zotero/storage/EYUD8GXV/.zotero-ft-cache

Under consideration for publication in J. Plasma Phys.

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arXiv:1903.07829v2 [physics.plasm-ph] 29 Aug 2019

Dynamo theories

Fran¸cois Rincon1,2†
1Universit´e de Toulouse; UPS-OMP; IRAP: Toulouse, France 2CNRS; IRAP; 14 avenue Edouard Belin, F-31400 Toulouse, France
(Received xx; revised xx; accepted xx)
These lecture notes are based on a tutorial given in 2017 at a plasma physics winter school in Les Houches. Their aim is to provide a self-contained graduate-student level introduction to the theory and modelling of the dynamo effect in turbulent fluids and plasmas, blended with a review of current research in the field. The primary focus is on the physical and mathematical concepts underlying different (turbulent) branches of dynamo theory, with some astrophysical, geophysical and experimental context disseminated throughout the document. The text begins with an introduction to the rationale, observational and historical roots of the subject, and to the basic concepts of magnetohydrodynamics relevant to dynamo theory. The next two sections discuss the fundamental phenomenological and mathematical aspects of (linear and nonlinear) smalland large-scale MHD dynamos. These sections are complemented by an overview of a selection of current active research topics in the field, including the numerical modelling of the geo- and solar dynamos, shear dynamos driven by turbulence with zero net helicity, and MHD-instability-driven dynamos such as the magnetorotational dynamo. The difficult problem of a unified, self-consistent statistical treatment of small and largescale dynamos at large magnetic Reynolds numbers is also discussed throughout the text. Finally, an excursion is made into the relatively new but increasingly popular realm of magnetic-field generation in weakly-collisional plasmas. A short discussion of the outlook and challenges for the future of the field concludes the presentation.

Contents‡

1. Introduction

4

1.1. About these notes

4

1.2. Observational roots of dynamo theory

4

1.3. What is dynamo theory about ?

8

1.4. Historical overview of dynamo research

10

1.5. An imperfect dichotomy

12

1.6. Outline

12

2. Setting the stage for MHD dynamos

13

2.1. Magnetohydrodynamics

13

2.1.1. Compressible MHD equations

13

2.1.2. Important conservation laws in ideal MHD

14

2.1.3. Magnetic-field energetics

15

2.1.4. Incompressible MHD equations for dynamo theory

16

2.1.5. Shearing sheet model of differential rotation

17

2.2. Important scales and dimensionless numbers

18

† Email address for correspondence: francois.rincon@irap.omp.eu ‡ Subsections marked with asterisks contain some fairly advanced, technical or specialised
material, and may be skipped on a first reading.

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2.2.1. Reynolds numbers

18

2.2.2. The magnetic Prandtl number landscape

18

2.2.3. Strouhal number

21

2.3. Dynamo fundamentals

21

2.3.1. Kinematic versus dynamical regimes

22

2.3.2. Anti-dynamo theorems

22

2.3.3. Slow versus fast dynamos

24

3. Small-scale dynamo theory

25

3.1. Evidence for small-scale dynamos

26

3.2. Zel’dovich-Moffatt-Saffman phenomenology

26

3.3. Magnetic Prandtl number dependence of small-scale dynamos

30

3.3.1. Small-scale dynamo fields at P m > 1

30

3.3.2. Small-scale dynamo fields at P m < 1

31

3.4. Kinematic theory: the Kazantsev model

33

3.4.1. Kazantsev-Kraichnan assumptions on the velocity field

34

3.4.2. Equation for the magnetic field correlator

34

3.4.3. Closure procedure in a nutshell*

35

3.4.4. Closed equation for the magnetic correlator

36

3.4.5. Solutions

36

3.4.6. Different regimes

37

3.4.7. Critical Rm

38

3.4.8. Selected results in the large-P m regime*

39

3.4.9. Miscellaneous observations

43

3.5. Dynamical theory

43

3.5.1. General phenomenology

44

3.5.2. Nonlinear growth

46

3.5.3. Saturation at large and low P m

47

3.5.4. Reconnecting dynamo fields

47

3.5.5. Nonlinear extensions of the Kazantsev model*

48

4. Fundamentals of large-scale dynamo theory

49

4.1. Evidence for large-scale dynamos

50

4.2. Some phenomenology

50

4.2.1. Coherent large-scale shearing: the Ω effect

50

4.2.2. Helical turbulence: Parker’s mechanism and the α effect

51

4.2.3. Writhe, twist, and magnetic helicity

52

4.3. Kinematic theory: mean-field electrodynamics

54

4.3.1. Two-scale approach

54

4.3.2. Mean-field ansatz

56

4.3.3. Symmetry considerations

56

4.3.4. Mean-field equation for pseudo-isotropic homogeneous flows

58

4.3.5. α2, αΩ and α2Ω dynamo solutions

58

4.3.6. Calculation of mean-field coefficients: First Order Smoothing

60

4.3.7. FOSA derivations of α and β for homogeneous helical turbulence 61

4.3.8. Third-order-moment closures: EDQNM and τ -approach*

64

4.4. Mean-field effects in stratified, rotating, and shearing flows

65

4.4.1. α effect in a stratified, rotating flow

66

4.4.2. Turbulent pumping*

67

4.4.3. R¨adler and shear-current effects*

68

4.5. Difficulties with mean-field theory at large Rm

70

4.5.1. The overwhelming growth of small-scale dynamo fields

70

Dynamo theories

3

4.5.2. Kazantsev model for helical turbulence*

71

4.5.3. P m-dependence of kinematic helical dynamos

73

4.6. Dynamical theory

74

4.6.1. Phenomenological considerations

74

4.6.2. Numerical results

75

4.6.3. Magnetic helicity perspective on helical dynamo quenching

76

4.6.4. Dynamical saturation in the helical Kazantsev model*

82

4.6.5. Quenching of turbulent diffusion

84

4.7. Overview of mean-field dynamo theory applications

85

4.7.1. Low-dimensional nonlinear mean-field models

85

4.7.2. Mean-field electrodynamics as a numerical analysis tool

87

5. The diverse, challenging complexity of large-scale dynamos

88

5.1. Dynamos driven by rotating convection: the solar and geo- dynamos

89

5.1.1. A closer look at the dynamo regimes of the Sun and the Earth

89

5.1.2. Global simulations of dynamos driven by rotating convection

92

5.2. Large-scale shear dynamos driven by turbulence with zero net helicity 96

5.2.1. Numerical simulations

96

5.2.2. Shear dynamo driven by a kinematic stochastic α effect

96

5.2.3. Shear dynamo driven by nonlinear MHD fluctuations*

100

5.3. Subcritical dynamos driven by MHD instabilities in shear flows

101

5.3.1. Numerical evidence, and a few puzzling observations

101

5.3.2. Self-sustaining nonlinear processes

103

5.3.3. Nonlinear dynamo cycles and subcritical bifurcations*

106

5.3.4. P m-dependence of the MRI dynamo transition*

107

5.3.5. From the MRI dynamo to large-scale accretion-disc dynamos*

110

5.3.6. Other instability-driven and subcritical dynamos

114

5.4. The large-Rm frontier

115

6. Dynamos in weakly-collisional plasmas

117

6.1. Kinetic versus fluid descriptions

118

6.2. Plasma dynamo regimes

119

6.2.1. Different regimes and orderings

119

6.2.2. Making compromises: the hybrid Vlasov-Maxwell model

120

6.2.3. The 3D-3V “small-scale” dynamo problem

121

6.3. Collisionless dynamo in the unmagnetised regime

121

6.4. Introduction to the dynamics of magnetised plasmas

123

6.4.1. Fluid-scale dynamics: pressure anisotropies and µ-conservation 123

6.4.2. Kinetic-scale dynamics: pressure-anisotropy-driven instabilities 126

6.4.3. Saturation of kinetic instabilities in a shearing magnetised plasma* 128

6.5. Collisionless dynamo in the magnetised regime

131

6.5.1. Is dynamo possible in the magnetised regime ?

131

6.5.2. How do magnetisation and kinetic effects affect dynamo growth ?* 132

6.6. Uncharted plasma physics

135

7. A subjective outlook for the future

136

7.1. Mathematical theory

136

7.2. Experiments and observations

137

7.3. The privileged position of numerics

138

Appendix A

139

A.1. MHD, astrophysical fluid dynamics and plasma physics textbooks

139

A.2. Dynamo theory books and reviews

139

A.3. Astrophysical and planetary dynamo reviews

140

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F. Rincon

1. Introduction

1.1. About these notes
These lecture notes expand (significantly) on a two-hour tutorial given at the 2017 Les Houches school “From laboratories to astrophysics: the expanding universe of plasma physics”. Many excellent books and reviews have already been written on the subjects of dynamo theory, planetary and astrophysical magnetism. Most of them, however, are either quite specialised, or simply too advanced for non-specialists seeking a general entrypoint into the field. The multidisciplinary context of this school, taking place almost a century after Larmor’s original idea of self-exciting fluid dynamos, provided an ideal opportunity to craft a self-contained, wide-ranging, yet relatively accessible introduction to the subject.
One of my central preoccupations in the writing process has been to attempt to distill in clear and relatively concise terms the essence of each of the problems covered, and to highlight to the best of my abilities the successes, limitations and connections of different lines of research in logical order. Although I may not have entirely succeeded, my sincere hope is that this review will nevertheless turn out to be generally useful to observers, experimentalists, theoreticians, PhD students, newcomers and established researchers in the field alike, and will foster new original research on dynamos of all kinds. It is quite inevitable, though, that such an ideal can only be sought at the expense of total exhaustivity and mathematical rigour, and necessitates making difficult editorial choices. To borrow Keith Moffatt’s wise words in the introduction of his 1973 Les Houches lecture notes on fluid dynamics and dynamos, “it will be evident that in the time available I have had to skate over certain difficult topics with indecent haste. I hope however that I have succeeded in conveying something of the excitement of current research in dynamo theory and something of the general flavour of the subject. Those already acquainted with the subject will know that my account is woefully one-sided”. Suggestions for further reading on the many different branches of dynamo research discussed in the text are provided throughout the document and in App. A to mitigate these limitations.
Finally, while the main focus of the notes is on the physical and practical mathematical aspects of dynamo theory in general, contextual information is provided throughout to connect the material presented to astrophysical, geophysical and experimental dynamo problems. In particular, a selection of astrophysical and planetary dynamo research topics, including the geo-, solar, and accretion-disc dynamos, is highlighted in the most advanced parts of the review to give a flavour of the diversity of research and challenges in the field.

1.2. Observational roots of dynamo theory
Dynamo theory finds its roots in the human observation of the Universe, and in the quest to understand the origin of magnetic fields observed or inferred in a variety of astrophysical systems. This includes planetary magnetism (the Earth, other planets and their satellites), solar and stellar magnetism, and cosmic magnetism (galaxies, clusters and the Universe as a whole). We will therefore start with a brief overview of the main features of astrophysical and planetary magnetism.
Consider first solar magnetism, whose evolution on human timescales and day-today monitoring make it a more intuitive dynamical phenomenon to apprehend than other forms of astrophysical magnetism. For the purpose of the discussion, we can single out two “easily” observable dynamical magnetic timescales on the Sun. The first one is the eleven-year magnetic cycle timescale over which the large-scale solar magnetic field reverses. The solar cycle shows up in many different observational records, the

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5

Figure 1. Top: large-scale solar magnetism. The eleven-year magnetic solar cycle (#21 − 22) observed in the chromosphere through the Hα spectral line (full solar discs), and historical sunspot-number record (Credits: NOAA/Zu¨rich/RDC/CNRS/INSU/Ondresjov Observatory/HAO). Bottom: local and global solar magnetic dynamics. The rapidly-evolving small-scale magnetic carpet, spicules and sunspot arches imaged near the limb in the lower chromosphere through the CaH spectral line (Credits: SOT/Hinode/JAXA/NASA).
most well-known of which is probably the number of sunspots as a function of time, see Fig. 1 (top) (note that the eleven-year cycle is also chaotically modulated on longer timescales). Large-scale solar magnetism is characterised by an average (mean) field of only a few tens of Gauss (see e.g. the review by Charbonneau 2014), however the field

