as arising due tothe fact that the shearing rate now is (V/L)(r/L)−2/3 at any scale r in theinertial range, and for the marginal mode, this is balanced by dissipation atr = rd with occurs at a rate η/r2d.
In Kolmogorov turbulence, the cut-off scale eddies have Rm(ld) = Pm, i.e.the magnetic Prandtl number. So, if Pm > Rcrit these eddies are themselvescapable of small scale dynamo action. The fastest growing modes then havea growth rate ∼ vd/ld, the eddy turn around time associated with the cut-offscale eddies, a time scale much smaller than the turn around time of outerscale eddies. Also, for the fastest growing mode w(r) is peaked about a radiuscorresponding to the diffusive scales associated with these eddies, and changessign across r ∼ ld.
We have taken the scale-dependent turbulent diffusion coefficient TL(0) −TL(r) ∝ rn with n = 4/3 to model Kolmogorov turbulence. The index nmeasures how ‘rough’ the velocity field is, with n = 2 corresponding to asmooth velocity field. It turns out that for growing modes one requires n > 1at least . Also in a small Pm flow the closer n is near unity, the larger couldbe the critical Rm needed to excite the small scale dynamo . This may beof relevance for the small Pm dynamo simulations which are described later.
5.2.2 Kazantsev dynamo in Fourier space
It is also instructive to study the Kazantsev problem in Fourier space. In kspace the differential equation (5.6) becomes an integro-differential equationfor the magnetic spectrum M(k, t), which is in general difficult to solve. How-ever, if the magnetic spectrum is peaked on scales much smaller than theflow, as is likely in the kinematic regime, then one can provide an approxi-mate treatment for the large k regime, k ≫ kf . Here kf is the forcing scale incase of a single scale random flow, or the viscous scale in case of Kolmogorovturbulence (assuming that eddies at the cut-off scale can also induce dynamoaction). In this large k ≫ kf limit the Kazantsev equation becomes 
− 2ηk2M, (5.11)
where γ = −(1/6)[∇2Tii(r)]r=0, is a measure of the rate of shearing by theflow. In terms of TL, we have γ = 7T ′′
L (0) + 8T ′L(0)/r. The evolution of the
magnetic spectrum was analyzed in some detail by , and is summarizednicely in Ref. .
Suppose the initial magnetic spectrum is peaked at some k = k′ ≪ kfR1/2m ,
i.e. at a wavenumber much smaller than the resistive wavenumber, then theamplitude of each Fourier mode grows exponentially in time at the rate 3
Meanwhile, the peak of excitation moves to larger k, with kpeak = k′ exp(35γt),
leaving behind a power spectrum M(k) ∝ k3/2. These features can of coursebe qualitatively understood as due to the effects of random stretching. Oncethe peak reaches the resistive scale, one has to solve again an eigenvalue prob-lem to determine the subsequent evolution of M(k, t). Substituting M(k, t) =expλγt Φ(k/kη), where kη = (γ/10η)1/2, into (5.11) and demanding that Φ → 0as k → ∞, one gets the solution ,
Φ(k/kη) = const× k3/2Kν(λ)(k/kη), ν(λ) =√
5(λ− 34). (5.12)
Here Kν is the Macdonald function and the eigenvalue λ must be determinedfrom the boundary condition at small k. This is a bit more tricky in the k-space analysis since the equations were simplified by taking the large k limit;fortunately the results seem independent of the exact form of the boundarycondition at small k in the large Rm limit. In Ref.  a zero flux boundarycondition is imposed at some k = k∗ ≪ kη and it is shown that this fixesλ ≈ 3/4 and ν ≈ 0. In case we adopt TL(r) = 1
3V L(1 − r2/L2), for r < L
and zero otherwise, one gets γ = 53V/L and so a growth rate Γ = 3
4γ = 5
which agrees with the WKBJ analysis of the Kazantsev equation obtained inSection 5.2.1.
5.2.3 Further results on the Kazantsev dynamo
A number of other interesting results have also been found for the Kazantsevproblem, particularly for the case of a single scale flow. In this case one canapproximate the velocity to be a linear random shear. In the regime when thefield has not yet developed small enough scales, resistivity is unimportant, andone can assume the field to be frozen into the flow. The result of such passiverandom advection is that the magnetic field strength develops a log-normalprobability distribution function (PDF) [208,209]. This log-normal form forthe PDF of B can be understood from the induction equation without thediffusion term Bi = Bjui,j where the time derivative is a Lagrangian one.Since the RHS is linear in Bi and is multiplied by a constant random matrixui,j, the gaussianity of logB is expected from the central limit theorem. Thisimplies that the magnetic field becomes highly intermittent. To what extentthis PDF is altered by resistivity is examined in Ref. . They find that evenwhen resistivity is included, the field is still intermittent and may be thoughtof as being concentrated into narrow strips.
This seems to be also borne out by a study of the behavior of higher order k-space correlators . By examining their late time evolution, these authorsalso find that the log-normality of the magnetic field PDF persists in thedissipative regime. An interpretation of their k-space scalings, suggests thatthe magnetic structures in physical space could look like ‘ribbons’ with thefield directed along these ribbons. Some recent numerical work , however,finds that even in the kinematic regime, the PDF of |B| changes character,perhaps due to having a finite box in the simulation.
Motivated by the need to understand the eventual nonlinear saturation of thesmall scale dynamo, work has been done by looking at the statistical propertiesof the Lorentz force, in particular the component B ·∇B [207,209,213]. Theidea is that in an incompressible flow, the effects of the Lorentz force willbe dominated by the magnetic curvature rather than the magnetic pressuregradient. In Ref.  it was shown that though both magnetic energy andmean-square curvature of field lines grow exponentially, the field strength andthe curvature are anti-correlated. Thus, regions of strong field are relativelystraight, while sharply curved fields are relatively weak. Such a result was infact found earlier in simulations of dynamo generation of magnetic fields dueto convection [165,214]. In these simulations the magnetic field was found tobe intermittent with relatively stronger field concentrated in structures whichare relatively straight. Furthermore, the saturation of the dynamo was tracedback to the component of B · ∇B that is perpendicular to the direction ofthe field; its field-aligned component (i.e. the tension force) was found to beunimportant for saturation .
The anti-correlation between the field strength and the curvature is interpreted
by [207,209] as showing that the field lies in folds, which are curved on theflow scale and with rapid reversals on the diffusive scale. To what extent thesmall scale dynamo generated field is indeed structured in this way is stilla matter of debate. Also, one should keep in mind that all the above semi-analytic results on the structure of the field pertain to the kinematic regime.The recent simulations of  discussed in Section 5.4 do not show evidencefor such folded structures.
5.3 Saturation of the small scale dynamo
How does the small scale dynamo saturate? The answer to this question willbe crucial to understand both the effect of the small scale dynamo generatedfields on the large scale dynamo, as well of the relevance of these fields toexplain say cluster fields. From the discussion of the kinematic regime, wesaw that if one were to wait long enough, the field would probably becomeintermittent and also concentrated into structures whose thickness, in at leastone dimension, is the resistive scale. However nonlinear effects may interveneto alter this kinematic result. There have been a number of attempts to modelthe nonlinear saturation of the small scale dynamo, none of which are entirelycompelling. We discuss these below.
Some nonlinear effects can arise just due to plasma effects like ambipolardrift in a partially ionized gas , anisotropic viscosity  or collision-less damping . For example, collisionless processes can become importantfor scales smaller than the ion mean free path and could prevent the mag-netic field to be concentrated below such a scale . Ambipolar diffusion isparticularly relevant in the galactic context, where there could be significantneutral component to the interstellar medium. It changes the effective diffusiv-ity, by adding an ambipolar diffusion component [198,202]; see Section 3.8.2and Appendix A. The effective magnetic Reynolds number of the interstel-lar medium, taking ambipolar diffusion into account, decreases from about3× 1019 to RAD ∼ 106 , as the field grows to microgauss strengths. Thiswill lead then lead to an important increase of the effective diffusive scale,which due to nonlinear ambipolar drift will become of order L/R
1/2AD ∼ 10−3L.
However since RAD ≫ Rcrit still, the small scale dynamo does not saturatedue to this mechanism.
5.3.1 Saturation via artificial nonlinear drifts
The above discussion of ambipolar drift motivates also a model nonlinear prob-lem which may give pointers to how the small scale dynamo can in principlesaturate, even in a fully ionized gas . Suppose one assumes that as mag-
netic field grows, and the Lorentz force pushes on the fluid, the fluid almostinstantaneously responds to develop an extra ‘drift’ component to the velocityproportional to the instantaneous Lorentz force. So we take a model velocityfield in the induction equation to be u = v + vN, the sum of an externallyprescribed stochastic field v, and a drift component vN = aJ ×B. For am-bipolar drift, a would be related to the properties of the partially ionized gas.Suppose we adopt instead a = τ/ρ, where τ is some response time, treatedas a phenomenological free parameter and ρ is the fluid density. This givesa model nonlinear problem, where the nonlinear effects of the Lorentz forceare taken into account as simple modification of the velocity field. This ofcourse also leads to a problem of closure, since in the equation for ML, thenonlinear drift term brings in a fourth order magnetic correlation. In [198,219]a gaussian closure is assumed (cf. Appendix A).
The back reaction in the form of a nonlinear drift, then simply replaces η byan effective, time dependent ηD = η + 2aML(0, t), in the ηT(r) term of Eq.(5.6)  (cf. Appendix A). We get
+GML +K, (5.13)
K(r, t) = 4aML(0, t)1
Define an effective magnetic Reynolds number (just like RAD), for fluid motionon scale L, by RD(t) = V L/ηD(t). Then as the energy density in the fluctuatingfield, say EM(t) = 3
2ML(0, t), increases, RD decreases. In the final saturated
state, with ∂ML/∂t = 0 (obtaining say at time ts), ML, and hence the effectiveηD in Eq. (5.13) become independent of time. Solving for this stationary statethen becomes identical to solving for the marginal (stationary) mode of thekinematic problem, except that Rm is replaced by RD(ts). The final saturatedstate is then the marginal eigenmode which obtains, when EM has grown (andRD decreased) such that RD(ts) = V L/[η + 2aML(0, ts)] = Rcrit ∼ 30–60(depending on the nature of the velocity field). Also for this saturated state,we predict that w(r) is peaked within a region r ≈ L(Rcrit)
−1/2 about theorigin, changes sign across r ∼ L and then rapidly decays for larger r/L.Further, from the above constraint, ML(0, ts) = vL/(2aRcrit), assuming η ≪2aML(0, ts). So the magnetic energy at saturation, EM(ts) = 3
2ML(0, ts) =
c . Of course since τ is an unknown model parameter, onecannot unambiguously predict EM. If we were to adopt τ ∼ L/v, that is theeddy turnover time, then EM at saturation is a small fraction ∼ R−1
crit ≪ 1, ofthe equipartition energy density.
This model for saturation may be quite simplistic, but gives a hint of onepossible property of the saturated state: It suggests that the final saturated
state of the small scale dynamo could be (i) universal, in that is doesn’t dependon the microscopic parameters far away from the resistive/viscous scales and(ii) have the properties similar to the marginal eigenmode of the correspondingkinematic small scale dynamo problem. It is of interest to check whether sucha situation indeed obtains.
The nonlinear drift velocity assumed above is not incompressible. Indeed am-bipolar drift velocity in a partially ionized medium, need not have a vanishingdivergence. And it is this extra diffusion that causes the dynamo to saturate.One may then wonder what happens if we retain incompressibility of the mo-tions induced by the Lorentz force? Is there still increased nonlinear diffusion?This has been examined in  by adopting u = v+vN, with an incompress-ible vN = a(B ·∇B −∇p). Here p includes the magnetic pressure, but canbe projected out in the usual way using ∇ · vN = 0. Such a model of theeffects of nonlinearity is very similar to the quasilinear treatments mean fielddynamo saturation (Section 9.2). For this form of nonlinearity, one gets anintegro-differential equation for the evolution of ML, which in general is notanalytically tractable. One can however make analytic headway in two limitsr = |x− y| ≫ l, and r ≪ l, where l(t) is the length scale over which ML(r, t) ispeaked. For example, during kinematic evolution, ML(r, t) is strongly peakedwithin a radius l = rd ∼ L/R1/2
m , where L is the correlation length associatedwith the motions. One gets 
K(r, t) = 2aML(0, t)1
L)2, r ≪ l
, r ≫ l. (5.15)
So the nonlinear back reaction term K in this model problem, is like a non-linear diffusion for small r ≪ l (yet partially compensated by a constant),transiting to nonlinear hyperdiffusion for r ≫ l . In both regimes thedamping coefficients are proportional to EM(t). So as the small scale fieldgrows and EM increases, the damping increases, leading to a saturated state.Note that both diffusion and hyperdiffusion would lead to an increase in theeffective resistivity, just as in the case of ambipolar drift; so this propertyobtains even if one demands vN to be incompressible.
A somewhat different model of the nonlinear backreaction can be motivated,if the fluid is highly viscous with the Re ≪ 1 ≪ Rm [221,222]. In this casevN is assumed to satisfy the equation ν∇2vN + J × B − ∇p = 0, and ∇ ·vN = 0. Again the equation for ML becomes an integro-differential equation.The extra nonlinear term is again simplified near r = 0. One gets K(0, t) =−M2
L/3−∫∞0 r(M ′
L(r))2 . This can be interpreted as a nonlinear reduction
of the G term governing the stretching property of the flow. Saturation of the
small scale dynamo will then result in this model by Lorentz forces reducingthe random stretching, rather than increased diffusion as in the models of[219,220].
5.3.2 Modifying the Kazantsev spectral equation
Another approach to the nonlinear small scale dynamo, with a large Prandtlnumber Pm has been explored by [207,223]. The idea is to modify the coeffi-cients of the Kazantsev equation in a phenomenologically motivated mannerand then examine its consequences. The motivation arises from the expectationthat, as the magnetic field grows, it suppresses the dynamo action of eddieswhich have energies smaller than the field. Only eddies which have energieslarger than the field are able to still amplify the field. Also in the kinematicregime of small scale dynamo action with large Pm, the magnetic spectrumis peaked at large k = kη (the resistive wavenumber). With these features inmind,  modify the γ in the Kazantsev k-space equation Eq. (5.11) bytaking it to be proportional to turn over rate of the smallest ‘unsuppressededdy’ and study the resulting evolution, assuming the fluid has a large Pm.Specifically, they adopt
γ(t) = c1
E(k)dk = EM(t), (5.16)
where E(k) is the kinetic energy spectrum, c1, c2 are constants, and ks(t) isthe wavenumber at which the magnetic energy equals the kinetic energy of allthe suppressed eddies.
In this model, after the magnetic energy has grown exponentially to the energyassociated with the viscous scale eddies, its growth slows down, and becomeslinear in time, with EM(t) = ǫt, where ǫ = v3/L is the rate of energy transferin Kolmogorov turbulence. This phase proceeds until the energy reaches thatof the outer scale eddies. However the peak in the magnetic spectrum evolvesto smaller k than the initial kη only on the resistive time scale, which is verylong for realistic astrophysical systems. This implies that the saturated smallscale fields could have energies comparable to the energy of the motions, butthe field will be still largely incoherent. More recently, another model has beenexplored in which saturation is achieved as a result of the velocity statisticsbecoming anisotropic with respect to the local direction of the growing field.
Clearly more work needs to be done to understand the saturation of small scaledynamos. A general feature however in all the above models is that the smallscale dynamo saturates because of the ‘renormalization’ of the coefficients
governing its evolution: increased nonlinear diffusion , increased diffusionplus additional hyperdiffusion , or reduced stretching [207,221–223]. Thenature of the saturated fields is still not very clear from these simplified modelsand one needs guidance also from simulations, to which we now turn.
Since the pioneering work of Meneguzzi, Frisch, and Pouquet , small scaledynamo action has frequently been seen in direct simulations of turbulence.In the early years the Kazantsev result was not yet well known and it camesomewhat as a surprise to many that kinetic helicity was not necessary fordynamo action. Until then, much of the work on dynamos had focused onthe α effect. Indeed, the work of Meneguzzi et al.  was rather seen asbeing complementary to the global simulations of Gilman and Miller  inthe same year. In those global dynamo simulations the flow was driven bythermal convection in a spherical shell in the presence of rotation, so therewas helicity and therefore also an α effect. The fact that dynamos do notrequire kinetic helicity was perhaps regarded as a curiosity of merely academicinterest, because turbulence in stars and galaxies is expected to be helical.
Only more recently the topic of nonhelical MHD turbulence has been followedup more systematically. One reason is that in many turbulence simulationskinematic helicity is often found to be rather weak, even if there is stratifica-tion and rotation that should produce helicity [165,215,226]. Another reasonis that with the advent of large enough simulations nonhelical dynamo actionhas become a topic of practical reality of any electrically conducting flow. Lo-cal and unstratified simulations of accretion disc turbulence also have shownstrong dynamo action . The fact that even completely unstratified nonro-tating convection can display dynamo action has been used to speculate thatmuch of the observed small scale magnetic field seen at the solar surface mightbe the result of a local dynamo acting only in the surface layers of the sun. The other possible source of small scale magnetic fields in the sun couldbe the shredding of large scale field by the turbulence [229,230].
When the magnetic field is weak enough, but the magnetic Reynolds numberlarger than a certain critical value, the magnetic energy grows exponentially.The growth rate of the magnetic energy is always a small fraction (∼ 20%) ofthe turnover time, (kfurms)
−1, of the turbulence at the energy carrying scale.The critical magnetic Reynolds number for dynamo action is around 35 whenPm = 1; see Ref. . This is around 30 times larger than the critical valuefor helical dynamos which is only around 1.2 . At early times the magneticenergy spectrum follows the k3/2 Kazantsev law, but then it reaches saturation.The kinetic energy spectrum is decreased somewhat, such that the magnetic
Fig. 5.1. Magnetic and kinetic energy spectra from a nonhelical turbulence simu-lation. The kinetic energy is indicated as a dashed line (except for the first timedisplayed where it is shown as a thin solid line). At early times the magnetic energyspectrum follows the k3/2 Kazantsev law while the kinetic energy shows a shortk−5/3 range. The Reynolds number is urms/(νkf) ≈ 660. 5123 meshpoints .
energy exceeds the kinetic energy at wavenumbers k > 5; see Fig. 5.1.
Kazantsev’s theory is of course not strictly applicable, because it assumes adelta correlated flow. Also, the k3/2 spectrum in the kinematic regime is ob-tained only for scales smaller than that of the flow, whereas in the simulationsthe velocity is not concentrated solely at the largest scale. This may be thereason why according to Kazantsev’s theory the growth rate is always overes-timated: in the simulations the actual growth rate of the magnetic energy isonly ∼ 0.18urmskf . On the other hand, the estimate for the ‘Kazantsev cutoffwavenumber’ where the kinematic spectrum peaks is fairly accurate.
Meanwhile simulations of nonhelical dynamos have been carried out at a re-spectable resolution of 10243 meshpoints . Such simulations are nowadaysdone on large parallel machines using the message passing interface (MPI) forthe communication between processors. Often, spectral methods are used tocalculate derivatives and to solve a Poisson-type equation for the pressure. Al-ternatively, high order finite difference schemes can be used, which are moreeasily parallelized, because only data of a small number of neighboring mesh-points need to be communicated to other processors. In such cases it is advan-tageous to solve the compressible equations, whose solutions approximate theincompressible ones when the rms velocity is small compared with the soundspeed. One such code, that is well documented and publicly available, is is thePencil Code . Many of the simulations presented in this review havebeen done using this code. Details regarding the numerical method can alsobe found in Ref. .
Fig. 5.2. Magnetic, kinetic and total energy spectra. 10243 meshpoints. TheReynolds number is urms/(νkf) ≈ 960 (from Ref. ).
In the simulations of Haugen et al.  the velocity field is forced randomlyat wavenumbers between 1 and 2, where k = 1 is the smallest wavenumber in abox of size (2π)3. These simulations begin to show an inertial range beyond thewavenumber k ≈ 8; see Fig. 5.2. The magnetic energy in the saturated stateis also peaked at about this wavenumber. Note that the semi-analytic closuremodels which lead to a renormalization of the diffusion coefficient [198,219],
suggest a peak of the saturated spectrum at a wavenumber kp ∼ kfR1/2crit. For
kf ≈ 1.5 and Rcrit ∼ 35, this predicts kp ∼ 8, which indeed seems to matchthe value obtained from the simulation.
The fact that the magnetic energy spectrum peaks at k ≈ 8 (or less) impliesthat in simulations there is not much dynamical range available before dissi-pation sets in. Furthermore, just before the dissipative subrange, turbulencealways exhibits a ‘bottleneck effect’ , i.e. a shallower spectrum (or excesspower) at large k, because the range of available wavenumber interactionsthrough which energy can freely be transported to smaller scales is restricted.While this effect is not very pronounced in laboratory wind tunnel turbulence[235,236], it has now become a very marked effect in fully three-dimensionalspectra available from high resolution simulations [237–239]. The reason forthe discrepancy has now been identified as being a mathematical consequenceof the transformation between three-dimensional and one-dimensional spectra.
In Fig. 5.3 we show a visualization of the magnetic field vectors at those pointswhere the magnetic field exceeds a certain field strength. The structures dis-played represent a broad range of sizes even within one and the same structure:the thickness of the structures is comparable to the resistive scale, their widthis a bit larger (probably within the inertial range) and their length is compa-rable to the box size (subinertial range). Although large scales are involved in
Fig. 5.3. Magnetic field vectors shown at those locations where |B| > 4Brms. Notethe long but thin arcade-like structures extending over almost the full domain. Thestructures are sheet-like with a thickness comparable to the resistive scale (fromRef. ).
this simulations, we must distinguish them from the type of large scale mag-netic fields seen in simulations with kinetic helicity that will be discussed inmore detail in Section 8 and that are invoked to explain the solar cycle.
For comparison with the analytic theory we plot in Fig. 5.4 the correlationfunction (5.9) of the magnetic field at saturation (and similarly for the ve-locity). Similar autocorrelation functions have also been seen in simulationsof convective dynamos . It turns out that the velocity correlation lengthis ∼ 3 while the magnetic field correlation length is ∼ 0.5; see Fig. 5.4. (Werecall that the box size is 2π.) Clearly, the magnetic correlation length is muchshorter than the velocity correlation length, but it is practically independent ofRe (= Rm) and certainly much longer than the resistive scale, ∼ 2π/kd ≈ 0.04.The fact that w(r) in the saturated state is independent of the microscopicRm agrees with the corresponding prediction of the closure model involvingartificial nonlinear drifts [198,219], mentioned in Section 5.3.1.
In contrast to large scale dynamos with helicity, which are now generally be-lieved to have a resistively limited saturation phase, nonhelical small scaledynamos are expected to have a dynamically limited saturation phase. How-ever, arguments in favor of an intermediate nonlinear regime, i.e. slower thanexponential growth, have been offered [207,240]. The simulation results shown
Fig. 5.4. Autocorrelation functions of magnetic field and velocity. Note that the au-tocorrelation functions are nearly independent of resolution and Reynolds number.The velocity correlation length is ∼ 3 while the magnetic correlation length is ∼ 0.5(from Ref. ).
Fig. 5.5. Saturation behavior of the spectral magnetic energy at wavenumbers k = 1(solid line) and k = 16 (dashed line). The average forcing wavenumber is kf = 1.5and the resolution is 5123 meshpoints. Note the slow saturation behavior for k = 1(from Ref. ).
in Fig. 5.5 seem compatible with a slow saturation behavior for some interme-diate time span (80 < urmskft < 200), but not at later times (urmskft > 200).
As the small scale dynamo saturates, various magnetic length scales in thesimulation increase quite sharply–some of them almost by a factor of two. Forexample, in convective dynamo simulations of a layer of depth d the magnetic
5〈B2〉/〈J2〉, increased from 0.04d to 0.07d during satu-
ration (see Fig. 4b of Ref. ). This increase of the characteristic length
Fig. 5.6. Evolution of characteristic wavenumbers. Note that the characteristic par-allel wavenumber of the field, k‖, becomes comparable with the scale of the box,while the so-called rms wavenumber, krms, drops by almost a factor of two as sat-uration sets in. Here, k2‖ = 〈|B · ∇B|2〉/〈B4〉 and k2rms = 〈|∇B|2〉/〈B2〉, whereangular brackets denote volume averages. (The wavenumbers kB×J and kB·J aredefined similarly to k‖, but with B × J and B · J , respectively, and follow a trendsimilar to that of krms.) The rms wavenumber of the flow, kλ, is proportional to theinverse Taylor microscale and decreases only slightly during saturation. (Adaptedfrom Ref. .)
scale is in qualitative agreement with the analytic theory of the nonlinearsaturation of the small scale dynamo (Section 5.3). In particular, during thekinematic stage these length scales remain unchanged. This can be seen fromFig. 5.6 where we show the evolution of various wavenumbers in a simulationof Schekochihin et al. . The approximate constancy of the characteristicwavenumbers during the kinematic stage illustrates that the simple-mindedpicture of the evolution of single structures (Section 5.1) is different from thecollective effect for an ensemble of many structures that are constantly newlygenerated and disappearing. We also note that in the saturated state the char-acteristic wavenumbers are approximately unchanged, suggesting that in thenonlinear regime there is no slow saturation phase (unlike the helical case thatwill be discussed later). During saturation, the drop of various characteristicwavenumbers is comparable to the drop of the inverse Taylor microscale seen
in the convective dynamo simulations .
5.5 Comments on the Batchelor mechanism
In an early attempt to understand the possibility of small scale dynamo action,Batchelor  appealed to the formal analogy between the induction equationand the vorticity equation,
Dt=W ·∇U + ν∇2W , (5.17)
Dt=B ·∇U + η∇2B. (5.18)
These equations imply the following evolution equations for enstrophy andmagnetic energy,
=WiWjsij − ν〈(∇×W )2〉, (5.19)
=BiBjsij − η〈(∇×B)2〉. (5.20)
Assuming that the rate of enstrophy dissipation, ν〈(∇ × W )2〉, is approxi-mately balanced by the rate of enstrophy production, WiWjsij, and that thisrate is similar to the rate of magnetic energy production, Batchelor arguedthat magnetic energy would grow in time provided η < ν, i.e.
Pm ≡ ν/η > 1. (5.21)
By now there have been several numerical investigations of dynamos that oper-ate in a regime where the magnetic Prandtl number is less than unity. For non-helical forced turbulence the dynamo with the smallest magnetic Prandtl num-ber, where dynamo action has still been found, is Pm ≈ 0.125; see Ref. .Simulations published so far are compatible with a weak dependence of thecritical value of Rm on Pm, possibly of power law form with Rm ≈ 35P−1/2
m forsmall values of Pm; see Fig. 5.7. This question certainly deserves further at-tention, because conflicting statements can be found in the literature, rangingfrom no dynamo action for Pm < Pm,crit  to no Pm-dependence at all .These different results may be explained by differences in the turbulence, inparticular the dependence of the correlation time on the wavenumber .
Comparing with Batchelor’s argument, the main reason why it may not ap-ply is that the W and B fields are in general not identical, and they do in
Fig. 5.7. Dependence of Rcrit on Pm for different values of kf (from Ref. ).
general show quite different statistics [164,165]. Establishing the asymptoticdependence of Rcrit on Pm is important because, even though the computerpower will increase, it will still not be possible to simulate realistic values ofPm in the foreseeable future.
Finally, we mention one property where the W and B fields do seem to showsome similarity. For a k−5/3 spectrum of kinetic energy the enstrophy spectrumis proportional to k1/3, so one may expect a similar spectrum for the magneticenergy in the wavenumber range where feedback from the Lorentz force can beneglected. Such results have indeed been reported in the context of convection. For forced turbulence, a k1/3 spectrum has only been seen in the range
k1 < k < kp, where kp ≈ kfR1/2crit is the wavenumber where the magnetic energy
spectrum peaks; see Fig. 5.2 and Section 5.4. We should emphasize, however,that it is not clear that this result is really a consequence of the imperfectanalogy between W and B.
5.6 Small and large scale dynamos: a unified treatment
We have so far discussed the case of small scale dynamos where the gener-ated field has correlation lengths of order or smaller than the forcing scaleof the flow. In the next section and thereafter we will discuss large scale ormean field dynamos. The large and small scale dynamo problems are usu-ally treated separately. However this separation is often artificial; there is noabrupt transition from the field correlated on scales smaller than L and thatcorrelated on larger scales. If we consider both large and small scale fields tobe random fields, it turns out that the equations for the magnetic correlationfunctions, which involve now both the longitudinal and helical parts, are al-ready sufficiently general to incorporate both small and large scale dynamos.
They provide us with a paradigm to study the dynamics in a unified fashion,which could be particularly useful to study the inverse cascade of magneticfields to scales larger than L. We elaborate below:
We now add a helical piece to the two point correlation of the velocity fieldfor the Kazantsev-Kraichnan flow, so we have
TN (r) +rirjr2
TL(r) + ǫijkrk F (r), (5.22)
where F (r) represents the helical part of the velocity correlations. At r = 0,we have
−2F (0) = −13
〈v(t) ·∇× v(t′)〉 dt′, (5.23)
indicating that F (r) is related to the kinetic helicity of the flow.
Consider a system of size S ≫ L, for which the mean field averaged over anyscale is zero. Of course, the concept of a large scale field still makes sense, asthe correlations between field components separated at scales r ≫ L, can inprinciple be non-zero.
Since the flow can be helical, we need to allow the magnetic field to also havehelical correlations. So, the equal-time, two point correlation of the magneticfield, Mij(r, t), is now given by
ML + ǫijkrk C, (5.24)
where C(r, t) represents the contribution from current helicity to the two-pointcorrelation.
The Kazantsev equation can now be generalized to describe the evolution ofboth ML and C [198,219,243,244] (see Appendix A). We get
+GML + 4αC, (5.25)
∂t= −2ηTC + αML, C = −
H ′′ +4H ′
where ML = ML(r, t) and C = C(r, t), while α = α(r) and ηT = ηT(r), andwe have defined the magnetic helicity correlation function H(r, t). Note that
H(0, t) = 16A ·B, whereas C(0, t) = 1
6J ·B. Also,
α(r) = −2[F (0)− F (r)], (5.27)
and so represents the effect of the helicity in the velocity field on the magneticfield. [We will see later in Section 8.10 that α(r → ∞) is what is traditionallycalled the α effect]. This new term has some surprising consequences.
Suppose α0 = −2F (0) 6= 0. Then one can see from Eqs. (5.25) and (5.26),that new generation terms arise at r ≫ L, or for scales much larger thanthe correlation scales of the flow, due to the α effect. These are in the formML = ....+4α0C and H = ...+α0M . These couple ML and C and lead to thegrowth of large scale correlations. There is also decay of the correlations forr ≫ L due to diffusion with an effective diffusion coefficient, ηT0 = η+ TL(0).From dimensional analysis, the effective growth rate is ΓR ∼ α0/R − ηT0/R
for correlations on scale ∼ R. This is exactly as in the large scale α2 dynamoto be discussed in the next section. This also picks out a special scale R0 ∼ηT0/α0 for a stationary state (see below). Further, as the small scale dynamo,is simultaneously leading to a growth of ML at r < L, the growth of large scalecorrelations can be seeded by the tail of the small scale dynamo eigenfunctionat r > L. Indeed, as advertised, both the small and large scale dynamosoperate simultaneously when α0 6= 0, and can be studied simply by solvingfor one ML(r, t).
The coupled time evolution of H and ML for a non-zero α0 requires numericalsolution, which we discuss later in Section 8.10. But interesting analyticalinsight into the system can be obtained for the marginal, quasi-stationarymode, with ML ≈ 0, H ≈ 0. Note that for η 6= 0, we will find that the abovesystem of equations always evolves at the resistive time scale. But for timescales much shorter than this, one can examine quasi-stationary states. ForH ≈ 0, we get from Eq. (5.26) C ≈ [α(r)/2ηT(r)]M . Substituting this intoEq. (5.25) and defining once again Ψ = r2
√ηTML, we get
= 0, (5.28)
where U0 is the potential defined earlier in Eq. (5.8). We see that the problemof determining the magnetic field correlations for the quasi-stationary modeonce again becomes the problem of determining the zero-energy eigenstate ina modified potential, U = U0 − α2/ηT. (Note that in the U in this subsectionis not to be confused with velocity in other sections). The addition to U0, dueto the helical correlations, is always negative definite. So helical correlationstend to make bound states easier to obtain. When F (0) = 0, and there is nonet α effect, the addition to U0 vanishes at r ≫ L, and U → 2ηT/r
2 at large
Γ = E = 0
2 η / r 2
/ r 2 − α0
r > L
Fig. 5.8. Schematic illustration of the potential U(r) for the marginal mode in helicalturbulence. A non-zero α0 allows the tunneling of the zero-energy state to producelarge scale correlations.
r, as before. The critical magnetic Reynolds number, for the stationary state,will however be smaller than when F (r) ≡ 0, because of the negative definiteaddition to U0 (see also Ref. ).
When α0 = −2F (0) 6= 0, a remarkable change occurs in the potential. Atr ≫ L, where the turbulence velocity correlations vanish, we have U(r) =2ηT0/r
2−α20/ηT0. So the potential U tends to a negative definite constant value
of −α20/ηT0 at large r (and the effective mass, 1/2ηT → 1/2ηT0, a constant
with r.) So there are strictly no bound states, with zero energy/growth rate,for which the correlations vanish at infinity. We have schematically illustratedthe resulting potential U in Fig. 5.8, which is a modification of figure 8.4 ofZeldovich et al. . In fact, for a non-zero α0, U corresponds to a potentialwhich allows tunneling (of the bound state) in the corresponding quantummechanical problem. It implies that the correlations are necessarily non-zeroat large r > L. The analytical solution to (5.28) at large r ≫ L, is easilyobtained. We have for r ≫ L, ML(r) = ML(r) ∝ r−3/2J±3/2(µr), where µ =α0/ηT0 = R−1
0 . This corresponds to
w(r) = w(r) = µr−1[C1 sin µr + C2 cosµr]; r ≫ L, (5.29)
where C1, C2 are arbitrary constants. Clearly for a non-zero α0, the correlationsin steady state at large r, are like ‘free-particle’ states, extending to infinity!In fact this correlation function is also the one which obtains if we demand
the random field to be force free with, ∇ ×B = µB. We see therefore thathaving helicity in the flow opens up the possibility of generating large scalefields, of scales much larger than that of the flow. This is the feature we willelaborate much more in the following sections.
6 Large scale turbulent dynamos
As the simulation results of Section 5.4 and the unified treatment in Section 5.6have shown, the distinction between small and large scale dynamos is some-what artificial. The so-called small scale dynamo may well generate magneticfield structures extending all the way across the computational domain; seeFig. 5.3. On the other hand, the overall orientation of these structures is stillrandom. This is in contrast to the spatio-temporal coherence displayed by thesun’s magnetic field, where the overall orientation of flux tubes at a certainlocation in space and time follows a regular rule and is not random. Dynamomechanisms that explain this will be referred to as large scale dynamos.
In this section we discuss the basic theory of large scale dynamos in terms ofthe alpha effect and discuss simple models. A discussion of large scale dynamosin terms of the inverse cascade is presented in the next section.
6.1 Phenomenological considerations
Important insights into the operation of the solar dynamo have come fromclose inspection of magnetic fields on the solar surface . One importantingredient is differential rotation. At the equator the sun is rotating about30% faster than at the poles. This means that magnetic field lines tend to bealigned with the azimuthal direction by the shear, which is mathematicallydescribed by the stretching term in the induction equation; see Eq. (3.19), i.e.
dt= Bpol ·∇Utor + ... . (6.1)
This term describes the generation of magnetic field Btor in the direction ofthe flow Utor from a cross-stream poloidal magnetic field Bpol. To an order ofmagnitude, the amount of toroidal field generation from a 100G poloidal fieldin a time interval ∆t is
∆Btor = Bpol ∆Ω⊙∆t ≈ 100G× 10−6 × 108 = 104G, (6.2)
where we have used Ω⊙ = 3 × 10−6 s−1 for the solar angular velocity, and∆Ω⊙/Ω⊙ = 0.3 for the relative latitudinal differential rotation. So, a 10 kGtoroidal field can be regenerated completely from a 100G poloidal field inabout 3 years.