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F. Rincon

itself can exceed kiloGauss strengths in large-scale features like sunspots†. There is also a lot of dynamical, small-scale, disordered magnetism in the solar surface photosphere and chromosphere, evolving on short time and spatial scales comparable to those of thermal convective motions at the surface (from a minute to an hour, and from a few kilometres to a few thousands of kilometres). This so-called “network” and “internetwork” small-scale magnetism, depicted in Fig. 1 (bottom), was discovered much more recently (Livingston & Harvey 1971). Its typical strength ranges from a few to a few hundred Gauss, and does not appear to be significantly modulated over the course of the global solar cycle (see e.g. Solanki et al. (2006); Stenflo (2013) for reviews). Large-scale stellar magnetic fields, including time-dependent ones, have been detected on many other stars (e.g. Donati & Landstreet 2009), but only for the Sun do we have accurate, temporally and spatiallyresolved direct measurements of small-scale stellar magnetism.
The second major natural, human-felt phenomenon that inspired the development of dynamo theory is of course the Earth’s magnetic field, whose strength at the surface of the Earth is of the order of 0.1 Gauss (10−5 T, Finlay et al. 2010). The dynamical evolution and structure of the field, including its many irregular reversals over a hundred-thousand to million-year timescale, is established through paleomagnetic and archeomagnetic records, marine navigation books, and is now monitored with satellites, as shown in Fig. 2 (top). While the terrestrial field is probably highly multiscale and multipolar in the liquid iron part of the core where it is generated, it is primarily considered as a form of large-scale dynamical magnetism involving a north and south magnetic pole. Several other planets of the solar system also exhibit large-scale, low-multipole surface magnetic fields and magnetospheres. Figure 2 (bottom) shows auroral emissions on Jupiter, whose magnetic field has a typical surface strength of a few Gauss (a few 10−4 T, Khurana et al. 2004). Just as in the Earth’s case, the large-scale external field of the other magnetic planets is almost certainly not representative of the structure of the field in the interior.
Moving further away from the Earth, we also learned in the second part of the twentieth century that galaxies, including our own Milky Way, host magnetic fields with a typical strength of the order of a few 10−5 Gauss (Beck & Wielebinski 2013). For a long time, observations would only reveal the ordered large-scale, global magnetic structure whose projection in the galactic plane would often take the form of spirals, see Fig. 3 (top). But recent high-resolution observations of polarised dust emission in our galaxy, displayed in Figure 3 (bottom), have now also established that the galactic magnetic field has a very intricate multiscale structure, of which a large-scale ordered field is just one component.
Magnetic fields of the order of a few 10−6 Gauss are also measured in the hot intracluster medium (ICM) of galaxy clusters (e. g. Carilli & Taylor 2002; Bonafede et al. 2010). The large-scale global structure and orientation of cluster fields, if any, is not well-determined (it should be noted in this respect that global differential rotation is not thought to be very important in clusters, unlike in individual galaxies, stars and planets). On the other hand, synchrotron polarimetry measurements in the radio-lobes of active galactic nuclei (AGN), such as that shown in Fig. 4 (top), suggest that there is a significant “small-scale”, turbulent ICM field component on scales comparable to or even smaller than a kiloparsec (Vogt & Enßlin 2005). Visible-light observations of the ICM, including in the Hα spectral line, also reveal the presence of colder gas structured into magnetised filaments, see Fig. 4 (bottom).
There has been as yet no direct detection of magnetic fields on even larger, cosmological scales. Magnetic fields in the filaments of the cosmic web and intergalactic medium are

† 1 Gauss= 10−4 Tesla is the most commonly encountered magnetic-field unit in astrophysics. Gaussian c.g.s. units are used throughout most of the text.

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Figure 2. Top: direct satellite measurements of the Earth’s magnetic field strength (in nano Teslas) in 2014 at an altitude of 450 km (Credits: Swarm/CNES/ESA). Bottom: ultra-violet emission of a 1998 Jupiter aurora (Credits: J. Clarke/STIS/WFPC2/HST/NASA/ESA).
thought to be of the order of, but no larger than a few 10−9 Gauss at Megaparsec scale. This upper bound can be derived from a variety of observational constraints, including on the cosmic microwave background (Planck Collaboration et al. 2016). Note however that a lower bound on the typical intergalactic magnetic-field strength, of the order of a few 10−16 Gauss, has been derived from high-energy γ-ray observations (Neronov & Vovk 2010). A detailed discussion of the current observational bounds on the scales and amplitudes of magnetic fields in the early Universe can be found in the review by Durrer & Neronov (2013).

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Figure 3. Top: large-scale spiral magnetic structure (line segments) of the M51 galaxy established from radio observations of polarised synchrotron emission by cosmic rays (Credits: MPIfR Bonn and Hubble Heritage Team. Graphics: Sterne and Weltraum). Bottom: map of the microwave galactic dust emission convolved with galactic magnetic-field lines reconstructed from polarisation maps of the dust emission (Credits: M. A. Miville-Deschˆenes/CNRS/ESA/Planck collaboration).
1.3. What is dynamo theory about ?
The dynamical nature, spatial structure and measured amplitudes of astrophysical and planetary magnetic fields strongly suggest that they must in most instances have been amplified to, and are further sustained at significant levels by internal dynamical mechanisms. In the absence of any such mechanism, calculations of magnetic diffusion notably show that “fossil” fields present in the early formation stages of different objects should decay over cosmologically short timescales, see e.g. Weiss (2002); Roberts &

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Figure 4. Top: Faraday rotation measure map (a proxy for the line-of-sight component of the magnetic field) in the synchrotron-illuminated radio-lobes of the Hydra A cluster (Credits: Taylor & Perley/VLA/NRAO). Bottom: visible-light observations of magnetised filaments in the core of the Perseus cluster (Credits: Fabian et al./HST/ESA/NASA).

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King (2013) for geomagnetic estimates. Besides, even in relatively high-conductivity environments such as stellar interiors, the fossil-field hypothesis cannot easily explain the dynamical evolution and reversals of large-scale magnetic fields over a timescale of the order of a few years either. So, what are these field-amplifying and field-sustaining mechanisms ? Most astrophysical objects (or at least some subregions within them) are fluids/plasmas in a dynamical, turbulent state. Even more importantly for the problem at hand, these fluids/plasmas are electrically conducting. This raises the possibility that internal flows create an electromotive force leading to the inductive self-excitation of magnetic fields and electrical currents. This idea of self-exciting fluid dynamos was first put forward a century ago by Larmor (1919) in the context of solar (sunspot) magnetism.
From a fundamental physics perspective, dynamo theory therefore generally aims at describing the amplification and sustainment of magnetic fields by flows of electrically conducting fluids and plasmas – most importantly turbulent ones. Important questions include whether such an excitation and sustainment is possible at all, at which rate the growth of initially very weak seed fields can proceed, at what magnetic energy such processes saturate, and what the time-dependence and spatial structure of dynamogenerated fields is in different regimes. At the heart of these questions lies a variety of difficult classical linear and nonlinear physics and applied mathematics problems, many of which have a strong connection with more general (open) problems in turbulence theory, including closure problems.
While fundamental theory is a perfectly legitimate object of study on its own, there is also a strong demand for “useful” or applicable mathematical models of dynamos. Obviously, researchers from different backgrounds have very different conceptions of what a useful model is, and even of what theory is. Astronomers for instance are keen on phenomenological, low-dimensional models of large-scale astrophysical magnetism with a few free parameters, as these provide an intuitive framework for the interpretation of observations. Solar and space physicists are interested in more quantitative and fine-tuned versions of such models to predict solar activity in the near future. Experimentalists need models that can help them minimise the mechanical power required to excite dynamos in highly-customised washing-machines filled with liquid sodium or plasma. Another major challenge of dynamo theory, then, is to build meaningful bridges between these different communities by constructing conceptual and mathematical dynamo models that are physically-grounded and rigorous, yet tractable and predictive. The overall task of dynamo theoreticians therefore appears to be quite complex and multifaceted.

1.4. Historical overview of dynamo research
Let us now give a very brief overview of the history of the subject as a matter of context for the main theoretical developments of the next sections. More detailed historical accounts are available in different reviews and books, including the very informative Encyclopedia of Geomagnetism and Paleomagnetism (Gubbins & HerreroBervera 2007), and the book by Molokov, Moreau & Moffatt (2007) on historical trends in magnetohydrodynamics.
Dynamo theory did not immediately take off after the publication of Larmor’s original ideas on solar magnetism. Viewed from today’s perspective, it is clear that the intrinsic geometric and dynamical complexity of the problem was a major obstacle to its development. This complexity was first hinted by the demonstration by Cowling (1933) that axisymmetric dynamo action is not possible (§2.3.2). Cowling’s conclusions were not particularly encouraging† and apparently even led Einstein to voice a pessimistic outlook
† “The theory proposed by Sir Joseph Larmor, that the magnetic field of a sunspot is

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on the subject (Krause 1993). The first significant positive developments only occurred after the second world war, when Elsasser (1946, 1947), followed by Bullard & Gellman (1954), set about formulating a spherical theory of magnetic field amplification by nonaxisymmetric convective motions in the liquid core of the Earth. In the same period, Batchelor (1950) and Schlu¨ter & Biermann (1950) started investigating the problem of magnetic field amplification by generic three-dimensional turbulence from a more classical statistical hydrodynamic perspective. In the wake of Elsasser’s and Bullard’s work, Parker (1955a) published a seminal semi-phenomenological article describing how differential rotation and small-scale cyclonic motions could combine to excite large-scale magnetic fields (§4.2.2). Parker also notably showed how such a mechanism could excite oscillatory dynamo modes (now called Parker waves) reminiscent of the solar cycle. The spell of Cowling’s theorem was definitely broken a few years later when Herzenberg (1958) and Backus (1958) found the first mathematical working examples of fluid dynamos.
The 1960s saw the advent of statistical dynamo theories. Braginskii (1964a,b) first showed how an ensemble of non-axisymmetric spiral wavelike motions could lead to the statistical excitation of a large-scale magnetic field. Shortly after that, Steenbeck, Krause & R¨adler (1966) published their mean-field theory of large-scale magnetic-field generation in flows lacking parity/reflectional/mirror invariance (§4.3). These and a few other pioneering studies (e.g. Moffatt 1970a; Vainshtein 1970) put Parker’s mechanism on a much stronger mathematical footing. In the same period, Kazantsev (1967) developed a quintessential statistical model describing the dynamo excitation of small-scale magnetic fields in non-helical (parity-invariant) random flows (§3.4). Interestingly, Kazantsev’s work predates the observational detection of “small-scale” solar magnetic fields. This golden age of dynamo research extended into the 1970s with further developments of the statistical theory, and the introduction of the concept of fast dynamos by Vainshtein & Zel’dovich (1972), which offered a new phenomenological insight into the dynamics of turbulent dynamo processes (§2.3.3). “Simple” helical dynamo flows that would later prove instrumental in the development of experiments were also found in that period (Roberts 1970, 1972; Ponomarenko 1973).
It took another few years for the different theories to be vindicated in essence by numerical simulations, as the essentially three-dimensional nature of dynamos made the life of numerical people quite hard at the time. In a very brief but results-packed article, Meneguzzi, Frisch & Pouquet (1981) numerically demonstrated both the excitation of a large-scale magnetic field in small-scale homogeneous helical fluid turbulence, and that of small-scale magnetic fields in non-helical turbulence. These results marked the beginnings of a massive numerical business that is more than ever flourishing today. Experimental evidence for dynamos, on the other hand, was much harder to establish. Magnetohydrodynamic (MHD) fluids are not easily available on tap in the laboratory and the properties of liquid metals such as liquid sodium create all kinds of powersupply, dissipation and safety problems. Experimental evidence for helical dynamos was only obtained at the dawn the twenty-first century in the Riga (Gailitis et al. 2000) and Karlsruhe experiments (Stieglitz & Mu¨ller 2001) relying upon very constrained flow geometries designed after the work of Ponomarenko (1973) and Roberts (1970, 1972). Readers are referred to an extensive review paper by Gailitis et al. (2002) for further details. Further experimental evidence of fluid dynamo action in a freer, more homogeneous turbulent setting has since been sought by several groups, but has so far

maintained by the currents it induces in moving matter, is examined and shown to be faulty ; the same result also applies for the similar theory of the maintenance of the general field of Earth and Sun.”

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only been reported in the Von K´arm´an Sodium experiment (VKS, Monchaux et al. 2007). The decisive role of soft-iron solid impellers in the excitation of a dynamo in this experiment remains widely debated (see short discussion and references in §2.1.4). Overall, the VKS experiment provides a good flavour of the current status, successes and difficulties of the liquid metal experimental approach to exciting a turbulent dynamo. For broader reviews and perspectives on experimental dynamo efforts, readers are referred to Stefani et al. (2008); Verhille et al. (2010); Lathrop & Forest (2011).

1.5. An imperfect dichotomy
The historical development of dynamo theory has roughly proceeded along the lines of the seeming observational dichotomy between large and small-scale magnetism, albeit not in a strictly causal way. We usually refer to the processes by which flows at a given scale statistically produce magnetic fields at much larger scales as large-scale dynamo mechanisms. Global rotation and/or large-scale shear usually (though not always) plays an important role in this context. As we shall discover, large-scale dynamos also naturally produce a significant amount of small-scale magnetic field, however magnetic fields at scales comparable to or smaller than that of the flow can also be excited by independent small-scale dynamo mechanisms if the fluid/plasma is sufficiently ionised. Importantly, the latter are usually much faster and can be excited even in the absence of system rotation or shear.
The dichotomy between small- and large-scale dynamos has the merits of clarity and simplicity, and will therefore be used in this tutorial as a rough guide to organise the presentation. However it is not as clear-cut and perfect as it looks at first glance, for a variety of reasons. Most importantly, large-scale and small-scale magnetic-field generation processes can take place simultaneously in a given system, and the outcome of these processes is entirely up to one of the most dreaded words in physics: nonlinearity. In fact, most astrophysical and planetary magnetic fields are in a saturated, dynamical nonlinear state: they can have temporal variations such as reversals or rapid fluctuations, but their typical strength does not change by many orders of magnitudes over long periods of time; their energy content is also generally not small comparable to that of fluid motions, which suggests that they exert dynamical feedback on these motions. Therefore, dynamos in nature involve strong couplings between multiple scales, fields, and dynamical processes, including distinct dynamo processes. Nonlinearity significantly blurs the lines between large and small-scale dynamos (and in some cases also other MHD instabilities), and adds a whole new layer of dynamical complexity to an already difficult subject. The small-scale/large-scale “unification” problem is currently one of the most important in dynamo research, and will accordingly be a recurring theme in this review.

1.6. Outline
The rest of the text is organised as follows. Section 2 introduces classic MHD material and dimensionless quantities and scales relevant to the dynamo problem, as well as some important definitions, fundamental results and ideas such as anti-dynamo theorems and the concept of fast dynamos. The core of the presentation starts in §3 with an introduction to the phenomenological and mathematical models of small-scale MHD dynamos. The fundamentals of linear and nonlinear large-scale MHD dynamo theory are then reviewed in §4. These two sections are complemented in §5 by essentially phenomenological discussions of a selection of advanced research topics including large-scale stellar and planetary dynamos driven by rotating convection, large-scale dynamos driven by sheared turbulence with vanishing net helicity, and dynamos mediated by MHD instabilities such

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as the magnetorotational instability. Section 6 provides an introduction to the relatively new but increasingly popular realm of dynamos in weakly-collisional plasmas. The notes end with a concise discussion of perspectives and challenges for the field in §7. A selection of good reads on the subject can be found in App. A. Subsections marked with asterisks contain some fairly advanced, technical or specialised material, and may be skipped on first reading.

2. Setting the stage for MHD dynamos
2.1. Magnetohydrodynamics
Most of these notes, except §6, are about fluid dynamo theories in the non-relativistic, collisional, isotropic, single fluid MHD regime in which the mean free path of liquid, gas or plasma particles is significantly smaller than any dynamical scale of interest, and than the smallest of the particle gyroradii. We will also assume that the dynamics takes place at scales larger than the ion inertial length, so that the Hall effect can be discarded. The isotropic MHD regime is applicable to liquid metals, stellar interiors and galaxies to some extent, but not quite to the ICM for instance, as we will discuss later. Accretion discs can be in a variety of plasma states ranging from hot and weakly collisional to cold and multifluid.

2.1.1. Compressible MHD equations
Let us start from the equations of compressible, viscous, resistive magnetohydrodynamics. First, we have the continuity (mass conservation) equation

∂ρ ∂t

+

∇

·

(ρU)

=

0

,

(2.1)

where ρ is the gas density and U is the fluid velocity field, and the momentum equation

ρ

∂U ∂t

+

U

·

∇

U

=

−∇P

+

J

× c

B

+

∇

·

τ

+

F(x,

t)

,

(2.2)

where P is the gas pressure, τij = µ (∇iUj + ∇jUi − (2/3) δij∇ · U) is the viscous stress tensor (µ is the dynamical viscosity and ν = µ/ρ is the kinematic viscosity), F is aforce per unit volume representing any kind of external stirring mechanism (impellers, gravity, spoon, supernovae, meteors etc.), B is the magnetic field, J = (c/4π)∇×B is the current density, and J × B/c, the Lorentz force, describes the dynamical feedback exerted by the magnetic field on fluid motions. The evolution of B is governed by the induction equation

∂B ∂t

=

∇×(U

×

B)

−

∇×(η∇×B)

,

supplemented with the solenoidality condition

(2.3)

∇·B=0 .

(2.4)

Equation (2.3) is derived from the Maxwell-Faraday equation and a simple, isotropic Ohm’s law for collisional electrons,

J=σ

E

+

U

× c

B

,

(2.5)

where σ is the electrical conductivity of the fluid. The first term E = U × B on the r.h.s. of equation (2.3) is called the electromotive force (EMF) and describes the induction of magnetic field by the flow of conducting fluid from an Eulerian perspective. The second

14

F. Rincon

term describes the diffusion of magnetic field in a non-ideal fluid of magnetic diffusivity η = c2/(4πσ). Both the Lorentz force and EMF terms in equations (2.2)-(2.3) play a
very important role in the dynamo problem, but so do viscous and resistive dissipation.
Finally, we have the internal energy, or entropy equation

ρT

∂S ∂t

+U·∇S

= Dµ + Dη + ∇ · (K∇T ) ,

(2.6)

where T is the gas temperature, S ∝ P/ργ is the entropy (γ is the adiabatic index), Dµ and Dη stand for the viscous and resistive dissipation, K is the thermal conductivity and the last term on the r.h.s. stands for thermal diffusion (we could also have added an inhomogeneous heat source, or explicit radiative transfer). An equation of state for the thermodynamic variables, like the ideal gas law P = ρRT , is also required in order to close this system (R here denotes the gas constant).
The compressible MHD equations describe the dynamics of waves, instabilities, turbulence and shocks in all kinds of astrophysical fluid systems, including stratified and/or (differentially) rotating fluids, and accommodate a large range of dynamical magnetic phenomena including dynamos and (fluid) reconnection. The ideal MHD limit corresponds to ν = η = K = 0. The reader is referred to the astrophysical fluid dynamics lecture notes of Ogilvie (2016), published in this journal, for a very tidy derivation and presentation of ideal MHD.

2.1.2. Important conservation laws in ideal MHD

There are two particularly important conservation laws in the ideal MHD limit that

involve the magnetic field and are of primary importance in the context of the dynamo

problem. To obtain the first one we combine the continuity and ideal induction equations

into

D Dt

B ρ

=

B ρ

·

∇U

,

(2.7)

where D/Dt = ∂/∂t + U · ∇ is the Lagrangian derivative. Equation (2.7) for B/ρ has the

same form as the equation describing the evolution of the Lagrangian separation vector

δr between two fluid particles,

Dδr Dt

=

δr

·

∇U

.

(2.8)

Hence, magnetic-field lines in ideal MHD can be thought of as being “frozen into” the

fluid just as material lines joining fluid particles. This is called Alfv´en’s theorem. Using

this equation and equation (2.4), it is also possible to show that the magnetic flux through

material surfaces δS (deformable surfaces moving with the fluid) is conserved in ideal

MHD,

D Dt

(B

·

δS)

=

0

.

(2.9)

If a material surface δS is deformed under the effect of either shearing or compres-

sive/expanding motions, the magnetic field threading it must change accordingly so that

B · δS remains the same. Alfv´en’s theorem enables us to appreciate the kinematics of

the magnetic field in a flow in a more intuitive geometrical way than by just staring at

equations, as it is relatively easy to visualise magnetic-field lines advected and stretched

by the flow. This will prove very helpful to develop an intuition of how small and large-

scale dynamo processes work.

A second important conservation law in ideal MHD in the context of dynamo theory is

the conservation of magnetic helicity Hm = A · B d3r, where A is the magnetic vector

Dynamo theories

15

potential. To derive it, we first write the Maxwell-Faraday equation for A,

1 c

∂A ∂t

=

−E

−

∇ϕ

,

(2.10)

where ϕ is the electrostatic potential. Combining equation (2.10) with equation (2.3)

gives

where

∂ ∂t

(A

·

B)

+

∇ · FHm

=

−2η (∇×B)

·

B

,

(2.11)

FHm = c (ϕB + E × A)

(2.12)

is the total magnetic-helicity flux. In the ideal case, we see that equation (2.11) reduces

to an explicitly conservative local evolution equation for A · B,

∂ ∂t

(A

·

B)

+

∇

·

[cϕB

+

A

×

(U

×

B)]

=

0

,

(2.13)

where equation (2.5) with η = 0 has been used to express the magnetic-helicity flux. Note that both Hm and FHm depend on the choice of electromagnetic gauge and are therefore not uniquely defined. Qualitatively, magnetic helicity provides a measure of the linkage/knottedness of the magnetic field within the domain considered and the conservation of magnetic helicity in ideal MHD is therefore generally understood as a conservation of magnetic linkages in the absence of magnetic diffusion or reconnection (see e.g. Hubbard & Brandenburg (2011); Miesch (2012); Blackman (2015); Bodo et al. (2017) for discussions of magnetic helicity dynamics in different astrophysical dynamo contexts).

2.1.3. Magnetic-field energetics

What about the driving and energetics of the magnetic field ? An enlighting equation in that respect is that describing the local Lagrangian evolution of the magnetic-field strength B associated with a fluid particle in ideal MHD (η = ν = 0),

1 B

DB Dt

=

Bˆ Bˆ

:

∇U

−

∇

·

U

,

(2.14)

where Bˆ = B/B is the unit vector defining the orientation of the magnetic field attached to the fluid particle, and we have used the double dot-product notation Bˆ Bˆ : ∇U = BˆiBˆj∇iUj. Equation (2.14) follows directly from equations (2.1)-(2.7), and shows that any increase of B results from either a stretching of the magnetic field along itself by a flow, or from a compression, and that the rate at which ln B changes is proportional to the local shearing or compression rate of the flow. Note that incompressible motions with no component parallel to the local original/initial background field do not affect the field strength at linear order, and only generate magnetic curvature perturbations (these are shear Alfv´en waves). Going back to full resistive MHD, the global evolution equation for the total magnetic energy within a fixed volume, derived for instance in the classic textbook of Roberts (1967), is

d dt

|B|2 8π

dV

=−

U·

(J

× c

B) dV

−

c 4π

(E × B) · dS −

|J|2 σ

dV

,

(2.15)

where the surface integral is taken over the boundary of the volume, oriented by an outward normal vector. The first term on the r.h.s. is a volumetric term equal to the opposite of the work done by the Lorentz force on the flow, the second surface term is the Poynting flux of electromagnetic energy through the boundaries of the domain

16

F. Rincon

under consideration, and the last term quadratic in J corresponds to Ohmic dissipation of electrical currents into heat. In the absence of a Poynting term (for instance in a periodic domain), we see that magnetic energy can only be generated at the expense of kinetic (mechanical) energy. In other words, we must put in mechanical energy in order to drive a dynamo.

2.1.4. Incompressible MHD equations for dynamo theory
Starting from compressible MHD enabled us to show that compressive motions, which are relevant to a variety of astrophysical situations, can formally contribute to the dynamics and amplification of magnetic fields. However, much of the essence of the dynamo problem can be captured in the much simpler framework of incompressible, viscous, resistive MHD, which we will therefore mostly use henceforth (further assuming constant kinematic viscosity and magnetic diffusivity). In the incompressible limit, ρ is uniform and constant, and the distinction between thermal and magnetic pressure disappears. The magnetic tension part of the Lorentz force provides the only relevant dynamical magnetic feedback on the flow in that case†. Rescaling F and P by ρ and B by (4πρ)1/2, so that B now stands for the Alfv´en velocity

UA

=

√B 4πρ

,

(2.16)

and introducing the total pressure Π = P + B2/2, we can write the incompressible momentum equation as

∂U ∂t

+

U

·

∇

U

=

−∇Π

+

B

·

∇

B

+

ν∆U

+

F(x,

t)

.

The induction equation is rewritten as

(2.17)

∂B ∂t

+

U

·

∇

B

=

B

·

∇

U

+

η∆B

.