The 100 kG field referred to earlier in connection with the tilt of emerging fluxtubes (Section 2.1.1) can probably be explained as the result of stretching ofmore localized ∼ 1 kG field patches that stay coherent over a time span forabout 3 yr.
In the bulk of the solar convection zone the turnover time, τturnover = urms/Hp
(where Hp is the pressure scale height) is only about 10 days. 2 In order that atoroidal field can be generated, a mean poloidal field needs to be maintainedand, in the case of the sun and other stars with cyclic field reversals, thepoloidal field itself needs to change direction every 11 years.
In the case of the sun, magnetic flux frequently emerges at the surface as bipo-lar regions. The magnetic field in sunspots is also often of bipolar nature. Itwas long recognized that such bipolar regions are tilted. This is now generallyreferred to as Joy’s law [36,246]. The sense of average tilt is clockwise in thenorthern hemisphere and anti-clockwise in the southern. This tilt is consistentwith the interpretation that a toroidal flux tube rises from deeper layers ofthe sun to upper layers where the density is less, so the tube evolves in an ex-panding flow field which, due to the Coriolis force, attains a clockwise swirl inthe northern hemisphere and anti-clockwise swirl in the southern hemisphere;see Fig. 6.1.
Observations suggest that once a tilted bipolar region has emerged at the solarsurface, the field polarities nearer to the poles drift rapidly toward the poles,producing thereby new poloidal field [172,247]. Underneath the surface, thefield continues as before, but there it is also slightly tilted, although necessarilyin the opposite sense (see Fig. 6.2). Because of differential rotation, the pointsnearest to the equator move faster, helping so to line up similarly oriented fields. As is evident from Fig. 6.2, a toroidal field pointing east in the northernhemisphere and west in the southern will develop into a global northwardpointing field above the surface.
2 The thermal flux of the sun is 4×1010 erg cm−2 s−1 A great portion of this energyflux is caused by convection and, according to mixing length theory, Fconv ≈ ρu3rms.Roughly, ρ = 0.1 g cm−3 in the lower part of the convection zone, so u ≈ (4 ×1010/0.1)1/2 = 70m/ s. The pressure scale height isRT/(µg) = 50Mm, so τturnover =10days.
Fig. 6.1. Solar magnetogram showing bipolar regions, their opposite orientationnorth and south of the equator, and the clockwise tilt in the northern hemisphereand the anti-clockwise tilt in the southern hemisphere. Note that the field orientationhas reversed orientation at the next cycle (here after 10 years). Courtesy of the HighAltitude Observatory.
Ω bipolar region(Joy’s law)
to the poles
later during the cycle
Fig. 6.2. Sketch of the Babcock-Leighton dynamo mechanism. As bipolar regionsemerge near the surface, they get tilted in the clockwise sense in the northernhemisphere and anti-clockwise in the southern hemisphere. Beneath the surfacethis process leaves behind a poloidal field component that points here toward thenorth pole on either side of the equator. Once the remaining subsurface field getssheared by the surface differential rotation, it points in the opposite direction asbefore, and the whole process starts again.
6.2 Mean-field electrodynamics
Parker  first proposed the idea that the generation of a poloidal field,arising from the systematic effects of the Coriolis force (Fig. 6.3), could be
Fig. 6.3. Production of positive writhe helicity by an uprising and expanding blobtilted in the clockwise direction by the Coriolis force in the southern hemisphere,producing a field-aligned current J in the opposite direction to B. Courtesy of A.Yoshizawa .
described by a corresponding term in the induction equation,
αBtor + ...)
It is clear that such an equation can only be valid for averaged fields (denotedby overbars), because for the actual fields, the induced electromotive force(EMF) U ×B, would never have a component in the direction of B. Whilebeing physically plausible, this approach only received general recognition andacceptance after Roberts and Stix  translated the work of Steenbeck,Krause, Radler  into English. In those papers the theory for the α effect,as they called it, was developed and put on a mathematical rigorous basis.Furthermore, the α effect was also applied to spherical models of the solar cycle(with radial and latitudinal shear)  and the geodynamo (with uniformrotation) .
In mean field theory one solves the Reynolds averaged equations, using eitherensemble averages, toroidal averages or, in cases in cartesian geometry withperiodic boundary conditions, two-dimensional (e.g. horizontal) averages. Wethus consider the decomposition
U = U + u, B = B + b. (6.4)
These averages satisfy the Reynolds rules,
U1 +U2 = U 1 +U 2, U = U , Uu = 0, U U = U U , (6.5)
∂U/∂t = ∂U/∂t, ∂U/∂xi = ∂U/∂xi. (6.6)
Some of these properties are not shared by several other averages; for gaussian
filtering U 6= U , and for spectral filtering U U 6= U U , for example. Note
that U = U implies that u = 0. Averaging Eq. (3.11) yields the mean fieldinduction equation,
U ×B + E − ηJ)
E = u× b (6.8)
is the mean EMF. Finding an expression for the correlator E in terms of themean fields is a standard closure problem which is at the heart of mean fieldtheory. In the two-scale approach  one assumes that E can be expandedin powers of the gradients of the mean magnetic field. This suggests the rathergeneral expression
Ei = αij(g, Ω,B, ...)Bj + ηijk(g, Ω,B, ...)∂Bj/∂xk, (6.9)
the tensor components αij and ηijk are called turbulent transport coefficient.They depend on the stratification, angular velocity, and mean magnetic fieldstrength. The dots indicate that the transport coefficients may also depend oncorrelators involving the small scale magnetic field, for example the currenthelicity of the small scale field, as will be discussed in Section 9.4. We have alsokept only the lowest large scale derivative of the mean field; higher derivativeterms are expected to be smaller (cf. Section 7.2 in Moffatt ).
The general subject has been reviewed in many text books [129,253], but theimportance of the small scale magnetic field (or rather the small scale currenthelicity) has only recently been appreciated [254,255], even though the basicequations were developed much earlier [9,256]. This will be discussed in moredetail in Section 8.7.
The general form of the expression for E can be determined by rather generalconsiderations . For example, E is a polar vector and B is an axial vector,so αij must be a pseudo-tensor. The simplest pseudo-tensor of rank two thatcan be constructed using the unit vectors g (symbolic for radial density orturbulent velocity gradients) and Ω (angular velocity) is
αij = α1δij g · Ω+ α2giΩj + α3gjΩi. (6.10)
Note that the term g · Ω = cos θ leads to the co-sinusoidal dependence of αon latitude, θ, and a change of sign at the equator. Additional terms that arenonlinear in g or Ω enter if the stratification is strong or if the body is rotatingrapidly. Likewise, terms involving U , B and b may appear if the turbulencebecomes affected by strong flows or magnetic fields. In the following subsec-tion we discuss various approaches to determining the turbulent transportcoefficients.
One of the most important outcomes of this theory is a quantitative formulafor the coefficient α1 in Eq. (6.10) by Krause ,
α1 g · Ω = −1615τ 2coru
2rmsΩ ·∇ ln(ρurms), (6.11)
where τcor is the correlation time, urms the root mean square velocity of theturbulence, and Ω the angular velocity vector. The other coefficients are givenby α2 = α3 = −α1/4. Throughout most of the solar convection zone, theproduct ρurms decreases outward. 3 Therefore, α > 0 throughout most of thenorthern hemisphere. In the southern hemisphere we have α < 0, and α varieswith colatitude θ like cos θ. However, this formula also predicts that α reversessign very near the bottom of the convection zone where urms → 0. This iscaused by the relatively sharp drop of urms .
The basic form and sign of α is also borne out by simulations of stratifiedconvection [259,260]. One aspect that was first seen in simulations is the factthat in convection the vertical component of the α effect can have the oppositesign compared with the horizontal components [259,260]. The same result haslater been obtained in analytic calculations of the α effect in supernova-driveninterstellar turbulence  and in FOSA calculations .
6.3 Calculation of turbulent transport coefficients
Various techniques have been proposed for determining turbulent transportcoefficients. Even in the kinematic regime, where the changes in the veloc-ity field due to Lorentz forces are ignored, these techniques have some severeuncertainties. Nevertheless, the various techniques produce similar terms, al-though the so-called minimal τ approximation (MTA) does actually predictan extra time derivative of the electromotive force . This will be discussed
3 This can be explained as follows: in the bulk of the solar convection zone theconvective flux is approximately constant, and mixing length predicts that it isapproximately ρu3rms. This in turn follows from Fconv ∼ ρurmscpδT and u2rms/Hp ∼gδT/T together with the expression for the pressure scale height Hp = (1− 1
Thus, since ρu3rms ≈ const, we have urms ∼ ρ−1/3 and ρu3rms ∼ ρ2/3.
in full detail in Section 10. We only mention here that in MTA the triplecorrelations are not neglected, like in the first order smoothing approximation(FOSA); see Section 6.3.1. Instead, the triple correlations are approximatedby quadratic terms. This is similar in spirit to the usual τ approximation usedin the Eddy Damped Quasi-Normal Markovian (EDQNM) closure approxima-tion (see Section 7.2), where quartic correlations are approximated by triplecorrelations. Other approaches include direct simulations [259,260,263] (seeSection 6.4), calculations based on random waves [129,264,265] or individualblobs [261,266] (see Section B.2.1), or calculations based on the assumption ofdelta-correlated velocity fields [13,197,267] (Section B.2). A summary of thedifferent approaches, their properties and limitations is given in Table 6.1.
Perhaps, one of the most promising approaches is indeed MTA, because thereis now some numerical evidence from turbulent transport experiments thatMTA may be valid even when the fluctuations are large; see Section 9.3, andRef. . This is of course beyond the applicability regime of FOSA. Earlyreferences to MTA include the papers of Vainshtein and Kitchatinov  andKleeorin and collaborators [270–272], but since MTA is based on a closurehypothesis, detailed comparisons between theory and simulations [4,8,268,275]have been instrumental in giving this approach some credibility.
In the following we discuss FOSA in some detail, and we defer the maindiscussion of MTA to Section 10.
6.3.1 First order smoothing approximation
The first order smoothing approximation (FOSA) or, synonymously, the quasi-linear approximation, or the second order correlation approximation is the sim-plest way of calculating turbulent transport coefficients. The approximationconsists of linearizing the equations for the fluctuating quantities and ignor-ing quadratic terms that would lead to triple correlations in the expressionsfor the quadratic correlations. This technique has traditionally been appliedto calculating the turbulent diffusion coefficient for a passive scalar or theturbulent viscosity (eddy viscosity).
Suppose we consider the induction equation. The equation for the fluctuatingfield can be obtained by subtracting Eq. (6.7) from Eq. (3.11), so
U × b+ u×B + u× b− E − ηj)
where j = ∇ × b ≡ J − J is the fluctuating current density. The first ordersmoothing approximation consists of neglecting the term u× b, because it isnonlinear in the fluctuations. This can only be done if the fluctuations are
Table 6.1Summary of various approaches to calculate turbulent transport coefficients.
Technique Refs. linear? homog.? b2/B2
EDQNM  no yes large
MTA [8,272] no no large?
simulations [259,260,263] no no modest/large
FOSA [129,253] no no small
stochastic  yes no large
random waves [129,264,265] no no small
individual blobs [261,266] no no small
small, which is a good approximation under rather restrictive circumstances,for example if Rm is small. The term E is also nonlinear in the fluctuations,but it is not a fluctuating quantity and gives therefore no contribution, andthe U × b is often neglected because of simplicity (but see, e.g., Ref. ).The neglect of the U term may not be justified for systems with strong shear(e.g. for accretion discs) where the inclusion of U could lead to new effects. In the case of small Rm, one can neglect both the G term and the timederivative of b, resulting in a linear equation
η∇2b = −∇ ×(
This can be solved for b, if u is given. E can then be computed relativelyeasily [129,253].
However, in most astrophysical applications, Rm ≫ 1. In such a situation,FOSA is thought to still be applicable if the correlation time τcor of the turbu-lence is small, such that τcorurmskf ≪ 1, where urms and kf are typical velocityand correlation wavenumber, associated with the random velocity field u. Un-der this condition, the ratio of the nonlinear term to the time derivative of b isargued to be ∼ (urmskfb)/(b/τcor) = τcorurmskf ≪ 1, and so G can be neglected (but see below). We then get
To calculate E , we integrate ∂b/∂t to get b, take the cross product with u,
and average, i.e.
E = u(t)×t∫
For clarity, we have suppressed the common x dependence of all variables.Using index notation, we have
E i(t) =
′) + ηilp(t, t′)Bp,l(t
with αip(t, t′) = ǫijkuj(t)uk,p(t′) and ηilp(t, t
′) = ǫijpuj(t)ul(t′), where we haveused Bl,l = 0 = ul,l, and commas denote partial differentiation. In the statis-tically steady state, we can assume that αip and ηilp depend only on the timedifference, t− t′. Assuming isotropy (again only for simplicity), these tensorsmust be proportional to the isotropic tensors δip and ǫilp, respectively, so wehave
α(t− t′)B(t′)− ηt(t− t′)J(t′)]
where α(t − t′) = −13u(t) · ω(t′) and ηt(t − t′) = 1
3u(t) · u(t′) are integral
kernels, and ω = ∇× u is the vorticity of the velocity fluctuation.
If we assume the integral kernels to be proportional to the delta function,δ(t− t′), or, equivalently, if B can be considered a slowly varying function oftime, one arrives at
E = αB − ηtJ (6.18)
α = −13
u(t) · ω(t′) dt′ ≈ −13τcoru ·ω, (6.19)
u(t) · u(t′) dt′ ≈ 13τcoru2. (6.20)
When t becomes large, the main contribution to these two expressions comesonly from late times, t − t′ ≪ t, because then the contributions from early
times are no longer strongly correlated with u(t). By using FOSA we have thussolved the problem of expressing E in terms of the mean field. The turbulenttransport coefficients α and ηt depend, respectively, on the helicity and theenergy density of the turbulence.
One must however point out the following caveat to the applicability of FOSAin case of large Rm. Firstly note that even if uτ/L ≪ 1, one can have Rm =(uτ/L)(L2/ητ) ≫ 1, because the diffusion time L2/η can be much largerthan the correlation time of the turbulence. As we have already discussed inSection 5.2, when Rm > Rcrit ∼ 30, small scale dynamo action will take placeto produce exponentially growing fluctuating fields, independent of the meanfield. So the basic assumption of FOSA of small b relative to B will be rapidlyviolated. Nevertheless, the functional form of the expressions for the turbulenttransport coefficients obtained using FOSA seem to be not too different fromthat found in simulations. It is likely that strong fluctuations produced bysmall scale dynamo action do not correlate well with u in u× b, so theywould not contribute to E . This interpretation will be developed further inSection 6.3.2 on the τ approximation, which works specifically only with thoseparts that do correlate.
6.3.2 MTA – the ‘minimal’ τ approximation
The ‘minimal’ τ approximation is a simplified version of the τ approximationas it has been introduced by Orszag  and used by Pouquet, Frisch andLeorat  in the context of the Eddy Damped Quasi Normal Markovian(EDQNM) approximation. In that case a damping term is introduced in orderto express fourth order moments in terms of third order moments. In the τapproximation, as introduced by Vainshtein and Kitchatinov  and Klee-orin and Rogachevskii [270,271], one approximates triple moments in termsof quadratic moments via a wavenumber-dependent relaxation time τ(k). The‘minimal’ τ approximation (MTA), as it is introduced by Blackman and Field, is applied in real space in the two-scale approximation. We will refer toboth the above types of closures (where triple moments are approximatedin terms of quadratic moments and a relaxation time τ) as the ’minimal’ τapproximation or MTA.
There are some technical similarities between FOSA and the minimal τ ap-proximation. The main advantage of the τ approximation is that the fluctu-ations do not need to be small and so the triple correlations are no longerneglected. Instead, it is assumed (and this can be and has been tested us-ing simulations) that the one-point triple correlations are proportional to thequadratic correlations, and that the proportionality coefficient is an inverserelaxation time that can in principle be scale (or wavenumber) dependent.
In this approach, one begins by considering the time derivative of E [8,272],
∂t= u× b+ u× b, (6.21)
where a dot denotes a time derivative. For b, we substitute Eq. (6.12) and foru, we use the Euler equation for the fluctuating velocity field,
∂t= − 1
ρ0∇p+ f + Fvis +H , (6.22)
where H = −u ·∇u+u ·∇u is the nonlinear term, f is a stochastic forcingterm (with zero divergence) and uncorrelated with b, and Fvis is the viscousforce. We have also assumed for the present that there is no mean flow (U = 0),and have considered the kinematic regime where the Lorentz force is set to zero(the latter assumption will be relaxed in Section 9.3). All these restrictions canin principle be lifted (see below). For an incompressible flow, the pressure termcan be eliminated in the standard fashion in terms of the projection operator.Now since f does not correlate with b, the only contribution to u× b, is the
small viscous term and the triple correlation involving b and H . The u× b
term however has non-trivial contributions. We get
∂t= αB − ηt J − E
where the last term subsumes the effects of all triple correlations, and
α = −13u · ω and ηt =
are coefficients that are closely related to the usual α and ηt coefficients inEq. (6.18). We recall that in this kinematic calculation the Lorentz force hasbeen ignored. Its inclusion (Section 9.3) turns out to be extremely important:it leads to the emergence of a small scale magnetic correction term in theexpression for α; see Eq. (9.8) below.
One normally neglects the explicit time derivative of E [272,271], and arrivesthen at almost the same expression as Eq. (6.18), except that now one dealsdirectly with one-point correlation functions and not only via an approxima-tion. Furthermore, the explicit time derivative can in principle be kept ,although it becomes unimportant on time scales long compared with τ ; seealso Refs. [268,275] for applications to the passive scalar problem and numer-ical tests. In comparison with Eq. (6.17), we note that if one assumes α(t− t′)and ηt(t − t′) to be proportional to exp[−(t − t′)/τ ] for t > t′ (and zero oth-erwise), one recovers Eq. (6.23) with the relaxation time τ playing now the
role of a correlation time. We will discuss the τ approximation further belowin connection with calculating nonlinear effects of the Lorentz force.
6.4 Transport coefficients from simulations
The main advantage of using simulations is that no approximations need tobe made other than the restriction to only moderate values of the magneticReynolds number. Most notably, this approach allows the determination oftransport coefficients in inhomogeneous systems in the presence of boundaries,for example in the sun where the convection zone changes into the lower orupper overshoot layer. As the Reynolds number is increased, however, thefluctuations in E become large and long integration times are necessary .A related problem is that for large values of Rm, small scale dynamo actionbecomes possible and, unless the imposed field is strong enough, the resultingvalues of E are only weakly linked to B0.
An obvious method for calculating the α effect is to use a simulation with animposed magnetic field, B0, and to determine numerically the resulting elec-tromotive force, E, so α = E ·B0/B
20 . Generalization to tensorial formulations
of α are straightforward. For example, the turbulent diamagnetic effect canbe determined as γ = (E ×B0)/B
20 , and the elements of the full α tensor can
simply be calculated as αij = E i/B0j .
The main conclusions obtained from such approaches is that the functionalform of Eq. (6.11) is basically verified. In particular, α has a positive maximumin the upper part of the convection zone (in the northern hemisphere), changessign with depth [259,260] and varies with latitude as expected from this equa-tion and is largest at high latitudes . There are however additional termssuch as a turbulent diamagnetic effect that was discovered later . Thiseffect, which causes a downward pumping of mean field, has also been seen insimulations of stratified convection. Simulations of a convective dynamo withstratification showed that flux tubes are regularly being entrained by strongdowndrafts, then pushed downward and amplified as the result of stretchingand compression [165,215,277]. The end result is a strong magnetic field at thebottom of the convection zone where the field is expected to undergo furtherstretching by differential rotation . More recent studies have allowed amore quantitative description of the pumping effect and the associated pumpvelocity [279,280].
Two surprising results emerged from the simulations. In convection the α ef-fect is extremely anisotropic with respect to the vertical direction such thatthe diagonal components of αij can even change sign. While the horizontalcomponent, αxx, shows the expected sign (roughly a negative multiple of the
Fig. 6.4. Vertical and horizontal components of the α effect. The gray lines repre-sent different times, the thick line is the time average, and the surrounding thincurves indicate the error of the mean. The simulation is carried out on the southernhemisphere. The convection zone proper is in the range 0 < z < 1, where z denotesdepth. The lower overshoot layer is in z > 1 and the top overshoot layer is in z < 0.In the upper parts of the domain (0 < z < 0.6) the vertical component of αV ≡ αzz
is positive (upper panels) while the horizontal component, αH ≡ αxx is negative(lower panels). The left and right hand columns are for simulations with differentangular velocity: slower on the left (the Taylor number is 2× 103) and faster on theright (the Taylor number is 104). Adapted from Ref. .
kinetic helicity), the vertical component, αzz, shows invariably the oppositesign [259,260]; see Fig. 6.4. This peculiar result can be understood by notingthat vertical field lines are wrapped around each other by individual down-drafts, which leads to field line loops oppositely oriented than if they werecaused by an expanding updraft.
In Fig. 6.5 we show the field line topology relevant to the situation in thesouthern hemisphere. The degree of stratification is weak, so the downdraftsat the top look similar to the updrafts at the bottom (both are indicatedby two swirling lines). At the top and bottom surfaces the magnetic field isconcentrated in the intergranular lanes which correspond to downdrafts at thetop and updrafts at the bottom (i). This leads to a clockwise swirl both at thetop and at the bottom (but anti-clockwise in the northern hemisphere); seethe second panel (ii). This in turn causes left-handed current helicity in theupper parts and right-handed current helicity in the lower parts, so one wouldthink that α is negative in the upper parts and positive in the lower parts.This is however not the case. Instead, what really matters is E = u×b, whereu is dominated by converging motions (both at the top and the bottom).
Fig. 6.5. Sketch showing the twisting of vertical magnetic field lines by downdrafts.The resulting electromotive force, E = u × b, points in the direction of the meanfield, giving a positive αV in the southern hemisphere and a negative αV in thenorthern hemisphere. Adapted from Ref. .
This, together with b winding in the counter-clockwise direction around thedowndraft and in the clockwise direction in the updraft causes u×b to point inthe direction of B at the top (so αzz is positive) and in the opposite directionat the bottom (so αzz is negative). Originally, this result was only obtainedfor weak stratification , but meanwhile it has also been confirmed forstrong stratification . We reiterate that a qualitatively similar result hasalso been obtained in analytic calculations of the α effect in supernova-driveninterstellar turbulence  and in FOSA calculations .
The other surprising result is that the turbulent pumping is not necessarilyrestricted to the vertical direction, but it can occur in the other two directions
Fig. 6.6. The latitudinal dependence of the three components of the pumping veloc-ity (θ = 0 corresponds to the south pole and θ = 90 to the equator). The verticalpumping velocity (γz , right-most panel) is positive, corresponding to downwardpumping, and almost independent of latitude. The two horizontal components of γvanish at the poles. The longitudinal pumping velocity (γy, middle panel) is neg-ative, corresponding to retrograde pumping, and the latitudinal pumping velocity(γx, left panel) is positive, corresponding to equatorward pumping. Adapted fromRef. .
as well. In general one can split the αij tensor into symmetric and antisymmet-
ric components, i.e. αij = α(S)ij + α
(A)ij , where the antisymmetric components
can be expressed in the form
α(A)ij = −ǫijkγk (6.25)
where γ is the pumping velocity. We recall that α(A)ij Bj = (γ×B)i, which looks
like the induction termU×B, so γ acts like an advection velocity. Simulationsshow that γ has a component pointing in the retrograde direction; see Fig. 6.6and Ref. . We shall return to the theory of this term in Section 10.3; seeEq. (10.64).
The latitudinal component of the pumping velocity points toward the equatorand has been invoked to explain the equatorward migration seen in the but-terfly diagram . The equatorward pumping was found to act mostly onthe toroidal component of the mean field while the poloidal field was found toexperience a predominantly poleward pumping velocity. This result has alsobeen confirmed using simulations of turbulent convection .
6.5 α2 and αΩ dynamos: simple solutions
For astrophysical purposes one is usually interested in solutions in sphericalor oblate (disc-like) geometries. However, in order to make contact with tur-bulence simulations in a periodic box, solutions in simpler cartesian geometry
Fig. 6.7. Mutual regeneration of poloidal and toroidal fields in the case of the αΩdynamo (left) and the α2 dynamo (right).
can be useful. Cartesian geometry is also useful for illustrative purposes. Inthis subsection we review some simple cases.
Mean field dynamos are traditionally divided into two groups; αΩ and α2
dynamos. The Ω effect refers to the amplification of the toroidal field by shear(i.e. differential rotation) and its importance for the sun was recognized veryearly on. Such shear also naturally obtains in disk galaxies, since they aredifferentially rotating systems . However, it is still necessary to regeneratethe poloidal field, in both stars and galaxies, for which the α effect is theprime candidate. This explains the name αΩ dynamo; see the left hand panelof Fig. 6.7. However, large scale magnetic fields can also be generated by theα effect alone, so now also the toroidal field has to be generated by the αeffect, in which case one talks about an α2 dynamo; see the right hand panelof Fig. 6.7. (The term α2Ω model is discussed at the end of Section 6.5.2.)
6.5.1 α2 dynamo in a periodic box
We assume that there is no mean flow, i.e. U = 0, and that the turbulence ishomogeneous, so that α and ηt are constant. The mean field induction equationthen reads
∂t= α∇×B + ηT∇2B, ∇ ·B = 0, (6.26)
where ηT = η+ηt is the sum of microscopic and turbulent magnetic diffusivity.We can seek solutions of the form
B(x) = Re[
B(k) exp(ik · x+ λt)]
Fig. 6.8. Dispersion relation for α2 dynamo, where kcrit = α/ηT.
This leads to the eigenvalue problem λB = αik × B − ηTk2B, which can be
written in matrix form as
−ηTk2 −iαkz iαky
iαkz −ηTk2 −iαkx
−iαky iαkx −ηTk2
This leads to the dispersion relation, λ = λ(k), given by
(λ+ ηTk2)2 − α2k2
= 0 (6.29)
with the three solutions
λ0 = −ηTk2, λ± = −ηTk
2 ± |αk|. (6.30)
The eigenfunction corresponding to the eigenvalue λ0 = −ηTk2 is propor-
tional to k, but this solution is incompatible with solenoidality and has to bedropped. The two remaining branches are shown in Fig. 6.8.
Unstable solutions (λ > 0) are possible for 0 < αk < ηTk2. For α > 0 this
corresponds to the range
0 < k < α/ηT ≡ kcrit. (6.31)
For α < 0, unstable solutions are obtained for kcrit < k < 0. The maximumgrowth rate is at k = 1
2kcrit. Such solutions are of some interest, because they
have been seen as an additional hump in the magnetic energy spectra fromfully three-dimensional turbulence simulations (Section 8).
Linear theory is only applicable as long as the magnetic field is weak, butqualitatively one may expect that in the nonlinear regime α becomes reduced(quenched), so kcrit decreases and only larger scale magnetic fields can bemaintained. This is indeed seen in numerical simulations ; see also Section 8.
6.5.2 αΩ dynamo in a periodic box
Next we consider the case with linear shear, and assume U = (0, Sx, 0), whereS = const. This model can be applied as a local model to both accretion discs(x is radius, y is longitude, and z is the height above the midplane) and to stars(x is latitude, y is longitude, and z is radius). For keplerian discs, the shear isS = −3
2Ω, while for the sun (taking here only radial differential rotation into
account) S = r∂Ω/∂r ≈ +0.1Ω⊙ near the equator.
For simplicity we consider axisymmetric solutions, i.e. ky = 0. The eigenvalueproblem takes then the form
−ηTk2 −iαkz 0
iαkz + S −ηTk2 −iαkx
0 iαkx −ηTk2
where ηT = η + ηt and k2 = k2x + k2
z . The dispersion relation is now
(λ+ ηTk2)2 + iαSkz − α2k2
= 0, (6.33)
with the solutions
λ± = −ηTk2 ± (α2k2 − iαSkz)
Again, the eigenfunction corresponding to the eigenvalue λ0 = −ηTk2 is not
compatible with solenoidality and has to be dropped. The two remainingbranches are shown in Fig. 6.9, together with the approximate solutions
Reλ± ≈ −ηTk2 ± |1
Imλ± ≡ −ωcyc ≈ ±|12αSkz|1/2, (6.36)
where we have made use of the fact that i1/2 = (1 + i)/√2.
Sometimes the term α2Ω dynamo is used to emphasize that the α effect is notneglected in the generation of the toroidal field. This approximation, which issometimes also referred to as the αΩ approximation, is generally quite goodprovided αkz/S ≪ 1. However, it is important to realize that this approxima-tion can only be applied to axisymmetric solutions .
Fig. 6.9. Dispersion relation for α2Ω dynamo. The dotted line gives the result forthe αΩ approximation Eqs. (6.35) and (6.36), for αkz/S = 0.25.
Table 6.2Summary of propagation directions and phase relation for αΩ dynamos.
object α S ϕ c wave propagation |ωcyc|∆t
disc − − −3π/4 + away from midplane −3π/4
disc/star? + − +3π/4 − equatorward −3π/4
star? − + −π/4 − equatorward +π/4
star + + +π/4 + poleward +π/4
6.5.3 Eigenfunctions, wave speed, and phase relations
We now make the αΩ approximation and consider the marginally excitedsolution (Reλ = 0), which can be written as
Bx = B0 sin kz(z − ct), By =√2B0
∣∣∣∣ sin[kz(z − ct) + ϕ], (6.37)
where B0 is the amplitude (undetermined in linear theory), and c = ωcyc/kzis the phase speed of the dynamo wave, which is given by
|2αSkz|1/2= ±ηTkz, (6.38)
where the upper (lower) sign applies when αS is positive (negative). The signof c gives the direction of propagation of the dynamo wave; see Table 6.2 fora summary of the propagation directions in different settings.
An important property of the αΩ dynamo solutions that can be read off fromthe plane wave solutions is the phase shift of ±3
4π (for S < 0) and ±π/4 (for
S > 0) between the poloidal and toroidal fields. It is customary [284,285] toquote instead the normalized time delay |ωcyc|∆t = ϕ sgn(c), by which thetoroidal field lags behind the radial field. These values are given in the lastcolumn of Table 6.2. Note that the temporal phase shift only depends on thesign of the shear S and not on α.
6.5.4 Excitation conditions in a sphere
For applications to stars it is essential to employ spherical geometry. Over thepast three decades, a number of two-dimensional and three-dimensional mod-els have been presented [251,286–288]. The dynamo is generally characterizedby two dynamo numbers,
Cα = α0R/ηT, CΩ = ∆ΩR2/ηT, (6.39)
where α0 and ∆Ω are typical values of α and angular velocity difference acrossthe sphere, and R is the outer radius of the sphere. In Fig. 6.10 we show thecritical values of Cα (above which dynamo action is possible) for differentvalues of CΩ using error function profiles,
f±(r; r0, d) =121± erf [(r − r0)/d] , (6.40)
for α(r, θ) = α0f+(r; rα, dα) cos θ and Ω(r) = ∆Ωf−(r; rΩ, dΩ), just as in theearly work of Steenbeck and Krause , who used rα = 0.9, rΩ = 0.7, anddα = dΩ = 0.075. On r = R the field is matched to a potential field. A detailedpresentation of the induction equation in spherical harmonics with differentialrotation, meridional circulation, anisotropic α effect and a number of othereffects is given by Radler .
The solutions are classified by the symmetry properties about the equator (Sand A for symmetric and antisymmetric fields, respectively), supplemented bythe spherical harmonic degree m characterizing the number of nodes in theazimuthal direction. Note that for axisymmetric modes (S0 and A0) the criticalvalue of Cα decreases if CΩ increases, while for the nonaxisymmetric modes(e.g. S1 and A1) Cα is asymptotically independent of CΩ. This behavior forS0 and A0 is understandable because for axisymmetric modes the excitationcondition only depends on the product of α effect and shear; see Eq. (6.35). ForS1 and A1, on the other hand, differential rotation either makes the dynamoharder to excite (if CΩ is small) or it does not affect the dynamo at all (largerCΩ). This is because when differential rotation winds up an nonaxisymmetricfield, anti-aligned field lines are brought close together . For sufficientlylarge values of CΩ the field is expelled into regions with no differential rotation(α2 dynamo) where the dynamo is essentially independent of CΩ.
Fig. 6.10. Critical values of Cα versus Cω for a dynamo in a spherical shell. Notethat near Cω = 300 the nonaxisymmetric modes S1 and A1 are more easily excitedthan the axisymmetric modes S0 and A0. Here, Cω (called CΩ in the rest of thisreview) is defined such that it is positive when ∂Ω/∂r is negative, and vice versa.Adapted from .
Generally, axisymmetric modes are easier to excite than nonaxisymmetricmodes. There can be exceptions to this just at the junction between α2 andαΩ dynamo behavior. This is seen near Cω = 300; see Fig. 6.10. Such be-havior was first reported by Robert and Stix , and may be importantfor understanding the occurrence of nonaxisymmetric fields in active stars;see Section 11.3. Other potential agents facilitating nonaxisymmetric fieldsinclude anisotropic [290,291] and nonaxisymmetric  forms of α effect andturbulent diffusivity.
Finally we note that for strong shear (large values of |CΩ|), and α > 0 in thenorthern hemisphere, the most easily excited modes are A0 (when ∂Ω/∂r isnegative, i.e. Cω > 0 in Fig. 6.10), and S0 (when ∂Ω/∂r is positive). Thisbehavior changes the other way around, however, when the dynamo operatesin a shell whose ratio of inner to outer shell radius exceeds a value of about0.7 . This is approximately the appropriate value for the sun, and it hasindeed long been recognized that the negative parity of the solar dynamo isnot always obtained from model calculations [294,355,356].
6.5.5 Excitation conditions in disc-like geometries
Another important class of astrophysical bodies are galactic discs as well asdiscs around young stars, compact objects, or supermassive black holes in thenuclei of active galaxies; see Section 2.2. Suppose we consider the simplestform of the axisymmetric mean field dynamo equations appropriate for a thin
disc of ,
BR = −(αBφ)′ + ηTB
′′R, Bφ = SBR + ηTB
Here, radial derivatives have been neglected, so the problem has become one-dimensional and can be solved separately at each radius. Primes and dotsdenote z and t derivatives, respectively, S = RdΩ/dR is the shear in thedisc, and (BR, Bφ, Bz) are the components of the mean field B in cylindricalcoordinates. On z = ±H one assumes vacuum boundary conditions which, inthis one-dimensional problem, reduce to BR = Bφ = 0. One can also imposeboundary conditions on the mid-plane, z = 0, by selecting either symmetric(quadrupolar) fields, BR = B
′φ = 0, or antisymmetric (dipolar) fields, B′
Bφ = 0. One can again define two dimensionless control parameters,
CΩ = Sh2/ηT, Cα = αh/ηT, (6.42)
which measure the strength of the shear and α effects, respectively, where h isa measure of the disc scale height. (Note that CΩ and Cα used here are akinto that defined in Section 6.5.4 and are identical to the symbols RΩ and Rα
commonly used in galactic dynamo literature). In spiral galaxies, the typicalvalues are CΩ ≈ −10 and Cα ≈ 1, and so |CΩ| ≫ Cα.
Since |CΩ| ≫ Cα, dynamo generation of axisymmetric solutions is controlledby the dynamo number D = CΩCα. Exponential growth of the fields is possi-ble, in the kinematic stage, provided |D| > Dcrit, where the critical dynamonumber Dcrit ∼ 6 − 10, depending on the exact profile adopted for the α(z).For negative ∂Ω/∂R and positive values of α in the upper disc plane (northern‘hemisphere’), the most easily excited modes are no longer A0, but S0 .This case is believed to be relevant to galaxies and one expects that the mosteasily excited solutions for galaxies, are modes with steady quadrupole (S0 st)symmetry in the meridional (Rz) plane . For these modes, the growth ratein the kinematic regime can be approximated by,
γ ≈ ηTh2
In most spiral galaxies the dynamo is supercritical for a large range of radii,and so galactic fields can indeed grow exponentially from small seed values.A detailed discussion of the properties of solutions in the galactic context isgiven in Ref. .
Further, as discussed in Section 6.5.4, axisymmetric modes are easier to excitethan nonaxisymmetric ones. Although, the observed nonaxisymmetric largescale mean field structures in some galaxies, can also be explained by invoking
nonaxisymmetric forms of α effect , turbulent diffusivity, or streamingmotions [425,432,433].