(2.18)

This form separates the physical effects of the electromotive force into two parts: advection/mixing represented by U · ∇ B on the left, and induction/stretching represented by B · ∇ U on the right. Magnetic-stretching by shearing motions is the only way to amplify magnetic fields in an incompressible flow of conducting fluid. In order to formulate the problem completely, equations (2.17)-(2.18) must be supplemented with

∇·U=0 , ∇·B=0 ,

(2.19)

and paired with an appropriate set of initial conditions, and boundary conditions in space. The latter can be a particularly tricky business in the dynamo context. Periodic boundary conditions, for instance, are a popular choice among theoreticians but may be problematic in the context of the saturation of large-scale dynamos (§4.6). Certain types of magnetic boundary conditions are also problematic for the definition of magnetic helicity. The choice of boundary conditions and global configuration of dynamo problems is not just a problem for theoreticians either: as mentioned earlier, the choice of soft-iron vs. steel propellers has a drastic effect on the excitation of a dynamo effect in the VKS experiment (Monchaux et al. 2007), raising the question of whether this dynamo is a pure fluid effect or a fluid-structure interaction effect (see e.g. Gissinger et al. 2008b; Giesecke et al. 2012; Kreuzahler et al. 2017; Nore et al. 2018).

† In the compressible case, magnetic pressure exerts a distinct dynamical feedback on the flow. This becomes important if the magnetic energy is locally amplified to a level comparable to the thermal pressure and can notably lead to density evacuation.

Dynamo theories

17

z (vertical or spanwise)
x (radial or shearwise)

⌦

US =

Sx ey y (azimuthal

or

streamwise)

Figure 5. The Cartesian shearing sheet model of differentially rotating flows.

2.1.5. Shearing sheet model of differential rotation

Differential rotation is present in many systems that sustain dynamos, but can take

many different forms depending on the geometry and internal dynamics of the system at

hand. As we will discover in §5.1, working in global cylindrical or spherical geometry is

particularly valuable if we seek to understand how large-scale dynamos like the solar or

geodynamo operate at a global level, because these systems happen to have fairly complex

differential rotation laws and internal shear layers. On the other hand, we do not in

general need all this geometric complexity to understand how rotation and shear affect

dynamo processes at a fundamental physical level. In fact, any possible simplification

is most welcome in this context, as many of the basic statistical dynamical processes

that we are interested in are usually difficult enough to understand at a basic level. In

what follows, we will therefore make intensive use of a local Cartesian representation

of differential rotation, known as the shearing sheet model (Goldreich & Lynden-Bell

1965), that will make it possible to study some essential effects of shear and rotation on

dynamos in a very simple and systematic way.

Consider a simple cylindrical differential rotation law Ω = Ω(R) ez in polar coordinates (R, ϕ, z) (think of an accretion disc or a galaxy). To study the dynamics around a

particular cylindrical radius R0, we can move to a frame of reference rotating at the local angular velocity, Ω ≡ Ω(R0), and solve the equations of rotating MHD locally (including Coriolis and centrifugal accelerations) in a Cartesian coordinate system (x, y, z) centred

on R0, neglecting curvature effects (all of this can be derived rigorously). Here, x corresponds to the direction of the local angular velocity gradient (the radial direction in

an accretion disc), and y corresponds to the azimuthal direction. In the rotating frame,

the differential rotation around R0 reduces to a simple a linear shear flow US = −Sx ey, where S ≡ −R0 dΩ/dR|R0 is the local shearing rate (Fig. 5).
This model enables us to probe a variety of differential rotation regimes by studying

the individual or combined effects of a pure rotation, parametrised by Ω, and of a

pure shear, parametrised by S, on dynamos. For instance, we can study dynamos

in non-rotating shear flows by setting Ω = 0 and varying the shearing rate S with

respect to the other timescales of the problem, or we can study the effects of rigid

rotation on a dynamo-driving flow (and the ensuing dynamo) by varying Ω while

setting S = 0. Cyclonic rotation regimes, for which the vorticity of the shear flow is

aligned with the rotation vector, have negative Ω/S in the shearing sheet with our

convention, while anticyclonic rotation regimes correspond to positive Ω/S. In particular,

aMn∗ti,cΩyc(lRon)i=c K√eGpMler∗i/aRn 3r/o2t,aitsiocnhatyrapcitcearlisoefdacbcyreΩtio=n

discs orbiting (2/3)S in this

around model.

a

central

mass

18

F. Rincon

The numerical implementation of the local shearing sheet approximation in finite domains is usually referred to as the “shearing box”, as it amounts to solving the equations in a Cartesian box of dimensions (Lx, Ly, Lz) much smaller than the typical radius of curvature of the system. In order to accommodate the linear shear in this numerical problem, the x coordinate is usually taken shear-periodic†, the y coordinate is taken periodic, and the choice of the boundary conditions in z depends on whether some stratification is incorporated in the modelling (if not, periodicity in z is usually assumed).

2.2. Important scales and dimensionless numbers

2.2.1. Reynolds numbers

Let us now consider some important scales and dimensionless numbers in the dynamo problem based on equations (2.17)-(2.18). First, we define the scale of the system under consideration as L, and the integral scale of the turbulence, or the scale at which energy is injected into the flow, as 0. Depending on the problem under consideration, we will have either L ∼ 0, or L 0. Turbulent velocity field fluctuations at scale 0 are denoted by u0. The kinematic Reynolds number

Re

=

u0 ν

0

(2.20)

measures the relative magnitude of inertial effects compared to viscous effects on the flow. The Kolmogorov scale ν ∼ Re−3/4 0 is the scale at which kinetic energy is dissipated in Kolmogorov turbulence, with uν ∼ Re−1/4u0 the corresponding typical velocity at that
scale. The magnetic Reynolds number

Rm

=

u0 η

0

(2.21)

measures the relative magnitude of inductive (and mixing) effects compared to resistive effects in equation (2.18), and is therefore a key number in dynamo theory.

2.2.2. The magnetic Prandtl number landscape

The ratio of the kinematic viscosity to the magnetic diffusivity, the magnetic Prandtl

number

Pm

=

ν η

=

Rm Re

,

(2.22)

is also a key quantity in dynamo theory. Unlike Re and Rm, P m in a collisional fluid is

an intrinsic property of the fluid itself, not of the flow. Figure 6 shows that conducting

fluids and plasmas found in nature and in the lab have a wide range of P m. One reason

for this wide distribution is that P m is very strongly dependent on both temperature and

density. For instance, in a pure, collisional hydrogen plasma with equal ion and electron

temperature,

Pm

2.5

×

103

n

T4 (ln Λ)2

,

(2.23)

where T is in Kelvin, ln Λ is the Coulomb logarithm and n is the particle number density in m−3. This collisional formula gives P m ∼ 1025 or larger for the very hot ICM of
galaxy clusters (although it is probably not very accurate in this context given the weakly

† A detailed description of a typical implementation of shear-periodicity in the popular pseudospectral numerical MHD code SNOOPY (Lesur & Longaretti 2007) can be found in App. A of Riols et al. (2013).

Dynamo theories

19

Figure 6. A qualitative representation of the magnetic Prandtl number landscape. The grey area depicts the range of Re and Rm (based on r.m.s. velocities) thought to be accessible in the foreseeable future through either numerical simulations or plasma experiments.
collisional nature of the ICM). The much denser and cooler plasmas in stellar interiors have much lower P m, for instance P m ranges approximately from 10−2 at the base of the solar convection zone to 10−6 below the photosphere. Accretion-disc plasmas can have all kinds of P m, depending on the nature of the accreting system, closeness to the central accreting object, and location with respect to the disc midplane.
Liquid metals like liquid iron in the Earth’s core or liquid sodium in dynamo experiments have very low P m, typically P m ∼ 10−5 or smaller. This has proven a major inconvenience for dynamo experiments, as achieving even moderate Rm in a very low P m fluid requires a very large Re and therefore necessitates a lot of mechanical input power, which in turns implies a lot of heating. To add to the inconvenience, the turbulence generated at large Re enhances the effective diffusion of the magnetic field, which makes it even harder to excite interesting magnetic dynamics. As a result, the experimental community has started to shift attention to plasma experiments in which P m can in principle be controlled and varied in the range 0.1 < P m < 100 by changing either the temperature or density of the plasma, as illustrated by equation (2.23). Finally, due to computing power limitations implying finite numerical resolutions, most virtual MHD fluids of computer simulations have 0.1 < P m < 10 (with a few exceptions at large P m). Hence, it is and will remain impossible in a foreseeable future to simulate magnetic-field amplification in any kind of regime found in nature. The best we can hope for is that simulations of largish or smallish P m regimes can provide glimpses of the asymptotic dynamics.

Spectral power

20

F. Rincon

Forcing scales

k 5/3
Kinetic energy spectrum

Pm 1 k⌘/k⌫ 1

Magnetic energy spectrum tail

k

k0

k⌫ ⇠ Re3/4k0

k⌘ ⇠ P m1/2k⌫

Figure 7. Ordering of scales and qualitative representation of the kinetic and magnetic energy spectra in k (wavenumber) space at large P m.

The large and small P m MHD regimes are seemingly very different. To see this,

consider first the ordering of the resistive scale η, i.e. the typical scale at which the

magnetic field gets dissipated in MHD, with respect to the viscous scale ν.

Large magnetic Prandtl numbers. For P m > 1, the resistive cut-off scale η is smaller

than the viscous scale. This suggests that a lot of the magnetic energy resides at scales

well below any turbulent scale in the flow. The situation is best illustrated in Fig. 7 by

taking a spectral point of view of the dynamics in wavenumber k ∼ 1/ space, introducing

the kinetic and magnetic energy spectra associated with the the Fourier transforms in

space of the velocity and magnetic field, and the viscous and resistive wavenumbers

kν ∼ 1/ ν and kη ∼ 1/ η (this kind of representation will be frequently encountered in

the rest of the review). To estimate η more precisely in this regime, let us consider the

case of Kolmogorov turbulence for which the rate of strain of eddies of size goes as

u / ∼ −2/3. For this kind of turbulence, the smallest viscous eddies are therefore also

the fastest at stretching the magnetic field. To estimate the resistive scale η, we balance

the stretching rate of these eddies uν/ ν ∼ Re1/2u0/ 0 with the ohmic diffusion rate at

the

resistive

scale

η/

2 η

.

This

gives

η ∼ P m−1/2 ν , P m 1 .

(2.24)

Low magnetic Prandtl numbers. For P m < 1, we instead expect the resistive scale η to

fall in the inertial range of the turbulence. This is illustrated in spectral space in Fig. 8.

To estimate η in this regime, we simply balance the turnover/stretching rate uη/ η of

the eddies at scale

η

with

the

magnetic

diffusion

rate

η/

2 η

.

Equivalently,

this

can

be

formulated as Rm( η) = u η η/η ∼ 1. The result is

η ∼ P m−3/4 ν , P m 1 .

(2.25)

Intuitively, the large-P m regime seems much more favourable to dynamos. In particular, the fact that the magnetic field “sees” a lot of turbulent activity at low P m could create many complications. However, and contrary to what was for instance argued in

Spectral power

Forcing scales

Dynamo theories

21

k 5/3
Magnetic energy spectrum tail

Pm ⌧ 1
k⌘/k⌫ ⌧ 1
Kinetic energy spectrum

k

k0

k⌘ ⇠ P m3/4k⌫

k⌫ ⇠ Re3/4k0

Figure 8. Ordering of scales and qualitative representation of kinetic and magnetic energy spectra at low P m.

the early days of dynamo theory by Batchelor (1950), we will see in the next sections that dynamo action is possible at low P m. Besides, the large-P m regime has a lot of non-trivial dynamics on display despite its seemingly simpler ordering of scales.

2.2.3. Strouhal number

Another important dimensionless quantity arising in dynamo theory is the Strouhal

number

St =

τc τNL

.

(2.26)

This number measures the ratio between the correlation time τc and the nonlinear turnover time τNL ∼ u/u of an eddy with a typical velocity u at scale u. A similar number appears in all dynamical fluid and plasma problems involving closures and,

despite being of order one in many physical systems worthy of interest (including fluid

turbulence), is usually used as a small parameter to derive perturbative closures such as

those described in the next two sections. Krommes (2002) offers an illuminating discussion

of the potential problems of perturbation theory applied to non-perturbative systems,

many of which are directly relevant to dynamo theory.

2.3. Dynamo fundamentals
Most of the material presented so far is relevant to a much broader MHD context than just dynamo theory. We are now going to introduce a few important definitions, and outline several general results and concepts that are specific to this problem: antidynamo theorems and fast/slow dynamos. A more in-depth and rigorous (yet accessible) presentation of these topics can notably be found in Michael Proctor’s contribution to the collective book on “Mathematical aspects of Natural dynamos” edited by Dormy & Soward (2007).

22

F. Rincon

2.3.1. Kinematic versus dynamical regimes
The question of the amplification and further sustainment of magnetic fields in MHD is fundamentally an instability problem with both linear and nonlinear aspects. The first thing that we usually need to assess is whether the stretching of the magnetic field by fluid motions can overcome its diffusion. The magnetic Reynolds number Rm provides a direct measure of how these two processes compare, and is therefore the key parameter of the problem. Most, albeit not all, dynamo flows have a well-defined, analytically calculable or at least computable Rmc above which magnetic-field generation becomes possible.
In the presence of an externally prescribed velocity field (independent of B), the induction equation (2.18) is linear in B. The kinematic dynamo problem therefore consists in establishing what flows, or classes of flows, can lead to exponential growth of magnetic energy starting from an initially infinitesimal seed magnetic field, and in computing Rmc of the bifurcation and growing eigenmodes of equation (2.18). The velocity field in the kinematic dynamo problem can be computed numerically from the forced Navier-Stokes equation with negligible Lorentz force†, or using simplified numerical flow models, or prescribed analytically. This linear problem is relevant to the early stages of magneticfield amplification during which the magnetic energy is small compared to the kinetic energy of the flow.
The dynamical, or nonlinear dynamo problem, on the other hand, consists in solving the full nonlinear MHD system consisting of equations (2.1)-(2.6) (or the simpler equations (2.17)-(2.18) in the incompressible case), including the magnetic back-reaction of the Lorentz force on the flow. This problem is obviously directly relevant to the saturation of dynamos, but it is more general than that. For instance, some systems with linear dynamo bifurcations exhibit subcritical bistability, i.e. they have pairs of nonlinear dynamo modes involving a magnetically distorted version of the flow at Rm smaller than the kinematic Rmc. There is also an important class of dynamical magneticfield-sustaining MHD processes, referred to as instability-driven dynamos, which do not originate in a linear bifurcation at all, and have no well-defined Rmc. These different mechanisms will be discussed in §5.3.

2.3.2. Anti-dynamo theorems
Are all flows of conducting fluids dynamos ? Despite the seemingly simple nature of induction illustrated by equation (2.14), there are actually many generic cases in which magnetic fields cannot be sustained by fluid motions in the limit of infinite times, even at large Rm. Two of them are particularly important (and annoying) for the development of theoretical models and experiments: axisymmetric magnetic fields cannot be sustained by dynamo action (Cowling’s theorem, 1933), and planar, two-dimensional motions cannot excite a dynamo (Zel’dovich’s theorem, 1956).
In order to give a general feel of the constraints that dynamos face, let us sketch qualitatively how Cowling’s theorem originates in an axisymmetric system in polar (cylindrical) geometry (R, ϕ, z). Assume that B is an axisymmetric vector field

B = ∇×(χeϕ/R) + Rψ eϕ

(2.27)

where χ(R, z, t) is a poloidal flux function and Rψ is the toroidal magnetic field. Similarly, assume that U is axisymmetric with respect to the same axis of symmetry as B, i.e.

U = Upol + RΩ eϕ ,

(2.28)

† Or, in dynamo problems involving thermal convection, the Rayleigh-B´enard or anelastic systems including equation (2.6).

Dynamo theories

23

where Upol(R, z, t), is an axisymmetric poloidal velocity field in the (R, z) plane and Ω(R, z, t) eϕ is an axisymmetric toroidal differential rotation. The poloidal and toroidal components of equation (2.18) respectively read

∂χ ∂t

+

Upol

·

∇χ

=

η

∆

−

2 R

∂ ∂R

χ

(2.29)

and

∂ψ ∂t

+

Upol

·

∇ψ

=

Bpol

·

∇Ω

+

η

∆

+

2 R

∂ ∂R

ψ.

(2.30)

Equation (2.30) has a source term, Bpol · ∇Ω, which describes the stretching of poloidal field into toroidal field by the differential rotation, and is commonly referred to as the Ω effect in the astrophysical dynamo community (more on this in §4.2.1). However, there is no similar back-coupling between ψ and χ in equation (2.29), and therefore there is no converse way to generate poloidal field out of toroidal field in such a system. The problem is that there is no perennial source of poloidal flux χ in equation (2.29). The advection term on the l.h.s. describes the redistribution/mixing of the flux by the axisymmetric poloidal flow in the (R, z) plane and can only amplify the field locally and transiently. The presence of resistivity on the r.h.s. then implies that χ must ultimately decay, and therefore so must the Bpol · ∇Ω source term in equation (2.30), and ψ. Overall, the constrained geometry of this system therefore makes it impossible for the magnetic field to be sustained†.
Cowling’s theorem is one of the main reasons why the solar and geo- dynamo problems are so complicated, as it notably shows that a magnetic dipole strictly aligned with the rotation axis cannot be sustained by a simple combination of axisymmetric differential rotation and meridional circulation. Note however that axisymmetric flows like the Dudley & James (1989) flow or von K´arm´an flows (Mari´e et al. 2003), on which the designs of several dynamo experiments are based, can excite non-axisymmetric dynamo fields with dominant equatorial dipole geometry (m = 1 modes with respect to the axis of symmetry of the flow). There are also mechanisms by which nearly-axisymmetric magnetic fields can be generated in fluid flows with a strong axisymmetric mean component (Gissinger et al. 2008a). We will find out in §4 how the relaxation of the assumption of flow axisymmetry gives us the freedom to generate large-scale dynamos, albeit generally at the cost of a much-enhanced dynamical complexity.
Many other anti-dynamo theorems have been proven using similar reasonings. As mentioned above, the most significant one, apart from Cowling’s theorem, is Zel’dovich’s theorem that a two-dimensional planar flow (i.e. with only two components), U2D(x, y, t), cannot excite a dynamo. A purely toroidal flow cannot excite a dynamo either, and a magnetic field of the form B(x, y, t) alone cannot be a dynamo field. All these theorems are a consequence of the particular structure of the vector induction equation, and imply that a minimal geometric complexity is required for dynamos to work. But what does “minimal” mean ? As computer simulations were still in their infancy, a large number of applied mathematician brain hours were devoted to tailoring flows with enough dynamical and geometrical complexity to be dynamos, yet simple-enough mathematically to remain tractable analytically or with a limited computing capacity.

† From a mathematical point of view, the linear induction operator for a pure shear flow is not self-adjoint. In a dissipative system, this kind of mathematical structure generically leads to transient secular growth followed by exponential or super-exponential decay, rather than simple exponential growth or decay of normal modes (see Trefethen et al. (1993) and Livermore & Jackson (2004) for a discussion in the dynamo context).

24

F. Rincon

2.5D (or 2D-3C) flows of the form U(x, y, t) with three non-vanishing components (i.e., including a z component) are popular configurations of this kind, that make it possible to overcome anti-dynamo theorems at a minimal cost. Some well-known examples are 2D-3C versions of the Roberts flow (Roberts 1972), and the GallowayProctor flow (GP, Galloway & Proctor 1992). The original version of the latter is periodic in time and therefore has relatively simple kinematic dynamo eigenmodes of the form B(x, y, z, t) = R {B2D−3C(x, y, t) exp (st + ikzz)}, where kz, the wavenumber of the magnetic perturbation along the z direction, is a parameter of the problem, B2D−3C(x, y, t) has the same time-periodicity as the flow, and s is the (a priori complex) growth rate of the dynamo for a given kz. While such flows are very peculiar in many respects, they have been instrumental in the development of theoretical and experimental dynamo research and have taught us a lot on the dynamo problem in general. They retain some popularity nowadays because they can be used to probe kinematic dynamos in higher-Rm regimes than in the fully 3D problem by concentrating all the numerical resolution and computing power into just two spatial dimensions. A contemporary example of this kind of approach will be given in §5.4.

2.3.3. Slow versus fast dynamos

Dynamos can be either slow or fast. Slow dynamos are dynamos whose existence hinges

on the spatial diffusion of the magnetic field to couple different field components. These

dynamos therefore typically evolve on a large, system-scale Ohmic diffusion timescale

τη,0 =

2 0

/η

and

their

growth

rate

tends

to

zero

as

Rm

→

∞,

for

instance

(but

not

necessarily) as some inverse power of Rm. For this reason, they are probably not relevant

to astrophysical systems with very large Rm and dynamical magnetic timescales much

shorter than τη,0. A classic example is the Roberts (1970, 1972) dynamo, which notably

served as an inspiration for one of the first experimental demonstrations of the dynamo

effect in Karlsruhe (Stieglitz & Mu¨ller 2001; Gailitis et al. 2002). Fast dynamos, on the

other hand, are dynamos whose growth rate remains finite and becomes independent of

Rm as Rm → ∞. Although it is usually very hard to formally prove that a dynamo is

fast, most dynamo processes discussed in the next sections are thought to be fast, and

so is for instance the previously mentioned Galloway & Proctor (1992) dynamo. The

difficult analysis of the Ponomarenko dynamo case illustrates the general trickiness of

this question (Gilbert 1988). A more detailed comparative discussion of the characteristic

properties of classic examples of slow and fast dynamo flows is given by Michael Proctor

at p. 186 of the Encyclopedia of geomagnetism and paleomagnetism edited by Gubbins

& Herrero-Bervera (2007).

The standard physical paradigm of fast dynamos, originally due to Vainshtein &

Zel’dovich (1972), is the stretch, twist, fold mechanism pictured in numerous texts,

including here in Fig. 9. In this picture, a loop of magnetic field is stretched by shearing

fluid motions so that the field strength increases by a typical factor two over a turnover

time through magnetic flux conservation. If the field is (subsequently or simultaneously)

further twisted and folded by out-of-plane motions, we obtain a fundamentally 3D

“double tube” similar to that shown at the bottom of Fig. 9. In that configuration,

the magnetic field in each flux tube has the same orientation as in the neighbouring

tube. The initial geometric configuration can then be recovered by diffusive merging of

two loops, but with almost double magnetic field compared to the original situation. If

we think of this cycle as being a single iteration of a repetitive discrete process (a discrete

map), with each iteration corresponding to a typical fluid eddy turnover, then we have all

the ingredients of a self-exciting process, whose growth rate in the ideal limit of infinite

Rm is γ∞ = ln 2 (inverse turnover times). Only a tiny magnetic diffusivity is required

Dynamo theories Stretch

25 Twist

Merge (requires a tiny bit of magnetic di↵usion)
3D essential

Fold

Figure 9. The famous stretch-twist-fold dynamo cartoon, adapted from Vainshtein & Zel’dovich (1972) and many others.
for the merging, as the latter can take place at arbitrary small scale. The overall process is therefore not diffusion-limited.
One of the fundamental ingredients here is the stretching of the magnetic field by the flow. More generally, it can be shown that an essential requirement for fast dynamo action is that the flow exhibits Lagrangian chaos (Finn & Ott 1988), i.e. trajectories of initially close fluid particles must diverge exponentially, at least in some flow regions. This key aspect of the problem, and its implications for the structure of dynamo magnetic fields, will become more explicit in the discussion of the small-scale dynamo phenomenology in the next section. Many fundamental mathematical aspects of fast kinematic dynamos have been studied in detail using the original induction equation or simpler idealised discrete maps that capture the essence of this dynamics. We will not dive into this subject any further here, as it quickly becomes very technical, and has already extensively been covered in dedicated reviews and books, including the monograph of Childress & Gilbert (1995) and a chapter by Andrew Soward in a collective book of lectures on dynamos edited by Proctor & Gilbert (1994).

3. Small-scale dynamo theory
Dynamos processes exciting magnetic fields at scales smaller than the typical integral or forcing scale 0 of a flow are generically referred to as small-scale dynamos, but can be very diverse in practice. In this section, we will primarily be concerned with the statistical theory of small-scale dynamos excited by turbulent, non-helical velocity fluctuations u driven randomly by an external artificial body force, or through natural hydrodynamic instabilities (e.g. Rayleigh-B´enard convection). We will indistinctly refer to such dynamos as fluctuation or small-scale dynamos. The first question that we would like to address, of course, is whether small-scale fluctuation dynamos are possible at all. We know that “small-scale” fields and turbulence are present in astrophysical objects, but is there actually a proper mechanism to generate such fields from this turbulence ? In particular,

26

F. Rincon

can vanilla non-helical homogeneous, isotropic incompressible fluid turbulence drive a fluctuation dynamo in a conducting fluid, a question first asked by Batchelor (1950) ?