For accretion discs, α might be negative in the northern hemisphere  andone therefore expects oscillatory quadrupoles (S0 osc) . In Fig. 6.11 weshow the growth rate of different modes, obtained from solving Eq. (6.41) forboth signs of the dynamo number . In order to find all the modes, eventhe unstable ones, one can easily solve Eq. (6.41) numerically as an eigenvalueproblem, λq = Mq, where the complex growth rate λ is the eigenvalue.
For illustrative purposes, we discuss the numerical technique in detail. Weintroduce mesh points, zi = i δz (excluding the boundaries at z = 0 andz = H), where δz = H/(N + 1) is the mesh spacing, and i = 1, 2, ..., Ndenotes the position on the mesh. The discretized eigenvector is
q = (BR1, Bφ1, BR2, Bφ2, ..., BRN , BφN)T . (6.44)
It is convenient to introduce the abbreviations ai = −αi/(2δz) and b =ηT/(δz)
2, so then the second-order accurate discretized form of Eq. (6.41)reads
BR i = ai+1Bφ i+1 − ai−1Bφ i−1 + b(BR i+1 − 2BR i +BR i−1), (6.45)
Bφ i = SBR i + b(Bφ i+1 − 2Bφ i +Bφ i−1), (6.46)
where α1, α2, ..., are the values of α at the different mesh points. Usingsymmetric boundary conditions on z = 0, we have BR 0 = 0 and B′
φ 0 =(−3Bφ 0 + 4Bφ1 − Bφ2)/(2δz) = 0, which is just the second order one-sidedfirst derivative formula . With this we can eliminate the boundary pointBφ 0 =
13(4Bφ1 − Bφ2), and have, in matrix form,
−2b −43a0 b 1
3a0 + a2 0 0 0 0 ...
S −23b 0 2
3b 0 0 0 0 ...
b a1 −2b 0 b a3 0 0 ...0 b S −2b 0 b 0 0 ...0 0 b a2 −2b 0 b a4 ...0 0 0 b S −2b 0 b ...... ... ... ... ... ... ... ... ...
In the lower right corner, no modification has to be applied, which then cor-responds to the vacuum boundary condition BR = Bφ = 0 on z = ±H . Forthe results shown in Fig. 6.11 we have simply assumed a linear profile, i.e.α = α0z/H , and ηT and S are constant. The eigenvalues of the matrix M canbe found using standard library routines.
Fig. 6.11. Eigenvalues of the dynamo equations with shear in slab geometry withradial shear. The dynamo number, in this figure, is defined positive when the shearin negative and α positive (opposite in sign to that in the text). Note that for α > 0the solution that is most easily excited is nonoscillatory (‘steady’) and has evenparity (referred to as S st) whilst for α < 0 it is oscillatory (S osc). Adapted from.
7 Magnetic helicity conservation and inverse cascade
The mean field approach in terms of α effect and other turbulent transportcoefficients has been quite popular in modeling the magnetic fields in variousastrophysical bodies. Its main shortcoming is the rather simplistic treatmentof turbulence. A very different approach has been pioneered by Frisch andcoworkers  and explored quantitatively in terms of the Eddy DampedQuasi-Normal Markovian (EDQNM) closure approximation , which willbe reviewed briefly below. In this approach, the main mechanism producinglarge scale fields is the inverse cascade of magnetic helicity toward larger scales.The α effect emerges in a self-consistent manner and, more importantly, the αeffect is amended by a correction term proportional to the small scale currenthelicity which plays a crucial role in modern mean field theory (Section 9).We begin with an illustration of the inverse cascade mechanism using a simple3-mode interaction model.
7.1 Inverse cascade in 3-mode interactions
The occurrence of an inverse cascade can be understood as the result of twowaves (wavenumbers p and q) interacting with each other to produce a wave ofwavenumber k. The following argument is due to Frisch et al. . Assumingthat during this process magnetic energy is conserved together with magnetichelicity, we have
Mp +Mq = Mk, (7.1)
|Hp|+ |Hq| = |Hk|, (7.2)
where we are assuming that only helicity of one sign is involved. Suppose theinitial field is fully helical and has the same sign of magnetic helicity at allscales, we have
2Mp = p|Hp| and 2Mq = q|Hq|, (7.3)
and so Eq. (7.1) yields
p|Hp|+ q|Hq| = 2Mk ≥ k|Hk|, (7.4)
where the last inequality is just the realizability condition (3.59) applied tothe target wavenumber k after the interaction. Using Eq. (7.2) we have
p|Hp|+ q|Hq| ≥ k(|Hp|+ |Hq|). (7.5)
In other words, the target wavevector k after the interaction of wavenumbersp and q satisfies
k ≤ p|Hp|+ q|Hq||Hp|+ |Hq|
The expression on the right hand side of Eq. (7.6) is a weighted mean of p andq and thus satisfies
min(p, q) ≤ p|Hp|+ q|Hq||Hp|+ |Hq|
≤ max(p, q), (7.7)
k ≤ max(p, q). (7.8)
In the special case where p = q, we have k ≤ p = q, so the target wavenum-ber after interaction is always less or equal to the initial wavenumbers. Inother words, wave interactions tend to transfer magnetic energy to smallerwavenumbers, i.e. to larger scale. This corresponds to an inverse cascade. Therealizability condition, 1
2k|Hk| ≤ Mk, was the most important ingredient in
this argument. An important assumption that we made in the beginning wasthat the initial field be fully helical; see Ref. [137,297] for the case of fractionalhelicity.
7.2 The EDQNM closure model
One of the earliest closure schemes applied to the MHD dynamo problem wasthe eddy-damped quasi-normal Markovian (EDQNM) approximation. Thiswas worked out in detail by Pouquet, Frisch and Leorat (PFL) . EDQNMhas been used frequently when dealing with fluid turbulence and is described,for example, in Refs. [274,298]. We do not describe it in great detail, but outlinehere just the basic philosophy, and the crucial insights gained from this closure.PFL assumed that the b and u fields were homogeneous, isotropic (but helical)random fields. This is similar to the unified treatment presented in Section 5.6.The large and small scale fields would again be distinguished by whether thewavenumber k is smaller or greater than the wavenumber of the forcing. Itwas also assumed that the initial b field is statistically invariant under signreversal (b → −b); the MHD equations then preserve this property and thecross helicity 〈u · b〉 is then always zero. Now suppose the MHD equationsfor the fluctuating fields are written symbolically as u = uu, where u standsfor some component of u or b and 〈u〉 = 0. This notation is used only toillustrate the effects of the quadratic nonlinearity; the linear dissipative andforcing terms have been dropped since they do not pose any specific closureproblem and can be re-introduced at the end. Then we obtain for the secondand third moments, again in symbolic form,
= 〈uuu〉, d〈uuu〉dt
= 〈uuuu〉. (7.9)
The quasi-normal approximation consists of replacing the fourth moment〈uuuu〉 by its gaussian value, that is the sum of products of second ordermoments. 4 It turns out that such an approximation leads to problems as-sociated with unlimited growth of the third moment, and the violation of
4 Note that the quasi-normal approximation may be more problematic when ap-plied to MHD since the dynamo generated B-field could initially be much moreintermittent than the velocity field (see for example the discussion of the kinematicsmall scale dynamo in Section 5). Therefore the validity of EDQNM applied toMHD is somewhat more questionable than when applied to pure hydrodynamic
the positivity constraint for the energy spectra. This is cured by assumingthat the neglected irreducible part of the fourth moment in (7.9) is in theform of a damping term, which is a suitable linear relaxation operator of thetriple correlations (a procedure called eddy-damping). One also carries out‘Markovianization’, by assuming that the third moment responds to the in-stantaneous product of the second moments. The resulting third moment issubstituted back on the RHS of the equation for second moments in (7.9).This results in a closed equation for the equal-time second-order moments
= θ(t)〈uu〉〈uu〉, (7.10)
where θ(t) essentially describes a relaxation time for a given triad of interactingmodes. It is given by,
τµ(s) ds dτ. (7.11)
Here µ(s) is the eddy damping operator and was written down in Ref. using phenomenological considerations. It depends on the kinetic and magneticspectrum and incorporates the damping effects on any mode due to nonlinearinteractions, Alfven effect and microscopic diffusion. (In the stationary case,where µ = constant, θ → µ−1). The derived evolution equations for the energyand helicity spectra of the random velocity and magnetic fields, are shownto preserve the quadratic invariants of total energy and magnetic helicity,just as in full MHD. The complete spectral equations under the EDQNMapproximation are give in Table I of Ref. , and will not be reproducedhere (see also the Appendix in Ref. ).
However several crucial insights resulted from this work about how the Lorentzforce affects the large scale dynamo. From the EDQNM evolution equationsfor the kinetic and magnetic spectra, PFL identified three important effects,all of which involve the coupling between widely separated scales. Suppose Mk
and Hk are the magnetic energy and helicity spectra and Ek, Fk the corre-sponding kinetic energy and helicity spectra. And suppose we are interestedin the dynamics of the magnetic energy at wavenumber k due to velocity andmagnetic fields at much larger scales (wavenumbers ≤ a0k) and much smallerscales (wavenumber ≥ k/a0), where a0 ≪ 1. Then, in concrete terms, PFLfound that the nonlocal contributions to the evolution are
˙(Mk)NLoc = kΓk(Ek −Mk) + αRk2Hk − 2νVk k
2Mk + . . . , (7.12)
turbulence. Nevertheless, there are indications (see Fig. 8.5 in Section 8.4 below)that once the large scale dynamo generation takes place, the probability distributionfunction of the field becomes more gaussian.
˙(Hk)NLoc = (Γk/k)(Fk − k2Hk) + αREk − 2νVk k
2Hk + . . . , (7.13)
˙(Ek)NLoc = −kΓk(Ek −Mk)− 2(25νVk + νM
k + νRk )k
2Ek + . . . , (7.14)
˙(Fk)NLoc = −kΓk(Fk − k2Hk)− 2(25νVk + νM
k + νRk )k
2Fk + . . . , (7.15)
Γk = 43k
αRk = −4
Fk − k2Hk
dq, νVk = 2
νMk = 2
θkqqMqdq, νRk = 2
∂q(Eq −Mq)dq. (7.18)
The first term on the RHS of (7.12), referred to by PFL as the Alfven effect,leads to equipartition of the kinetic and magnetic energies, at any scale dueto magnetic energy at larger scales. This happens on the Alfven crossing timeof the larger scale field. The second term, very important for what follows,shows that the growth of large scale magnetic energy is induced by the smallscale ‘residual’ helicity, which is the difference between the kinetic helicityand a current helicity contribution, due to the small scale magnetic field. Thepart of αR
k depending on the kinetic helicity corresponds closely to the usualα effect, once one realizes that θ is like a correlation time. Over and abovethis term PFL discovered for the first time the current helicity contribution,the k2Hk term in αR
k . The third term gives the turbulent diffusion of thelarge scale magnetic field. Surprisingly, this term does not get affected, to theleading order, by nonlinear effects of the Lorentz force (although the smallscale magnetic field does affect the diffusion of the large scale velocity field).PFL also gave a heuristic derivation of the last two results, because of theirpotential importance. This derivation, has since been reproduced in variousforms by several authors [184,300,301] and also extended to include higherorder corrections . We outline it in Section 9.2 and also use it to discussthe nonlinear saturation of the large scale dynamo in Section 9.4.
The EDQNM equations were numerically integrated by PFL to study thedynamo growth of magnetic energy. Most important in the present context isthe case when both kinetic energy and helicity are continuously injected. Inthis case PFL found what they described as an inverse cascade of magnetichelicity, which led to the appearance of magnetic energy and helicity in ever
Fig. 7.1. Inverse cascade in a numerical solution of the EDQNM equations showingthe magnetic energy spectrum (here called EM
k ) at two different times. Kineticenergy and helicity are injected at wavenumber k = 1. Note the peak of magneticenergy propagating toward smaller values of k. Adapted from .
increasing scales (limited only by the size of the system). Fig. 7.1 shows theresulting magnetic energy spectrum at two times. One can see the build upof large scale magnetic energy on scales much larger than the injection scale(k = 1). PFL have also argued that this inverse cascade of magnetic energyresulted from a competition between the helicity (residual α) and the Alfveneffect. We shall return to this question in subsequent sections.
8 Simulations of large scale dynamos
For astrophysical applications, the inverse cascade approach using the PFLmodel seemed too idealized compared to the α2 and αΩ dynamo models thathave been studied intensively in those years when the PFL model was pro-posed. Furthermore, there seems little scope for generalizing EDQNM to in-homogeneous systems with rotation. Nevertheless, the basic idea of an inversecascade was well established and verified by several groups [4,224,302–304].Only recently, however, there have been serious attempts to bridge the gapbetween the PFL approach and mean field models. In this section we reviewrecent efforts  to study helically driven hydromagnetic turbulence and to
compare with the associated α2 dynamo model that is applicable in the samesituation with fully periodic boundary conditions. After that, in Section 9, weconsider in detail the implications of the conservation of magnetic helicity tomean field models.
8.1 The basic equations
In order to simulate the effect of cyclonic turbulence without actually includingthe physical effects that contribute to cyclonic turbulence one can substitutethe buoyancy term by an explicit body force. The effects of stratification androtation are therefore neglected.
A compressible isothermal gas with constant sound speed cs, constant dynam-ical viscosity µ, and constant magnetic diffusivity η is considered. To makesure the magnetic field stays solenoidal, i.e. ∇ ·B = 0, B is expressed in termsof the magnetic vector potential A, so the field is written as B = ∇×A. Thegoverning equations for density ρ, velocity u, and magnetic vector potentialA, are given by
D ln ρ
Dt= −∇ · u, (8.1)
Dt= −c2s∇ ln ρ+
3∇∇ · u) + f , (8.2)
∂t= u×B + η∇2A, (8.3)
where D/Dt = ∂/∂t + u ·∇ is the advective derivative. The current density,J = ∇×B/µ0, is obtained in the form µ0J = −∇2A+∇∇ ·A. The gaugeφ = −η∇ · A for the electrostatic potential is used, and η = constant isassumed, so the magnetic diffusion term is just η∇2A. Details regarding thenumerical solution of these equations are analogous to the nonhelical case andhave been discussed elsewhere . Many of the simulations presented herehave been done using the Pencil Code .
For the following it is useful to recall that each vector field can be decomposedinto a solenoidal and two vortical parts with positive and negative helicity, re-spectively. These are also referred to as Chandrasekhar-Kendall functions (cf.Section 3.7). It is often useful to decompose the magnetic field into positiveand negative helical parts. Here, instead, we use the helical fields (with positivehelicity) as forcing function f of the flow. We restrict ourselves to functionsselected from a finite band of wavenumbers around the wavenumber kf , butdirection and amplitude are chosen randomly at each timestep. Details can
be found in Ref. . Similar work was first carried out by Meneguzzi et al., but at the time one was barely able to run even until saturation. Sincethe nineties, a lot of work has been done on turbulent dynamos with ABCflow-type forcing [188,189,191,304]. In none of these investigations, however,the saturation behavior has been studied and so the issue of resistively slowmagnetic helicity evolution past initial saturation remained unnoticed. It isexactly this aspect that has now become so crucial in understanding the satu-ration behavior of nonlinear dynamos. We begin by discussing first the linear(kinematic) evolution of the magnetic field.
8.2 Linear behavior
Dynamo action occurs once the magnetic Reynolds number exceeds a certaincritical value, Rcrit. For helical flows, Rcrit is between 1 and 2 (see Table 1 ofRef. ), where the magnetic Reynolds number is defined as Rm = urms/(ηkf).In the supercritical case, Rm > Rcrit, the field grows exponentially with growthrate λ, and λ is proportional to the inverse turnover time, urmskf . The resis-tively limited saturation behavior that will be discussed below in full detail hasno obvious correspondence in the kinematic stage when the field is too weakto affect the motions by the Lorentz force . Nevertheless, there is actuallya subtle effect on the shape of the eigenfunction as Rm increases: the magneticenergy spectrum has two bumps, each being governed by opposite magnetichelicity. In the weakly supercritical case the two bumps are far apart fromeach other, but as Rm increases, the two bumps in the spectra move closertogether, decreasing thus the net magnetic helicity. But before we can fullyappreciate this phenomenon, we need to discuss the effect the kinetic helicityhas on the magnetic field.
A helical velocity field tends to drive helicity in the magnetic field as well, butin the nonresistive limit magnetic helicity conservation dictates that 〈A ·B〉 =const = 0 if the initial field (or at least its helicity) was infinitesimallyweak. (Here and elsewhere, angular brackets denote volume averages.) Thus,there must be some kind of magnetic helicity cancelation. Under homoge-neous isotropic conditions there cannot be a spatial segregation in positiveand negative helical parts. Instead, there is a spectral segregation: there is abump at the forcing wavenumber and another ‘secondary’ bump at somewhatsmaller wavenumber. The two bumps have opposite sign of magnetic helicitysuch that the net magnetic helicity is close to zero (and it would be exactlyzero in the limit Rm → ∞. At the forcing wavenumber, the sign of magnetichelicity agrees with that of the kinetic helicity, but at smaller wavenumberthe sign of magnetic helicity is opposite. At small Rm, this secondary peakcan be identified with the wavenumber where the corresponding α2-dynamohas maximum growth rate; see Section 6.5.1. Simulations seem to confirm
Fig. 8.1. Power spectra of magnetic energy of Run 3 of Ref. . During the initialgrowth phase the field saturates at small scales first and only later at large scales (lefthand panel). Later, when also the large scale field saturates, the field at intermediatescales (k = 2, 3, and 4) becomes suppressed. In the second panel, intermediate timesare shown as dotted lines, t = 700 is shown in solid and t = 1600 is shown as a thicksolid line. The forcing wavenumber is kf = 5. Adapted from Ref. .
that, as Rm increases, kmax approaches12kf , as one would expect from α2
dynamo theory. Since these two peaks have opposite magnetic helicity, theirmoving together in the high-Rm limit tends to lower the net magnetic helicity,thus confirming earlier results [185,186] that suggest that the total magnetichelicity approaches zero in the high-Rm limit.
8.3 Nonlinear behavior
Eventually, the magnetic energy stops increasing exponentially. This is dueto the nonlinear terms, in particular the Lorentz force J × B in Eq. (8.2),which begins to affect the velocity field. The temporal growth of the powerspectra becomes stagnant, but the spectra saturate only partially; see Fig. 8.1,where we show data from a run with forcing at wavenumber kf = 5. In theleft hand panel we see that, by the time t = 600, the power spectra havesaturated at larger wavenumbers, k >∼ 3. However, it takes until t ≃ 1600 forthe power spectra to be saturated also at k = 1 (right hand panel of Fig. 8.1).In order to see more clearly the behavior at large scales, we show in Fig. 8.2data from a run with kf = 27 and compare spectra in the linear and nonlinearregimes. In the linear regime, all spectra are just shifted along the ordinate,so the spectra have been compensated by the factor Mini exp(2λt), where λis the growth rate of the rms field and Mini is the initial magnetic energy. Inthe nonlinear regime the bump on the right stays at approximately the same
Fig. 8.2. Power spectra of magnetic energy of positively and negatively polarizedparts (M+
k and M−k ) in the linear and nonlinear regimes. The spectra in the linear
regime have been compensated by the exponential growth factor to make themcollapse on top of each other. Here the forcing wavenumber is in the dissipativesubrange, kf = 27, but this allows enough scale separation to see the inverse transferof magnetic energy to smaller k. The data are from Run B of Ref. .
wavenumber (the forcing wavenumber), while the bump on the left propagatesgradually further to the left. As it does so, and since, in addition, the amplitudeof the secondary peak increases slightly, the net magnetic helicity inevitablyincreases (or rather becomes more negative in the present case). But, becauseof the asymptotic magnetic helicity conservation, this can only happen on aslow resistive time scale. This leads to the appearance of a (resistively) slowsaturation phase past the initial saturation; see Fig. 8.3.
8.4 Emergence of a large scale field
In the simulations of Ref.  the flow was forced at an intermediate wavenum-ber, k ≈ kf = 5, while the smallest wavenumber in the computational domaincorresponds to k = k1 = 1. The kinetic energy spectrum peaks at k ≈ kf ,which is therefore also the wavenumber of the energy carrying scale. The tur-bulence is nearly fully helical with 〈ω · u〉/(kf〈u2〉) ≈ 0.7...0.9. The initialfield is random, but as time goes on it develops a large scale component atwavenumber k ≈ k1 = 1; see Fig. 8.4.
The large scale field, seen in Fig. 8.4, has only one preferred direction, whichis the wavevector k1 of B. Different initial conditions can produce differentdirections of k1; see Fig. 6 of Ref. . A suitable definition of the mean field isa two-dimensional average over the two directions perpendicular to k1. (This
Fig. 8.3. The three stages of the magnetic field growth: exponential growth untilinitial saturation (when 〈B2〉/µ0 = 〈ρu2〉), followed by a (resistively) slow satura-tion phase. In this plot we have used µ0 = 1. The energy of the large scale magnetic
field, 〈B2〉, is shown for comparison. The data are from Run 3 of Ref. .
can only be done a posteriori when we know the direction of k1, so in practiceone has to calculate all three possibilities and select the right one in the end.)The large scale field is one of the following three eigenfunctions of the curloperator whose wavevectors point along a coordinate direction, i.e.
cos kmzsin kmz
0cos kmxsin kmx
where km = k1 = 1. These fields are force-free and are also referred to asBeltrami fields. The large scale field is fully helical, but with opposite (here
negative) current helicity relative to the small scale field, i.e. J ·B = −kmB2<
0. This property alone allows us to estimate the saturation amplitude of thedynamo, as will be done in the next section.
In view of analytic theories such as EDQNM (see Section 7.2) and the uni-fied treatment presented in Section 5.6 it is important to note that the threemagnetic field components obey, to a good approximation, gaussian statistics.This is shown in the left hand panel of Fig. 8.5 where we plot probability
Fig. 8.4. Cross-sections of Bx(x, y, 0) for Run 3 of Ref.  at different times showingthe gradual build-up of the large scale magnetic field after t = 300. Dark (light)corresponds to negative (positive) values. Each image is scaled with respect to itsmin and max values. The final state corresponds to the second eigenfunction givenin Eq. (8.4), but with some smaller scale turbulence superimposed.
functions (PDF) of Bx and By for Run 3 of Ref.  at the time of saturation.(The PDF of Bz looks very similar to that of Bx and is not shown.) We recallthat in this particular simulation the x and z components of the magnetic fieldshow large scale variation in the y direction. One therefore sees two markedhumps in the PDFs of Bx and Bz, which can be fitted by a superposition oftwo gaussians shifted away from zero, together with another gaussian of lowerweight around zero. The y component of the field does not show a large scalefield and can be fitted by a single gaussian of the same width. In the absenceof large scale dynamo action the three components of the magnetic field arecertainly not gaussians but stretched exponentials (see Ref.  for such re-sults in the context of convection). For comparison we show in the right handpanel of Fig. 8.5 the corresponding PDFs for nonhelically forced turbulence.
8.5 Importance of magnetic helicity
In the following we show that for strongly helical dynamos the saturationamplitude and the saturation time can accurately be estimated from magnetichelicity considerations alone – without actually solving the induction equation
Fig. 8.5. Probability density functions of Bx/Brms and By/Brms for Run 3 of Ref. at the time of saturation (left). For comparison, the PDFs are also shown for nonheli-cally forced turbulence (right). In the left hand plot, the dotted lines give gaussianfits with a width of 0.27. By/Brms is fitted by a single gaussian around zero, whileBx/Brms is fitted by a superposition of three gaussians (one around zero with weight0.26, and two with weight 0.37 shifted by ±0.76 away from zero).
explicitly. The argument is similar in nature to the way how the descent speedof a free-falling body can be calculated based on the consideration of kineticand potential energies alone. A more accurate treatment of the saturationbehavior of helical dynamos will be presented in Sections 9.4–9.5.3.
8.5.1 Saturation amplitude
Even though the current helicity of the large and small scale fields are finite,the current helicity of the total (small scale plus large scale) field must ap-proach zero in the long time limit. This is evident from the magnetic helicityequation (3.34) which, for a closed a periodic domain, is simply
dt〈A ·B〉 = −2η〈J ·B〉. (8.5)
Thus, in the steady state (d/dt = 0) one has 〈J ·B〉 = 0. However, the timescale on which this can be achieved is the resistive one, as will be discussed inthe following.
Splitting the magnetic field and current density into mean and fluctuatingcomponents we have, using Eq. (6.4),
J ·B = J ·B + j · b, (8.6)
and therefore also 〈J ·B〉 = 〈J ·B〉+ 〈j · b〉. In the steady state we have
−〈J ·B〉 = 〈j · b〉 (steady state). (8.7)
Here, overbars denote suitably defined two-dimensional averages (Section 8.4)and angular brackets denote volume averages. If both mean and fluctuating
fields are fully helical we have 〈J · B〉 ≈ ∓km〈B2〉 and 〈j · b〉 ≈ ±kf〈b2〉,where kf is the typical wavenumber of the fluctuating field (which is close to
the wavenumber of the energy carrying scale). This yields km〈B2〉 = kf〈b2〉,and if the small scale field is in equipartition with the turbulent motions, i.e.if 〈b2〉 ≈ 〈µ0ρu
2〉 ≡ B2eq, we have 
〈B2〉 = kfkm
〈b2〉 ≈ kfkm
B2eq > B2
eq (steady state), (8.8)
so for a fully helical dynamo the large scale field at k = km is in generalin super-equipartition with the kinetic energy of the turbulence. This fact isindeed confirmed by simulations which also show strong large scale fields insuper-equipartition; see Fig. 8.6.
Obviously, the simulated values of 〈B2〉/B2eq fall somewhat short of this sim-
plistic estimate, although the estimate becomes more accurate in the case oflarger magnetic Reynolds number (Run 3 has Re = Rm ≡ urms/ηkf ≈ 18,Run 2 has Re = Rm ≈ 6, Run 1 has Re = Rm ≈ 2.4).
8.5.2 Saturation time
It turns out that the time dependence in Fig. 8.6 can well be described bya fit formula that can be derived from the magnetic helicity equation. Nearsaturation, |C1| ≈ |Cf |, but |H1| ≪ |Hf |, because C1 = k2
1H1 and Cf =k2f Hf together with k1/kf ≪ 1. At late times we can therefore, to a good
approximation, neglect the time derivative of Hf relative to the time derivativeof H1 in Eq. (8.5), so
H1 = −2ηk21H1 − 2ηk2
fHf . (8.9)
At small scales up to the energy carrying scale of the turbulence (k ≤ kf) themagnetic field evolves on a dynamical time scale and saturates quickly. Thus,there is a time, t = tsat, after which the small scale field has saturated andthus the small scale magnetic helicity can be considered fixed, so for givenHf = const we can solve for H1(t), i.e.
H1(t) = Hfk2f
Fig. 8.6. Late saturation phase of fully helical turbulent dynamos for three differentvalues of the magnetic Reynolds number:Rm ≡ urms/ηkf = 2.4, 6, and 18 for Runs 1,2, and 3 respectively; see Ref. . The mean magnetic field, B, is normalized withrespect to the equipartition value, Beq =
√µ0ρ0urms, and time is normalized with
respect to the kinematic growth rate, λ. The dotted lines represent the fit formula(8.10) which tracks the simulation results rather well.
The time tsat is determined simply by the strength of the initial seed magneticfield, Bini; for weaker fields it takes somewhat longer to reach saturation.Since the growth is exponential, the dependence is only logarithmic, so tsat =λ−1 ln(Beq/Bini), where λ is the growth rate of the field of the rms field. Theexpression (8.10) describes the evolution of B quite well, as shown in Fig. 8.6.
Clearly, Eq. (8.10) is not applicable too close to t = tsat. This is also evidentfrom Fig. 8.6, which shows that in the simulations (solid lines) there is a finitemean field already at t = tsat. In order to describe this phase correctly, onehas to retain the time derivative of Hf , as will be done in Sections 9.4 and 9.5in the framework of the dynamical quenching model. At early times resistiveeffects have not yet played any role, so the total magnetic helicity must beapproximately zero, and this means that a helical large scale field must bysmaller than Beq; see Section 9.5.2 for details.
8.6 Alpha effect versus inverse cascade
The process outlined above can be interpreted in two different ways: inversecascade of magnetic helicity and/or α effect. The two are similar in that theytend to produce magnetic energy at scales larger than the energy-carryingscale of the turbulence. As can be seen from Figs. 8.1 and 8.2, the presentsimulations support the notion of nonlocal inverse transfer . However, thisis not really an inverse cascade in the usual sense, because there is no sustainedflux of energy through wavenumber space as in the direct Kolmogorov cascade.Instead, there is just a bump traveling in wavenumber space from a k thatis already smaller than kf to even smaller values of k. In that respect, thepresent simulations seem to differ from the interpretation of PFL based onthe EDQNM closure approximation .
The other interpretation is in terms of the α effect. We recall that for α2
dynamos, there is a wavenumber kmax where the growth of the large scale fieldis fastest; see Section 6.5.1. For reasonable estimates, kmax coincides with theposition of the secondary bump in the spectrum; see Ref. , Sect. 3.5. Thiscan be taken as evidence in favor of the α effect. In the nonlinear regime, thesecondary bump travels to the left in the spectrum (i.e. toward smaller k). Inthe EDQNM picture this has to do with the equilibration of kinetic and currenthelicities at progressively smaller wavenumbers, which then leads to saturationat that wavenumber, but permits further growth at smaller wavenumbers untilequilibration occurs, and so forth. Another interpretation is simply that after αis quenched to a smaller value, kmax = α/(2ηT) peaks at smaller wavenumberswhere the growth has not yet saturated, until equilibration is attained also atthat scale.
8.7 Nonlinear α effect
It is quite clear that αmust somehow depend on the strength of the mean field,B. By considering the effect of B on the correlation tensor of the turbulenceit has been possible to derive an α quenching formula of the form [307–309]
α = αK
(for |B| ≪ Beq). (8.11)
For practical applications, to make sure that |α| decreases with increasing fieldstrength and to prevent α from changing sign, Eq. (8.11) has been replacedby the fit formula 
(conventional quenching). (8.12)
However, it has long been noted that in the astrophysically relevant case,Rm ≫ 1, the magnitude of the fluctuating field is likely to exceed that of the
mean field, i.e. b2/B2 ≫ 1. Indeed, kinematic mean field theory predicts that
〈b2〉/〈B2〉 = Rm (kinematic theory). (8.13)
This result is a direct consequence of flux freezing (Section 3.3) during thecompression of a uniform field of strength B and scale L into a sheet of ‘skin’thickness d = LR1/2
m . A superficial argument might suggest that the quenchingformula (8.12) should take the small scale field into account. Using Eq. (8.13),this then leads to
(catastrophic quenching). (8.14)
This formula was first suggested by Vainshtein and Cattaneo . Only re-cently it has become clear that, even though Eq. (8.13) is actually no longervalid in the nonlinear regime, Eq. (8.14) can indeed emerge in a more rigorousanalysis under certain circumstances .
The problem with Eq. (8.14) is that α becomes strongly suppressed already
eq ≪ 1. Conversely, for the sun where B2/B2
eq ≈ 1, this means thatα would be negligibly small. Equation (8.14) is therefore sometimes referredto as catastrophic quenching formula.
Given the potentially catastrophic outcome of mean field theory when applyingEq. (8.14) to astrophysically relevant situations, the problem of α quenchinghas begun to attract significant attention in the last few years. Considerableprogress has recently been made by restricting attention to the simplest possi-ble system that still displays an α effect, but that is otherwise fully nonlinear.
8.8 Alpha quenching in isotropic box simulations
The issue of (catastrophic) α quenching was preceded by the related issue ofηt quenching. Indeed, already 30 years ago concerns have been expressed that turbulent diffusion might not work when the magnitude of the field isstrong. A serious argument against catastrophic ηt-quenching came from themeasurements of decay times of sunspots where the magnetic field is strongand yet able to decay almost on a dynamical time scale. Estimates for theturbulent magnetic diffusivity in sunspots suggest ηt ≈ 1011 cm2 s−1 [315–318].
Quantitatively, catastrophic (i.e. Rm-dependent) ηt quenching was first sug-gested based on two-dimensional simulations with an initially sinusoidallymodulated large scale magnetic field in the plane of the motions . How-ever, these results have to be taken with caution, because constraining thefield to be in the plane of the motions is artificial, and one would not expectcatastrophic quenching to be present if this constraint was relaxed, as can beseen both from closure models  and from three-dimensional simulations.
The issue of catastrophic α quenching was originally motivated by analogywith catastrophic ηt quenching , but then backed up by simulations withan imposed field , so α is calculated as the ratio of the resulting electromo-tive force and the imposed magnetic field; cf. Section 6.4. Another techniqueis to modify or remove a component of the mean field in an otherwise self-consistent simulation and to describe the response of the system in terms of αeffect and turbulent diffusion . This technique is easily explained by looking,for example, at the x-component of the α2 dynamo equation (Section 6.5.1),
∂z+ (η + ηt)
If, at some point in time, the x component of the mean field is removed, i.e.Bx → Bx − Bx, then Eq. (8.15) describes the immediate recovery of Bx. Therate of recovery is proportional to α. This method allows an estimate not onlyof α, but also of ηt by measuring the simultaneous temporary reduction of By.It turns out  that both methods give comparable results and confirm thecatastrophic quenching results (8.14). This result will later be understood asa special case of the dynamical quenching formula in the nearly steady limitfor force-free fields; see Eq. (9.20).
Yet another method is to impose a nonuniform field that is a solution of themean field dynamo equations – for example a Beltrami field in the case ofperiodic box. By changing the wavelength of the Beltrami field, one can deter-mine both α and ηt simultaneously. This method has been used in connectionwith the underlying flow field of the Karlsruhe dynamo experiment [180,181].
8.9 Dynamo waves in simulations with shear
The fact that dynamos exhibit cyclic behavior when there is shear is notsurprising if one recalls the type of solutions that are possible for αΩ dynamos(Section 6.5.2). On the other hand, until only a few years ago the conceptof mean field theory was only poorly tested and there was enough reason todoubt its validity especially in the nonlinear regime. Even in the linear regime
Fig. 8.7. Space-time diagram of the mean toroidal field at x = −π (negative localshear) and x = 0 (positive local shear). Dark (light) shadings refer to negative (pos-itive) values. Note the presence of dynamo waves traveling in the positive (negative)z-direction for negative (positive) local shear (from Ref. ).
the relevance of mean field theory has been doubtful because the mean fieldcan be much weaker than the fluctuating field [124,321].
Given all these reservations about the credibility of mean field theory it was asurprise to see that cyclic dynamo action does actually work . As in thesimulations without shear, the emergence of a large scale field is best seen inthe nonlinear regime; see Fig. 8.7. The reason is that prior to saturation a largenumber of different modes are excited, while in the nonlinear regime most ofthem are suppressed by the most dominant mode. Nonlinearity therefore hasa ‘self-cleaning’ effect [4,244].
Fig. 8.8. Evolution of Bx and By at x = −π and z = 0. Note that Bx has beenscaled by a factor −100. Here the overbars denote only a one-dimensional averageover the direction of shear, y. (Adapted from Ref. .)
The dynamo exhibits a certain phase relation between poloidal and toroidalfields (see Fig. 8.8). This phase relation is quite similar to what is expectedfrom a corresponding mean field model (see Section 6.5.3). Comparison with αquenching models in the same geometry (see Fig. 9 of ) produces similarlyanharmonic oscillations, as seen in Fig. 8.8. This strongly suggests that thesimulation results can basically be described in terms of the mean field concept.
An important question for astrophysical applications is which of the proper-ties of the dynamo depend on resistivity. Certainly the late saturation be-havior of the field does depend on resistivity and satisfies the ‘magnetic he-licity constraint’ embodied by Eq. (8.10); see Fig. 8 of Ref. . Subse-quent simulations for different values of the magnetic Reynolds number alsoseem to confirm that the cycle frequency, ωcyc, scales resistively and thatωcyc/(ηk
21) ≈ const = O(10) for Rm ranging from 30 to 200; see Table 5 of
Ref. . On the other hand, in all cases the large scale magnetic energyexceeds the kinetic energy by a factor that is between 20 (for smaller Rm) and60 (for larger Rm). It is therefore clear that these models are in a very differentparameter regime than the solar dynamo where the energy of the mean fieldis at most comparable to the kinetic energy of the turbulence. Furthermore,all turbulent transport coefficients are necessarily strongly suppressed by suchstrong fields even if the actual suppression is not explicitly dependent on themagnetic Reynolds number. A more detailed interpretation of these simula-tion data has been possible by using a dynamical quenching model that willbe discussed in Section 9.5.3.