3.1. Evidence for small-scale dynamos
Direct experimental observations of small-scale fluctuation dynamos have only recently been reported in laser experiments (Meinecke et al. 2015; Tzeferacos et al. 2018), although the reported magnetic-field amplification factor of ∼ 25 is relatively small by experimental standards. The most detailed evidence (and interactions with theory) so far has been through numerical simulations. In order to see what the basic evidence for small-scale dynamos in a turbulent flow looks like, we will therefore simply have a look at the original numerical study of Meneguzzi et al. (1981), which served as a template for many subsequent simulations†.
The Meneguzzi et al. experiment starts with a three-dimensional numerical simulation of incompressible, homogeneous, isotropic, non-helical Navier-Stokes hydrodynamic turbulence forced randomly at the scale of a (periodic) numerical domain. This is done by direct numerical integration of equation (2.17) at Re = 100 with a pseudo-spectral method. After a few turnover times ensuring that the turbulent velocity field has reached a statistically steady state, a small magnetic field seed is introduced in the domain and both equations (2.17)-(2.18) are integrated from there on (P m = 1 in the simulation). The time-evolution of the total kinetic and magnetic energies during the simulation is shown in Fig. 10 (left). After the introduction of the seed field, magnetic energy first grows, and then saturates after a few turnover times by settling into a statistically steady state. Figure 10 (right) shows the kinetic and magnetic energy spectra in the saturated regime. The magnetic spectrum has a significant overlap with the velocity spectrum, but peaks at a scale significantly smaller than the forcing scale of the turbulence. Also, its shape is very different from that of the velocity spectrum. We will discuss this later in detail when we look at the theory.
To summarise, this simulation captures both the kinematic and the dynamical regime of a small-scale dynamo effect at Rm = 100, P m = 1 (although the dynamical impact of the magnetic field on the flow in the saturated regime is not obvious in this particular simulation), and it provides a glimpse of the statistical properties of the magnetic dynamo mode through the shape of the magnetic spectrum. Interestingly, it does not take much resolution to obtain this result, as 64 Fourier modes in each spatial direction are enough to resolve the dynamo mode in this regime. Keep in mind, though, that performing such a 3D MHD simulation was quite a technical accomplishement in 1981, and required a massive allocation of CPU time on a CRAY supercomputer !

3.2. Zel’dovich-Moffatt-Saffman phenomenology
Having gained some confidence that a small-scale dynamo instability is possible, the next step is to understand how it works. While the general stretch, twist, fold phenomenology provides a qualitative flavour of how such a dynamo may proceed, it would be nice to be able to make sense of it through a more quantitative, yet physically transparent analysis. Such an analysis was conducted by Zel’dovich et al. (1984) for an idealised time-dependent flow model consisting of a linear shear flow “renovated” (refreshed) randomly at regular time intervals, and by Moffatt & Saffman (1964) for the
† Pragmatic down-to-earth experimentalists feeling uneasy with a primarily numerical and theoretical perspective on physics problems may or may not find some comfort in the observation that essentially the same dynamo has since been reported in many MHD “experiments” performed with different resolutions and numerical methods.

Dynamo theories

27

Figure 10. The first simulations of small-scale dynamo action, conducted with a pseudo-spectral MHD code and 643 numerical resolution with dealiasing, Rm = 100 and P m = 1 (time is measured in multiples of the turnover time 0/u0 at the turbulent forcing scale). Left: time-evolution of the kinetic (EV ) and magnetic (EM ) energies. Right: corresponding kinetic and magnetic energy spectra in the saturated stage (adapted from Meneguzzi et al. 1981).

c1 > 0 > c2 > c3

e3

e2

e1

c1 > c2 > 0 > c3

e3

e2

e1

Rope

Pancake

Figure 11. Possible deformations of a fluid particle (or magnetic field lines) under an incompressible strain.

simpler case of time-independent linear shear, based on earlier hydrodynamic work by Pearson (1959).
Let us consider the incompressible, kinematic dynamo problem (2.18) paired with a simple non-uniform but spatially “smooth” incompressible random flow model u(r) = C r, where C is a random matrix with Tr C = 0, and assume that the magnetic field at t = 0, B(r, 0) = B0(r), has finite total energy, no singularity and limr→∞ B0(r) = 0. The evolution of the separation vector δr connecting two fluid particles is given by

dδri dt

=

Cik δ rk

.

(3.1)

We first analyse the situation considered by Moffatt & Saffman (1964) where C is
constant. In an appropriate basis (e1, e2, e3), C = diag(c1, c2, c3) with c1 + c2 + c3 = 0. There are two possible ways to stretch and squeeze the magnetic field, namely we can
form magnetic ropes if c1 > 0 > c2 > c3, or magnetic pancakes if c1 > c2 > 0 > c3. Both cases are depicted in Fig. 11. We will only analyse the rope case in detail here, but
will also give the results for the pancake case. In both cases, we expect the stretching of the magnetic field along e1 to lead to magnetic amplification as B2 ∼ exp(2c1t) in ideal MHD. However, the perpendicular squeezing implies that even a tiny magnetic diffusion

28

F. Rincon

matters. Is growth still possible in that case ? To answer this question, we decompose B into shearing Fourier modes

B(r, t) = Bˆ k0 (t) exp (ik(t) · r)d3k0 ,

(3.2)

where k0, the initial lagrangian wavenumbers, can be thought of as labelling each evolving mode, and the different k(t) ≡ k(k0, t) are time-evolving wavenumbers with k(t = 0) = k0 (Bˆ k0 (t) in this context should not be confused with the unit vector, introduced in §2.1.3, defining the orientation of B). Replacing B by this expression in the induction
equation, we have

dBˆ k0 dt

= C Bˆ k0 − ηk2Bˆ k0

for each k0, and

dk dt

=

−CT k

.

with k(t) · Bˆ k0 (t) = 0 at all times. The diffusive part of the evolution goes as

(3.3) (3.4)

t
exp −η k2(s)ds
0

(3.5)

and leads to the super-exponential decay of most Fourier modes because k3 ∼ k03 exp(|c3|t). At any given time t, the “surviving” wavenumbers live in an exponentially narrower cone of Fourier space such that

t
η k2(s)ds = O(1) .
0

(3.6)

In the rope case, the initial wavenumber of the modes still surviving at time t is given by

k02 ∼ exp(−|c2|t), k03 ∼ exp(−|c3|t) .

(3.7)

Accordingly, the initial magnetic field of these surviving modes is

[Bk0 (0) · e1] ∼ k02/k01 [Bk0 (0) · e2] ∼ exp(−|c2|t)

(3.8)

(from the solenoidality condition for B). As the field is stretched along e1, we then find that the amplitude of the surviving rope modes at time t goes as

Bˆ k0 (t) ∼ exp (c1t) exp (−|c2|t).

(3.9)

We can now estimate from equation (3.2) the total magnetic field in physical space. The
first term in the integral goes as exp [(c1 − |c2|)t] from equation (3.9), and the wavevector space element as exp [−(|c2| + |c3|)t] from equation (3.7), so that overall

B(r, t) ∼ exp (−|c2|t) .

(3.10)

Hence the magnetic field is stretched and squeezed into a rope that decays pointwise asymptotically. But what about the total magnetic energy

Em = B2(t, r)d3r

(3.11)

in the volume of fluid ? |B|2 ∼ exp (−2|c2|t), but the volume occupied by the field goes as exp (c1t). Importantly, there is no shrinking of the volume element along the second and third axis because magnetic diffusion sets a minimum scale in these directions.

Dynamo theories

29

Figure 12. A sequence of random linear shearing events (to be thought of in 3D).

Regrouping everything, we obtain

Em ∼ exp [(c1 − 2|c2|)t] ∼ exp [(|c3| − |c2|)t] .

(3.12)

The second twiddle equality only applies in 3D. Similar conclusions hold for the pancake case, except that Em ∼ exp [(c1 − c2)t]. Overall, we see that the total magnetic energy of magnetic ropes decays in 2D, because |c2| = c1 in that case. This is of course expected from Zel’dovich’s anti-dynamo theorem. On the other hand, the magnetic energy grows in 3D because |c3| > |c2| and the volume occupied by the magnetic field grows faster than the pointwise decay rate of the field itself (Moffatt & Saffman 1964).
Zel’dovich et al. (1984) generalised these results to random, time-dependent shears. They considered a shear flow “renovating” every time-interval τ , such as shown in Fig. 12. This generates a succession of random area-preserving stretches and squeezes, which can be described in multiplicative matrix form. More precisely, the matrix Tt ≡ T(t0, t) such that k(k0, t) = Ttk0 is put in Volterra multiplicative integral form

t
Tt = I − CT(s)ds ,
s=0

(3.13)

where I is the unit tensor. From there, the formal solution of the linear induction equation

is

B(r, t) =

t
exp iTt (k0 · r) − η (Tsk0)2ds (TTt )−1Bk0 (0)d3k0 .
0

(3.14)

The hard technical work lies in the calculation of the properties of the multiplicative

integral. Zel’dovich et al. managed to show that the cumulative effects of any random

sequence of shear can be reduced to diagonal form. In particular, they proved the

surprising result that for any such infinite sequence there is always a net positive

Lyapunov exponent γ1 corresponding to a stretching direction

lim
n→∞

1 nτ

ln k(k0, nτ )

≡

γ1

>

0

.

(3.15)

The underlying Lyapunov basis is a function of the full random sequence, but it is independent of time. This is a form of spontaneous symmetry breaking: while the system has no privileged direction overall, any particular infinite sequence of random shears will generate a particular eigenbasis. Moreover, as a particular sequence of random shears unfolds, this Lyapunov basis crystallises exponentially fast in time, while the exponents converge as 1/t (Goldhirsch, Sulem & Orszag 1987).
The random problem therefore reduces to that of the constant strain matrix described earlier. This establishes that magnetic energy growth is possible in a smooth, 3D chaotic velocity field even in the presence of magnetic diffusion, and shows that the exponential

30

F. Rincon

`⌘

e3 e1

e2

`

Figure 13. Organisation of the magnetic field in folds perpendicular to the local compressive direction e3 of a shearing velocity field (c2 < 0 case). The typical flow scale over which the folds develop is denoted by here.
separation of initially nearby fluid trajectories is critical to the dynamo process. The linear shear assumption can be relaxed to accommodate the case where the flow has large but finite size. The main difference in that case is that magnetic field can also be constantly reseeded in wavenumbers outside of the cone described by equation (3.7) through wavenumber couplings/scattering associated with the u×B induction term, and this effect facilitates the dynamo.
Overall, what makes this dynamo possible in 3D but not in 2D is the existence of an (almost) “neutral” direction e2 in 3D. In 2D, c1 + c2 = 0 and the field must be squeezed as much along e2 as it is stretched along e1. In that case, decays ensues according to the first twiddle inequality in equation (3.12). In 3D, on the other hand, this exact constraint disappears and some particular field configurations can survive the competition between stretching and diffusion. More precisely, the surviving fields are organised into folds in (e1, e2) planes perpendicular to the most compressive direction e3, with reversals occurring along e2 at the resistive scale η. This is illustrated in Fig. 13.
3.3. Magnetic Prandtl number dependence of small-scale dynamos
3.3.1. Small-scale dynamo fields at P m > 1
A linear shear flow has a spatially uniform gradient and is therefore the ultimate example of a large-scale shear flow. The magnetic mode that results from this kind of dynamo, on the other hand, has typical reversals at the resistive scale η, which of course becomes very small at large Rm. The problem described above is therefore implicitly typical of the large-P m MHD regime introduced in §2.2.2. The fastest shearing eddies at large P m in Kolmogorov turbulence are spatially-smooth, yet chaotic viscous eddies, and take on the role of the random linear shear flow in the Zel’dovich model. Interestingly, because this dynamo only requires a smooth, chaotic flow to work, there should be no problem with exciting it down to Re = O(1) (random Stokes flow). On the other hand, Rm must be large enough for stretching to win over diffusion. There is therefore