8.10 The magnetic helicity constraint in a closure model
The magnetic helicity constraint is quite general and independent of any meanfield or other model assumptions. All that matters is that the flow possesseskinetic helicity. The generality of the magnetic helicity constraint allows oneto eliminate models that are incompatible with magnetic helicity conserva-tion. In previous sections we discussed a simple unified model of large andsmall scale dynamo, generalizing the Kazantsev model to include helical ve-locity correlations. In [198,219] this model was further extended to includeambipolar diffusion as a model nonlinearity. This is a useful toy model sincethe magnetic field still obeys helicity conservation even after adding ambipolardrift. So it is interesting to see how in this model the α effect is being regulatedby the change in magnetic helicity, and whether there are similarities to thedynamical quenching model under full MHD.
This issue was examined in Ref.  by solving the moment equations derivedin [198,219] numerically. We have given in the Appendix A the derivation of themoment equations for the magnetic correlations in the presence of ambipolardrift. In deriving the moment equations with ambipolar drift nonlinearity, oneencounters again a closure problem. The equations for the second moment,fortunately, contains a fourth order correlator of the magnetic field. This wasclosed in [198,219] assuming that the fourth moment can be written as aproduct of second moments. The equations for the longitudinal correlationfunction ML(r, t) and the correlation function for magnetic helicity density,H(r, t), then are the same as Eqs. (5.25) and (5.26), except for additions tothe coefficients ηT (r) and α(r). We have
+GML + 4αN(r)C, (8.16)
∂t= −2ηNC + αNML, (8.17)
αN = α(r) + 4aC(0, t), ηN = ηT (r) + 2aM(0, t). (8.18)
Note that at large scales
α∞ ≡ αN(r → ∞) = −13τ〈ω · u〉+ 1
3τAD〈J ·B〉/ρ0, (8.19)
η∞ ≡ ηN(r → ∞) = 13τ〈u2〉+ 1
where τAD = 2aρ0. Here, angular brackets denote volume averages over all
space. This makes sense because the system is homogeneous. Expression (8.19)for α∞ is very similar to the α suppression formula due to the current helicitycontribution first found in the EDQNM treatment by  (see Section 7.2).The expression for η∞ has the nonlinear addition due to ambipolar diffusion. 5
It is important to point out that the closure model, even including the abovenonlinear modifications, explicitly satisfies helicity conservation. This can beseen by taking r → 0 in (8.17). We get
H(0, t) = −2ηC(0, t), d〈A ·B〉/dt = −2η〈J ·B〉, (8.21)
where we have used the fact that 〈A ·B〉 = 6H(0, t), and 〈J ·B〉 = 6C(0, t).It is this fact that makes it such a useful toy model for full MHD (see below).
Adopting functional forms of TL(r) and F (r) constructed from the energyand helicity spectra resembling Run 3 of Ref. , Eqs. (8.16) and (8.17) weresolved numerically . In the absence of kinetic helicity, F = 0, and withoutnonlinearity, a = 0, the standard small scale dynamo solutions are recovered.The critical magnetic Reynolds number based on the forcing scale is around60. In the presence of kinetic helicity this critical Reynolds number decreases,confirming the general result that kinetic helicity promotes dynamo action(cf. [219,245]). In the presence of nonlinearity the exponential growth of themagnetic field terminates when the magnetic energy becomes large. After thatpoint the magnetic energy continues however to increase nearly linearly. Unlikethe case of the periodic box the magnetic field can here extend to larger andlarger scales; see Fig. 8.9. The corresponding magnetic energy spectra,
EM(k, t) =1
(kr)3ML(r, t) j1(kr) dk, (8.22)
are shown in Fig. 8.10.
The resulting magnetic field is strongly helical and the magnetic helicity spec-tra (not shown) satisfy |HM(k, t)| <∼ (2/k)EM(k, t). One sees in Fig. 8.10 thedevelopment of a helicity wave traveling toward smaller and smaller k. This isjust as in the EDQNM closure model  (see Fig. 7.1) and in the simulationsof the full MHD equations [4,306] (see Fig. 8.2).
In the following we address the question of whether or not the growth of thislarge scale field depends on the magnetic Reynolds number (as in ). To a
5 A corresponding term from the small scale magnetic field drops out as a conse-quence of the requirement that the turbulent velocity be solenoidal. By contrast,the ambipolar drift velocity does not obey this restriction.
Fig. 8.9. Evolution of magnetic correlation function ML (denoted by M(r, t) in thisfigure) for different times, for η = 10−3. The correlation function of the magnetichelicity (denoted in this figure by N(r, t)), is shown in the inset. η = 10−3.
Fig. 8.10. Evolution of magnetic energy spectra. Note the propagation of magnetichelicity and energy to progressively larger scales. The k−2 slope is given for orien-tation. Note the similarities with Figs. 7.1 and 8.2.
good approximation the wavenumber of the peak is given by
kpeak(t) ≈ α∞(t)/η∞(t). (8.23)
This result is familiar from mean field dynamo theory (see also Ref. ),
where the marginal state with zero growth rate, has k = α/ηt, while the fastestgrowing mode has k = α/2ηt. In our model problem, the large scale field, seemsto go to the quasi static state (which has no 1/2 factor) rather than the fastestgrowing mode, and then evolve slowly (on the resistive time scale), through asequence of such states. This evolution to smaller and smaller wavenumbers isalso consistent with simulations (, Sect. 3.5). Note that here kpeak decreaseswith time because α∞ tends to a finite limit and η∞ increases; see Section 6.5.1.(This is not the case in the box calculations where kpeak ≥ 2π/L.)
As we saw from Eq. (8.21) the magnetic helicity, 〈A ·B〉 = 6H(0, t), can onlychange if there is microscopic magnetic diffusion and finite current helicity,〈J ·B〉 = 6C(0, t). In Fig. 8.11 we show that, after some time t = ts, 〈J ·B〉reaches a finite value. This value increases somewhat as η is decreased. In allcases, however, 〈J ·B〉 stays below 〈ω ·u〉(2τ/a), so that |α∞| remains finite;see (8.19). A constant 〈J ·B〉 implies from (8.21) that 〈A ·B〉 grows linearlyat a rate proportional to η. However, since the large scale field is helical, andsince most of the magnetic energy is by now (after t = ts) in the large scales,the magnetic energy is proportional to 〈B2〉 ≈ kpeak〈A ·B〉, and can thereforeonly continue to grow at a resistively limited rate, see Fig. 8.11. It is to beemphasized that this explanation is analogous to that given in Ref.  andSection 8.3 for the full MHD case; the helicity constraint is independent of thenature of the feedback!
These results show that ambipolar diffusion (AD) provides a useful modelfor nonlinearity, enabling analytic (or semi-analytic) progress to be made inunderstanding nonlinear dynamos. There are two key features that are sharedboth by this model and by the full MHD equations: (i) large scale fields arethe result of a nonlocal inverse cascade as described by the α effect, and (ii)after some initial saturation phase the large scale field continues to grow ata rate limited by magnetic diffusion. This model also illustrates that it ishelicity conservation that is at the heart of the nonlinear behavior of largescale dynamos; qualitatively similar restrictions arise even for very differentnonlinear feedback provided the feedback obeys helicity conservation.
9 Magnetic helicity in mean field models
9.1 General remarks
We have seen from both numerical simulations and closure models that mag-netic helicity conservation strongly constrains the evolution of large scalefields. Due to helicity conservation, large scale fields are able to eventuallygrow only on the resistive time scale. Note that the magnetic helicity evo-
Fig. 8.11. (a) Evolution of 〈J ·B〉 for different values of η. The corresponding valueof α∞ is shown on the right hand side of the plot. (b) The evolution of magneticenergy for the same values of η.
lution does not depend on the detailed nonlinear backreaction due to theLorentz force. It merely depends on the induction equation. It therefore pro-vides a strong constraint on the nonlinear evolution of the large scale field.The effects of this constraint need to be incorporated into any treatment ofmean field dynamos. One also needs to elucidate to what extent one can thenunderstand the properties of the observed large scale magnetic fields in starsand galaxies. This will be the aim of the present section, where we solve si-multaneously the mean field dynamo equation (with the turbulent coefficientsdetermined by the nonlinear backreaction) together with helicity conservationequations.
We will see that the magnetic helicity conservation equations for the meanand the turbulent fields, as well as of course the mean field dynamo equation,involve understanding the mean turbulent EMF, E , in the nonlinear regime. Sowe will need to model the nonlinear effects on E, taking into account nonlinearbackreaction effects of the Lorentz force due to both mean and fluctuatingfields.
There are two quite different forms of feedback that can arise. One is theeffect of the dynamo-generated mean field on the correlation tensor of theturbulence, uiuj. The growing mean field could cause the suppression of the αeffect and turbulent diffusion. Such modifications of the turbulent transport
coefficients have been calculated since the early seventies [308,309], adopt-ing usually an approximation (random waves or FOSA), which linearizes therelevant equations in the fluctuations. This approach missed an important in-gredient in the nonlinear back reaction due to Lorentz forces: modificationsto E that involve the fluctuating fields themselves. These arise in calculatingu× b from terms involving the correlation of b and the Lorentz force in themomentum equation (see below). In particular, the α effect gets renormalizedin the nonlinear regime by the addition of a term proportional to the currenthelicity of the fluctuating field. This is an important effect which is crucialfor the correct description of the dynamo saturation, but one which has beenmissed in much of the earlier work.
The full problem of solving the induction equation and the momentum (Navier-Stokes) equation including the Lorentz force simultaneously, is a formidableone. One either takes recourse to numerical simulations or uses rather moreuncertain analytic approximations in numerical mean field models. The ana-lytic treatments of the backreaction typically involve the quasi-linear approx-imation or a closure scheme to derive corrections to the mean field dynamocoefficients. For example the EDQNM closure suggests that the α effect isrenormalized by the addition a term proportional to the current helicity ofthe small scale field j · b, while the turbulent diffusion is left unchanged. Itwould be useful to understand this result in a simpler context. For this pur-pose we examine first a simple heuristic treatment of the effects of the backreaction.
9.2 A heuristic treatment for the current helicity term
Originally, when the magnetic field is weak, the velocity is a sum of meanvelocity U and a turbulent velocity u = u(0) (which may be helical). Thevelocity field u(0) can be thought of as obeying the momentum (Navier-Stokes)equation without the Lorentz force. Then suppose we assume an ansatz that inthe nonlinear regime the Lorentz force induces an additional nonlinear velocitycomponent uN, that is, U = U + u(0) + uN, where ∇ · uN = 0 and
ρuN = µ−10 (B ·∇b + b ·∇B)−∇p′ +O(uu, bb, νuN), (9.1)
where the dots denote partial time differentiation, and O(uu, bb, νuN) indicatesall the neglected terms, those nonlinear in b and u = u(0) + uN, and theviscous dissipation. Also, p′ is the perturbed pressure including the magneticcontribution. PFL assumed that the mean field, B, was strong enough that thenonlinear and viscous terms could be neglected. Zeldovich et al.  arguedthat when multiplied by b and averaged, the resulting triple correlations couldbe replaced by double correlations, fairly close in spirit to the minimal τ
approximation; but they did not actually perform a calculation akin to thecalculation as done below. Gruzinov and Diamond [300,301] also neglectedthis term, calling this a quasi-linear dynamo. This ‘quasilinear’ treatment issomewhat approximate; nevertheless it helps in heuristically understandingthe result of the EDQNM closure model, in an analytically tractable fashion.
Due to the addition of uN, the turbulent EMF now becomes E = u(0) × b +uN × b, where the correction to the turbulent EMF is
(B ·∇b + b ·∇B −∇p′)× b(t) dt′, (9.2)
where (as usual) µ0 = ρ0 = 1 has been assumed. One again assumes thatthe small scale magnetic field b(t) has a short correlation time, say τb, andthat τb is small enough that the time integration can be replaced by a simplemultiplication by τb. We can calculate EN either in coordinate space or in k-space. The calculation, following , is given in Appendix C. To the lowestorder one gets
EN = 13τb j · b B, (9.3)
that is a correction to E akin to the magnetic contribution found by PFL. Further, one verifies the result of PFL that to the lowest order, theturbulent diffusion of the large scale field is not affected by nonlinear effectsof the Lorentz force. However, if one goes to higher order in the derivativesof B, one gets additional hyperdiffusion of the mean field and higher orderadditions to the α effect .
The above heuristic treatment is not very satisfactory, since it neglects possiblyimportant nonlinear terms in the momentum equation. At the same time theEDQNM treatment has the limitation that one assumes the random u andb field to be homogeneous and isotropic. We will now return to MTA, whichallows one to remove some of these limitations.
9.3 The minimal tau approximation: nonlinear effects
In this section we discuss the main aspects of the minimal tau approximation(MTA). Again, for the purpose of clarity, we restrict ourselves to the assump-tion of isotropy [8,268,275], but the method can readily be and has beenapplied to the anisotropic case [272,323]. A full treatment of inhomogeneousand anisotropic turbulence is given in Section 10.
In order to incorporate the evolution equations for the fluctuation in the ex-
pression for E one can just calculate its time derivative, rather than calculatingE itself . This way one avoids the approximate integration in Eqs. (6.19)and (6.20) or Eq. (9.2). Thus,
∂t= u× b+ u× b, (9.4)
where dots denote partial differentiation with respect to t. The dominantcontributions in these two terms are (using ρ0 = µ0 = 1)
u× b = 13j · b B + (j × b)× b− (ω × u)× b−∇p× b+ . . . , (9.5)
u× b = −13ω · u B − 1
3u2 J + u×∇× (u× b) + . . . , (9.6)
where the first term on the right hand side of Eq. (9.5) and the first two termson the right hand side of Eq. (9.6) are the usual quadratic correlation terms;all other terms are triple correlations. (For a detailed derivation Section 10below). Thus, we can write
∂t= αB − ηtJ + T , (9.7)
where T are the triple correlation terms, and
α = −13
ω · u− j · b)
, and ηt =13u2, (9.8)
are turbulent transport coefficients that are related to α and ηt, as used inEq. (6.18), via α = τα and ηt = τ ηt. Note that there are no free parameters inthe expression for α, neither in front of ω · u nor in front of j · b. Comparingwith the result of the heuristic treatment, Eq. (9.3), it is now plausible thatτb and τ should be the same. The full derivation for the more general case ofslow rotation and weak stratification is given the Section 10.
As we already emphasized earlier, the crucial step is now that, unlike the caseof FOSA or the heuristic treatment, the triple correlators are not neglected,but they are assumed to be a negative multiple of the second order correlator,i.e. T = −E/τ . This assumption has been checked numerically (see below) anda similar assumption has recently been verified for the case of passive scalardiffusion, cf. Refs. [268,275]. Using MTA, one arrives then at an explicitlytime-dependent equation for E ,
∂t= αB − ηtJ − E
where the last term subsumes the effects of all triple correlations.
In order to show that the assumption of a correlation between quadratic andtriple moments is actually justified we compare, using data from Run 3 ofRef. , the dependence of the triple moments and the mean field on position.We have calculated the triple correlation as
T = (j × b− ω × u−∇p)× b+ u×∇× (u× b). (9.10)
We recall that in Run 3 of Ref.  the mean field varied in the y direction withcomponents pointing in the x and z directions [third example in Eq. (8.4)].Thus, Bx(y) and Bz(y) exhibit a sinusoidal variation as shown in Fig. 9.1by the solid and dashed lines, respectively. Since α is negative (i.e. oppositeto the helicity of the forcing, which is positive), we expect a positive corre-lation between T and B. This is indeed the case, as shown by the full andopen symbols in Fig. 9.1. This demonstrates that the triple correlations areimportant and cannot be neglected, as is done in FOSA. Furthermore, sincethe correlation between T and B is positive, and α < 0, this implies thatτ > 0, which is necessary for τ to be interpreted as a relaxation time. Todemonstrate that T or B also correlate with E requires long time averaging[263,281], whereas here we have only considered a single snapshot, so no timeaveraging was involved.
In order to determine the value of τ , it is necessary to calculate E to high pre-cision, so time averaging is now necessary. The procedure we use is equivalentto that used in the passive scalar case  where a mean concentration gra-dient was imposed. Here we imposed instead a gradient in the magnetic vectorpotential or, what is equivalent, a uniform B0. In the steady case, a consistentway of calculating τ is from the relation α = τα, where α = 〈E ·B0〉t/B2
0 isthe usual alpha effect, and α is given by Eq. (9.8). The resulting normalizedvalues of τ are shown in Fig. 9.2 as a function of imposed field strength, B0,for different values of the magnetic Reynolds number.
The relaxation time is here normalized by the local turnover time. In thecontext of mean field dynamo theory this ratio is called the Strouhal number. (In other contexts, the Strouhal number is used differently and is also ameasure of an externally imposed oscillation frequency.) So, here we define
St = τurmskf (9.11)
as our nondimensional measure of the relaxation time.
In the passive scalar case, St was found to converge to a value of about 3 inthe limit of large Reynolds number and small values of kf . In the present casewe find that St is approximately unity for small field strengths, but decreases
Fig. 9.1. Comparison of the spatial dependence of two components of the meanmagnetic field and the triple correlation in Run 3 of Ref. . The magnetic fieldis normalized by the equipartition field strength, Beq, and the triple correlation isnormalized by k1B
3eq, but scaled by a factor of 2.5 make it have a similar amplitude
as the mean field. Note that Bx (solid line) correlates with Tx (filled dots) and Bz
(dashed line) correlates with Tz (open dots).
like B−20 once B0 becomes comparable with the equipartition field strength,
Beq. This result seems to be independent of magnetic Reynolds number; seeFig. 9.2.
To summarize, the effect of the Lorentz forces is, to leading order, the additionof a current helicity contribution to α and hence to α. There is an additionalsuppression effect due to the dependence of τ on the strength of the meanfield. This suppression seems however to be independent of magnetic Reynoldsnumber and hence not the main limiting factor for the growth of large scalefields when the magnetic Reynolds number is large.
An immediate difficulty with Eq. (9.3) is that this expression cannot directlybe used in a mean field model, because one only has information about themean fields, J and B, and not the fluctuations, j and b. The solution tothis problem is to invoke the magnetic helicity equation as an auxiliary equa-tion to couple j · b to the mean field equations. In other words, one has tosolve both the large scale dynamo equation and the magnetic helicity equationsimultaneously. We turn to this issue in the next section.
Fig. 9.2. Strouhal number as a function of field strength for three different valuesof the Reynolds number. Note the suppression of St proportional to B−2
0 , for theRm = 60 run (resolution 2563, kf/k1 = 5.1).
9.4 The dynamical quenching model
While conventional mean field theory is suitable to capture the structure ofthe mean field correctly, it has become clear that simple quenching expressionsof the form (8.12) or (8.14) are unable to reproduce correctly the resistivelylimited saturation phase. Instead, they would predict saturation on a dynam-ical time scale, which is not only in contradiction with simulations , but itwould also violate magnetic helicity conservation. The key to modeling the latesaturation behavior correctly is to ensure that the magnetic helicity equationEq. (8.5) is satisfied exactly at all times.
We begin with the usual mean field equation (6.7), which has to be solvedfor a given form of E . For α effect and possibly other mean field dynamos,the E term produces magnetic helicity of the large scale field. The evolutionequation of magnetic helicity of the mean field B can be obtained in the usualfashion from the mean field induction equation. We restrict ourselves here tothe case of a closed domain. In that case one obtains
dt〈A ·B〉 = 2〈E ·B〉 − 2η〈J ·B〉, (9.12)
which is independent of U . [Here, as earlier, angular brackets denote averag-ing over all space, while overbars denote suitably defined averages that couldbe two-dimensional (cf. Section 8.4) or one-dimensional if there is shear (cf.Section 8.9)]. One sees that there is a source term for the mean field helicity〈A ·B〉, due to the presence of E . The 〈J ·B〉 term can be approximated byk21〈A ·B〉 and corresponds to a damping term. This merely reflects the fact
that the operation of a mean field dynamo automatically leads to the growthof linkages between the toroidal and poloidal mean fields. Such linkages mea-sure the helicity associated with the mean field. One then wonders how thismean field helicity arises? To understand this, we need to consider also theevolution of the small scale helicity 〈a · b〉. Since the Reynolds rules apply, wehave 〈A ·B〉 = 〈A ·B〉+ 〈a · b〉, so the evolution equation for 〈a · b〉 can bededuced by subtracting Eq. (9.12) from the evolution equation (8.5) for thetotal helicity. The fluctuating field then obeys the equation
dt〈a · b〉 = −2〈E ·B〉 − 2η〈j · b〉, (9.13)
so the sum of Eqs. (9.12) and (9.13) gives Eq. (8.5) without any involvement ofthe E term. We see therefore that the term 〈E ·B〉 merely transfers magnetichelicity between mean and fluctuating fields conserving the total helicity!
In order to guarantee that the total helicity evolution Eq. (8.5) is alwaysobeyed, Eq. (9.13) has to be solved as an auxiliary equation along with themean field dynamo equation (6.7). From the previous two sections we haveseen that E is now determined by replacing the kinematic α effect by theresidual α effect,
α = −13τ〈ω · u〉+ 1
3τ〈j · b〉 ≡ αK + αM, (9.14)
and with no immediate modifications to ηt. As mentioned before, subsequentmodifications (i.e. quenching) of αK and ηt occur as a direct consequence of thedecrease of the turbulence intensity and/or the relaxation time (see Fig. 9.2)[129,307,309], but this is not a particularly dramatic effect, because it doesnot depend on Rm. More important is the 〈j · b〉 term in αM. This term canbe related to 〈a · b〉 in Eq. (9.13) for any given spectrum of magnetic helicity.One can write 〈j · b〉 = k2
f 〈a · b〉 quite generally for some given kf ; in casethe small scale field is dominated by a single scale, like the forcing scale, thenkf would correspond to this scale. The relaxation time τ can be expressed interms of B2
eq = u2rms using ηt =
rms. Furthermore, τ , and therefore also ηt,
may still depend on |B| in a way as shown in Fig. 9.2. (We recall that we haveused ρ0 = µ0 = 1 throughout.)
The final set of equations can then also be summarized in the more compactform [6,7]
U ×B + αB − (η + ηt)J]
α〈B2〉 − ηt〈J ·B〉
where Rm = ηt/η is a modified definition of themicroscopicmagnetic Reynoldsnumber; cf. Eq. (3.18). Simulations of forced turbulence with a decaying largescale magnetic field  suggest ηt ≈ (0.8...0.9) × urms/kf . Thus, Rm =(0.8...0.9) × Rm, but for all practical purposes the two are so close togetherthat we assume from now on Rm = Rm. (As discussed in Section 3.4, η cannotbe replaced by a turbulent value, so Rm is really very large!) We can now ap-ply these equations to discuss various issues about the dynamical quenchingof mean field dynamos.
9.4.1 Comparison with algebraic α quenching
In order to appreciate the nature of the solutions implied by Eqs. (9.15)and (9.16), consider first the long-time limit of a nonoscillatory dynamo. Inthis case the explicit time dependence in Eq. (9.16) may be neglected (adia-batic approximation ). Solving the resulting equation for α yields [9,254]
α =αK + ηtRm〈J ·B〉/B2
(for dα/dt = 0). (9.17)
Curiously enough, for the numerical experiments with an imposed large scalefield over the scale of the box , where B is spatially uniform and thereforeJ = 0, one recovers the ‘catastrophic’ quenching formula (8.14),
1 + Rm〈B2〉/B2eq
(for J = 0), (9.18)
which implies that α becomes quenched when 〈B2〉/B2eq = R−1
m ≈ 10−8 for thesun, and for even smaller fields for galaxies.
On the other hand, if the mean field is not imposed but maintained by dynamoaction, B cannot be spatially uniform and then J is finite. In the case of a
Beltrami field , 〈J · B〉/〈B2〉 ≡ km is some effective wavenumber of thelarge scale field with km ≤ km. Since Rm enters both the numerator and thedenominator, α tends to ηtkm, i.e.
α → ηtkm (for J 6= 0 and J ‖ B). (9.19)
Compared with the kinematic estimate, αK ≈ ηtkf , α is only quenched bythe scale separation ratio. More importantly, α is quenched to a value that isjust slightly above the critical value for the onset of dynamo action, αcrit =(η + ηt)km ≡ ηTkm.
It remains possible, however, that ηt is quenched via a quenching of τ (seeFig. 9.2). Thus, the question of how strongly α is quenched in the sun or thegalaxy, has been diverted to the question of how strongly ηt is quenched .Note that quasi-linear treatments or MTA do not predict ηt quenching at thelowest order and for weak mean fields [325,326]. One way to determine ηt andits possible quenching is by looking at numerical solutions of cyclic dynamoswith shear (αΩ-type dynamos), because in the saturated state the cycle fre-quency is equal to ηtk
2m. The best agreement between models and simulations
is achieved when ηt begins to be quenched when 〈B2〉/B2eq is around 0.3; see
Ref.  for details. This means that ηt is only quenched non-catastrophically.This is consistent with the quenching of τ ; see Fig. 9.2. However, more detailedwork at larger magnetic Reynolds numbers needs to be done – preferentiallyin more realistic geometries that could be more readily applied to stars andgalaxies.
9.4.2 α2 dynamos
For α2 dynamos in a periodic box a special situation arises, because thenthe solutions are degenerate in the sense that J and B are parallel to each
other. Therefore, the term 〈J · B〉B is the same as 〈B2〉J , which meansthat in the mean EMF the term αB, where α is given by Eq. (9.17), has acomponent that can be expressed as being parallel to J . In other words, theroles of turbulent diffusion (proportional to J) and α effect (proportional toB) cannot be disentangled. This is the force-free degeneracy of α2-dynamos ina periodic box . This degeneracy is also the reason why for α2-dynamos thelate saturation behavior can also be described by an algebraic (non-dynamical,
but catastrophic) quenching formula proportional to 1/(1+Rm〈B2〉) for bothα and ηt, as was done in Ref. . To see this, substitute the steady statequenching expression for α, from Eq. (9.17), into the expression for E . Wefind
E = αB − (η + ηt)J =αK +Rmηt〈J ·B〉/B2
B − ηtJ
which shows that in the force-free case the adiabatic approximation, togetherwith constant (unquenched) turbulent magnetic diffusivity, becomes equal tothe pair of expressions where both α and ηt are catastrophically quenched.This force-free degeneracy is lifted in cases with shear or when the large scalefield is no longer fully helical (e.g. in a nonperiodic domain, and in particularin the presence of open boundaries).
The dynamical quenching approach seems to be quite promising given that itdescribes correctly the α2 dynamo found in the simulations. There are how-ever severe limitations that have to be overcome before it can be used in morerealistic mean field models. Most importantly, the case of an inhomogeneoussystem, possibly one with boundaries, is not solved, although several promis-ing approaches have been suggested [327–329,331,332]. The difficulty is thatEq. (9.13) cannot straightforwardly be generalized to the non-homogeneouscase of a mean magnetic helicity density, because a · b would not be gauge-invariant. One possibility is to consider the evolution equation of j · b instead(as we do below). The other problem is that of boundaries that would requirea microscopic theory for the small scale losses of magnetic helicity throughthe boundaries [327,328,331]. In any case, a more sophisticated theory shouldstill reproduce the homogeneous case, for which we now have a fairly accurateand quantitative understanding.
9.5 Saturation behavior of α2 and αΩ dynamos
In the case of homogeneous αΩ dynamos, a major fraction of the toroidal field
can be generated by shear – independently of helicity. Therefore, 〈B2〉 can beenhanced without producing much 〈J ·B〉. We discuss below the strength ofthe fields, both when the final resistive saturation limit has been reached andthe case when Rm is so large that a quasi-static, non-resistive limit is morerelevant (for example if galaxies).
9.5.1 Final field strength
In case one waits long enough, i.e. longer than the resistive time scale, wenoted that the final field strength is determined by the condition 〈J ·B〉 = 0;
〈J ·B〉 = −〈j · b〉. (9.21)
In order to connect the current helicities with magnetic energies, we proceedas follows. First, one can quite generally relate the current and magnetic he-licities by defining characteristic wavenumbers, km and kf , for the mean andfluctuating fields via
k2m = 〈J ·B〉/〈A ·B〉, (9.22)
k2f = 〈j · b〉/〈a · b〉. (9.23)
For a fully helical field, the same wavenumbers will also relate the currenthelicity and energy in the field. On the other hand, if the field is not fullyhelical, one can introduce efficiency factors ǫm and ǫf , which characterize thethe fractions to which the mean and fluctuating fields are helical, respectively,and write
〈J ·B〉/〈B2〉 = kmǫm ≡ km, (9.24)
〈j · b〉/〈b2〉 = −kfǫf ≡ −kf . (9.25)
Here, km and kf are are ‘effective wavenumbers’ for mean and fluctuatingfields. In the final state, km will be close to the smallest wavenumber in thecomputational domain, k1. In the absence of shear, ǫm is of order unity, butit can be less if there is shear or if the boundary conditions do not permitfully helical large scale fields (see below). In the presence of shear, ǫm turnsout to be inversely proportional to the magnitude of the shear. The value ofǫf and kf , on the other hand, is determined by small scale properties of theturbulence and is assumed known.
Both km and kf are positive. However, ǫm can be negative which is typicallythe case when αK < 0. The sign of ǫf is defined such that it agrees with thesign of ǫm, i.e. both change sign simultaneously and hence kmkf ≥ 0. In moregeneral situations, km can be different from k1. Using Eqs. (9.24) and (9.25)together with Eq. (8.7) we have
km〈B2〉 = 〈J ·B〉 = −〈j · b〉 = kf〈b2〉, (9.26)
and so in the final saturated state,
〈B2〉/〈b2〉 = κf/κm, (9.27)
which generalizes Eq. (8.8) to the case with fractional helicities; see alsoEq. (79) of Ref. . Of course, this analysis only applies to flows with he-licity. In the nonhelical case, km = kf = 0, so Eq. (9.26) and Eq. (9.27) do notapply.
In the helical (or partially helical) case we can further determine the finalfield strength of the mean and fluctuating fields. This is possible because inperiodic geometry with homogeneous α effect the amplitude and energy of thedynamo wave is in general constant in time and does not vary with the cycle.We now use the mean field dynamo equation (9.15) to derive the evolutionequation for the magnetic helicity of the large scale field, and apply it to thesaturated state, so
0 = α〈B2〉 − (η + ηt)〈J ·B〉. (9.28)
We also use the evolution equation (9.16) for the magnetic contribution to theα effect, applied again to the saturated state,
0 =α〈B2〉 − ηt〈J ·B〉
Note both equations apply also when U 6= 0, because the U × B term hasdropped out after taking the dot product with B. We now eliminate α fromEqs. (9.28) and (9.29) and thus derive expressions for the final steady state
values of 〈B2〉 ≡ B2fin in terms of B2
eq and 〈b2〉 ≡ b2fin, using Eq. (9.27). We get,
=αK − ηTkm
=αK − ηTkm
In models where ηt is also quenched, both small scale and large scale fieldstrengths increase as ηt is more strongly quenched. (Of course, regardless ofhow strongly ηt may be quenched, we have always from (9.27), B2
κf/κm in the final state).
One can also write Bfin in an instructive way, using the relevant dynamo controlparameters for α2 and αΩ dynamos, defined in Section 6.5.3. We generalizethese for the present purpose and define
Cα = αK/ηTkm, CΩ = S/ηTk2m, (9.31)
where αK is the initial value of α due to the kinetic helicity and S = ∆Ωis a typical value of the shear in an αΩ dynamo. From Eq. (9.28), the final
steady state occurs when α = ηTkm ≡ αcrit or a critical value of the dynamoparameter Cα,crit = αcrit/ηTkm. One can then write, using (9.30),
(1 +R−1m )1/2Beq. (9.32)
For an αΩ dynamo, the relevant dynamo number is given by D0 = CαCΩ
initially and in the final steady state, by Dcrit = Cα,critCΩ, since the sheardoes not get affected by the nonlinear effects we consider. In this case thefinal mean field strength can be written as
(1 +R−1m )1/2Beq. (9.33)
Note once again that the above analysis only applies to flows with helicity.In the nonhelical case we have αK = km = kf = 0, so Eq. (9.30) cannotbe used. Nevertheless, without kinetic helicity one would still expect a finitevalue of 〈b2〉 because of small scale dynamo action. Furthermore, even in thefully helical case there can be substantial small scale contributions. Closerinspection of the runs of Ref.  reveals, however, that such contributions areparticularly important only in the early kinematic phase of the dynamo.
In summary, the saturation field strength depends only on the scale separationratio, see Eq. (9.27), and not, for example, on the intensity of the turbulence.
9.5.2 Early time evolution
During the early growth phase the magnetic helicity varies on time scalesshorter than the resistive time, so the last term in the dynamical α quenchingequation (9.16), which is proportional to R−1
m → 0, can be neglected and so αevolves then approximately according to
2f (α− ηtkm)
This equation can be used to describe the end of the kinematic time evolution
when 〈B2〉 grows exponentially. (We refer to this phase as late ’kinematic’because the slow resistive saturation has not yet set in, although of course αis already becoming suppressed due to growth of the current helicity). We seefrom (9.34) that the −2ηtk
2f α term leads to a reduction of α. This leads to
a dynamical reduction of α until it becomes comparable to ηtkm, shutting offany further reduction. Therefore, the early time evolution leads to a nearly
Fig. 9.3. Evolution of 〈B2〉 and 〈b2〉 (solid and dashed lines, respectively) in a dou-bly-logarithmic plot for an α2 dynamo with ηt = const for a case with kf/k1 = 10.Note the abrupt initial saturation after the end of the kinematic exponential growth
phase with 〈B2〉/〈b2〉 ∼ 0.1, followed by a slow saturation phase during which the
field increases to its final super-equipartition value with 〈B2〉/〈b2〉 ∼ 10 .
Rm-independent growth phase. At the end of this growth phase a fairly signifi-cant large scale field should be possible [5–7]. The basic physical reason is thatthe suppression of α occurs due to the growth of small scale current helicity,hence the small scale magnetic helicity; since helicity is nearly conserved forRm ≫ 1 this implies the corresponding growth of opposite signed large scalemagnetic helicity and hence the large scale field. We mention, however, thatnumerical simulations have not been able to confirm a sharp transition fromexponential to linear (resistively limited) growth, as seen in Fig. 9.3, whichshows a numerical solution of Eqs. (9.15) and (9.16); see Refs. [6,333]. Theabsence of a sharp cross-over from exponential to linear growth in the turbu-lence simulations could be related to the fact that several large scale modesare competing, causing an extra delay in the selection of the final mode .
In the following discussion we restrict ourselves to the case where η is smallor Rm ≫ 1. Magnetic helicity is then well conserved and this conservationrequires that
〈A ·B〉 ≈ −〈a · b〉 (for t ≤ tkin). (9.35)
Here the time t = tkin marks the end of the exponential growth phase (andthe ‘initial’ saturation time tsat used in Ref. ). This time is determined bythe condition that the term in parenthesis in Eq. (9.34) becomes significantly
reduced, i.e. α becomes comparable to ηtkm. We would like to estimate thefield strengths of the large and small scale fields by this time. The helicityconservation constraint (9.35), and the definitions in (9.22), (9.23) and (9.24),give
αM = 13τ〈j · b〉 = 1
f 〈a · b〉 = −13τk2
f 〈A ·B〉 = −ηtk2f
We use this in (9.14) for α, and demand that α governed by the dynamicalquenching equation (9.16) settles down to a quasi steady state by t = tkin. Wethen get for the mean squared strength of the large scale field at t = tkin,
=αK − ηtkm
− R−1M ≈ αK − ηtkm
where the effect of a finite Rm has been included through an approximatecorrection factor 
ι = 1 +R−1m
The strength of the small scale field at t = tkin is given by using (9.35)
αK − ηtkm
Not surprisingly, at the end of the kinematic phase the small scale magneticenergy is almost the same as in the final state; see Eq. (9.30). However, thelarge scale magnetic energy is still by a factor k2
m/k2f smaller than in the final
state (ǫm may be somewhat different in the two stages). This result was alsoobtained by Subramanian  using a similar approach.
These expressions can be further clarified by noting that the fractional helicityof the small scale field is likely to be similar to that of the forcing velocityfield. We can then write 〈ω ·u〉 ≈ ǫfkf〈u2〉; so αK ≈ ηtǫfkf , and one arrives at
which shows that Bkin can be comparable to and even in excess of Beq, espe-cially when ǫm is small (i.e. for strong shear). This is of interest in connection
with the question of why the magnetic field in so many young galaxies canalready have equipartition field strengths.
As emphasized in Ref. , even for an α2 dynamo, the initial evolution toBkin is significantly more optimistic an estimate than what could have beenexpected based on lorentzian α quenching. In the case of an αΩ dynamo [6,7],one could have km ≪ km, and so Bkin can be correspondingly larger. In fact,for
ǫm/ǫf ≤ km/kf , (9.41)
the large scale field begins to be of order Beq and exceed the small scale fieldalready during the kinematic growth phase.
It is once again instructive to express Bkin in terms of the dynamo controlparameters, Cα and CΩ Note that for small η or Rm ≫ 1, αcrit = ηTkm ≈ ηtkmand ι ≈ 1. One can then write
For an αΩ dynamo with dynamo number D0 = CαCΩ and a critical valueDcrit = Cα,critCΩ, we have (assuming shear does not get affected) 
We can estimate how strong the shear has to be for the large scale field tobe comparable to Beq. Since αK ≈ ηtǫfkf , Cα ≈ ǫf(kf/km), or km/kf ≈ ǫf/Cα.Using D0 = CαCΩ, we can rewrite (9.43) as
Now km/kf ≪ 1 in general. So Bkin can become comparable to Beq providedthe shear is strong enough that CΩǫ
2f > CαDcrit. For dynamo action to be
possible, one also requires CΩCα > Dcrit or Cα > Dcrit/CΩ. Combining thesetwo inequalities, we see that large scale fields can become comparable to theequipartition fields if CΩ >∼ Dcrit/ǫf . The critical dynamo number Dcrit willdepend on the physical situation at hand; Dcrit = 2 for a one dimensionalαΩ dynamo with periodic boundary conditions and homogeneous αK. Fur-ther, in simulations of rotating convection, ǫf ≈ 0.03 ; assuming thatthis relatively low value of ǫf is valid in more realistic simulations, we have
CΩ >∼ 2/ǫf ≈ 60 for the condition for which the large scale field becomes com-parable to Beq and the small scale field during the kinematic growth phase.(Note that when the large scale field becomes comparable to Beq quenchingdue to the large scale field itself will become important, a process we haveso far ignored). This condition on the shear, could be satisfied for stellar dy-namos, but not likely to be satisfied for galactic dynamos, where typicallyD0/Dcrit ∼ 2 (cf. Section 11.5).
In summary during the subsequent resistively limited saturation phase theenergy of the large scale field grows first linearly,
〈B2〉 ≈ B2kin + 2ηk2
m(t− tkin) (for t > tkin), (9.45)
and saturates later in a resistively limited fashion; see Eq. (8.10).
9.5.3 Comparison with simulations
In order to compare the dynamical quenching model with simulations of tur-bulent oscillatory dynamos with shear , it is important to consider thesame geometry and shear profile, i.e. sinusoidal shear, U = (0, Sk−1
1 cos k1x, 0),and a mean field B = B(x, z, t). We discuss here the results of Ref. , wheresimulations of Ref.  are compared with dynamical quenching models us-ing either a fixed value of ηt or, following earlier suggestions [4,5], a turbulentdiffusivity that is quenched in the same way as α, i.e. ηt ∝ α.
Both simulation and model show dynamo waves traveling in the positive z-direction at x = ±π and in the negative z-direction at x = 0, which is consis-tent with the three-dimensional simulations using negative values of αK, whichaffects the direction of propagation of the dynamo waves.
It turns out that in all models the values of b2fin are smaller than in the sim-ulations. This is readily explained by the fact that the model does not takeinto account small scale dynamo action resulting from the nonhelical compo-nent of the flow. Comparing simulations with different values of Rm, the cyclefrequency changes by a factor compatible with the ratio of the two magneticReynolds numbers. This is not well reproduced by a quenching expression forηt that is independent of Rm. On the other hand, if ηt is assumed to be propor-tional to α, then ωcyc becomes far smaller than what is seen in the simulations.A possible remedy would be to have some intermediate quenching expressionfor ηt. We should bear in mind, however, that the present model ignores thefeedback from the large scale fields. Such feedback is indeed present in thesimulations, which also show much more chaotic behavior than the model .
In conclusion the dynamical quenching model predicts saturation amplitudes
of the large scale field that are smaller than the equipartition field strength bya factor that is equal to the ratio of the turbulent eddy size to the system size,and hence independent of the magnetic Reynolds number. The field strengthscan be much larger if the dynamo involves shear and is highly supercritical.This may be relevant to explain the amplitudes of fields in stars, but the effectof shear is not likely to be strong enough to explain large scale galactic fieldscompletely, without a small scale helicity flux (see below and Section 11.5).One the other hand, for the sun and sun-like stars, the main issue is the cycleperiod. No conclusive answer can be given until there is definitive knowledgeabout the quenching of the turbulent magnetic diffusivity, ηt. The more ηt isquenched, the longer the cycle period can become, unless there are significantlosses of magnetic helicity through the open boundaries that all these bodiesmust have. Before discussing this in Section 9.7 we first address the questionof how close to being resistively limited the solar cycle might be. The answeris somewhat surprising: not much!
9.6 How close to being resistively limited is the 11 year cycle?
In this section we present an estimate of the amount of magnetic helicity, H ,that is expected to be produced and destroyed during the 11 year cycle .We also need to know which fraction of the magnetic field takes part in the11-year cycle. Here we are only interested in the relative magnetic helicity inone hemisphere. Following an approach similar to that of Berger , onecan bound the rate of change of magnetic helicity, as given by Eq. (8.5), interms of the rate of Joule dissipation, QJoule, and magnetic energy, M , i.e.
∣∣∣∣∣≤ 2η〈J ·B〉V ≤ 2η
〈J2〉〈B2〉V ≡ 2√
where V is the volume, 〈ηJ2〉V = QJoule the Joule heat, and 〈12B2〉V = M the
magnetic energy. For an oscillatory dynamo, all three variables, H , M , andQJoule vary in an oscillatory fashion with a cycle frequency ωcyc of magneticenergy (corresponding to 11 years for the sun – not 22 years), so we estimate|dH/dt| <∼ ωcyc|H| and |dM/dt| <∼ QJoule <∼ ωcycM , so we get
ωcyc|H| ≤ 2√
which leads to the inequality [137,306]
|H|/(2M) ≤ ℓskin, (9.48)
where ℓskin =√
2η/ωcyc is the skin depth, here associated with the 11 yearfrequency ωcyc. Thus, the maximum magnetic helicity that can be generatedand dissipated during one cycle is characterized by the length scale |H|/(2M),which has to be less than the skin depth ℓskin.
For η we have to use the Spitzer resistivity [see Eq. (3.14)], so η increases fromabout 104 cm2/s at the base of the convection zone to about 107 cm2/ s nearthe surface layers and decreases again in the solar atmosphere, see Eq. (3.14).Using ωcyc = 2π/(11 yr) = 2× 10−8 s−1 for the relevant frequency at which Hand M vary, we have ℓskin ≈ 10 km at the bottom of the convection zone andℓskin ≈ 300 km at the top.
The value of ℓskin should be compared with the value |H|/(2M) that canbe obtained from dynamo models . For a sphere (or rather a half-sphere)with open boundary conditions and volume V (for example the northern hemi-sphere), one has to use the gauge-invariant relative magnetic helicity of Bergerand Field ; see Eq. (3.35) and Section 3.4. When applied to an axisym-metric mean field, which can be written as B = bφ +∇× (aφ), it turns outthat the relative magnetic helicity integral is simply 
H = 2∫
ab dV (axisymmetry). (9.49)
The results of model calculations show that , when the ratio of poloidalto toroidal field, Bpole/Bbelt, is in the range consistent with observations,Bpole/Bbelt = (1...3)×10−4, then the ratio HN/(2MNR) is around (2−5)×10−4
for models with latitudinal shear. (Here, the subscript ‘N’ refers to the north-ern hemisphere). This confirms the scaling with the poloidal to toroidal fieldratio ,
HN/(2MNR) = O(Bpol/Btor) >∼ Bpole/Bbelt. (9.50)
Given that R = 700Mm this means that HN/(2MN) ≈ 70...200 km, whichwould be comparable to the value of ℓskin near the upper parts of the solarconvection zone.
The surprising conclusion is that the amount of mean field helicity that needsto be generated in order to explain the large scale solar magnetic fields isactually so small, that it may be plausible that microscopic magnetic diffusioncould still play a role in the solar dynamo. In other words, although openboundary effects may well be important for understanding the time scale ofthe dynamo, the effect does not need to be extremely strong.
9.7 Open boundaries
In the preceding sections we made the assumption that no magnetic helicityflows through the boundaries. This is of course quite unrealistic, and strongmagnetic helicity losses are indeed observed; see Section 2.1.2. The questionis how important is this effect and can it possibly alleviate the magnetic he-licity constraint by allowing the α effect to be less strongly quenched .In order for this to happen, however, it is important that the losses of mag-netic helicity occur predominantly at small scales. This is unfortunately notwhat the simulations available so far have shown . Instead, most of themagnetic helicity is lost through the large scale magnetic field rather than thesmall scale field – and it is of course the small scale magnetic field that onewants to get rid of and which contributes to the quenching of the α effect. Itmust be emphasized, however, that there are no simulations yet that allow forlosses of small scale magnetic helicity in the form of coronal mass ejections,for example.
9.7.1 Open box simulations
A straightforward way of extending the helical turbulence simulation to anopen domain is to apply boundary conditions that permit a magnetic helicityflux. One possibility is the pseudo-vacuum boundary condition, i.e. the fieldis assumed vertical on the boundaries, so n ·B = 0.
An important issue is how to define a gauge invariant magnetic helicity of thelarge scale field, B(z, t). Because of periodic boundary conditions in the twohorizontal directions, the z-component of B must be constant to obey ∇·B =0, and since the mean flux is assumed to vanish, we have Bz = 0, and hencethere is no field component pointing through the boundaries. The Berger-Field formalism would therefore predict a vanishing reference field. Anotherpossibility is to define a gauge-invariant magnetic helicity by extrapolatingthe magnetic vector potential linearly to a periodic field. This corresponds toa reference field given by 
= 〈B〉, (9.51)
i.e. the reference field is given by the mean horizontal flux per unit area.
We note that Bref
is not constant in time, because the vertical boundariespermit a diffusive loss of horizontal magnetic flux. One unpleasant property ofa magnetic helicity defined with this reference field is that it is not additive,i.e. the sum of the magnetic helicities of two adjacent sub-volumes is not equalto the helicity of the whole volume [336,337]. It is not clear yet whether thisproperty disqualifies such a magnetic helicity for the mean field. This question
is particularly important when comparing with the magnetic helicity of thesmall scale field, for which the usual Berger and Field procedure can still beused. It would therefore be important to attempt an independent assessmentof the relative importance of large scale versus small scale magnetic helicitylosses in other (perhaps more realistic) systems, where the Berger-Field gaugeinvariant magnetic helicity can be defined equally for both components.
9.7.2 The vacuum cleaner experiment
The possible significance of open boundary conditions can be explained as fol-lows. We have seen that, at the end of the kinematic evolution the magneticfield can only change on a resistive time scale, if magnetic helicity conserva-tion is obeyed. The large scale field can therefore only increase if this doesnot involve the generation of magnetic helicity (for example via differentialrotation). Any increase due to the α effect automatically implies an increaseof small scale magnetic helicity, and hence also small scale field. However, oncethe small scale field has reached equipartition with the kinetic energy, it can-not increase much further. Any preferential loss of small scale magnetic energywould immediately alleviate this constraint, as has been demonstrated in a nu-merical experiment where magnetic field at and above the forcing wavenumberhas been removed in regular intervals via Fourier filtering .
In the experiment shown in Fig. 9.4 the flow is forced with helical wavesat wavenumber k = 5, giving rise to large scale dynamo action with slowsaturation at wavenumber k = 1. The magnetic field is then Fourier filteredin regular intervals (between a tenth and a quarter of the e-folding time ofthe kinematic dynamo) and the components above k = 4 are set to zero,which is why this is called the vacuum cleaner experiment. Of course, thesmall scale magnetic energy will quickly recover and reach again equipartitionfield strength, but there remains a short time interval during which the smallscale magnetic energy is in sub-equipartition, allowing the magnetic helicityto growth almost kinematically both at small and large scales. The effect fromeach such event is small, but the effect from all events together make up asizeable effect.
9.7.3 Speculations about boundary conditions
In order for the anticipated effect to happen in the sun, or at least in a morerealistic simulation, it is important that the losses of magnetic helicity occurpredominantly at small scales. Although simulations have not shown this sofar (Section 9.7.1), it is conceivable that preferential small scale magnetichelicity losses may occur in the presence of a more realistic modeling of thesurface, where magnetostatic equilibrium configurations may lose stability,
Fig. 9.4. The effect of removing small scale magnetic energy in regular time intervals∆t on the evolution of the large scale field (solid lines). The dashed line gives the
evolution of 〈B2〉 for Run 3 of Ref. , where no such energy removal was included.In all cases the field is shown in units of B2
eq = ρ0〈u2〉. The two solid lines show the
evolution of 〈B2〉 after restarting the simulation from Run 3 of Ref.  at λt = 20and λt = 80. Time is scaled with the kinematic growth rate λ. The curves labeled(a) give the result for ∆t = 0.12λ−1 and those labeled (b) for ∆t = 0.4λ−1. Theinset shows, for a short time interval, the sudden drop and subsequent recovery ofthe total (small and large scale) magnetic energy in regular time intervals (adaptedfrom ).
leading to the ejection of plasmoids and possibly magnetic helicity. Except forthe vacuum cleaner experiment discussed in Section 9.7.2, there is as yet nofoundation for this simple-minded speculation. In this context it is importantto recall that whenever magnetic eruptions of any form do occur, the field isalways found to be strongly twisted [45,46,48,338,339]. It is therefore plausiblethat such events are an important part of the solar dynamo.
A second comment is here in order. Looking at the coronal mass ejectionsdepicted in Fig. 2.3 it is clear that fairly large length scales are involved in thisphenomenon. This makes the association with the small scale field dubious.Indeed, the division between large and small scale fields becomes exceedinglyblurred, especially because small and large scale fields are probably associatedwith one and the same flux tube structure, as is clear from Fig. 9.5.
The sketch shown in Fig. 9.5 shows another related aspect. Once a flux tubeforms a twisted Ω-shaped loop (via thermal or magnetic buoyancy) it developsa self-inflicted internal twist in the opposite sense. (In the sketch, which ap-plies to the northern hemisphere, the tilt has positive sense, corresponding topositive writhe helicity, while the internal twist is negative.) These helicities
j x <B >
flux gainflux gain
initial fieldribboninitial field
Fig. 9.5. Schematic of kinematic helical αΩ dynamo in northern hemisphere is shownin (a) and (b), whilst the dynamic helical αΩ dynamo is shown by analogy in (c)and (d). Note that the mean field is represented as a line in (a) and (b) and as atube in (c) and (d). (a) From an initial toroidal loop, the α effect induces a risingloop of right-handed writhe that gives a radial field component. (b) Differentialrotation at the base of the loop shears the radial component, amplifying the toroidalcomponent, and the ejection of the top part of loop (through coronal mass ejections)allow for a net flux gain through the rectangle. (c) Same as (a) but now with thefield represented as a flux tube. This shows how the right-handed writhe of the largescale loop is accompanied by a left-handed twist along the tube, thus incorporatingmagnetic helicity conservation. (d) Same as (b) but with field represented as ribbonor tube. (e) Top view of the combined twist and writhe that can be compared withobserved coronal magnetic structures in active regions. Note the N shape of theright-handed large scale twist in combination with the left-handed small scale twistalong the tube. The backreaction force that resists the bending of the flux tube isseen to result from the small scale twist. Note that diffusing the top part of the loopsboth allows for net flux generation in the rectangles of (a)–(d), and alleviates thebackreaction that could otherwise quench the dynamo. Courtesy E. G. Blackman.
can be associated with the positive J · B and the negative j · b, which hasalso been confirmed by calculating Fourier spectra from a buoyant flux tubetilting and twisting under the influence of the Coriolis force and the verticaldensity gradient [48,340]; see Fig. 9.6. Instead of visualizing the magnetic fieldstrength, which can be strongly affected by local stretching, we visualize therising flux tube using a passive scalar field that was initially concentrated alongthe flux tube. This is shown in Fig. 9.7. This picture illustrates quite clearlythat the α effect cannot operate without (at the same time) twist helicity ofthe opposite sign.
Fig. 9.6. Magnetic helicity spectra (scaled by wavenumber k to give magnetic helicityper logarithmic interval) taken over the entire computational domain. The spectrumis dominated by a positive component at large scales (k = 1...5) and a negativecomponent at small scales (k > 5). Adapted from Ref. .
Fig. 9.7. Three-dimensional visualization of a rising flux tube in the presence ofrotation. The stratification is adiabatic such that temperature, pressure, and densityall vanish at a height that is about 30% above the vertical extent shown. (The actualcomputational domain was actually larger in the x and z directions.) Adapted fromRef. .
9.7.4 A phenomenological model of magnetic helicity losses
In the absence of a reliable and well tested theory for the small scale fluxes, weresort here to a phenomenological description of magnetic helicity losses. Weimagine that the loss of magnetic helicity across the boundary is accomplishedby the turbulent exchange of eddies across the boundary. The rate of magnetic
helicity loss is then proportional to some turbulent diffusivity coefficient, ηmor ηf , for the losses from mean and fluctuating parts, respectively. Again,we assume that the small and large scale fields are maximally helical (orhave known helicity fractions ǫm and ǫf) and have opposite signs of magnetichelicity at small and large scales. The details can be found in Refs. [48,137]The strength of this approach is that it is quite independent of mean fieldtheory.
We proceed analogously to Section 8.5.2 where we used the magnetic helic-ity equation (8.5) for a closed domain to estimate the time derivative of themagnetic helicity of the mean field, H1 (or Hm, which is here the same), byneglecting the time derivative of the fluctuating field, Hf . This is a good ap-proximation after the fluctuating field has reached saturation, i.e. t > tsat.Thus, we have
dt︸ ︷︷ ︸
= −2ηmk2mHm − 2ηfk
2f Hf , (9.52)
where ηm = ηf = η corresponds to the case of a closed domain; see Eq. (8.9) inSection 8.5.2. Assuming that surface losses can be modeled as turbulent diffu-sion terms, we expect the values of ηm and ηf to be enhanced, correspondingto losses from mean and fluctuating parts, respectively.
The phenomenological evolution equations are then written in terms of thelarge and small scale magnetic energies, Mm and Mf respectively, where weassume Mm = ±1
2kmHm and Mf = ∓1
2kfHf for fully helical fields (upper/lower
signs apply to northern/southern hemispheres). The phenomenological evolu-tion equation for the energy of the large scale magnetic field then takes theform
dt= −2ηmkmMm + 2ηfkfMf , (9.53)
The positive sign of the term involving Mf reflects the generation of large scalefield by allowing small scale field to be removed. After the time when the smallscale magnetic field saturates, i.e. when t > tsat, we have Mf ≈ constant. Afterthat time, Eq. (9.53) can be solved to give
Mm = Mfηfkfηmkm
, for t > tsat. (9.54)
This equation highlights three important aspects:
• The time scale on which the large scale magnetic energy evolves dependsonly on ηm, not on ηf (time scale is shorter when ηm is increased).
• The saturation amplitude diminishes as ηm is increased, which compen-sates the accelerated growth just past tsat , so the slope dMm/dt isunchanged.
• The reduction of the saturation amplitude due to ηm can be offset by havingηm ≈ ηf ≈ ηt, i.e. by having losses of small and large scale fields that areabout equally important.
The overall conclusions that emerge are: first, ηm > η is required if the largescale field is to evolve on a time scale other than the resistive one and, second,ηm ≈ ηf is required if the saturation amplitude is not to be catastrophicallydiminished. These requirements are perfectly reasonable, but so far they havenot been borne out by simulations; see Section 9.7.1.
An important limitation of the analysis above has been the neglect of dHf/dt,precluding any application to early times. Alternatively, one may directly useEqs. (9.12) and (9.13) and turn them into evolution equations for Mm and Mf ,respectively,
dt= 2(α− kmηt)Mm − 2ηmkmMm, (9.55)
dt= 2(α− kmηt)Mm − 2ηfkfMf , (9.56)
where, α = αK+αM is the total α effect and αM the contribution from the smallscale current helicity; see Eq. (9.14). As in Eq. (9.16), we can express the coef-ficient 1
3τ in terms of ηt/B
2eq, and have in terms of energies, αM = ±ηtkfMf/Ef
where Ef =12B2
eqV is the kinetic energy. Equations (9.55) and (9.56) can beapplied at all times, but they reduce to Eq. (9.53) in the limit of late timeswhen Mf = const.
Another caveat of the phenomenological approach is that we have modeledthe helicity fluxes with ηm and ηf as parameters which we can fix freely. Thephysical mechanism which causes these fluxes may not allow such a situa-tion. For example if initially Hm = −Hf so that total helicity is zero, andthen helicity fluxes involved mass fluxes which carried both small and largescale fields (cf. Section 11.2.2 below), then the flux of small and large scalehelicity could always be the same for all times. But from helicity conservation(neglecting microscopic diffusion), one will always have Hm = −Hf , and soMm/km = Mf/kf .
9.8 Mean-field models of magnetic helicity losses
An important problem associated with the dynamical quenching model, asformulated so far, is the fact that, unlike the volume average 〈A ·B〉 over aperiodic domain, the spatially dependent magnetic helicity density a · b is notgauge-invariant. It is then possible to formulate the theory directly in termsof the small scale current helicity, j · b. This leads to an extra divergenceterm of a current helicity flux, FC . Using the evolution equation for the b,∂b/∂t = −∇ × e, we can derive an evolution equation for the small scalecurrent helicity in the form
∂tj · b = −2 e · c−∇ ·FC , (9.57)
where c = ∇ × j is the curl of the small scale current density, j = J − J ,e = E −E is the small scale electric field, E = ηJ − E is the mean electricfield, and
FC = 2e× j + (∇× e)× b (9.58)
is the mean current helicity flux resulting from the small scale field.
Mean-field models trying to incorporate such magnetic helicity fluxes havealready been studied by Kleeorin and coworkers [327–329] in the context ofboth galactic and solar magnetic fields. An interesting flux of helicity whichone obtains even in nonhelical but anisotropic turbulence has also been pointedout by Vishniac and Cho . We will examine a general theory of such fluxesusing the MTA in Section 10.4 below.
In the isotropic case, e · c can be approximated by k2f e · b. Using FOSA or
MTA, this can be replaced by k2f (E · B + ηj · b); see also Eq. (9.16), which
then leads to the revised dynamical quenching formula
2 − ηtJ ·B + 12k−2f ∇ ·FC
where the current helicity flux, FC = 2e× j + (∇× e)× b, has entered asan extra term in the dynamical quenching formula (9.16).
Making the adiabatic approximation, i.e. putting the rhs of Eq. (9.59) to zero,one arrives at the algebraic quenching formula
α =αK +Rm
ηtJ ·B − 12k−2f ∇ ·FC
(dα/dt = 0). (9.60)
Considering again the Rm → ∞ limit, one has now
α → ηtkm − 12k−2f (∇ ·FC)/B
which shows that losses of negative helicity, as observed in the northernhemisphere of the sun, would enhance a positive α effect [327–329]. Here,
km = J · B/B2is a spatially dependent generalization of Eq. (9.24) to the
inhomogeneous case. If the current helicity flux is approximated by a Fickiandiffusive flux proportional to the gradient of the small scale current helicity,i.e. FC ≈ −2ηf∇j · b, where ηf is an effective turbulent diffusion coefficientfor the small scale current helicity (see Section 9.7.4), the second term ofEq. (9.61) becomes approximately ηf km, so α approaches a combination ofηtkm and ηf km, confirming again that α has increased.
10 Revisiting the microscopic theory using MTA
In this section we describe in full detail the technique to calculate turbulenttransport coefficients in the presence of slow rotation and weak inhomogeneity.The technique to incorporate slow rotation and weak inhomogeneity is thesame as that introduced by Roberts and Soward  long ago, except thathere we are using MTA. The present results are basically in full agreementwith a resent calculation by Radler, Kleeorin and Rogachevskii , exceptthat here we retain the j · b correction to α throughout, even when this termwas vanishing initially. Indeed, in the nonlinear regime there is no limit inwhich this term can be ignored, because it is self-generated by the α effect. Inthis section, overbars denote ensemble averages, but for all practical purposeswe may assume them to be equivalent to the spatial averages used before.
As we have emphasized before, the ignorance of the j · b feedback term inmuch of the mean field dynamo literature is one of the main reasons whythis theory has produced incorrect or unreliable results and has not been inagreement with numerical simulations of hydromagnetic turbulence . Weare now at a point where a lot of work has to be repeated with the correctfeedback in place. What needs to be done now is a careful calculation of theturbulent transport coefficients that include, in particular, inhomogeneity and
losses through boundaries. Here we are only able to present the initial stepsin that direction.
10.1 Transport coefficients in weakly inhomogeneous turbulence
We consider the case when the correlation tensor of fluctuating quantities(u and b) vary slowly on the system scale, say R. Consider the equal time,ensemble average of the product f(x1)g(x2). The common dependence of fand g on t is assumed and will not explicitly be stated. We have
f(x1)g(x2) =∫ ∫
f(k1)g(k2) ei(k1·x1+k2·x2) d3k1 d3k2, (10.1)
where f and g are the Fourier transforms of f and g respectively. (In generalthe Fourier transform of any function, say f , will be denoted by the samesymbol with a ‘hat’ i.e. f). We define the difference r = x1−x2 and the meanR = 1
2(x1 + x2), keeping in mind that all two point correlations will vary
rapidly with r but slowly with R . We can re-express this correlation as
f(x1)g(x2) =∫ ∫
f(k + 12K)g(−k + 1
2K) ei(K·R+k·r) d3K d3k
Φ(f , g,k,R) eik·r d3k, (10.2)
where we have defined the new wavevectors k = 12(k1−k2) and K = k1+k2.
Also for later convenience we define
Φ(f , g,k,R) =∫
f(k + 12K)g(−k + 1
2K) eiK·R d3K. (10.3)
In what follows we require the correlation tensors of the u and b fields andthe cross correlation between these two fields in Fourier space. The velocityand magnetic correlations, as well as the turbulent electromotive force E, aregiven by
vij(k,R) =Φ(ui, uj,k,R), u2(R) = δij
vij(k,R) d3k, (10.4)
mij(k,R) =Φ(bi, bj ,k,R), b2(R) = δij
mij(k,R) d3k, (10.5)
χjk(k,R) =Φ(uj, bk,k,R), E i(R) = ǫijk
χjk(k,R) d3k. (10.6)
In order to calculate these quantities we first consider the time derivative ofχjk,
∂t= Φ(uj ,
˙bk,k,R) + Φ( ˙uj , bk,k,R), (10.7)
where the dots on bj and ui denote a partial time derivative. The equation forb is given by Eq. (6.12). For the present we neglect the mean velocity termU . The Fourier transform of this equation is then,
˙bk(k) = ǫkpqǫqlmikp
ul(k − k′)Bm(k′) d3k′ + Gk(k)− ηk2bk(k), (10.8)
where we have included the nonlinear term, G = ∇× (u× b−u× b). (Notethat since u× b is a mean field quantity, it will give zero contribution to χjk,when multiplied by u and ensemble averaged.) Using the momentum equationfor the velocity field U and splitting the velocity into mean and fluctuatingparts, U = U + u, and neglecting for simplicity the mean flow, U = 0, wehave in a rotating frame,
∂t= −∇peff +B ·∇b+ b ·∇B − 2Ω× u+H + f + ν∇2u. (10.9)
Here we have assumed incompressible motions with ∇ ·u = 0, and a constantdensity; we have therefore adopted units such that µ0 = ρ0 = 1. We have de-fined the effective pressure peff which contains the perturbed pressure includingthe magnetic contribution, f is the random external force, and the terms non-linear in u and b are gathered in H = −u ·∇u+ u ·∇u+ b ·∇b− b ·∇b.In the rotating frame, we also have a Coriolis force contribution 2Ω×u. Theeffective pressure term can be obtained by solving a Poisson type equation de-rived by taking the divergence of Eq. (10.9). In Fourier space this correspondsto multiplying the equation by the projection operator, Pij(k) = δij − kikj ,
where k = k/|k| is the unit vector of k, so
(B ·∇b)s + (b ·∇B)s − 2ǫslmΩlum + fs + Ts − νk2us
Here we have dropped the argument k on all quantities, and introduced thenotation
(B ·∇b)s =∫
Bl(k − k′)(ik′l) bs(k
′) d3k′, (10.11)
(b ·∇B)s =∫
bl(k − k′)(ik′l) Bs(k
′) d3k′ (10.12)
for the Fourier transforms of the magnetic tension/curvature force. We cansimplify the rotational part of the Coriolis force by using the identity
ǫijk = kikpǫpjk + kjkpǫipk + kkkpǫijp, (10.13)
noting that Pijǫjkl = kkkpǫipl + klkpǫikp, and using incompressibility kkuk = 0,to get
Pjs(k) ǫslmΩlum = k ·Ω ǫjpmkpum. (10.14)
We are now in a position to calculate ∂χjk/∂t. Using the induction equation
for˙b given by (10.8), the first term in (10.7) is given by
Φ(uj ,˙bk,k,R) = χK
jk + ujGk − η(k + 12i∇)2χjk, (10.15)
where we have substituted K by −i∇ and have introduced the abbreviation
χKjk = ǫkpqǫqlm
uj(k + 12K)ul(−k + 1
2K − k′) Bm(k
× i(−kp +12Kp) d
3k′ eiK·R d3K. (10.16)
Further the triple correlations of the form ujGk, will be either ignored (FOSA)or replaced by the double correlation χjk, divided by a relaxation time (the τapproximation). To evaluate the above velocity correlation in ∂χK
jk/∂t, we need
to bring the uj ul term into a form similar to Eq. (10.4), so that we can replaceit by vjl. Thus, we define new wavevectors k1 = k+ 1
2K and k2 = −k+ 1
and transform to new variables, (k′,K ′), where K ′ = k1 + k2 = K − k′. Wethen have
χKjk = ǫkpqǫqlm
uj(k + 12k′ + 1
2K ′) ul(−(k + 1
2k′) + 1
i(−kp +12k′p +
′)·R d3K ′ d3k′ (10.17)
Using the definition of the velocity correlation function vij in Eqs. (10.3)and (10.4) and carrying out first the integral over K ′, and replace iK ′
p by∇p ≡ ∂/∂Rp, we can write
χKjk = ǫkpqǫqlm
p +12∇p) vjl(k + 1
′) eik′·R d3k′.(10.18)
Note that, since the mean field B varies only on large scales, B(k′) will benon-zero only for small |k′ |. This suggests expanding vjl above in a Taylor
series in k′, i.e.
vjl(k + 12k′,R) ≈ vjl(k,R) + 1
We will keep only terms which can contribute derivatives in R no higher thanthe first derivative. Integrating over k′, using ǫkpqǫqlm = δklδpm − δkmδpl, and
noting that the inverse Fourier transform of ik′sBm(k
′) is Bm,s ≡ ∂Bm/∂Rs,we get
χKjk = −ik ·B vjk +
12B ·∇vjk − vjl
Now turn to the second term in (10.7) for ∂χjk/∂t. This is given by
Φ( ˙uj , bk,k,R) = χMjk + χΩ
jk + Hj bk − ν(k − 12i∇)2χjk, (10.21)
d3k′ d3K eiK·R Pjs(k + 12K)×
[i(kl +12Kl − k′
l) Bl(k′) bs(k + 1
2K − k′)bk(−k + 1
+ (ik′l) Bs(k
′) bl(k + 12K − k′)bk(−k + 1
χΩjk = 2Ωmǫjlt
ul(k + 12K)bk(−k + 1
2K) eiK·R d3K. (10.23)
Here we have introduced p = k + 12K for brevity. Once again, the triple
correlations of the form Hj bk, will be either ignored (FOSA) or replaced bythe double correlation χjk, divided by a typical correlation time (the τ ap-
proximation). The term f · b does not contribute, because f is assumed tobe δ-correlated and does not correlate with b. We first simplify the χΩ
jk term,keeping only terms which are at most a first derivative in R. For this firstexpand pmpt/p
(km + 12Km)(kt +
(k + 12K)2
keeping terms that are at most linear in K. Then the integral over K can becarried out using the definition of χij. We get
χΩjk = −Ajlχlk − Bjlm
where the χlk without superscript is the full χlk, and
Ajl = −2ǫjltk ·Ωk2
Bjlm = iǫjlmk ·Ωk2
As shown in Appendix D, the simplification of χMjk leads to
χMjk = ik ·Bmjk +
12B ·∇mjk +Bj,lmlk
We can now add all the different contributions to get,
∂t= Ijk− Ajlχlk − Bjlm
+ujGk + Hj bk − (η + ν)k2χjk − i(η − ν)k ·∇χjk, (10.29)
Ijk =−ik ·B (vjk −mjk) +12B ·∇(vjk +mjk) +Bj,lmlk −Bk,lvjl
and we have only kept the terms in the microscopic diffusion and viscosity upto first order in the large scale derivative.
The first order smoothing approximation can be recovered by neglecting thetriple correlation terms 〈ujGk〉+ 〈Hj bk〉. Using MTA, one summarily approxi-
mates the triple correlations as a damping term, and takes 〈ujGk〉+ 〈Hj bk〉 ≈−χjk/τ(k), where in general the parameter τ could be k-dependent. (The mi-croscopic diffusion terms can either be absorbed into the definition of τ(k) orneglected for large Reynolds numbers). One then has
∂t= Ijk − Ajlχlk − Bjlm
∂t= Ijk −
τDjlχlk − Bjlm
where Djl = δjl + τAjl. On time scales long compared with τ , one can neglectthe time derivative of χjk, so one has 
Djlχlk + τBjlm∂χlk
∂Rm= τ Ijk. (10.33)
In the weakly inhomogeneous case that we are considering, this equation canbe solved iteratively. To zeroth order, one neglects the ∂/∂R terms on bothsides of Eq. (10.33). This gives a zeroth order estimate
χ(0)lk = D
−1lj τ I
(0)jk , (10.34)
I(0)jk = −ik ·B (vjk −mjk). (10.35)
Here the inverse matrix D−1jl is given by
δjl + Cokǫjlmkm + Co2kkjkl
1 + Co2k
where Cok ≡ 2Ωτ · k is the Coriolis number with respect to the component ofΩ that is aligned with k, and k ≡ k/k is the unit vector of k.