Dynamo theories

31

Figure 14. Left: 2D snapshot of |u| in a 3D simulation of non-helical, homogeneous, isotropic smooth random flow forced at the box scale 0 for Re = 1 ( ν = 0). Right: snapshot of the strength |B| of the kinematic dynamo magnetic field generated by this flow for Rm = P m = 1250, and corresponding magnetic field directions (arrows). The field in this large-P m regime has a strongly folded geometric structure: it is almost uniform along itself, but reverses on the very fine scale η/ 0 ∼ P m−1/2, ∼ 0.03 in that example. The brighter regions correspond to large field strengths (adapted from Schekochihin et al. 2004c).
always a minimal requirement to resolve resistive-scale reversals in numerical simulations (typically the 64 Fourier modes per spatial dimension of the Meneguzzi et al. simulation).
Many three-dimensional direct numerical simulations (DNS) of the kind conducted by Meneguzzi et al. have now been performed, that essentially confirm the Zel’dovich phenomenology and folded field structure of the small-scale dynamo in the P m > 1 regime. Snapshots of the smooth velocity field and particularly clean folded magnetic field structures in the relatively asymptotic large-P m regime Re = 1, P m = 1250, are shown in Fig. 14. The Fourier spectra of these two images (not shown) are obviously very different, which is of course reminiscent of the Meneguzzi et al. results. In fact, all simulations down to P m = O(1), including the Meneguzzi et al. experiment, essentially produce a dynamo of the kind described above. Figure 15 provides a map in the (Re, Rm) plane of the dynamo growth rate γ of the small-scale dynamo, and a plot of the critical magnetic Reynolds number Rmc,ssd above which the dynamo is excited (Rm here and in Fig. 14 and Fig. 16 is defined as Rm = urms 0/(2πη)). Rmc,ssd is found to be almost independent of P m and approximately equal to 60 for P m > 1. As P m decreases from large values to unity, the folded field structure gradually becomes more intricate, but for instance we can always spot very fast field reversals perpendicular to the field itself. This gradual change can be seen on the two leftmost 2D snapshots of Fig. 16 representing |B| in simulations at P m = 1250 and P m = 1.
3.3.2. Small-scale dynamo fields at P m < 1
What about the P m < 1 case ? Batchelor (1950) argued based on an analogy between the induction equation and the vorticity equation, that there could be no dynamo at all for P m < 1 (a concise account of Batchelor’s arguments on the small-scale dynamo can be found in Davidson (2013), § 18.3). As explained in §2.2.2, the magnetic field sees a very different, and much more irregular velocity field in the low-P m regime, and we would naturally expect this to have a negative impact on the dynamo. Whether the

nd turbulent induction at low magnetic Prandtl numbers

7

32

F. Rincon

a)

Figure 15. and growth

Critical magnetic Reynolds rates (colour squares) of the

kninuemm(bbaetr)icRsmmac,lsls-dsca(bleladcyknasomliod

line with excited by

full circles) non-helical,

homogeneous, isotropic turbulence forced at the box scale, as a function of Re. The parameter

range of the plot corresponds approximately to the grey box in Fig. 6. Rmc,ssd increases by a

1. (a) Growthfa/cdtoercaaylmroasttefooufr ⟨fo|Br P|2m⟩ <vs1. P(amdapftoerd fifrvoemvSaclhueeksocohfihRinmet(aRl.m20∼07)6. 0:

nd series H0; Rm ∼ 110: series A and HA ; Rm ∼ 230: series B and HB;

0: series C and HC; Rm ∼ 830: series D and HD). The round points were

using the code written by J. Maron, the square points using the code written

akov. The unit of the growth rate is approximately one inverse turnover time

ter scale (the precise units of time are set by ε = 1 and the box size = 1.) (b)

decay rates in the parameter space (Re, Rm). The filled points correspond to

h Laplacian viscosity, the empty ones to runs with 8th-order hyperviscosity.

rpolated stability curves Rmc(Re) based on the Laplacian and hyperviscous

hown separately. For comparison, we also plot the Rmc(Re) curve obtained by

al. (2006) for Fthigeutruerb16u.le2nDcesnwapitshhoatsmofeathneflsotrwen(gstehe|Bse|cotifotnhe2.k5infeomradtiicscduysnsaimono).magnetic field for 3D

simulations of non-helical, homogeneous, isotropic turbulence forced at the box scale. Left:

P m = Rm = 1250, Re = 1. Centre: P m = 1, Re = Rm = 440. Right: P m = 0.07, Rm = 430,

y,

Taylor,

RRMeeya=norl6od2ns0n0u&(mbbeceMaruinsceWththiisilsclaipasaemrtisiscduela2firn0es0dim4uu,slianStgicoanhneueksffeoesccthhivyiephevirinvsci,socsoiCtuysoddweitselseripmya,itnioedn

only, the kinetic statistically from

2004,

SchetakhnoedcsPhimmiuh=liant0io.0en7tdcaaatslae.)s.,2dN0eo0stp5eit)te.hReBmvoebrtyehindgicffeoesrdseenentstimaalalrygenthepteiscseafimuedeldoinsstbpruoetchctutsrriemasulb,laettiwoenesn(atdhaepPtemd

=1 from

ons ((1) andSch(e2k)o)chaihnindetuasle. 2t0h07e). same units of length and time, but

g, FFT anddpynaarmalolesulirsvaivteisoinn tahligsorergitimhemresm, aaisnewdealnl oapsenslaingdhstolymedwihffaetrceonnttroversial theoretical e random foarncdinnugm. eWricealhqauvesetiocnhefocrkmedancyoyneacrlsu(sVivaienlsyhttehina1t9A82.; NIsokvaikkoovve’tsal. 1983; Vainshtein uces the old&erKriechsautlitnsovo1b9t8a6i;nReodgawchitevhskJii. &MKalreoeonr’isn c1o99d7e;.Vincenzi 2002; Haugen et al. 2004;
Schekochihin et al. 2004a; Boldyrev & Cattaneo 2004; Schekochihin et al. 2005b). The first conclusive numerical demonstrations of kinematic dynamos at low P m in
e of the Dynnoanm-hoelical isotropic homogeneous turbulence were only performed a few years ago by
Iskakov et al. (2007). While the question of the optimal numerical configuration to reach
w the growtthhe/ldowe-cPamy drayntaems oorfegtihmee mis anogtneenttiicreleynseetrtgleyd,⟨t|hBe |m2a⟩invrse.suPltms of these Meneguzzi-

runs. The runs in each sequence have the same fixed value of η

proximately the same value of Rm. Thus, decreasing Pm in each

Dynamo theories

33

like simulations of homogeneous, isotropic turbulence are that the critical Rm of the dynamo increases significantly as P m becomes smaller than one (see Fig. 15), and that the nature of the low-P m dynamo is quite different from its large-P m counterpart. The rightmost snapshot of Fig. 16 shows for instance that the structure of the magnetic field at P m < 1 is radically altered in comparison to even the P m = 1 case. The disappearance of the folded field structure is perhaps not that surprising, given that we are completely outside of the domain of applicability of Zel’dovich’s smooth flow phenomenology for P m < 1. Unfortunately, a clear physical understanding of the lowP m small-scale kinematic dynamo process comparable to that of the large P m case is still lacking. As we will see in the next paragraph, though, the increase in Rmc,ssd at low P m can be directly tied to the roughness of the velocity field at the resistive scale, within the framework of the mathematical Kazantsev model.
Numerically, the problem with the low-P m regime is that one must simultaneously ensure that Rm is large enough to trigger the dynamo (Rmc,ssd at low P m appears to be at least a factor two larger than at large P m depending on how the problem is set-up), and that Re is significantly larger than Rm ! In practical terms, a resolution of 5123 is required to simulate such high Re turbulence in pseudo-spectral numerical simulations with explicit laplacian dissipation. Only now is this kind of MHD simulation becoming routine in computational fluid dynamics. Note finally that the excitation of small-scale dynamos at both P m > 1 and P m < 1 appears to be quite independent of the hydrodynamic turbulent-forcing mechanism, and even of the details of the turbulent flow. For instance, results similar to Fig. 15 have been obtained using hyperviscosity in DNS, MHD shell-models (Stepanov & Plunian 2006; Buchlin 2011) and DNS and largeeddy simulations of the (turbulent) Taylor-Green flow (Ponty et al. 2005). Of importance to astrophysics, small-scale fluctuation dynamo action is also known to be effective in P m > 1 simulations of turbulent thermal convection, Boussinesq and stratified alike (Nordlund et al. 1992; Cattaneo 1999; V¨ogler & Schu¨ssler 2007; Pietarila Graham et al. 2010; Moll et al. 2011; Bushby & Favier 2014). Low-P m turbulent convection is also widely thought to be the main driver of small-scale solar-surface magnetic fields, although clean, conclusive DNS simulations of a fluctuation dynamo driven by turbulent convection at P m significantly smaller than one have still not been conducted.

3.4. Kinematic theory: the Kazantsev model
Would it not be nice if we could calculate analytically the growth rate, energy spectrum, or probability density function of small-scale dynamo fields for different kinds of velocity fields ? Despite all the numerical evidence and data available on the kinematic small-scale dynamo problem, there is still no general quantitative statistical theory of the problem, for reasons that will soon become clear. Kazantsev (1967), however, derived a solution to the problem under the assumption that the velocity field is a random δ-correlated-in-time (white-noise) Gaussian variable. This particular statistical ensemble of velocity fields is commonly referred to as the Kraichnan ensemble, after Kraichnan (1968) independently introduced it in his study of the structure of passive scalars advected by turbulence.
At first glance, the Kazantsev-Kraichnan assumptions do not seem very fitting to solve transport problems involving Navier-Stokes turbulence, as the latter is intrinsincally nonGaussian and has a scale-dependent correlation time of the order of the eddy turnover time. The Kazantsev model has however proven extremely useful to calculate and even predict the kinematic properties of small-scale dynamos, and many of its results appear to be in very good quantitative agreement with Navier-Stokes simulations. The same can be said of the Kraichnan model for the passive scalar problem. It is also a very elegant piece of applied mathematics that provides a nice playground to acquaint oneself with

34

F. Rincon

turbulent closure problems, and offers a different perspective on the physics of smallscale dynamos. We will therefore go through the key points of the derivation of the Kazantsev model for the simplest three-dimensional, incompressible, non-helical case, and discuss some particularly important results that can be derived from the model. More detailed derivations of the model, including different variations in different MHD regimes, including compressible ones, can notably be found in the work of Kulsrud & Anderson (1992), Vincenzi (2002), Schekochihin, Boldyrev & Kulsrud (2002a), Boldyrev & Cattaneo (2004), and Tobias, Cattaneo & Boldyrev (2011a), all of which have largely inspired the following presentation.

3.4.1. Kazantsev-Kraichnan assumptions on the velocity field
We consider a three-dimensional, statistically steady and homogeneous fluctuating incompressible velocity field with two-point, two-time correlation function

ui(x, t)uj(x , t ) = Rij(x − x, t − t) .

(3.16)

We assume that u has Gaussian statistics,

P [u] = C exp

−

1 2

dt dt

d3x d3x Dij(t − t, x − x)ui(x, t)uj(x , t ) ,

(3.17)

where C is a normalisation factor and the covariance matrix Dij is related to Rij by

dτ d3yDik(t − τ, x − y)Rkj(τ − t, y − x) = δijδ(t − t)δ(x − x) .

(3.18)

We further assume that u is δ-correlated in time, ui(x, t)uj(x , t ) = Rij(x − x, t − t) = κij(r)δ(t − t) ,

(3.19)

where r = x − x is the spatial correlation vector. We restrict the calculation to the
isotropic, non-helical case for the time being (the helical case is also interesting in the context of large-scale dynamo theory and will be discussed in §4.5.2). In the absence of
a particular axis of symmetry in the system and of helicity, we are only allowed to use δij and r to construct κij(r)†, and the most general expression that we can form is

κij(r) = κN (r)

δij

−

rirj r2

+

κL

(r)

rirj r2

.

(3.20)

where r = |r| and κN and κL are the tangential and longitudinal velocity correlation functions. For an incompressible/solenoidal vector field, we have

κN = κL + (rκL)/2 .

(3.21)

3.4.2. Equation for the magnetic field correlator

Our goal is to derive a closed equation for the two-point, single-time magnetic correlation function (or, equivalently, for the magnetic energy spectrum, as the two are related by a Fourier transform)

Bi(x, t)Bj(x , t) = Hij(r, t) .

(3.22)

† If the turbulence is made helical (but remains isotropic), equation (3.20) must be supplemented by an extra term proportional to the fully anti-symmetric Levi-Cevita tensor εijk (see §4.5.2).

Dynamo theories

For the same reasons as above, we can write

Hij(r, t) = HN (r, t)

δij

−

rirj r2

+

HL

(r,

t)

rirj r2

,

35 (3.23)

HN = HL + (rHL)/2 .