To the next order, we keep first derivative terms and substitute χ(0)lk in the
∂χlk/∂Rm term. This gives
χjk = D−1jl τ Ilk − τ 2D−1
Further, in many situations one will be concerned with the slow rotation limit,where Ωτ ≪ 1. In this case one needs to keep only terms that are at mostlinear in Ωτ . We do this below; partial results to higher order in Ωτ can befound in Ref. . To linear order, D−1
jl = δjl + 2τ k · Ωǫjlmkm. Substitutingthis into Eq. (10.37), and noting that the Bplm term is already linear in Ω, weget
χjk = τ Ijk + 2τ 2k ·ΩǫjlmkmIlk − τ 2Bjlm∂I
We will work with this expression to evaluate E. The first term in (10.38)contributes to E, even in the case when Ω = 0. This contribution is given by
E(0)i = ǫijk
τ Ijkd3k (10.39)
− ik ·B(vAjk −mAjk) +
12B ·∇(vAjk +mA
+Bj,lmlk − Bk,lvjl − 12km Bm,s
Note that due to the presence of the antisymmetric tensor ǫijk only the anti-symmetric parts of vjk and mjk survive in some of the terms above, and theseare denoted by vAjk and mA
To proceed we need the form of the velocity and magnetic correlation tensors.We adopt the form relevant when these are isotropic and weakly inhomoge-neous, as discussed in detail in Refs. [272,307]. We take
2k2(ki∇j − kj∇i)
2i +k ·∇k2
− (kiǫjlm + kjǫilm)klk2
F (k,R), (10.40)
2k2(ki∇j − kj∇i)
2i +k ·∇k2
− (kiǫjlm + kjǫilm)klk2
Here 4πk2E and 4πk2M are the kinetic and magnetic energy spectra, respec-tively, and 4πk2F and 4πk2N are the corresponding helicity spectra. (Notethat in this section we use the symbol F (k,R) to denote the kinetic helicityspectrum; this should not be confused with the the coordinate space F (r)defined earlier, for the helical part of the velocity correlation function). Theyobey the relations
u2 (R) = 2∫
E(k,R)d3k, ω · u (R) = 2∫
b2 (R) = 2∫
M(k,R)d3k, j · b (R) = 2∫
Note that, so far, we have also made no assumptions about the origin ofthe velocity and magnetic fluctuations, apart from their general form. The
fluctuating velocity could be driven by random forcing and also be respondingto the effects of rotation and/or the nonlinear effects of the Lorentz forces.For example, suppose one were to assume that the turbulence were originallynonhelical, that is E = E(0) and F was originally zero. The helical parts of thevelocity correlations can still be generated due to rotation and stratification. Inthis case the helical part of the velocity correlation will no longer be isotropic,and will reflect the anisotropy induced by both rotation and stratification. Towork out the corresponding modification to vij one has to take into accountthe effect of the Coriolis force in some approximate way. This has been done in, by assuming that the velocity induced by rotation is very small comparedto the original turbulent velocity and using the τ approximation in .Interestingly, it turns out that, to the lowest order in R derivatives, rotationinduces a helical part to vij that can be described simply by adopting ananisotropic F = FΩ. It is shown in Ref.  that, under the τ approximation,this is given by
FΩ(k,R) = −2τ ∗[
(k ·Ω) (k ·∇)−Ω ·∇]
where τ ∗(k) is another correlation time, which could in principle be differentfrom τ(k). We use this for F when we discuss the effects of rotation, althoughone must keep in mind that it is likely to give only a crude and at mostqualitative estimate of the effects of rotation. 6
We substitute the velocity and magnetic correlations given in (10.40) and(10.41) into (10.40) for E (0). Note that the term proportional to vAjk + mA
gives zero contribution, as the antisymmetric parts of the correlations are oddin ki and so integrate to zero, while doing the k-integration. (This continuesto hold even if F = FΩ). Also the terms that have k derivatives, give zerocontribution after integrating by parts, and using ∇ ·B = 0. Further, in termswhich already have Ri derivatives of B, one need not include the Ri derivativecontributions of vjk and mjk. The remaining terms can be written as
E(0)i = ǫijk
k ·B[12(kj∇k − kk∇j)(E −M)− ǫjkmkm(F −N)
+Bj,lPlk(k)M −Bk,lPjl(k)E − 2kj ksBs,lPlk(k)M
6 We also point out that the velocity and magnetic correlations will becomeanisotropic when the mean field becomes strong and begins to influence the tur-bulence; throughout the discussion below we do not explicitly take this feature intoaccount, since the major feed back to the alpha effect due to the current helicity isalready important for weak mean fields. Also the anisotropy induced by rotation isalready important, even when the mean field is weak. For a discussion of the effectsof anisotropy induced by strong mean fields, see for example .
where, as before, commas denote partial differentiation with respect to R, i.e.Bj,l ≡ ∂Bj/∂Rl.
10.2 Isotropic, helical, nonrotating turbulence
In the first instance, suppose we were to assume that the turbulence is drivenby isotropic forcing which is also helical, so that all spectral functions dependonly on |k|. One can then do the angular parts of the k-integral in (10.44),using a relation valid for any F(k) of the form
kikjF(k)d3k = 13δij
E(0)i = ǫijk
(Bj∇k − Bk∇j)E − M
F − N
where we have defined
τ(k)E(k,R) d3k, M =∫
τ(k)F (k,R) d3k, N =∫
τ(k)N(k,R) d3k. (10.47)
For a constant τ(k) = τ0 say, we simply have E = 12τ0u2, M = 1
2τ0b2, F =
12τ0ω · u, N = 1
2τ0j · b. So in this case we have for the turbulent EMF
E(0) = αB − ηtJ + γ ×B, (10.48)
α = −13τ0(ω · u− j · b), ηt =
13τ0u2, γ = −1
6τ0∇(u2 − b2), (10.49)
and µ0 = ρ0 = 1 is still assumed. We see that the α effect has the advertisedcurrent helicity correction, but the turbulent diffusion is unaffected by thesmall scale magnetic fluctuations. There is also a turbulent diamagnetic effectand this is affected by magnetic fluctuations, vanishing for equipartition fields.We should also point out that the fluctuating velocity and magnetic fieldsabove are the actual fields and not that of some fiducial ‘original’ turbulenceas for example implied in . Of course, we have not yet taken account oftheir dynamics in any manner, for example the suppression effects from theLorentz force.
10.3 Turbulence with helicity induced by rotation
Now consider the case when helicity in the turbulence is induced by rotationand stratification. For this case, we adopt a velocity correlation function, whereF is related to E as in (10.43). For mjk, we retain the expression given by(10.41). Note that  have neglected the current helicity terms in theircalculation of the effects of rotation on E. There is no reason to make thisassumption, as even if such helical magnetic correlations were initially zero,they would be generated dynamically, during the operation of the large scaledynamo due to magnetic helicity conservation. So we have to keep this term aswell. We will also only keep terms linear in Ωτ and linear in the Ri derivative.
The first term in (10.38) also gives now an Ω dependent contribution to E ,instead of just the usual isotropic ω · u term. This contribution, can be cal-culated by substituting the anisotropic FΩ from Eq. (10.43) in place of theisotropic F into Eq. (10.44). We have now
E(0) = 1
3τ0j · b B − ηtJ + γ ×B + E
τ k ·BkmFΩd3k
klkmktkn∇n − klkm∇t
=−1615Bi(Ω ·∇)E∗ + 4
15Ω ·B ∇iE
∗ + 415Ωi B ·∇E∗, (10.51)
where we have used the relation valid for any F(k),
klkmktknF(k) d3k = 115(δlmδtn + δltδmn + δlnδmt)
F(k) d3k, (10.52)
d3k τ(k)τ ∗(k) E(0)(k,R)(
= 12τ 20u
the latter equality being valid for a constant τ = τ ∗ = τ0. Note that E (Ω1) inEq. (10.51) with E∗ = 1
(0)2 gives a turbulent emf identical to the α effectderived by Krause  for the case when there is no density stratification (cf.Eq. (6.10) and Eq. (6.11)). It would seem that Krause’s formula has actuallyassumed τ = τ ∗ and also missed the additional contributions to be derivedbelow.
The second and third terms in (10.38) also contribute to E for non-zero Ω.These contributions to E denoted as E (Ω2) and E (Ω3) respectively, are calcu-lated in Appendix E. In computing these terms we also need the followingintegrals
τ 2(k)E(k,R)d3k, M (2) =∫
τ 2(k)kE ′(k,R)d3k, M (3) =∫
τ 2(k)kM ′(k,R)d3k, (10.54)
where primes denote derivatives with respect to k. For constant τ(k) = τ0 wehave
E(2) = 12τ 20u
(0)2, M (2) = 12τ 20 b
together with E(3) = −3E(2) and M (3) = −3M (2).
The net turbulent EMF is obtained by adding all the separate contributions,E i = E
(0)i + E
(Ω2)i + E
(Ω3)i , so
E i = αijBj − ηijJ j + (γ ×B)i + (δ × J)i + κijkBj,k, (10.56)
where the turbulent transport coefficients, for k-independent correlation timesτ = τ ∗ = τ0, are given by,
αij =13τ0δijj · b− 12
δijΩ ·∇(u(0)2 − 13b2)
− 1124(Ωi∇j + Ωj∇i)(u(0)2 + 3
ηij =13τ0δij u(0)2, (10.58)
γ = −16τ0∇
u(0)2 − b2)
− 16τ 20Ω×∇
u(0)2 + b2)
δ = 16Ωτ 20
u(0)2 − b2)
κijk =16τ 20 (Ωjδik + Ωkδij)
u(0)2 + 75b2)
Note that the combination (δ × J)i + κijkBj,k reduces to
E = ... + 13τ 20(
u(0)2 + 15b2)
∇(Ω ·B) + 25τ 20 b
2Ω ·∇B, (10.62)
so for b2 = 0 (and constant Ω) this combination is proportional to ∇(Ω ·B),and hence, if u(0)2 + 1
5b2 is constant, it gives no contribution under the curl
– in agreement with earlier work . For a comparison with Ref.  seeAppendix F.
In general, u(0)2 + 15b2 is not constant, and so this term contributes to the α
effect. Instead of Eq. (10.57) we have then
αij =13τ0j · b− 12
15τ 20 δijΩ ·∇(u(0)2 − 1
+ 15τ 20 (Ωi∇j + Ωj∇i)(u(0)2 + 1
+ 16τ 20 (Ωi∇j − Ωj∇i)(u(0)2 + 1
The last term is antisymmetric and can therefore be included in the secondterm of the expression for γ in Eq. (10.59) which then becomes
γ = ...− 13τ 20Ω×∇
u(0)2 + 35b2)
This term corresponds to a longitudinal pumping term of the form discussedin Section 6.4; see the middle panel of Fig. 6.6. Since u(0)2 increases outward,the longitudinal pumping is in the retrograde direction, which is in agreementwith the simulations .
As mentioned earlier, even if the original turbulence is nonhelical, one cannotassume the magnetic part of the correlation functions to be nonhelical, (asdone in Ref. ), since the magnetic helical parts will automatically be gen-erated during the large scale dynamo operation, due to helicity conservation.So one still has a current helicity contribution to the α effect.
In Ref.  the j · b contribution was neglected, because the small scale fieldwas considered to be completely unaffected by the dynamo. However, such arestriction is not necessary and the approach presented above is valid even ifb2 and j · b are affected (or even produced) by the resulting dynamo action.The velocity term, on the other hand, is not quite as general, which is why wehave to keep the superscript 0 in the term u(0)2.
10.4 Nonlinear helicity fluxes using MTA
As emphasized above, the j · b term gives the most important nonlinear con-tribution to the α effect. In this section we present the general theory for thisterm. One of our aims is also to examine possible fluxes of helicity which arisewhen one allows for weak inhomogeneity in the system. Indeed, Vishniac andCho  derived an interesting flux of helicity, which arises even for nonheli-cal but anisotropic turbulence. We derive this flux using MTA, generalizingtheir original derivation to include also nonlinear effects of the Lorentz force
and helicity in the fluid turbulence . As we see, the Vishniac-Cho flux canalso be thought of as a generalized anisotropic turbulent diffusion. Further,due to nonlinear effects other helicity flux contributions arise which are dueto the anisotropic and antisymmetric part of the magnetic correlations.
Instead of starting with magnetic helicity, let us start with an equation forthe evolution of the small scale current helicity, j · b = ǫijkbi∂jbk, since this isexplicitly gauge invariant. As before we assume that the correlation tensor ofb varies slowly on the system scale R. We then have, in terms of the Fouriercomponents bi,
j(x) · b(x) = ǫijk
bi(k + 12K)bk(−k + 1
i(−kj +12Kj) e
iK·R d3K d3k. (10.65)
Here we have used the definition of correlation functions as given by Eq. (10.1),but evaluated at r = 0. The evolution of j · b is given by
∂tj · b = Ψ1 +Ψ2, (10.66)
Ψ1/2 = ǫijk
(1/2)ik (k,K) eiK·Rd3K d3k, (10.67)
˙bi(k + 1
2K)bk(−k + 1
M(2)ik = bi(k + 1
˙bk(−k + 1
As shown in Appendix G, the final result for Ψ1 and Ψ2 is
where the upper and lower signs apply to Ψ1 and Ψ2, respectively. Also T1
and T2 represent the triple correlations of the small scale u and b fields andthe microscopic diffusion terms that one gets on substituting Eq. (10.8) intoEq. (10.67) respectively (see Appendix G). Adding the Ψ1 and Ψ2 terms onegets
∂tj · b = 2ǫijkǫipqǫqlm
d3k + TC . (10.71)
Here we have defined∫
[T1(k) + T2(k)] d3k = TC . In principle, we could also
apply the MTA to write the nonlinear triple correlation terms and the mi-croscopic diffusion terms as a damping of the current helicity correlation, i.e.TC = −j · b/τc, where τc is some correlation time. But we will not requirethis at present, and so continue to write this term as Tc. Using ǫijkǫipqǫqlm =ǫljkδpm−ǫmjkδpl to simplify the above expression, and integrating the ∂χlk/∂ksterm by parts, yields
∂tj · b = ǫljk
2χlk kj(k ·B)− χlk∇j(ik ·B)
−ikj B ·∇χlk + 2ikjχpk∇pBl
d3k + TC . (10.72)
Note that only the antisymmetric parts of χlk contribute in the first threeterms above due to the presence of ǫljk. We can now use Eq. (10.72) combinedwith our results for χlk derived in the previous subsections to calculate thecurrent helicity evolution. We concentrate below on nonrotating turbulence.For such turbulence we have from Eq. (10.38), χlk = τ Ilk, where Ilk is givenby Eq. (10.30). We use this in what follows. Let us denote the four terms inEq. (10.72) by A1, A2, A3 and A4, respectively, with
A1 = 2ǫljk
χlkkj(k ·B) d3k, A2 = −ǫljk
χlk∇j(ik ·B) d3k, (10.73)
A3 = −ǫljk
ikjB ·∇χlk d3k, A4 = 2iǫljk
kjχpk∇pBl d3k. (10.74)
The first term, A1, is given by
A1 = 2τǫljk
− ik ·B(vAlk −mAlk) +
12B ·∇(vAlk +mA
−Bk,svls − 12km Bm,s
Due to the presence of ǫljk, only the antisymmetric parts of the tensors vlkand mlk survive, and these are denoted by vAlk and mA
lk respectively. Also notethat the last term above vanishes because it involves the product ǫljkklkj = 0.
All the other terms of Eq. (10.72) already have one R derivative, and soone only needs to retain the term in χlk = τ Ilk which does not contain R
derivatives. These terms are given by
A2 = −τǫljk
−ik ·B(vAlk −mAlk)]
A3 = −∇ ·(
−ik ·B(vAlk −mAlk)])
A4 = 2τǫljk
−ik ·B(vpk −mpk)]
where we have used ∇ ·B = 0 to write A3 as a total divergence. We now turnto specific cases.
10.4.1 Isotropic, helical, nonrotating turbulence
Let us first reconsider the simple case of isotropic, helical, nonrotating, andweakly inhomogeneous turbulence. For such turbulence we can again useEq. (10.40) and Eq. (10.41) for velocity and magnetic correlations. The kderivative terms in A1 in this case involve an integral over an odd number ofki and so vanishes. Only terms which involve integration over an even num-ber of ki survive. Also, in terms which already involve one Ri derivative, oneneeds to keep only the homogeneous terms in Eq. (10.40) and Eq. (10.41).With these simplifications we have from Eq. (10.75)
A1 = 2ǫljk
− ǫlknknk ·B(F −N)
+Bl,sPskM − Bk,sPlsE]
Carrying out the angular integrals over the unit vectors ki
k2(F −N) d3k + 23B · J
k2(M + E) d3k. (10.80)
In the case of isotropic turbulence, the second and third terms, A2 and A3, arezero because, to leading order in R derivatives, the integrands determining A2
and A3 have an odd number (3) of ki’s. The fourth term given by Eq. (10.78)is A4 = 2τǫljkBs∇pBl
k2kj ks[E −M ] d3k, or
A4 =23J ·Bτ
k2[E −M ] d3k. (10.81)
Adding all the contributions, A1 + A2 + A3 + A4, we get for the isotropic,helical, weakly inhomogeneous turbulence,
∂tj · b = 4
k2(F −N) d3k + 43J ·Bτ
k2E d3k + TC . (10.82)
We see that there is a nonlinear correction due to the small scale helical part
of the magnetic correlation to the term ∝ B2. But the nonlinear correction to
the term ∝ J ·B has canceled out, just as for turbulent diffusion. Recall alsothat for isotropic random fields, the spectra Hk of magnetic helicity a · b andCk of current helicity j · b are related by Hk = k−2Ck. So the first two termsof the current helicity evolution equation Eq. (10.82) give exactly the sourceterm −2E ·B for the magnetic helicity evolution. Also for this isotropic caseone sees that there is no flux which explicitly depends on the mean magneticfield.
10.4.2 Anisotropic turbulence
Let us now consider anisotropic turbulence. In the first term A1 in Eq. (10.72),given by Eq. (10.75), one cannot now assume the isotropic form for the velocityand magnetic correlations. But again, due to the presence of ǫljk, only theantisymmetric parts of the tensors vlk and mlk survive. Also the last term inEq. (10.75) vanishes because it involves the product ǫljkklkj = 0. One canfurther simplify the term involving k derivatives by integrating it by parts.Straightforward algebra, and a judicious combination of the terms then gives
2χlk kj(k ·B) d3k = τǫljk
kjkp(Bl,smsk − Bk,svls) d3k +BpBm,j
All the other terms A2, A3 and A4 cannot be further simplified. They areexplicitly given by
A2 = −τǫljkBpBs,j
lk) d3k, (10.84)
A3 = −∇s
A4 = 2τǫljkBpBl,s
kjkp(vsk −msk) d3k. (10.86)
Adding all the contributions, A1 + A2 + A3 + A4, we get
∂tj · b=2ǫjlkτ
3k − BpBs,j
d3k + TC . (10.87)
Here vSls =12(vls+vsl) is the symmetric part of the velocity correlation function.
Let us discuss the various effects contained in the above equation for currenthelicity evolution. The first term in Eq. (10.87) represents the anisotropicversion of helicity generation due to the full nonlinear alpha effect. In fact,for isotropic turbulence it exactly will match the first term in Eq. (10.82).The second term in Eq. (10.87) gives the effects on helicity evolution due toa generalized anisotropic turbulent diffusion. This is the term which containsthe Vishniac-Cho flux. To see this, rewrite this term as
∂j · b∂t
=∇ ·FV + 4τBkǫkljBp,s
Here the first term is in the form of the Vishniac-Cho flux term,
FVs = 4τǫjlkBpBk
3k = 4τ(B · ω)(B ·∇)us, (10.89)
while the second is the effect on helicity due to ‘anisotropic turbulent diffusion’.(We have not included the large scale derivative of vls to the leading order.) Ifwe recall that Hk = k−2Ck for homogeneous turbulence, then Eq. (10.89) forF
V leads to the magnetic helicity flux given in Eq. (18) and (20) of Vishniacand Cho .
This split into helicity flux and anisotropic diffusion may seem arbitrary; somesupport for its usefulness comes from the fact that, for isotropic turbulence,F
V vanishes, while the second term exactly matches with the correspondinghelicity generation due to turbulent diffusion, i.e. the J ·B term in Eq. (10.82).Of course, we could have just retained the non-split expression in Eq. (10.88),which can then be looked at as an effect of anisotropic turbulent diffusion onhelicity evolution. Also interestingly, there is no nonlinear correction to thisterm from mS
ls, just like there is no nonlinear correction to turbulent diffusionin lowest order!
Finally, Eq. (10.87) also contains terms (the last two) involving only the anti-symmetric parts of the magnetic correlations. These terms vanish for isotropicturbulence, but contribute to helicity evolution for nonisotropic turbulence.
The last term gives a purely magnetic contribution to the helicity flux, butone which depends only on the antisymmetric part of mlk. Note that suchmagnetic correlations, even if initially small, may spontaneously develop dueto the kinetic alpha effect or anisotropic turbulent diffusion and may againprovide a helicity flux. More work is needed to understand this last flux termbetter. 7
In summary, MTA allows a conceptually simple and mathematically straight-forward, although technically somewhat involved analytic treatment of themean field transport coefficients. The calculation of αij, ηij , γ, δ, and κijk
agrees in all important aspects with earlier treatments . The by far mostimportant new aspect is the inclusion of the j · b feedback term. This must becoupled to a dynamical calculation of j · b, which has hitherto been ignoredin the vast majority of dynamo models. The nonlinear feedback in the case ofhomogeneous dynamos in closed domains is now well understood (Section 9.4).However, in the case of open domains helicity fluxes need to be calculated.Here, only partial results are available. Agreement with earlier calculations iscurrently restricted to a particular term found earlier by Vishniac and Cho. Clearly, more work by the various groups is required before we cangenerate a coherent picture.
11 Discussion of dynamos in stars and galaxies
The applicability of the full set of mean field transport coefficients to modelsof stars and galaxies is limited by various restrictions: analytic approachesallow only weak anisotropies and suffer uncertainties by using approximationssuch as FOSA or MTA, while the results of numerical simulations to calculatetransport coefficients are difficult to parameterize and apply only to low mag-netic Reynolds numbers. This, combined with the general lack of confidencedue to partially conflicting results, has resulted in a rather fragmentary usageof various possible terms. Thus, only partial results can be reported here.
It is clear that all the interesting effects are controlled by the degree ofanisotropy in the turbulence. One of the important ones is rotation, without
7 We note that we have not got in our analysis any helicity flux term which isexactly like that advocated by Kleeorin and coworkers [327–329]. In the originaltreatment involving path integral methods, a flux proportional to the anisotropicpart of the alpha tensor and the anisotropic part of the magnetic helicity tensor werederived . This flux may arise only if we also adopt the stochastic methods thatwe used in Appendix A, but now generalized to an inhomogeneous system. However,the subsequent use of this flux in [327–329] involved adopting a phenomenologicalterm proportional to the mean field; such a term should have arisen automaticallyin our treatment using MTA.
Table 11.1Summary of angular velocities, estimated turnover times, and the resulting inverseRossby number for various astrophysical bodies.
Ω [s−1] τ Ro−1 = 2Ωτ
Solar convection zone (lower part) 3× 10−6 106 s 6
T Tauri stars 2× 10−5 106 s 40
Protostellar discs 2× 10−7 109 s 400
Proto-neutron stars 2× 103 10−3 s 2
Discs around neutron stars 10−2 104 s 200
Galaxy 10−15 107 yr 0.6
Jupiter 2× 10−4 106 s 200
which there would be no α effect. We begin with a discussion of the rela-tive importance of rotational effects in various astrophysical bodies and turnthen to a somewhat subjective assessment of what is the current consensus inexplaining the nature of magnetic fields in various bodies.
11.1 General considerations
How rapid does the rotation have to be in order that the anisotropy effecton the turbulence becomes important? A suitable nondimensional measureof Ω is the inverse Rossby number, Ro−1 = 2Ωτ , where τ corresponds tothe correlation time if FOSA is used, or to the relaxation time if MTA isused. In practice, τ is often approximated by the turnover time, τturnover. InTable 11.1 we give some estimates for Ro−1 for various astrophysical bodies.For the sun, Ro−1 is around 5 near the bottom of the convection zone (but goesto zero near the surface layers). In galaxies, but also in proto-neutron stars,Ro−1 is smaller (around unity). Accretion discs tend to have large values ofRo−1 (around 100). This is directly a consequence of the fact that here theturbulence is weak, as quantified by a small value of the Shakura-Sunyaevviscosity parameter (αSS ≈ 0.01 [164,227,342]). Planets also tend to have largevalues of Ro−1, because here the turbulence is driven by a weak convectiveflux, so the turnover time is long compared with the rotation period.
In the following we first discuss some issues connected with understanding andmodeling the solar dynamo, and then turn to dynamos in stars and planets,and their relation to laboratory liquid metal experiments. Finally, we turn todynamos in accretion discs, galaxies and galaxy clusters.
11.2 The solar dynamo problem
There are several serious shortcomings in our understanding of the sun’s mag-netic field. Although there have been many theoretical attempts, there is asyet no solution to the solar dynamo problem. With one exception , all themean field models presented so far have ignored the magnetic helicity issuealtogether, so it is not clear what significance such models still have. Neglect-ing magnetic helicity can only be considered a reasonable approximation if|j · b| ≪ |ω · u|, which would is probably not valid in the nonlinear regime.We have seen earlier the crucial constraints on the nonlinear dynamo imposedby helicity conservation, which have not yet been properly incorporated intomany of the solar dynamo models.
It should ultimately be possible to simulate the entire sun with its three-dimensional turbulence, the tachocline (see Section 11.2.6), and the resultingdifferential rotation. Several attempts have been made starting with the earlywork of Gilman  and Glatzmaier , and new high resolution calcula-tions are currently underway [346,347]. All these simulations have been suc-cessful in generating both small scale and large scale magnetic fields, althoughthey have not yet been convincingly demonstrated cyclic behavior. Neverthe-less, there is a tendency for the toroidal field belts to propagate away fromthe equator, rather than toward the equator as in the sun. It is tempting toassociate this with the sign of the α effect that these simulations generate(even though an explicit α effect is of course not invoked).
It should be emphasized that in none of the convection simulations cur-rently available (both in spherical shells and in cartesian boxes) the magneticReynolds numbers are large enough to see the effect of magnetic helicity con-servation (the dynamo growth rate should be much larger than the ohmicdecay rate ). We are only now at the threshold where magnetic helicityeffects begin to control the magnetic field evolution.
11.2.1 Magnetic helicity and cycle period
There is the worry that in large scale simulations magnetic helicity conser-vation could either prevent cyclic behavior or it might significantly prolongthe cycle period [48,137]. On the other hand, if in the sun the importance ofmagnetic helicity conservation is only marginally important (e.g., if magnetichelicity fluxes dominate over resistive losses), one could imagine a prolonga-tion of the cycle period by a factor of about ten. Such a factor would be neededto improve the results of conventional models, which invariably produce tooshort cycle periods if magnetic helicity conservation is ignored . The an-ticipated prolongation of the cycle period might therefore be regarded as a
step in the right direction.
The strength of the magnetic and current helicity fluxes, and hence the degreeof magnetic helicity conservation, is possibly self-regulating via losses throughthe outer surface, for example such that the magnetic helicity losses are justas strong as to affect the time scales only slightly. Again, this is at presentquite speculative, and there is no simulation that is able to demonstrate thatmagnetic helicity losses can alleviate the magnetic helicity problem that leadsto the slow saturation behavior and that might lead to prolonged cycle pe-riods. There are only partial results  suggesting that α might not bequenched catastrophically if there is shear combined with open boundaries. Itseems plausible that a significant portion of the magnetic helicity losses fromthe sun occur through coronal mass ejection (CMEs) . So far, however, noturbulence simulation has realistically been able to allow for such magnetichelicity losses, nor have simulations to our knowledge been able to producephenomena that can even remotely be associated with CMEs. It seems there-fore important to continue studying the large scale dynamo problem in morerealistic settings where CMEs are possible.
11.2.2 Does the sun eject bi-helical fields?
On theoretical grounds, if most of the helicity of the solar magnetic field is pro-duced by the α effect, one would expect the solar magnetic field to be bi-helical[47,48,349], i.e. the magnetic field has positive and negative magnetic helicityat different scales, but hardly any net magnetic helicity. On the other hand,if most of the sun’s helicity is caused by differential rotation, the field mightequally well not be bi-helical. (We recall that differential rotation causes seg-regation of magnetic helicity in physical space, i.e. between north and south,while the α effect causes a segregation of helicity in wavenumber space; seeSection 3.4.) So far, however, the solar magnetic field has not explicitly beenseen to be bi-helical. Indirectly, however, a bi-helical nature of the solar mag-netic field is indicated by the fact that bipolar regions are tilted according toJoy’s law [36,246] (see also Section 6.1), suggesting the presence of positivemagnetic helicity in addition to the negative magnetic helicity indicated bythe magnetic twist found in active regions.
What is missing is a quantitative assessment of the relative magnitudes of largeand small scale magnetic helicities. To get an idea about the sign and possiblythe magnitude of the relative magnetic helicity of the longitudinally averagedfield (denoted here by overbars), one should ideally calculate 2
AφBφ dV ,which is the gauge-invariant magnetic helicity of Berger and Field for axisym-metric fields in a sphere (see Section 9.6). As a first step in that direction, onecan determine Aφ at the solar surface from the observed radial component ofthe magnetic field, Br ; see Fig. 11.1. Empirically, we know that Br is ap-
Fig. 11.1. Mean dipole symmetry radial field, Br, reconstructed from the coefficientsof Stenflo [32,33] (upper panel). The corresponding toroidal component of the meanvector potential, Aφ, derived from Br (lower panel). Solid contours denote posi-tive values, dotted contours negative values. The solar cycle maximum of 1982 ishighlighted, as is the latitude of 10 where Br was then strongest. The signs of var-ious quantities at or around this epoch are also shown (see text for more details).Adapted from Ref. .
proximately in antiphase with Bφ. Comparison of the two panels of Fig. 11.1suggests that AφBφ was negative just before solar maximum (t < 1982 yr)and positive just after (t > 1982 yr). Thus, the present attempt to assess thesign of the magnetic helicity of the large scale field does not support nor ex-clude the possibility that the large scale field has positive magnetic helicity,as suggested by the tilt of bipolar regions.
11.2.3 Migration of activity belts and butterfly diagram
There are still a number of other rather long standing problems. Most impor-tant perhaps is the sense of migration of the magnetic activity belts on eitherside of the equator. This migration is traditionally believed to be associatedwith the phase velocity of the dynamo wave; see Section 6.5.2. However, thiswould be in contradiction with α > 0 (as expected in the northern hemisphere)
and ∂Ω/∂r > 0 (known from helioseismology ).
One of the currently favored models is the flux transport model where merid-ional circulation is assumed to be oriented such that, at the bottom of theconvection zone, meridional circulation would advect the dynamo wave equa-torward [351–357]. This proposal came originally quite as a surprise, becausemeridional circulation was normally always found to make dynamos nonoscilla-tory well before the anticipated advection effect could take place [253,286,290].
Whether or not the flux transport proposal is viable cannot be decided atpresent, because this model (and essentially all other solar dynamo modelsavailable so far) lack consistency with respect to the conservation of total(small and large scale) magnetic helicity. In addition, there are several otherproblems. With realistic profiles of differential rotation it is difficult to producesatisfactory butterfly diagrams . It is also difficult to produce fields ofdipolar rather ran quadrupolar parity .
11.2.4 The phase relation between poloidal and toroidal fields
Even if the problem of the migration direction of the toroidal activity beltswas solved (for example by meridional circulation; see Section 11.2.3), thereremains the problem of the phase relation between poloidal and toroidal field.For example, when Bφ < 0 (as seen from the orientation of bipolar regions;see Fig. 6.1 for Cycle 21 during the year 1982) we have Br > 0 (as seen fromsynoptic magnetograms, see Fig. 11.1 at t = 1982 yr), and vice versa [284,285].The basic problem is actually connected with the sense of the radial differentialrotation, i.e. the fact that ∂Ω/∂r > 0 in low solar latitudes. This always turnsa positive Br into a positive Bφ, and vice versa, regardless of the sign of the αeffect. No convincing solution to this problem has yet been offered, althoughit has been suggested  that the problem might disappear in more realisticsettings.
11.2.5 The flux storage problem
Since the early eighties it was realized that flux tubes with a strength similarto or in excess of that in sunspots would float up to the surface in a timeshort (∼ 50 days) compared with the dynamo time scale (∼ 10 yr). Therefore,magnetic buoyancy acts as a very efficient sink term of mean toroidal field.This led to the suggestion  that the dynamo may operate in the lowerpart of the convection zone or just below it. Model calculations [360,361] haveshown however that the dynamo regime becomes so thin that the dynamobecomes too inefficient  and that it tends to produce too many toroidalflux belts in each hemisphere .
An alternative viewpoint is that the dynamo might still work in the convectionzone proper, but that turbulent pumping brings the field continuously to thebottom of the convection zone [277–279,294], from where strong toroidal fluxbelts can rise to the surface and form bipolar regions (Fig. 6.1). The end resultis similar in the sense that in both cases bipolar regions would be caused byflux tubes anchored mostly in the overshoot layer. It should also be emphasizedthat magnetic buoyancy might not only act as a sink in a destructive sense,but it can also contribute to driving an α effect [363–366]
11.2.6 Significance of the tachocline
The tachocline is the layer where the latitudinal differential rotation turns intorigid rotation . This layer is now known to be quite sharp and to coincidewith the bottom of the convection zone . The reason the latitudinal differ-ential rotation does not propagate with time deeper into the radiative interioris probably connected with the presence of a weak primordial magnetic field[369–371].
The tachocline is likely to be the place where the vertical shear gradient playsan important role in amplifying the toroidal magnetic field. This is perhapsnot so much because the shear gradient is strongest at and just below thetachocline, but because the turbulent magnetic diffusivity is decreased, andbecause the magnetic field is pumped into this layer from above.
In and below the tachocline the total relaxation time is governed more stronglyby the thermal time scale. Thus, if the system is slightly thrown out of balance,it will take a thermal time scale to reestablish a new equilibrium. Furthermore,as long as a new equilibrium state has not yet been reached, fairly strong am-plitudes may develop, making the effective relaxation time even longer. Whilethis will not be a problem for the sun, it may be a problem for simulations.In practice, this means that one has to be more careful setting up initialconditions and, perhaps most importantly, one should deliberately chose pa-rameters whereby the thermal and acoustic time scales are not more disparatethan what can be handled in a simulation. Nevertheless, keeping at least somerepresentation of the tachocline, rather than neglecting it altogether, may becrucial.
11.3 Stellar dynamos
Looking at stars other than the sun allows us to test the dependence of thedynamo on radius, on the thickness of the convection zone and, in particular,on the angular velocity of the star. In this section we summarize a few suchdependencies and discuss whether they can be reproduced by dynamo models.
Fig. 11.2. Magnetic field lines inside and outside a model of a fully convectiveM-dwarf. Color coded is the radial field component on the lower hemisphere. Notethat field lines pass right through the inner parts of the star. Courtesy of W. Dobler.
11.3.1 Fully convective stars
Toward the less massive stars along the main sequence the thickness of the con-vection zone increases relative to the stellar radius (although in absolute unitsthe thickness remains around 200Mm) until the star becomes fully convective.Such stars would lack a lower overshoot layer which was often thought to bean important prerequisite of solar-like dynamos. On the other hand, turbulentpumping would in any case tend to concentrate the field toward the center ofthe star, and since the gravitational acceleration vanishes at the center, themagnetic field can probably still be stored for some time. Figure 11.2 shows avisualization of magnetic field lines of a self-consistent turbulence simulationof a fully convective dynamo. In this simulation the dynamo is weakly super-critical. Following similar approaches by other groups [372–375], the star isembedded in a sphere, which avoids computational difficulties associated withcoordinate singularities in explicit finite difference methods using sphericalcoordinates. The star’s radius is 27% of the solar radius and the mass is 21%of the mass of the sun.
Another class of fully convective stars are the T Tauri stars, i.e. stars thathave not yet settled on the main sequence. These stars are generally knownto spin very rapidly, so magnetic braking via magnetic field lines anchored inthe surrounding protoplanetary accretion disc is usually invoked to explain
the much slower rotation rate of evolved stars such as the sun .
11.3.2 Models of stellar cycles
Stars exhibiting cyclic behavior can cover a broad parameter range that allowsus to test whether the dependence of cycle properties on stellar parametersagrees with what is expected from dynamo models. For orientation one canconsider the marginally excited solution from linear theory and it is indeedcommon to look at plane wave solutions [376,377].
Several results have emerged from this type of analysis. Comparing modelswith different model nonlinearities (α quenching, feedback on the differentialrotation, and magnetic buoyancy) it turns out  that only models withmagnetic buoyancy as the dominant nonlinearity are able to produce a posi-
tive exponent σ in the relation ωcyc/Ω = c1Ro−σ; see Eq. (2.1). We emphasize
that it is not sufficient that ωcyc increases with Ω. For example, an increaseproportional to Ω0.5, which is found for classical dynamo models , is in-sufficient.
Meanwhile, there has been some uncertainty regarding the correct cycle fre-quency dependence on Ro−1. We recall that in Fig. 2.7 we found two separatebranches, both of which with positive slope σ. On the other hand, a negativeslope has been found when plotting ωcyc/Ω versus the dimensional form of Ω(rather than the nondimensional Ro−1 = 2Ωτturnover) . One may arguethat the nondimensional form is to be preferred, because it yields a smallerscatter .
While ordinarily quenched dynamos tend to produce negative exponents σ, ‘anti-quenched’ dynamos can produce the observed exponents when thedynamo alpha and the dissipation rate, τ−1 = ηk2
z , increase with field strength,e.g. like
α = α0|B/Beq|n, τ−1 = τ−10 |B/Beq|m, m, n > 0. (11.1)
The motivation behind anti-quenching lies in the realization that the motionsdriving turbulent transport coefficients can be caused by magnetic instabili-ties. Examples of such magnetic instabilities include the magnetic buoyancyinstability [364–366], the magneto-rotational instability [62,164], and globalinstabilities of the tachocline differential rotation [356,380].
Assuming that the dynamo operates in its fundamental mode (even in thenonlinear regime) one can apply the relations (6.35) and (6.36) for a fixedvalue kz. This leads to two complex algebraic relations 
τ−10 |B/Beq|m = |1
′|1/2 |B/Beq|n/2, (11.2)
ωcyc = τ−10 |B/Beq|m. (11.3)
Dividing both equations by Ω and making use of the relation 〈R′HK〉 ∝ |B/Beq|κ
(see Section 2.1.3), we have
2Ro 〈R′HK〉m/κ =
∣∣∣∣ = 2Ro 〈R′
For slow rotation α is proportional to Ω, while for rapid rotation it is inde-pendent Ω , and for Ω′ the possibilities range from being independentof Ω  to being proportional to Ω0.7 . To account for the differentpossibilities, we make the more general assumption
∣∣∣∣∣∼ Ro−q, (11.6)
where q is expected to be between 0 and −2. We thus have
Ro 〈R′HK〉m/κ ∼ Ro−q/2〈R′
∣∣∣∣ ∼ Ro 〈R′
Using the definitions (2.1), i.e. 〈R′HK〉 ∼ Ro−µ and |ωcyc/Ω| ∼ Ro−σ, together
with σ = µν, we have
Ro1−mµ/κ ∼ Ro−q/2−nµ/2κ, (11.9)
Ro−µν ∼ Ro1−mµ/κ, (11.10)
so µν = mµ/κ − 1 = q/2 + nµ/2κ, which yields the explicit results for thedynamo exponents m and n,
m = κ(ν + 1/µ), n = κ(2ν − q/µ). (11.11)
In Table 11.2 we summarize the results obtained from the subset of starsthat show cycles . The different branches might be associated with theoccurrence of different magnetic instabilities in different parameter regimes,but this is at present just speculation.
Table 11.2Summary of observable parameters characterizing stellar cycle properties (σ, ν, andµ) and the corresponding model parameters m and n, introduced in Eq. (11.1); seeRef. .
Stars σ ν µ m n
inactive 0.46 0.85 0.99 0.9 0.9-1.7
active 0.48 0.72 0.99 0.8 0.8-1.6
Although it is well recognized that single mode approximations are not suffi-cient to solve the dynamo equations as stated [see, e.g., Eq. (9.15)], the one-mode equations may still be closer to physical reality, because there is nowevidence that the spectral sensitivity of the turbulent transport coefficients ishighest at small wavenumbers . In other words, the multiplication αB inEq. (6.3) should be replaced by a convolution α B, where α would now bean integral kernel. In wavenumber space, this would correspond to a multipli-cation with a k-dependent α such that α(k) is largest for small values of k.One may hope that future simulations will shed more light on this possibility.
11.3.3 Rapidly rotating stars or planets
An important outcome of mean field calculations in the presence of rapidrotation is the prediction that the α effect becomes highly anisotropic andtakes asymptotically the form 
αij = α0
δij − ΩiΩj/Ω2)
which implies that, if the angular velocity points in the z direction, for ex-ample, i.e. Ω = Ωz, the vertical component of α vanishes, i.e. αzz → 0 forRo−1 ≡ 2Ωτ ≫ 1. For calculations of axisymmetric and nonaxisymmetricmodels using large but finite values of Ro−1 see Refs. [294,384].
In Eq. (11.12) we stated the asymptotic form of the α tensor in the limitof rapid rotation. Such a highly anisotropic α tensor is known to lead tostrong nonaxisymmetric magnetic field configurations [182,385]. Later, suchan α effect was applied to modeling the magnetic fields of the outer giantplanets that are known to have very large values of Ro−1. Simulations usingparameters relevant to the outer giant planets of our solar system show thatthese bodies may have a magnetic field that corresponds to a dipole lying inthe equatorial plane [384,386].
Fig. 11.3. Convection columns in a rapidly rotating spherical shell. Courtesy of A.Yoshizawa .
The biggest enemy of nonaxisymmetric fields is always differential rotation,because the associated wind-up of the magnetic field brings oppositely orientedfield lines close together, which in turn leads to enhanced turbulent decay .This is quite different to the case of an axisymmetric field, where the wind-up brings equally oriented field lines together, which leads to magnetic fieldenhancements; i.e. the Ω effect.
As the value of Ro−1 increases, the α tensor becomes not only highly anisotropic,but the magnitudes of all components decreases. This is a common phe-nomenon known as ‘rotational quenching’ that affects virtually all turbulenttransport coefficients. An important turbulent transport coefficient that wewill not say much about here is the so-called Λ effect (modeling the toroidalReynolds stress), is responsible for driving differential rotation in stars includ-ing the sun. Very rapidly rotating stars are therefore expected to have verylittle relative differential rotation, which is indeed observed . This meansthat the αΩ dynamo will stop working and one would therefore expect ananisotropic α2 dynamo mechanism to operate in rapidly rotating bodies withouter convection zones. This means that such stars should generate a pre-dominantly nonaxisymmetric field. There are indeed numerous observationalindications for this [388,389]. It should be noted, however, that already thelarge scale flow that is generated by the magnetic field and the Reynolds stress(i.e. the Λ effect) tend to make the field nonaxisymmetric . In addition,in rapidly rotating stars all motions tend to be mostly in cylindrical sur-faces (Taylor-Proudman theorem). This effect is believed to cause starspotsto emerge mostly at high latitudes in rapidly rotating stars [391,392]. Fur-thermore, the convective motions tend to be column-like . This led tothe proposal of the Karlsruhe dynamo experiment where the flow is similarlycolumn-like.
Fig. 11.4. The dynamo module (after ). The signs + and – indicate that thefluid moves in the positive or negative z–direction, respectively, in a given spingenerator.
11.3.4 Connection with the Karlsruhe dynamo experiment
In the Karlsruhe dynamo experiment, liquid sodium is pumped upward anddownward in alternating channels . Each of these channels consist of aninner and an outer pipe, and the walls of the other one are arranged such thatthe flow follows a swirling pattern; see Fig. 11.4. The swirl in all columns issuch that the associated kinetic helicity has the same sign everywhere. In fact,locally such a flow is strongly reminiscent of the Roberts flow (Section 4.2.2)that is known to generate a Beltrami field in the plane perpendicular to thedirection of the pipes, i.e. (cos kz, sin kz, 0) if z is the direction of the pipes andthe phase shift in z has been ignored; see Section 8.4. On a global scale sucha field corresponds to a dipole lying in the xy plane, and perpendicular to thez direction; see Fig. 11.4. Applied to the earth, it would therefore correspondto a nonaxisymmetric field and would not really describe the magnetic fieldof the earth, although it might be suitable for explaining the magnetic fieldsof the other giant planets Uranus and Neptune  that are indeed highlynonaxisymmetric .
11.3.5 Chaos and intermittency
The search for more complicated temporal and spatio-temporal patterns hasalways been popular in mean field dynamo research. In spatially resolved mod-els (as opposed to one-mode approximations ) it was for a long time dif-ficult to find spatially well resolved solutions that showed chaotic or even justquasiperiodic behavior. In fact, it was thought to be essential to invoke an ad-
ditional explicit time dependence of the form of dynamical quenching [10–12].More recently it turned out that quasiperiodic and chaotic solutions to themean field dynamo equations are possible when the dynamo number is raisedsufficiently [288,397]. Intermittent behavior has also been found when the feed-back on the mean flow is taken into account [398,399]. For moderately largedynamo numbers, such solutions require however small turbulent magneticPrandtl numbers, which is an assumption that is not confirmed by simula-tions . Nevertheless, intermittent solutions of this or some other kind arethought to be relevant in connection with understanding the intermittency ofthe solar cycle and the occurrence of grand minima [400,401].
The question is of course how much one can trust such detailed predictions ofmean field theory when we are still struggling to confirm the validity of meanfield theory in much simpler systems. One may hope that in the not too distantfuture it will be possible to compare mean field models with simulations insomewhat more extreme parameter regimes where quasi-periodic and chaoticbehaviors are expected to occur.
11.3.6 Dynamos in proto-neutron stars
The ∼ 1013G magnetic field of neutron stars is traditionally thought to be theresult of compressing the magnetic field of its progenitor star. The difficultywith this explanation is that in the early phase of a neutron star (proto-neutronstar) the neutrino luminosity was so immense and the neutrino opacity highenough that the young neutron star must have been convectively unstable.Although this phase lasts for only ∼ 20 seconds, this still corresponds to some104 turnover times, because gravity is so strong that the turnover time isonly of the order of milliseconds. This would be long enough to destroy theprimordial magnetic field and to regenerate it again by dynamo action .
If the field was initially generated by an α effect, and if the associated smallscale magnetic helicity has been able to dissipate or escape through the outerboundaries, the field must have attained some degree of magnetic helicity.Once the turbulence has died out, such a helical field decays much more slowlythan nonhelical fields , and it would have attained the maximum possiblescale available in a sphere. Such a field may well be that of an aligned orinclined dipole, as observed. There are also some interesting parallels betweendynamo action in decaying turbulence in neutron stars and that in the plannedtime-dependent dynamo experiment in Perm based on decaying turbulence inliquid sodium . In the Perm experiment, a rapidly spinning torus withliquid sodium will suddenly be braked, which leads to swirling turbulence dueto suitable diverters on the wall of the torus.
11.4 Accretion disc dynamos
When dynamo theory was applied to accretion discs, it seemed at first justlike one more example in the larger family of astrophysical bodies that hostdynamos [405–408]. Later, with the discovery of the magneto-rotational in-stability [61,62], it became clear that magnetic fields are crucial for main-taining turbulence in accretion discs [409,410], and that this can constitutea self-excited process whereby the magnetic field necessary for the magneto-rotational instability is regenerated by dynamo action.
Simulations in a local (pseudo-periodic shearing box) geometry have shownthat this self-excited system can act both as a small scale dynamo if there isno vertical density stratification [227,342] and as a large scale dynamo if thereis stratification [164,411].
From a turbulence point of view it is interesting to note that the flow is highlyanisotropic with respect to the toroidal direction. This is true even down to thesmallest scale in the simulations. From a mean field dynamo point of view thissystem (with stratification included) is interesting because it is an examplesof a simulation of a dynamo where the turbulence occurs naturally and is notdriven by an artificial body force. This simulation is also an example wherean α effect could be determined .
11.4.1 Dynamo waves in shearing sheet simulations
When the vertical field boundary condition is used, i.e. Bx = By = 0 onz = ztop and zbot, the horizontal flux through the box is no longer conserved,and hence the horizontal components of the horizontally averaged field may bedifferent from zero, i.e. Bx 6= 0 6= By, even though they may be zero initially.This is exactly what happened in the shearing box dynamo simulation whenthese boundary conditions where used. In Fig. 11.5 we show the space-timediagram (or butterfly diagram in solar physics) of the mean magnetic fieldof such a simulation [383,412]. In this particular simulation the symmetry ofthe magnetic field has been restricted to even parity about z = 0, so thecomputation has been carried out in the upper disc plane, 0 < z < Lz, whereLz = 2H is the vertical extent of the box, H is the gaussian scale height ofthe hydrostatic equilibrium density, and z = 0 corresponds to the equatorialplane.
It is interesting to note that the spatio-temporal behavior obtained from thethree-dimensional simulations resembles in many ways what has been obtainedusing mean field models; see Section 6.5.2. Note, however, that in Fig. 11.5dynamo waves propagate in the positive z direction, i.e. c > 0 in Eq. (6.38).This requires α < 0, which is indeed consistent with the sign of α obtained
Fig. 11.5. Horizontally averaged radial and toroidal magnetic fields, Bx andBy respectively, as a function of time and height, as obtained from the fullythree-dimensional simulation of a local model of accretion disc turbulence. Timeis given in units of rotational periods, Trot. (No smoothing in z or t is applied.)Dotted contours denote negative values. Adapted from Ref. .
earlier by means of correlating Ey with By [164,412], but it is opposite towhat is expected in the northern hemisphere (upper disc plane) from cyclonicevents.
Mean-field model calculations confirm that the space-time diagram obtainedfrom the simulations (Fig. 11.5) can be confirmed with a negative α effectof the magnitude found by correlating Ey with By and a turbulent magneticdiffusivity comparable to the turbulent kinematic viscosity obtained by mea-suring the total (Reynolds and Maxwell) stress . The directly determinedα effect is however so noisy, that it is hard to imagine that it can produce amean field that is as systematic as that in the simulation (Fig. 11.5). How-ever, model calculations show that, even when the noise level of α exceeds theaverage value by a factor of 10, the resulting mean field is still fairly coherent
Fig. 11.6. Resulting Bx and By from a mean field calculation with negative αeffect (α0 = −0.001ΩH), together with a 10 times stronger noisy components(αN = 0.01ΩH), and a turbulent magnetic diffusivity (η0 = 0.005ΩH2), producinga cycle period of about 30 orbits. Adapted from Ref. .
(Fig. 11.6) and in fact similar to the field obtained from the simulations .
A negative sign of the α effect in accretion discs may arise because of twoimportant circumstances. First, the vertical velocity fluctuation is governedby magnetic buoyancy and second, shear is important. Following a simpleargument of Ref. , the α effect is dominated by the contribution from themomentum equation and the toroidal magnetic field, By,
∼ uzbx ∼(
By ≡ αyyBy, (11.13)
where the vertical acceleration is assumed to be mostly due to magnetic buoy-ancy, i.e. uz ≈ −(δρ/ρ0) g and −δρ/ρ0 = δB2/(2p0) ≈ Byby/p0, where wehave linearized with respect to the fluctuations. In accretion discs the shear
is negative, i.e. ∂U y/∂x < 0, and therefore bybx < 0. If this effect does in-deed dominate, the α effect will be negative, i.e. αyy = ταyy < 0. Subsequentwork based on FOSA confirms the possibility of a negative value of αyy forsufficiently strong shear , as is indeed present in accretion discs. How-ever, because of the competing effect to drive a positive α effect from thermalbuoyancy, the sign of αyy changes to the conventional sign for weak shear.
11.4.2 Outflows from dynamo active discs
It is now commonly believed that all accretion discs have undergone a phasewith strong outflows. One mechanism for driving such outflows is magneto-centrifugal acceleration . The magnetic field necessary might be the fieldthat is dragged in from the embedding environment when the disc forms, butit might also be dynamo generated . In Fig. 11.7 we present a numericalsolution of such a model , where the dynamo is a mean field αΩ-dynamowith negative α effect . In the presence of open boundaries consideredhere, the most easily excited solution has dipolar symmetry about the equa-torial plane .
Three-dimensional simulations of dynamo action in accretion disc tori haveconfirmed that the magneto-rotational instability can sustain a magnetic fieldalso in such a global model [418–421]. So far, these simulations have not yetdeveloped outflows nor did a large scale magnetic field form. It is possible,however, that this is mainly a matter of time scales and that the simulationshave simply not yet been run for long enough to generate sufficiently stronga large scale field. Alternatively, it is also possible that these simulations arelacking other ingredients necessary to generate large scale fields, for examplebetter resolved small scale motions allowing flux tubes to form and to resolvethe possible twist in these.
11.5 Galactic dynamos
We saw in Section 2.3 that spiral galaxies have large scale magnetic fields ofthe order of a few 10−6G, coherent on scales of several kpc, and also highlycorrelated (or anti-correlated) with the optical spiral arms. Does the meanfield turbulent dynamo provide a viable model for understanding the originsuch fields? We discuss below a number of observed properties of galacticfields which favor a dynamo origin of disc galaxy magnetic fields. However thestrength of the field itself may not be easy to explain, in view of the helicityconstraint.
Fig. 11.7. Outflow from a dynamo active accretion disc driven by a combination ofpressure driving and magneto-centrifugal acceleration. The disc represents a proto-stellar disc, whose tenuous outer regions will are heated by the magnetic field. Thispressure contributes to hydrostatic support, which is changed only slightly by the fi-nite outflow velocities. The extent of the domain is [0, 8]×[−2, 30] in nondimensionalunits (corresponding to about [0, 0.8]AU × [−0.2, 3]AU in dimensional units). Leftpanel: velocity vectors, poloidal magnetic field lines and grey scale representation ofh in the inner part of the domain. Right panel: velocity vectors, poloidal magneticfield lines and normalized specific enthalpy h/|Φ| in the full domain. Adapted fromRef. .
11.5.1 Preliminary considerations
To begin with, all the ingredients needed for large scale dynamo action arepresent in disc galaxies. There is shear due to differential rotation and soany radial component of the magnetic field will be efficiently wound up andamplified to produce a toroidal component. A typical rotation rate is Ω ∼25 km s−1 kpc−1 (at r ≈ 10 kpc), and for a flat rotation curve (Uφ = ΩR =
constant) the shear rate, defined as S = RΩ′ ≡ RdΩ/dR, would be the same,i.e. S = −Ω. This corresponds to a rotation time of ∼ 2π/Ω ∼ 2.5 × 108 yr.So, in a Hubble time (∼ 1010 yr) there have been about 40 − 50 rotations.If only shear were involved in the generation of the galactic field, a fairlystrong primordial seed field of ∼ 10−7 G would be required just after thegalaxy formed. (We will also discuss other problems with such a primordialhypothesis below).
A mechanism to exponentiate the large scale field, like for example the αΩdynamo, is therefore desirable. An α effect could indeed be present in a discgalaxy. Firstly, the interstellar medium in disc galaxies is turbulent, mainlydue to the effect of supernovae randomly going off in different regions. In arotating, stratified medium like a disc galaxy, such turbulence becomes helical(Section 10.3). Therefore, we potentially have an α effect and so we expectαΩ mean field dynamo action. Typical parameters for this turbulence are, avelocity scale v ∼ 10 km s−1, an outer scale l ∼ 100 pc giving an eddy turnovertime τ ∼ l/v ∼ 107 yr. 8 This yields an estimate of the turbulent diffusioncoefficient, ηt ∼ 1
3vl ∼ 0.3 km s−1 kpc or ηt ∼ 1026 cm2 s−1. The inverse Rossby
(or Coriolis) number is ∼ 2Ωτ ∼ 0.6 (see Table 11.1, where we have takenΩ = 30 km s−1 kpc−1) and so we can use Eq. (10.57) to estimate α due torotation and stratification. This gives α ∼ τ 2Ω(v2/h) ∼ 0.75 km s−1, whereh ∼ 400 pc is the vertical scale height of the disc.
We gave a brief discussion of the simplest form of the mean field dynamo equa-tions appropriate for a thin galactic disc in Section 6.5.5, following Ref. ;here we just gather some important facts. We recall that two dimensionlesscontrol parameters,
CΩ = Sh2/ηt, Cα = αh/ηt, (11.14)
measure the strengths of shear and α effects, respectively. (Note that these areidentical to the RΩ and Rα commonly used in galactic dynamo literature). Forthe above galactic parameters, the typical values are CΩ ≈ −10 and Cα ≈ 1,and so |CΩ| ≫ Cα. Dynamo generation is controlled by the dynamo numberD = CΩCα, whose initial ‘kinematic’ value we denote by D0. Exponentialgrowth of the field is possible in the kinematic stage, provided |D0| > Dcrit,where the critical dynamo number Dcrit ∼ 6...10, depending on the exactprofile adopted for α(z). Modes of quadrupole symmetry in the meridional(Rz) plane are the easiest to excite in the case of a thin disc with D <
8 Numerical simulations of such a turbulent multiphase interstellar medium regu-lated by supernovae explosions, and including rotation, show  the multiphasegas in a state of developed turbulence, with the warm and hot phases having rmsrandom velocities of 10 and 40 km s−1 respectively, and with turbulent cell size ofabout 60 pc for the warm phase.
0. Further the growth rate in the kinematic regime given by (6.43) has anumerical value,
γ ≈ ηth2
≈ (1...10)Gyr−1 (11.15)
The dynamo is generally supercritical, and the value of γ evaluated locally atdifferent radial position is positive for a large range of radii, and so galacticfields can be indeed grow exponentially from small seed values. Furthermore, ina Hubble time scale of 1010 yr, one could exponentially grow the field by a fac-tor of about e30 ≈ 1013, provided the growth rate determined in the kinematicregime were applicable. In this case, even a small field of order 10−19 G wouldin general suffice as the initial seed magnetic field; with galaxies at higherredshift requiring a larger seed magnetic field. We shall discuss below howthe nonlinear restrictions, due to helicity conservation, may alter this picture.Before this, we first gather some of the observational indications which favora dynamo origin for the galactic field, nicely reviewed by Shukurov [423,424].
11.5.2 Observational evidence for dynamo action
The magnetic pitch angle
The simplest piece of evidence for dynamo action is just the form of themagnetic field lines as projected onto the disc. It is seen that the regularmagnetic field in spiral galaxies is in the form of a spiral with pitch anglesin the range p = −(10 . . . 30), where a negative p indicates a trailing spiral.Note that for dynamo action one needs to have non-zero BR and Bφ, or a non-zero pitch angle for the magnetic field lines projected onto the disc. A purelyazimuthal field, with BR = 0, will decay due to turbulent diffusion, and thesame is true for a purely radial field. A simple estimate of the pitch angle canbe obtained from the ‘no-z’ approximation  to the dynamo equations,whereby one replaces z derivatives by just a division by h. Substituting alsoBR, Bφ ∝ exp γt, this gives
BR = −αBφ
Bφ = SBR. (11.16)
One derives a rough estimate for the ratio 
tan p =BR
≈ − l
For l/h = 1/4 and a flat rotation curve with Ω = −S, one then gets p ∼−14. This is in the middle of the range of observed pitch angles in spiral
galaxies. More detailed treatments of galactic dynamos  confirm this simpleestimate. The above estimate is based on kinematic theory, but similar pitchangles are also obtained when one considers nonlinear dynamo models ;see also the case of models for M31 in Ref. . (The no-z approximationalso gives an estimate for γ as in Eq. (11.15) but with the crude estimateDcrit = 1).
An alternative hypothesis to the dynamo is that galactic magnetic fields aresimply strong primordial fields that are wound up by differential rotation (seeRefs. [428,429] and references therein). In this case, detailed analysis shows that differential rotation leads a field which rapidly reverses in radius(on scales ∼ 100 pc). More importantly the field is also highly wound upwith pitch angles of order p < 1, clearly much smaller than the observedvalues. In Ref. , it is argued that streaming motions in spiral arms wouldnevertheless lead to a field-aligned along the spiral; but away from the armsthe field will be nearly toroidal. Clearly, in the case of NGC6946 it is thefield in between the arms which has a spiral form with moderately large pitchangles with an average p ∼ 35 for several of the magnetic arms . Also inthe case of M31, although the field itself occupies a ring-like region, the fieldlines are still in the form of spirals with pitch angles ∼ 10 . . . 20 .
Another possibility is that the primordial field is not tightly wound up becauseturbulent diffusion compensates for the winding due to shear . In thiscase there would be a balance between shear and turbulent diffusion, andfrom the toroidal part of Eq. (11.16) one can estimate BR/Bφ ∼ 1/Cω; whichimplies p ∼ 6 for CΩ ∼ 10. This is larger than the pitch angle determinedneglecting turbulent diffusion, but may still not be large enough to accountfor the observed range of p in galaxies. Also one would need a constant sourceof BR, which could perhaps be due to accretion, since otherwise turbulentdiffusion (without an α effect) will cause BR to decay. The galactic dynamoon the other hand provides a natural explanation for the observed pitch anglesof the regular magnetic field.
Even vertical symmetry of the field in the Milky Way
Another argument in favor of the galactic dynamo theory is the observedsymmetry properties of the regular magnetic field about the galactic equator.Dynamo theory predicts that the toroidal and radial field components shouldbe symmetric about the equator. The even parity mode has a larger scale ofvariation in the vertical direction and is therefore subject to a weaker turbulentdiffusion than the odd parity mode . A wavelet analysis of the Faradayrotation measures of extragalactic sources indeed indicates that the horizontalcomponents of the regular magnetic field have even parity, that is they aresimilarly directed on both sides of the disc .
Such an even parity field can be produced from a primordial field if it isoriginally almost parallel to the disc plane. However, such a field would stillsuffer the excessive winding mentioned above. On the other hand, it is equallylikely that a primordial field has a large component parallel to the rotationaxis (entering through z < 0 and exiting through z > 0). In this case onewould have a dipole structure for the wound up primordial field, which is notsupported by the analysis of . (As mentioned earlier, the determinationof the magnetic field structure from Faraday rotation can be complicated bylocal perturbations).
The azimuthal structure of disc galaxy fields
The kinematic growth rates predicted by an axisymmetric galactic dynamoare also the largest for purely axisymmetric field structures. Of course thespiral structure induces nonaxisymmetry, and this can enhance the growthrates of nonaxisymmetric dynamo modes [292,425,432,433]. However this en-hancement is still not such that one expects a widespread dominance of non-axisymmetric magnetic structures in disc galaxies. Early interpretations ofFaraday rotation in spiral galaxies seemed to indicate a prevalence of bisym-metric structures , which would be difficult to explain in the frameworkof dynamo theory; but this has since not been confirmed. Indeed as discussedalso in Section 2.3.2, many galaxies have mostly distorted axisymmetric mag-netic structures, wherein the axisymmetric mode is mixed in with weakerhigher m modes. Only M81, among the nearby spirals, remains a case for apredominantly bisymmetric magnetic structure. The fact that it is physicallyinteracting with a companion may have some relevance for the origin of itsbisymmetric fields .
It would not be difficult to produce bisymmetric (m = 1) structure by thewinding up of a primordial field; more difficult to explain would be the dom-inance of nearly axisymmetric structures. Such configuration would requirean initial primordial field to be systematically asymmetric, with its maximumdisplaced from the disc center.
The observed nonaxisymmetric spiral magnetic structures are also aligned oranti-aligned with the optical spirals. This is intriguing because of the following:A given nonaxisymmetric mode basically rotates at nearly the local rotationfrequency, where the mode is concentrated. On the other hand, the optical spi-ral pattern rotates at a ‘pattern’ frequency very different in general comparedto the local rotation frequency. It has been shown [425,432,433,436] that suchstructures can be maintained for a range of radii of about a few kpc around thecorotation radius of the spiral pattern. Near the corotation radius a frozen-infield will rotate with the spiral pattern; and around this radius a moderateamount of the turbulent diffusion allows modes to still rotate with the pat-tern frequency, instead of with the local rotation frequency. More intriguing
are the ‘magnetic arms’ seen in NGC6946, and the anti-alignment betweenthese magnetic arms and optical spiral arms. Explaining this feature in a dy-namo model requires an understanding of how spiral structure influences thedifferent dynamo control parameters [437–439].
Other possible evidence for dynamo action include  radial reversals of themagnetic field in the Milky Way, the complicated magnetic structures thatare observed in M51, and the magnetic ring seen in M31. Shukurov  hasargued that all these features can be understood in dynamo models of thesesystems; whereas it is not clear how they may be explained in models involvingprimordial fields.
Strength of the regular magnetic field
The regular magnetic field in disc galaxies is close to energy equipartitionwith the interstellar turbulence. This feature directly indicates that the fieldis somehow coupled to the turbulent motions; although the exact way bywhich the regular field achieves equipartition with the turbulence is still tobe understood (see below). On the other hand a primordial origin of the reg-ular field in galaxies would put a special restriction on the strength of therequired pregalactic seed magnetic field. In models of the generation of pri-mordial magnetic fields based on processes occurring in the early universe (cf.Refs. [440–442] and references therein), the strength of the generated field ishighly uncertain.
11.5.3 Potential difficulties for galactic dynamos
Magnetic helicity constraint
The major potential difficulty for the galactic dynamo is still the restrictionimposed by magnetic helicity conservation. We have discussed these issues indetail in Section 9. The magnetic Reynolds number in galaxies is large enoughthat one would expect the total magnetic helicity to be well conserved. The factthat shear plays a major role in the galactic dynamo implies that the galacticfield is not maximally helical, and one can somewhat ease the restrictionsimposed by helicity conservation. Nevertheless, if there is negligible net smallscale helicity flux out of the galaxy, the mean galactic field is limited to  B ≈(km/kf)Beq[(D0/Dcrit) − 1]1/2 for moderately supercritical dynamo numbers;see also Section 9 and Eq. (9.43). If we adopt D0/Dcrit ∼ 2, km/kf ∼ l/h,with a turbulence scale l ∼ 100 pc, and h ∼ 400− 500 pc, then the mean fieldstrength would be 1/4 to 1/5 of equipartition at saturation.
Of course a preferential loss of small scale helicity could increase the satu-ration field strength further and this has indeed been invoked by Kleeorinand coworkers [327–329] and by Vishniac and Cho . However, it is not
at present clear whether such preferential loss of small scale helicity can ac-tually be obtained in galaxies; the Vishniac and Cho flux is not yet seen insimulations  and the theoretical underpinning of the phenomenologicallyimposed helicity fluxes [327–329] has not yet been fully clarified. This wouldbe one area to investigate further.
Small scale magnetic noise
As noted in Section 5, the small scale dynamo generates fluctuating magneticfields in the kinematic stage, at a rate faster than the large scale dynamo. Canthis rapidly generated magnetic noise suppress large scale galactic dynamo ac-tion? The analytical estimate of the nonlinear α effect using MTA presentedin Eq. (10.57) indicates that the presence of a equipartition strength fluctu-ating magnetic field does reduce the α effect, but only by a factor of ∼ 2/3for the diagonal components, and even enhances the off-diagonal components.Also, in both MTA and quasilinear treatments (FOSA) describing the effectof fluctuating fields, ηt is not renormalized at all. Furthermore, the nonlinearbehavior seen in direct simulations of large scale dynamos [4,137,322] can beunderstood using helicity conservation arguments alone, without further re-duction of the turbulent transport coefficients due to magnetic noise; althoughsimilar simulations at even higher values of the magnetic Reynolds number aredesirable. Overall, there is no strong evidence that magnetic noise due to smallscale dynamo action, catastrophically suppresses α and ηt; but this issue alsoneeds to be further investigated. For example, it would be useful to computethe turbulent diffusion of a large scale magnetic field in the presence of strongsmall scale MHD turbulence, like that described by Goldreich and Sridhar.
Magnetic fields in young galaxies?
The galactic dynamo exponentiates seed mean magnetic fields over a timescale of about (1...10) Gyr−1, depending on the dynamo parameters. For ayoung galaxy which is say formed at redshift z = 5 and is observed at aredshift of z = 2, its age is T ∼ 1.7 Gyr in the currently popular flat Lambdadominated cosmology (with cosmological constant contributing a density ΩΛ =0.7, matter density Ωm = 0.3 in units of the critical density and a Hubbleconstant H0 = 70 km s−1Mpc−1). For a mean field growth rate of say γ ∼3 Gyr−1, this corresponds to a growth by a factor eγT ∼ 75; and so even witha seed of ∼ 10−9 G, we would have a mean field of only ∼ 0.075µG. So, ifone sees evidence for strong microgauss strength mean fields in galaxies athigh redshifts, the galactic dynamo would have difficulties in accounting forthem. If the seed fields were much weaker, say ∼ 10−18G, like those producedin batteries, one would have difficulty in accounting for microgauss strengthmean fields even for moderate redshift objects, with z ∼ 0.5.
At present there is some tentative evidence for magnetic fields from Faradayrotation studies of high redshift quasars and radio galaxies. A Faraday rotationmap of the extended and polarized jet of the quasar PKS 1229-021 at z =1.038, has revealed fields ordered on kpc scales and with strengths of a fewmicrogauss at z = 0.395, corresponding to the position of an interveningabsorption system . Evidence for similarly ordered fields also comes fromrotation measure map of the quasar 3C191, which has an associated absorptionsystem at z = 1.945 similar to its emission redshift . A number of highredshift radio galaxies at z > 2  also show very high Faraday rotation of∼ 1000 radm−2, which could be indicative of strong ordered magnetic fields intheir host galaxy (although such fields could also arise from the magnetizationof a sheath around the radio lobe). There is a general difficulty that a givenRM could arise in several intervening systems including the source and ourGalaxy. This introduces difficulties in determining magnetic fields associatedwith high redshift systems . A promising approach would be to use radiogravitational lenses which show differences in Faraday rotation between theirdifferent images [104,447]. The lines of sight to the different images followvery similar paths in the source and our galaxy, but have large transverseseparations in an intervening object, and so could probe intervening magneticfields better.
We should remember that the small scale dynamo could itself produce strongmagnetic fields on time scales of ∼ 107 yr. However this would be correlatedat most on the scales of the turbulence and will not be able to explain fieldsordered on kpc scales. The fields generated by a small scale dynamo couldnevertheless provide a strong seed magnetic field for the large scale dynamo.
11.6 Cluster magnetic fields
We saw in Section 2.4 that clusters of galaxies appear to be magnetized,with fields ranging from several µG to tens of µG in some cluster centers,and with coherence scales l ∼ 10 kpc. How are such fields generated andmaintained? Note that a tangled magnetic field, left to evolve without anyforcing, would generate motions with a velocity of order the Alfven velocityvA = B/(4πρ)1/2, and result in decaying MHD turbulence with a characteristicdecay time τdecay ∼ l/vA [449,450]. For a µG field, in a cluster medium withdensities ∼ 10−3 cm−3 is vA ∼ 70 km s−1 and τdecay ∼ 1.4 × 108 yr. Althoughthe energy density in MHD turbulence decays as a power law, this time scaleis still much smaller than the typical age of a cluster, which is thought to beseveral billion years old. One therefore needs some mechanism to generate andmaintain such tangled cluster magnetic fields.
First, there are a number of sources of seed magnetic fields in galaxy clusters. Itis well known that the intracluster medium (ICM) has metals which must havebeen generated in stars in galaxies and subsequently ejected into the galacticinterstellar medium (ISM) and then into the ICM. Since the ISM is likely tobe magnetized with fields of order a few µG, this would lead to a seed fieldin the ICM. The exact manner in which the ISM from a galaxy gets mixedinto the cluster gas is uncertain; possibly involving tidal and ram pressurestripping of the galactic gas, together with galactic outflows . One canroughly estimate the seed field resulting from stripping the galactic gas, byusing flux conservation; that is Bseed ∼ (ρICM/ρISM)
2/3Bgal. For Bgal ∼ 3µG,and ρICM/ρISM ∼ 10−2...10−3, one gets Bseed ∼ 0.1...0.03µG. One may get evenlarger seed fields if cluster galaxies have substantial magnetized outflows: if∼ 103 galaxies have mass outflow with M ∼ 0.1M⊙ yr−1 lasting for 1Gyr,with a Poynting flux about 10% of the material flux, and the field gets mixedinto the cluster gas over a Mpc sized region, Bseed ∼ 0.3µG would result .
Another source of seed fields is likely to be also the outflows from earlier gen-eration of active galaxies (radio galaxies and quasars) [452–455]. Such outflowsmay leave behind magnetized bubbles of plasma in some fraction of the inter-galactic medium (typically ∼ 10% ), which when incorporated into theICM would seed the general cluster gas with magnetic fields. If one assumesthe cluster gas is about 103 times denser than the IGM, and blindly uses theenhancement of the bubble field due to compressions during cluster formation,one can get fields as large as 0.1 − 1µG in the ICM . However this is toignore the issue of how the field in the magnetized bubble, especially if it ispredominantly relativistic plasma from a radio galaxy, mixes with the non-magnetized and predominantly thermal gas during cluster formation, and theresulting effects on both the field strength and coherence scales (see Ref. for the related problem of getting cosmic rays out of radio cocoons). It islikely that, while AGNs and galaxies provide a potentially strong seed mag-netic field, there would still be a need for their subsequent amplification andmaintenance against turbulent decay.
In most astrophysical systems, like disc galaxies, stars and planets, rotationplays a very crucial role in this respect; both in providing strong shear and inmaking random flows helical, and hence leading to large scale dynamo action.Clusters on the other hand, are expected to have fairly weak rotation–if atall, so one has to take recourse to some other mechanism for understandingcluster magnetism. One possibility is small scale turbulent dynamo action;indeed most early work on the generation of cluster magnetic fields exploredthis possibility [457–461]. The turbulence was thought to be provided by wakesof galaxies moving through the intracluster medium. However galaxy wakesare probably inefficient. Firstly, the gas in the galaxy would get significantlystripped on the first passage through the cluster, and stop behaving as a ‘hard’sphere in producing a turbulent wake. Furthermore, the wakes generated by
the Bondi-Hoyle gravitational accretion are probably not pervasive enough, be-cause the accretion radius ∼ GM/c2s ∼ 0.5 kpc (M/1011M⊙)(cs/10
3 km s−1)−2
is much smaller than galaxy radii of order 10 kpc. Further, calculations us-ing the EDQNM equations  gave pessimistic estimates for the generatedfields when the turbulence is induced by galactic wakes. A different source forcluster turbulence and magnetic fields is probably required.
Turbulence in clusters can also arise in mergers between clusters [462–465]. Inhierarchical structure formation theories, clusters of galaxies are thought to beassembled relatively recently. They form at the intersection of filaments in thelarge scale structure, involving major mergers between comparable mass ob-jects and also the accretion of smaller mass clumps. In this process, it is likelythat clusters develop significant random flows, if not turbulence [464,465].These would originate not only due to vorticity generation in oblique accre-tion shocks and instabilities during the cluster formation, but also in the wakesgenerated during the merger with smaller mass subclumps. In a simulation ofcluster formation, it was found that the intracluster medium becomes ‘tur-bulent’ during cluster formation , and this turbulence persists even afterabout 5 Gyr after the last major merger. At this time peculiar velocities areof order 400 km s−1 within 1/2 the Virial radius rv; see Fig. 1 in . A visualinspection of the flow field reveals ‘eddies’ with a range of sizes of 50−500 kpc.Similar peculiar gas motions have also been found in other numerical simu-lations of cluster formation  and cluster mergers . In the mergersimulation  ram pressure effects during the merger are though to displacethe gas in the cluster core from the potential center, causing it to become un-stable. The resulting convective plumes produce large scale turbulent motionswith eddy sizes up to several hundred kiloparsecs. Again, even after about aHubble time (∼ 11 Gyr after the first core interaction), these motions persistsas subsonic turbulence, with velocities of order 10...20% of the sound speedfor equal mass mergers and twice as large for mergers with a mass ratio of1 : 3. (In all these cases, since there is limited spatial resolution, it may bebetter to call the flows random flows rather than turbulence.)
A quantitative assessment of the importance of such random shear flows andperhaps turbulence, for the generation of cluster magnetic fields has only be-gun recently, by doing direct simulations [468,469] and also using semi-analyticestimates . Simulations of cluster mergers  showed two distinct stagesof evolution of the field. In the first stage, the field becomes quite filamentaryas a result of stretching and compression caused by shocks and bulk flows dur-ing infall. Subsequently, as bulk flows are replaced by more turbulent motions,the magnetic energy increases by an average factor ∼ 3 to localized enhance-ments of ∼ 20. It is argued  that this increase is likely to be a lowerlimit, as one cannot resolve the formation of eddies on scales smaller than halfthe cluster core radius. Magnetic field evolution in a smooth particle hydro-dynamics simulation of cluster collapse has also been reported [469,471]; they
found that compression and shear in random flows during cluster formationcan increase the field strengths by factors of order 103.
Both these groups use ideal MHD and it would be useful to relax this assump-tion. It will be important to do further simulations which have the resolutionto follow also the development of turbulence and the nonlinear cascade tosmall scales. This is especially important given the current lack of consensus(cf. Section 5) about how the small scale dynamo saturates, especially in thehigh Prandtl number systems like the ICM. As discussed in connection withnonlinear small scale dynamos (Section 5.4), it is likely that, once the mag-netic field has reached equipartition field strength, the power spectrum shoulddecrease with increasing wavenumber. If this is not seen, the simulation maystill not be sufficiently well resolved or it may not have run for long enough,or both. Overall, it appears plausible that cluster magnetism is the result ofcompression, random shear and turbulent amplification of seed fields fromgalaxies and AGNs; but work on understanding the efficiency and details ofall these processes is still in its infancy.
12 Where do we stand and what next?
Significant progress has been made in clarifying and understanding mean fielddynamo theory in the nonlinear regime. Only a few years ago it was a com-pletely open question whether or not α is catastrophically quenched, for exam-ple. What is worse, simulations were not available yet that showed whether ornot mean field dynamos can work at all when the magnetic Reynolds numberis large. A lot has changed since then and we have now begun to develop andconfirm numerically the nonlinear mean field formalism for the case where themedium is statistically homogeneous.
One of the currently most pressing problems is to develop and test numericallya dynamical quenching theory that is valid also in the inhomogeneous case,where the strength of the α effect varies in space and changes sign, and tothe case with open boundaries allowing helicity flux to escape. This should bestudied in a geometry that is close enough to the real case of either the solarconvection zone or to galactic discs. On the other hand, there is still a casefor testing more thoroughly the dependence of turbulent transport coefficientson the magnetic Reynolds number even in cases with closed boundaries. Anissue that is not fully resolved is whether the turbulent magnetic diffusionis quenched in a way that depends on the magnetic Reynolds number. Oneway of clarifying this issue is by considering oscillatory large scale dynamoswith shear , which give the possibility of measuring directly the turbulentmagnetic diffusivity via the cycle frequency .
As far as solar dynamos is concerned, there remain clearly many problems tobe solved. The first and perhaps most severe one is simply the lack of a reliabletheory: the mean field theory as it stands at the moment and as it has beenused even in recent years does not reproduce what is known from simulations. Again, we have here in mind issues related to magnetic helicity conser-vation, which need to be dealt with using dynamical quenching theory. Evenfrom a more practical point of view, if one is able to argue that in the sun themagnetic helicity issue can be solved in such a way that conventional theoryremains applicable, there would still be many hurdles, as outlined above. Apopular model is the flux transport model with a suitable meridional circula-tion profile, but here the preferred modes have usually quadrupolar symmetryabout the equator rather than dipolar symmetry [355–357].
For the galactic dynamo, one probably still needs to identify mechanisms bywhich fluxes of small scale helicity can preferentially leave the system, so asto build up the regular field to the observed values. However, unlike the caseof the sun where we have direct evidence for losses of helical magnetic fluxthrough the surface, galaxies lack such direct evidence – even though theyallow direct inspection all they way to the midplane. Regarding the satura-tion strength of the large scale field, the problem of course is not as severeas it was once thought, since even in the kinematic phase, and conservinghelicity, a significant regular field can probably be built up by the mean fielddynamo. Further, for galaxies at least, the mean field dynamo theory seems toreasonably explain the structure of the observed fields. In the case of clustersof galaxies, the origin of turbulence which may be needed for dynamo actionis not yet settled. The structure of the field will also depend on an improvedunderstanding of how the small scale dynamo saturates.
Another important question is what role does the small scale dynamo playrelative to the large scale dynamo. In galaxies, where the magnetic Prandtlnumber is large, it has been argued that the magnetic field is dominated bysmall scale fields. This is an issue that clearly requires further clarification.In solar and stellar dynamos, on the other hand, the small scale dynamo maynot work at all any more, or the critical dynamo number may be much largerthan for unit magnetic Prandtl number. Whatever the answer, it is likely thatthe magnetic Prandtl number dependence of the critical magnetic Reynoldsnumber can soon be settled using dedicated high-resolution simulations.
We thank Rainer Beck, Eric Blackman, Leonid Kitchatinov, David Moss, Karl-Heinz Radler, Gunther Rudiger, and Anvar Shukurov for discussions, detailedcomments and corrections to earlier versions of this review. We acknowledge
the hospitality of IUCAA and Nordita during our mutual visits during whichmuch of this work has flourished. We thank the Danish Center for ScientificComputing for granting time on the Horseshoe cluster in Odense.
A Evolution of the correlation tensor of magnetic fluctuations
The derivation of these equations involves straightforward but rather tediousalgebra and follows the discussion in . We therefore only outline the stepsand the approximations below leaving out most of the algebraic details. Tomake the appendix fairly self contained we repeat some of the equations whichare also given in the main text. We start with the induction equation for themagnetic field, including a nonlinear ambipolar diffusion term, written as
ipqUpBq + η∇2Bi, (A.1)
where we have defined for later convenience, the operator
Rxipq = ǫilmǫmpq
Here U = U + v + vN, where U is the mean velocity field v is the stochasticvelocity which may be helical, and which is δ- correlated in time and vN =a(J ×B)×B is the nonlinear ambipolar diffusion component, which is usedas a model nonlinearity.
Recall that we assume v to be an isotropic, homogeneous, gaussian random ve-locity field with zero mean, and δ-correlated in time. That is 〈vi(x, t)vj(y, s)〉 =Tij(r)δ(t− s), with Tij as defined in (5.22),
TN (r) +rirjr2
TL(r) + ǫijkrk F (r), (A.3)
where TL, TN and F are the longitudinal, transverse and helical parts of thecorrelations respectively. The induction equation becomes a stochastic equa-tion. We split the magnetic field into mean field B = 〈B〉 and a fluctuatingfield b = B − B. The equation for the mean field, for the Kazantsev modelvelocity field is derived in Appendix B. Here we concentrate on the evolutionof the fluctuating field. We assume b also to be a homogeneous, isotropic, ran-dom field, with an equal time two point correlation 〈bi(x, t)bi(y, t)〉 = Mij(r, t),
where r = x− y, r = |r| and
Mij = MN
+ Cǫijkrk. (A.4)
Here ML(r, t) and MN(r, t) are the longitudinal and transverse correlationfunctions for the magnetic field while C(r, t) represents the (current) helicalpart of the correlations. The evolution of Mij(r, t) can be got from
∂t(〈bi(x, t)bj(y, t)〉) =
The second term in the square brackets is easy to evaluate using the equationfor the mean field (see below in Section B.2). The first term is
∂t(Bi(x, t)Bj(y, t)) = Bi(x, t)
∂tBj(y, t). (A.6)
Suppose we define the two-point product Bi(x, t)Bj(y, t) = Bij(x,y, t) fornotational convenience. Substitute (A.1) into (A.6) and and let the initial valueof the two-point product be Bi(x, 0)Bj(y, 0) = B0
ij . Then, at an infinitesimaltime δt later, this product is given by the formal integral solution:
Bij = B0ij +
jpqUypBiq] + δt[η∇2
xBij +∇2yBij ]. (A.7)
For clarity, the x and y and t′ dependencies of the fields has been suppressed(except in ∇2) in the integrand. We write down an iterative solution to thisequation to various orders in δt. To zeroth order, one ignores the integraland puts Bij(x,y, t
′) = B0ij To the next order one substitutes B0
ij for Bij in
Eq. (A.7), to get a first order iteration B(1)ij , and then B(1)
ij for Bij , to get B(2)ij .
The resulting equation is then averaged to get the 〈Bij〉 and the correspond-ing contribution from Bi(x, δt)Bj(y, δt) subtracted to get the equation forMij(δt) = 〈bi(x, δt)bj(y, δt)〉. The presence of the δ-correlated u implies thatone has to go up to second order iteration to get Mij(δt) correct to linear orderin δt. Then dividing by δt and taking the limit of δt → 0, we get
vp(y, t) Rxilm(vl(x, s)[Mmq +Bm(x)Bq(y)])
vp(x, t) Ryjlm(vl(y, s)[Mqm +Bq(x)Bm(y)])
vp(y, t) Ryqlm(vl(y, s)Mim)
vp(x, t) Rxqlm(vl(x, s)Mmj)
+ η[∇2yMij +∇2
xMij ] +Ryjpq
+Ryjpq (〈vNp(y)bi(x)Bq(y)〉) +Rx
ipq (〈vNp(x)Bq(x)bj(y)〉) . (A.8)
The first two terms on the RHS of Eq. (A.8) represent the effect of velocitycorrelations on the magnetic fluctuations (Mij) and the mean field (B). Thenext two terms give the ‘turbulent transport’ of the magnetic fluctuations bythe turbulent velocity, the 5th and 6th terms the ’microscopic diffusion’. The7th and 8th terms the transport of the magnetic fluctuations by the meanvelocity. The last two nonlinear terms give the effects of the back reaction dueto ambipolar drift on the magnetic fluctuations.
For the discussions of the small scale dynamo in Section 5, the unified treat-ment of the large scale dynamo in Section 5.6 or the toy closure model inSection 8.10 we do not keep the mean field terms. (The coupling to the meanfield can be important in discussions of helicity flux). This also means that wecan continue to treat the statistical properties of the magnetic fluctuations asbeing homogeneous and isotropic, and use Mij(x,y, t) = Mij(r, t).
All the terms in the above equation, can be further simplified by using theproperties of the magnetic and velocity correlation functions. In order to ob-tain equations for ML and C, we multiply Eq. (A.8) by (rirj)/r2 and ǫijkr
and use the identities
ML(r, t) = Mij(rirj/r2), C(r, t) = 1
We consider some steps in simplifying the first two terms. The first term inEq. (A.8 ) is given by
Ryjpq (vp(y, t)R
xilm(vl(x, s)Mmq)) ds
= −ǫituǫulmǫjrsǫspq [TlpMmq],rt ,(A.10)
where f, l denotes partial derivative ∂f∂rl. For examining the evolution ofML one needs to multiply the above equation by rirj/r2. We can simplify theresulting equation by using the identity
∂rt− δjtδirA, (A.11)
where A = T lpMmq. Then using ǫituǫulm = δilδtm− δimδtl, and the definition ofTL, TN and F , straightforward algebra gives the contribution of the first term
ML|1st = − 1
2ML + 4FC. (A.12)
The second term of (A.8) gives an identical contribution.
To derive the evolution of H due to these terms multiply (A.10) by ǫijkrk .
Using the fact that the turbulent velocity and small scale field have vanishingdivergence, we have Mij,j = 0 and Tij,j = 0. This allows one to simplify thecontribution from the first term to (∂C/∂t)
C|1st = −ǫijkrk
2r2(Tij,trMtr + TtrMij,tr − Tir,tMtj,r − Ttj,rMir,t) . (A.13)
The first two terms on the RHS of Eq. (A.13) can be further simplified bynoting that ǫijkTij = 2Frk and ǫijkMij = 2Crk . We have then
2r2(Tij,trMtr + TtrMij,tr) = −
TLC′′ +T ′
Here prime denotes a derivative with respect to r. To evaluate the contribu-tion of the last two terms on the RHS of (A.13), it is convenient to split upthe tensors Mij and Tij into symmetric and antisymmetric parts (under theinterchange of (ij) ). We put a superscript S on the symmetric part and A onthe antisymmetric part. Then we can write after some algebra
2r2(Tir,tMtj,r + Ttj,rMir,t] =
Atj,r + TA
CT ′′L + FM ′′
L + T ′LC
′ +M ′LF
′ +4CT ′
Adding the contributions from (A.14) and (A.15) gives
C|1st = − 1
∂r(TLLC + FML)
The second term of (A.8) gives an identical contribution. The third and fourthterms add to give a contribution
(3rd + 4th) = 4F (0)ǫjqm(∂Mim/∂rq) + 2TL(0)∇2Mij (A.17)
to the RHS of (A.8), hence justifying their being called ‘turbulent transport’of Mij (compare this to the microscopic diffusion term 2η∇2Mij).
The last two nonlinear terms give the effects of the back reaction due toambipolar drift on the magnetic fluctuations. They involve 4−th order corre-lations of b. In evaluating this term, we make the gaussian closure approxi-mation that the fourth order moment of the fluctuating field can be writtenas a product of 2nd moments. In this case the nonlinear terms add to give acontribution
nonlinear terms = −8aC(0, t)ǫjqm(∂Mim/∂rq) + 4aML(0, t)∇2Mij (A.18)
to the RHS of (A.8). The gaussian assumption of the magnetic correlationsresults in the nonlinearity of this term appearing as a nonlinearity in thecoefficient, rather than the correlation function itself. Gathering together allthe terms, we get for the coupled evolution equations for ML and C :
+GML + 4αNC, (A.19)
∂r(2ηNC − αNML)
We can also write (A.20) in terms of the magnetic helicity correlation functionH(r, t), which is related to the current helicity correlation C(r, t) by
C = − 1
∂t= −2ηNC + αNML. (A.22)
Here we have also defined
ηN = η + TLL(0)− TLL(r) + 2aML(0, t),
αN = −[2F (0)− 2F (r)] + 4aC(0, t),
G = −2(T ′′L + 4T ′
These equations generalize the Kazantsev equation to the helical case and alsoinclude a toy nonlinearity in the form of ambipolar diffusion. If we take thelimit r → 0 in Eq. (A.22), we get H(0, t) = −2ηC(0, t), which is exactly theequation for conservation (evolution) of magnetic helicity. So our nonlinearclosure model incorporates also the important constraint provided by helicity
conservation. This set of equations is used to discuss the nonhelical small scaledynamo (Section 5), a unified treatment of small and large scale dynamos(Section 5.6) and also the helicity constraint in this toy model (Section 8.10).
B Other approaches to calculating α and ηt
B.1 The lagrangian approach to the weak diffusion limit
In the limit of large Rm ≫ 1, one may also think of neglecting magneticdiffusion completely. The time evolution of the magnetic field can then besolved exactly using the Cauchy solution,
Bi(x, t) =Gij(x0, t)
where Gij = ∂xi/∂x0j is the lagrangian displacement matrix,B0(x0) = B(x, 0)is the initial condition, and
x(x0, t) = x0 +
uL (x0, t′) dt′ (B.2)
is the position of an advected test particle whose original position was at x0.We have also defined the Lagrangian velocity uL(x0, t) = U(x(x0, t)). Usingthe Cauchy solution, Moffatt  showed that
α(t) = −13
uL(x0, t) · ωL(x0, t′) dt′, (B.3)
where ωL = ∇L × uL is the vorticity of the lagrangian velocity fluctuationwith respect to x0. So α(t) = 0 at the initial time t = 0, but the expectation isthat for times much longer than the correlation time of the turbulence, t ≫ τ ,α(t) settles to a constant value. One could then take the limit t → ∞ in theabove integral. 9
9 The convergence of the integral to a constant when t → ∞ is not guar-anteed because the above integral has derivatives of the form ∂ui/∂x0j =(∂ui/∂xm)(∂xm/∂x0j). Although ∂ui/∂xm is statistically stationary in time,∂xm/∂x0j is in general not, since particles initially separated by some δx0, tendto wander further and further apart in a random flow, and |δx| ∝ |δx0|t1/2. So sec-ularly growing terms may in principle contribute to the integral determining α(t)making the limit t → ∞ meaningless .
Deriving an expression for the ηt coefficient is more complicated. A naiveexpectation is that one will have
uL(x0, t) · uL(x0, t′) dt′, (B.4)
analogous to the case of the effective turbulent diffusivity of a scalar field.One does get this term, but there are additional terms for the magnetic field,derived for example in Moffatt’s book . For these terms convergence tofinite values at large times is even more doubtful. Simulations by Kraichnan suggested that α(t) and ηt(t) converge to finite values of order u and uLrespectively, as t → ∞, for statistically isotropic velocity fields with gaussianstatistics. However, numerical simulations  using a frozen velocity fieldsuggest that in the limit of large magnetic Reynolds numbers α tends to zero.
B.2 Delta-correlated velocity fields
One of the few situations for which the kinematic mean field dynamo equationscan be derived exactly is for a random flow that is δ-correlated in time, asintroduced by Kazantsev . Such a flow was also used by Kraichnan to discuss passive scalar evolution. Such flows are of course artificial and nota solution of the momentum equation, but they serve as an excellent examplewhere the mathematics can be treated exactly.
To discuss mean field dynamo action, we have to add a helical piece tothe correlation function of the stochastic velocity field v driving the flow.So now we adopt U = U + v in the induction equation where, as before〈vi(x, t)vj(y, s)〉 = Tij(r)δ(t − s), with Tij as defined in (5.22). At r = 0, wehave
−2F (0) = −13
〈v(t) · (∇× v(t′))〉 dt′[
≈ −13τv · (∇× v)
〈v(t) · v(t′)〉 dt′[
where the last expressions in parenthesis would apply if we had assumed asmall but finite correlation time τ .
The induction equation is a stochastic equation and we would like to convert itinto an equation for the mean magnetic field B (see also Zeldovich et al. 1983,
Chapter 8). Let the magnetic field at an initial time, say t = 0, be B(x, 0).Then, at an infinitesimal time δt later, the field is given by the formal integralsolution:
B(δt) = B(0) +
dt′ ∇× [U(t)×B(t′)]− η∇×B(t′) . (B.7)
For clarity, the common x dependence of U and B, has not been explic-itly displayed above. We write down an iterative solution to this equationto various orders in δt. To zeroth order, one ignores the integral and putsB(0)(x, δt) = B(x, 0). To the next order one substitutesB0 forB in Eq. (B.7),to get a first order iteration B(1), and then B(1) for B, to get B(2). The re-sulting equation is then averaged to get the B(x, δt). The presence of theδ-correlated v implies that one has to go up to second order iteration to getB(x, δt) correct to linear order in δt. This procedure yields
B(δt) =B(0)− δt[∇× (U ×B(0))− η∇×B(0)]
v(t′)× [∇× (v(t′′)×B(0))]
where the overbar denotes averaging over an ensemble of the stochastic velocityfield v, as before. On using the fact that v at time t is not correlated with theinitial magnetic field B(x, 0) and taking the limit δt → 0 we get after somestraightforward algebra,
U ×B + αB − (η + ηt)∇×B]
The effect of the turbulent velocity is again to introduce the standard extraterms representing the α effect with α = −2F (0) and an extra turbulent con-tribution to the diffusion ηt = TL(0). If one allows for weakly inhomogeneousturbulence, one could also discover a turbulent diamagnetic effect due to thegradient of ηt, which expels magnetic fields from regions of strong turbulence.
B.2.1 Transport coefficients from random waves and individual blobs
We mention two additional approaches that have been important in under-standing the origin and nature of the α effect. One is based on the superpo-sition of random waves that are affected and modified through the presenceof rotation, stratification, and magnetic fields [264,265,308,474,363], and theother is based on the detailed analysis of individual convection cells  or
blobs from supernova explosions . The latter approach has to an α tensorof the form
αR −Vesc 0Vesc αφ 00 0 αZ
in cylindrical (R, φ, Z) coordinates. The effect of individual supernova explo-sions is slightly different from the effect of the so-called superbubble where oneexplosion has triggered several others nearby. The latter leads to structuresmore symmetric about the midplane causing αZ to be negligibly small. Forindividual supernova bubbles, on the other hand, the sign of αZ is found tobe opposite to the sign of αφ. There is also a vertical pumping effect away
from the midplane, corresponding to the antisymmetric components of the αtensor,
γi = −12ǫijkαjk = δiZVesc. (B.11)
The lack of downward pumping is possibly an artifact of neglecting the returnflow. The motivation for neglecting the return flow is the somewhat unjustifiedassumption that the magnetic field will have reconnected to their originalposition by the time the return flows commences.
C Calculating the quasi-linear correction EN
We calculate EN here in coordinate space representation. We can eliminate thepressure term in uN using the incompressibility condition. Defining a vectorF = a[B ·∇b + b ·∇B], with a = τ/(µ0ρ0), one then gets
EN = 〈F × b〉 − 〈[∇(∇−2∇ · F )]× b〉, (C.1)
where ∇−2 is the integral operator which is the inverse of the Laplacian,written in this way for ease of notation. We will write down this integralexplicitly below, using −(4πr)−1 to be the Greens function of the Laplacian.We see that EN has a local and nonlocal contributions.
To calculate these, we assume the small scale field to be statistically isotropicand homogeneous, with a two-point correlation function 〈bi(x, t)bi(y, t)〉 =Mij(r, t), given by (5.24),
Mij = MN
+ Cǫijkrk. (C.2)
Recall that ML(r, t) and MN(r, t) are the longitudinal and transverse corre-lation functions for the magnetic field while C(r, t) represents the (current)helical part of the correlations. Since ∇ · b = 0,
We also will need the magnetic helicity correlation, H(r, t) which is given byC = −(H ′′ + 4H ′/r), where a prime ′ denotes derivative with respect to r. Interms of b, we have
ML(0, t) =13〈b2〉, 2C(0, t) = 1
3〈j · b〉, 2H(0, t) = 1
3〈a · b〉, (C.4)
where b = ∇ × a and j = ∇ × b. The local contribution to EN is easilyevaluated,
ELN ≡ 〈F × b〉 = −aML(0, t)J + 2aC(0, t)B, (C.5)
where J = ∇×B.
At this stage (before adding the nonlocal contribution) there is an importantcorrection to the alpha effect, with α = αK + αM, where
αM = 2aC(0, t) = 13τ〈j · b〉. (C.6)
(We again set µ0 = 1 and ρ0 = 1 above and in what follows). There is alsoa nonlinear addition to the diffusion of the mean field (the −aML(0, t)∇×B
term), which as we see gets canceled by the nonlocal contribution!
Let us now evaluate the nonlocal contribution. After some algebraic simplifi-cation, this is explicitly given by the integral
(ENLN )i(x, t)≡−(〈[∇(∇−2
∇ · F )]× b〉)i
where y = r+x. Note that the mean field B will in general vary on scales Rmuch larger than the correlation length l of the small scale field. We can thenuse the two-scale approach to simplify the integral in Eq. (C.7). Specifically,assuming that (l/R) < 1, or that the variation of the mean field derivativein Eq. (C.7), over l is small, we expand ∂Bl(y, t)/∂y
m, in powers of r, about
r = 0,
∂xm∂xp∂xq+ . . . , (C.8)
where we have defined the unit vector ni = ri/r (we will soon see why we havekept terms beyond the first term in the expansion). Simplifying the derivative∂Mmk(r, t)/∂r
l using Eq. (5.24) and noting that ǫijkrjrk = 0, we get
r−1(ML −MN)nmδkl +M ′Nnlδmk
+Cǫmkl + rC ′nf nlǫmkf
We substitute (C.8) and (C.9) into (C.7), use
15[δijδkl + δikδjl + δilδjk], (C.10)
to do the angular integrals in (C.7), to get
ENLN = aML(0, t)J +
5H(0, t)∇2B +
∇2J . (C.11)
The net nonlinear contribution to the turbulent EMF is EN = ELN + ENL
N ,got by adding Eq. (C.5) and Eq. (C.11). We see firstly that the nonlineardiffusion term proportional to ∇ × B has the same magnitude but oppositesigns in the local (Eq. (C.5)) and nonlocal (Eq. (C.11)) EMF’s and so exactlycancels in the net EN. This is the often quoted result [256,254,300,301] that theturbulent diffusion is not renormalized by nonlinear additions, in the quasi-linear approximation. However this does not mean that there is no nonlinearcorrection to the diffusion of the mean field. Whenever the first term in anexpansion is exactly zero it is necessary to go to higher order terms. This iswhat we have done and one finds that EN has an additional hyperdiffusionEHD = ηHD∇2(∇×B) = ηHD∇2J , where
dr rML(r, t). (C.12)
Taking the curl of EN, the nonlinear addition to the mean field dynamo equa-tion then becomes,
∇× EN = (αM + hM∇2)∇×B − ηHD∇4B. (C.13)
Here αM = 13a〈j · b〉 is the standard nonlinear correction to the α effect
[254,256], and hM = 15a〈a · b〉 is an additional higher order nonlinear helical
correction derived here.
D Derivation of Eq. (10.28)
Now turn to the simplification of the χMjk term. We again define wavevectors,
k1 = k + K/2 − k′ and k2 = −k + K/2, and transform to new variables,(k′,K ′), where K ′ = k1 + k2 = K − k′. Note that since all Fourier variableswill be non-zero only for small |K | and |k′|, they will also only be non-zerofor small |K ′|. The equation (10.22) becomes
d3K ′ d3k′ ei(K′+k′)·R Pjs(k + 1
2K ′ + 1
[i(−kl +12K ′
l − 12k′l) Bl(k
′) Msk(k − 12k′;K ′)
+ (ik′l) Bs(k
′) Mlk(k − 12k′;K ′)], (D.1)
where, for notational convenience, we have defined
Msk(k − 12k′;K ′) = bs(k − 1
2k′ + 1
2K ′)bk(−(k − 1
2k′) + 1
2K ′). (D.2)
(Note that the terms involving k′lBl(k
′) above are zero, since ∇ ·B = 0). Wenow expand Pjs(k + 1
2K ′ + 1
2k′) about K ′ = 0 and both Pjs and Mlk about
k′ = 0, keeping at most terms linear in K ′ and k′ respectively to get
+Pjs(k) i(k′l +
′) + Bs(k′) Mlk(k;K
− i(k′s +K ′
(k · B) Msk(k;K′)
d3K ′ d3k′ ei(K′+k′)·R . (D.3)
We have also used here the fact that ksmsk = 12i(∂vsk/∂Rs) above, to neglect
terms of the form k′jksmsk and K ′
jksmsk, which will lead to terms with twoderivatives with respect to R. (Note that in homogeneous turbulence with∇ · b = 0 one will have ksmsk = 0).
Integrating over K ′ and k′, and using the definition of mij we finally get
∂t= ik ·Bmjk +
12B ·∇mjk +Bj,lmlk
E Calculation of the second and third terms in (10.38)
The contribution due to the second term is:
τ 2 k ·Ω kmIlk d3k
τ 2 k ·Ω km
− ik ·B (v(0)lk −mlk) +
+Bl,smsk − Bk,sv(0)ls − 1
∂v(0)lk /∂kα + ∂mlk/∂kα
Note that due to the presence of the antisymmetric tensor ǫjlm the last term
of Ilk does not contribute to E(Ω2)i . Further note that to linear order in Ω, the
velocity correlation vlk can be replaced by the nonhelical, isotropic, velocitycorrelation of the original turbulence v
(0)lk . Substituting the general form of the
magnetic and velocity correlations, and noting that only terms which have aneven number of ki survive the angular integrations, we have
τ 2 k ·Ω km
− k ·B 12kk∇l(E
(0) −M) + k ·BǫlknknN
+ 12B ·∇[δlk(E
(0) +M)] +Bl,sPskM − Bk,sδlsE(0)
(0) +M)′ − δlαkk(E(0) +M)]
where primes denote derivatives with respect to k. The N dependent termvanishes on doing the angular integrals, and the other terms in (E.2) simplifyto
E(Ω2)i =− 1
15Bi(Ω ·∇)(E(2) − M (2)) + 4
15Ω ·B ∇i(E
(2) − M (2))
+ 35Ωi B ·∇(E(2) + 11
9M (2)) + Ωl(E
(2) − M (2))[16B[l,i] +
(3) + M (3)). (E.3)
Here, B(l,i) ≡ Bl,i + Bi,l, B[l,i] ≡ Bl,i − Bi,l, and E(2), ..., M (3) have beendefined in Eq. (10.54).
For a constant τ(k) = τ0 say, we simply have E(2) = 12τ 20u
(0)2, M (2) = 12τ 20 b
while E(3) and M (3) depend on the velocity and magnetic spectra.
Now consider the third term in (10.38). Using Eqs. (10.27) and (10.35) this isgiven by,
τ 2 BjlmI(0)lk,md
ǫjlmk ·Ω+ ǫjltktΩm − 2ǫjltk ·Ω ktkm
k ·B(v(0)lk −mlk)
Using Eqs. (10.40) and (10.41), only those terms in v(0)lk and mlk which contain
an even number of ki survive after doing the angular integrals over k in (E.4).Further, those terms which contain a spatial derivative do not contribute, since(E.4) already contains a spatial derivative, and we are keeping only terms upto first order in the R derivatives. Also, in those terms that have ǫjlt, the
part of the velocity and magnetic correlations proportional to kl give zerocontribution. Using these properties, and doing the resulting angular integrals,we get
E(Ω3)i =− 7
(2) − M (2))]− 215Ωl∇i[Bl(E
(2) − M (2))]
(2) − M (2))]. (E.5)
F Comparison with the paper by Radler, Kleeorin, and Rogachevskii
If one were to assume a power law form for the spectrum of velocity andmagnetic correlations in the range k0 < k < kd (as done in Ref. ),
4πk2E(0) = (q − 1)u(0)2
, 4πk2M = (q − 1)b2
and also take τ(k) = τ ∗(k) = 2τ0(k/k0)1−q, as in Ref. , then
E(2) = 43τ 20
2, M (2) = 4
and hence all the τ 20 terms in Eqs. (10.57)–(10.61), would have to be multipliedby 4/3. In this case one also has E(3) = −(q+2)E(2) and M (3) = −(q+2)M (2),so one has for the last term in Eq. (10.61)
E(3) + M (3)]
= − 845(q + 2)τ 20
u(0)2 + b2
This leads to
αij =13τ0j · b− 16
δijΩ ·∇(u(0)2 − 13b2)
− 1124(Ωi∇j + Ωj∇i)(u(0)2 + 3
ηij =13τ0δij u(0)2, (F.5)
γ = −16τ0∇
u(0)2 − b2)
− 29τ 20Ω×∇
u(0)2 + b2)
δ = 29Ωτ 20
u(0)2 − b2)
κijk =29τ 20 (Ωjδik + Ωkδij)
[u(0)2 + 75b2] + 2
5(q − 1)[u(0)2 + b2]
Our results for the rotation dependent terms in α, γ, δ (and κ when q = 1), allagree exactly with the result of ; however we get a different coefficient (2/5instead of −4/5) in front of the term proportional to (q− 1) in the expressionfor κijk.
G Calculation of Ψ1 and Ψ2
In order to calculate Ψ1 and Ψ2 we use the induction equation Eq. (10.8) forthe fluctuating field. We then get for Ψ1
∫ ∫ ∫
ul(k + 12K − k′)bk(−k + 1
i(kp +12Kp)i(−kj +
iK·R d3K d3k′ + T1(k)
Here, T1(k) subsumes the triple correlations of the small scale u and b fieldsand the microscopic diffusion terms that one gets on substituting Eq. (10.8)into Eq. (10.67). We can later, if needed, approximate this term as a dampingterm under the MTA. We transform from the variables (k′,K) to a new set(k′,K ′) where K ′ = K − k′, use the definition of the velocity-magnetic fieldcorrelation χlk(k,R), and carry out the integral over K ′ to write
d3k′ eik′·R Bm(k
′) (ikp +12ik′
j +12∇j)χlk(k − 1
2k′,R) + T1(k)
Once again, since B varies only on large scales, B(k′) only contributes at smallk′. One can then make a small k′ expansion of χlk, and do the k′ integral,retaining only terms which are linear in the large scale derivatives, to get
∂ks) + 1
−kj∇pBm)χlk +12i(kp∇jχlk − kj∇pχlk)Bm
This agrees with the result given in Eq. (10.70) for the upper sign.
We now turn to Ψ2. Again, using the induction equation (10.8) for the fluctu-ating field, we get for Ψ2
∫ ∫ ∫
ul(−k + 12K − k′)bi(k + 1
i(−kp +12Kp)i(−kj +
iK·R d3K d3k′ + T2(k)
where T2(k) represents the triple correlations and the microscopic diffusionterms that one gets on substituting Eq. (10.8) into Eq. (10.67). We transformfrom the variables (k′,K) to a new set (k′,K ′) where K ′ = K − k′. It isalso convenient to change from the variables (k,k′) to (−k,−k′). Under thischange we have
ul(k + 12k′ + 1
2K ′)bi(−k − 1
2k′ + 1
′) i[kp +12(K ′
p − k′p)]i(kj +
j − k′j))
ei(K′−k′)·R d3K ′ d3k′ + T2(k)
We can now carry out the integral over K ′ using the definition of the velocity-magnetic field correlation χli(k,R), to get
′) (ikp − 12ik′
(ikj − 12ik′
j +12∇j)χli(k + 1
2k′,R) + T2(k)
Once again, since B varies only on large scales, B(k′) only contributes at smallk′. One can then make a small k′ expansion of χlk, and do the k′ integral,retaining only terms which are linear in the large scale derivatives, to get
− kpkj(Bmχli +12i∇sBm
∂ks) + 1
+kj∇pBm)χli +12i(kp∇jχli + kj∇pχli)Bm
We have used here∫
′·Rd3k′ = Bm(R), (G.8)
′)eik′·Rd3k′ = ∇jBm(R). (G.9)
Interchanging the indices i and k in Ψ2 yields the result given in Eq. (10.70)for the lower sign.
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