(3.24)

Taking the i-th component of the induction equation at point (x, t) and multiplying it by

Bj(x , t), then taking the j-th component at point (x , t) and multiplying it by Bi(x, t) we find, after adding the two results, the evolution equation for Hij,

∂ H ij ∂t

=

∂ ∂x k

Bi(x, t)Bk(x , t)uj(x , t) − Bi(x, t)Bj(x , t)uk(x , t)

+

∂ ∂xk

Bk(x, t)Bj(x , t)ui(x, t) − Bi(x, t)Bj(x , t)uk(x, t)

+η

∂2 ∂ xk 2

+

∂2 ∂x k2

Hij ,

(3.25)

where ∂/∂x k · = −∂/∂xk · = ∂/∂rk · because of statistical homogeneity. Equation (3.25) is exact, but we are now faced with an important difficulty: the timederivative of the second-order magnetic correlation function depends on mixed thirdorder correlation functions, and we do not have explicit expressions for these correlators. We could write down evolution equations for them too, but their r.h.s. would then involve fourth-order correlation functions, and so on. This is a familiar closure problem.

3.4.3. Closure procedure in a nutshell*
In order to make further progress, we have to find a (hopefully physically) way to truncate the system of equations by replacing the higher-order correlation functions with lower-order ones. This is where the Kazantsev-Kraichnan assumptions of a random, δcorrelated-in-time Gaussian velocity field come into play. The assumption of Gaussian statistics implies that nth-order mixed correlation functions involving u can be expressed in terms of (n − 1)th-order correlation functions using the Furutsu-Novikov (Gaussian integration) formula:

ui(x, t)F [u] = dt

d3x ui(x, t)ul(x , t )

δF [u] δul(x , t )

,

(3.26)

where F [u] stands for any functional of u, and the δF/δui are functional derivatives. We can use this formula in equation (3.25) to replace all the third-order moments appearing in the r.h.s. by integrals of products of second-order moments. To illustrate how the closure procedure proceeds, let us isolate just one of these terms,

t
ui(x, t)Bk(x, t)Bj(x , t) = dt
0

d3x ui(x, t)ul(x , t )

δ Bk(x, t)Bj(x , t) δul(x , t )

.

(3.27)

Applying the Gaussian integration formula is a critical first step, but more work is needed.

In particular, equation (3.27) involves a time-integral encapsulating the effects of flow

memory. For a generic turbulent flow for which the correlation time is not small compared

the relevant dynamical timescales of the problem, the problem is non-perturbative and

there is no known method to calculate such an integral exactly. However, as a first step we

could still assume that it is small, and perform the integral perturbatively (the expansion

parameter will be the Strouhal number). The Kazantsev-Kraichnan assumption of zero

correlation-time corresponds to the lowest-order calculation. Using equation (3.19) in

36

F. Rincon

equation (3.27) removes the time-integral and leaves us with the task of calculating the equal-time functional derivative δ Bk(x, t)Bj(x , t) /δul(x , t) . This expression can be
explicitly calculated using the expression of the formal solution of the induction equation,

Bk(x, t) =

t
dt

Bm(x, t

)

∂

uk (x, ∂xm

t

)

−

um(x,

t

)

∂ B k (x, ∂xm

t

)

+

η∆Bk(x,

t

)

.

(3.28)

Functionally differentiating this equation (and that for Bj(x , t)) with respect to δul(x , t) introduces δ(x − x ) and δ(x − x ), which makes the space-integral in equation (3.27) trivial and completes the closure procedure.
The end result of the full calculation outlined above are expressions of all the mixed third-order correlation functions in terms of a combination of products of the two-point second-order correlation function of the magnetic field with the (prescribed) second-order correlation function of the velocity field (and their spatial derivatives).

3.4.4. Closed equation for the magnetic correlator
From there, it can be shown easily using appropriate projection operators and isotropy that the original, complicated unclosed equation (3.25) reduces to the much simpler closed scalar equation for HL(r, t),

∂HL ∂t

=

κHL

+

4 r

κ

+

κ

HL +

κ

+

4 r

κ

HL ,

(3.29)

where

κ(r) = 2η + κL(0) − κL(r)

(3.30)

can be interpreted as (twice) the “turbulent diffusivity”. If we now perform the change

of variables

HL(r, t) =

ψ(r, t) r2κ(r)1/2

,

(3.31)

we find that equation (3.29) reduces to a Schr¨odinger equation with imaginary time

∂ψ ∂t

=

κ(r)ψ

− V (r)ψ ,

(3.32)

which describes the evolution of the wave function of a quantum particle of variable mass

m(r)

=

1 2κ(r)

(3.33)

in the potential

V (r) =

2 r2

κ(r)

−

1 2

κ

(r) −

2 r

κ

(r)

−

κ (r)2 4κ(r)

.

(3.34)

3.4.5. Solutions We can then look for solutions of equation (3.32) of the form

ψ = ψE(r)e−Et .

(3.35)

Keeping in mind that ψ stands for the magnetic correlation function, we see that exponentially growing dynamo modes correspond to discrete E < 0 bound states. The existence of such modes depends on the shape of the Kazantsev potential, which is entirely determined by the statistical properties of the velocity field encapsulated in the function

Dynamo theories κ(r). The variational result for E is

E=

2mV ψE2dr + ψE2dr
. 2mψE 2 dr

37 (3.36)

If we are just interested in the question of whether a dynamo is possible, we can equivalently solve the equation

ψE + [E − Veff (r)] ψE = 0 .

(3.37)

where

Veff (r)

=

V (r) κ(r)

.

(3.38)

The ground state describes the fastest growing mode, and therefore the long-time asymp-

totics in the kinematic regime of the dynamo.

3.4.6. Different regimes
After all this hard work, we are now almost in a position to answer much more specific questions. For instance, we would like to predict whether dynamo action is possible in a smooth flow (something we already know from §3.2), and in a turbulent flow with Kolmogorov scalings (something the Zel’dovich et al. model does not tell us). Let us assume that the velocity field has power-law scaling,

κL(0) − κL(r) ∼ rξ ,

(3.39)

where ξ is called the roughness exponent. Introducing the resistive scale η = (2η)1/ξ, the effective Kazantsev potential as a function of ξ is

Veff (r)

=

2 r2

for

r

η,

=

2 − 3 ξ/2 − 3 ξ2/4 r2

for

r

η.

(3.40) (3.41)

The overall shape of the potential is illustrated in Fig. 17 for different values of ξ. The potential becomes attractive at ξ = ξc = 11/3 − 1, but growing bound modes only exist for ξ > 1.
In order to find out which kind of flows are dynamos according to the Kazantsev model, we have to make a small handwaving argument in order to relate the scaling properties of flows with finite correlation times to those of Kazantsev-Kraichnan velocity fields. Recalling that ui(x, t)uj(x , t ) = κij(r)δ(t − t), and that a δ function is dimensionally the inverse of a time, we write

κ(r) ∼ δu(r)2τ (r) ∼ rδu(r) ,

(3.42)

where δu(r) is the typical velocity difference between two points separated by r, and we have assumed that the relevant timescale in the dimensional analysis is the scaledependent eddy turnover time τ (r) ∼ r/δu(r) (this twiddle-algebra analysis also clarifies the association of κ with a turbulent diffusivity). Using equation (3.42), we find that ξ = 2 for a smooth flow whose velocity increments scale as δu ∼ r (the linear shears of the Zel’dovich phenomenology). Qualitatively, it is tempting to associate this ξ = 2 case with a large-P m regime because all the magnetic field sees at large P m is a large-scale, smooth viscous flow. On the other hand, for a flow with Kolmogorov scaling, δu ∼ r1/3

38
Ve↵ (r)

F. Rincon

p ⇠ < ⇠c = 11/3 1
r
`⌘ ⇠ > ⇠c
Bound modes (⇠ > 1)

Figure 17. Kazantsev potential as a function of r for different roughness exponents ξ. An attractive potential forms at ξc. Bound (growing) dynamo modes require ξ > 1.

and ξ = 1 + 1/3 = 4/3. This case would instead correspond to a low-P m regime in which the magnetic field sits in the middle of the inertial range.
According to the Kazantsev model, a necessary condition for small-scale dynamo action to be possible is that ξ > 1. This condition is satisfied in both smooth flows (ξ = 2, P m 1 regime) and rough flows, including Kolmogorov-like turbulence (ξ = 4/3, P m 1 regime). The study of the low-P m-like case was not part of the original Kazantsev (1967) article, and only appeared in later work by various authors (Vainshtein (1982); Vainshtein & Kichatinov (1986); Vincenzi (2002); Boldyrev & Cattaneo (2004), see also Eyink (2010); Eyink & Neto (2010) for a Lagrangian perspective on the problem). In this respect, we should also mention the work of Kraichnan & Nagarajan (1967) contemporary to that of Kazantsev. Kraichnan & Nagarajan arrived at a closed equation for the evolution of the magnetic-energy spectrum similar to that derived by Kazantsev using a different closure procedure, and found exponential growth of magnetic energy by solving this equation numerically as an initial value problem for a prescribed Kolmogorov velocity spectrum.

3.4.7. Critical Rm

Another interesting result that can be derived from the Kazantsev model is the existence of a critical Rm (Ruzmaikin & Sokolov 1981). A finite scale separation 0/ η is introduced in the problem by assuming the existence of an integral scale 0 beyond which the velocity field decorrelates. The shape of the Kazantsev potential in that case is shown in Fig. 18, and the asymptotic form is

Veff (r)

=

2 r2

for

r

η,

=

2 − 3 ξ/2 − 3 ξ2/4 r2

for

η

r

0,

=

2 r2

for

0

r.

(3.43) (3.44) (3.45)

Dynamo theories

39

Ve↵ (r)

Rm < Rmc

r

`⌘

`0

Rm > Rmc

Figure 18. Critical Rm effect in the Kazantsev model. The existence of an attractive potential (growing dynamo modes) requires a large-enough scale separation between the integral scale 0 and the resistive scale η.

The potential is now repulsive at both small and large scales. As a result, the existence of an attractive range of scales and bound modes now depends on the existence of a largeenough scale separation 0/ η, or large enough Rm. If the scale separation is too low, magnetic diffusion, whose stabilizing effects translate as a repulsive potential at smallscales, always wins over stretching, and no dynamo action is possible. This argument is independent of ξ, and therefore predicts the existence of a critical Rm in both large- and low-P m regimes.

3.4.8. Selected results in the large-P m regime*

The Kazantsev model can be used to derive a variety of quantitative results when the magnetic field is excited at scales much smaller than the spatial correlation scale of the flow. One particularly interesting such case is the large-P m regime encountered earlier, for which the dynamo is driven by a smooth viscous-scale flow. This is called the Batchelor regime, in reference to Batchelor, Howells & Townsend’s 1959 work on the transport of passive scalars. Note that this problem is not of mere applied mathematics interest, and has notably been at the core of important developments on the theory of astrophysical galactic dynamos (Kulsrud & Anderson 1992).
In the Batchelor regime, the velocity field correlator (3.20)-(3.21) can be expanded in Taylor series by writing κL = κ0 − κ2r2/4 + · · ·, κN = κ0 − κ2r2/2 + · · ·, resulting in

κij (r)

=

κ0δij

−

κ2

r2 2

δij

−

1 2

rirj r2

+··· ,

(3.46)

A technical digression is in order: while it is possible to make quantitative calculations in physical space starting from equation (3.46), the easiest and most popular route to calculate magnetic eigenfunctions and the magnetic energy spectrum goes through Fourier-space. This was in fact the primary method used by Kazantsev (1967). We will take that route in the next paragraphs, and find that the results can be expressed in

40

F. Rincon

terms of the coefficients of the Taylor expansion (3.46). The correspondence between the two descriptions can be found in Schekochihin et al. (2002a), App. A.

Fokker-Planck equation. At scales much smaller than the viscous scale (small-scale ap-

proximation), the spectral equivalent of the Schr¨odinger equation (3.32) for the magnetic

correlation function is a Fokker-Planck equation for the one-dimensional magnetic energy

spectrum

M (k, t)

=

1 2

k2|Bˆ k(t)|2dΩk ,

(3.47)

where Bˆ k(t) denotes the Fourier transform in space of B (once again not to be confused in this context with the unit vector in the direction of B). Namely,

∂M ∂t

=

γ 5

k2

∂2M ∂k2

−

2k

∂M ∂k

+ 6M

− 2ηk2M ,

(3.48)

with

γ

=

5 4

κ2

=

5 2

|κL(0)|

∼

δuν /

ν

,

(3.49)

the typical growth rate of the magnetic energy, of the order of the viscous shearing rate

represented by the κ2 parameter in the smooth flow expansion (3.46). These results are

only quantitatively valid in the incompressible 3D case.

Kazantsev spectrum and growth rate. Equation (3.48) has a diffusion term in wavenum-

ber space. If we start with magnetic perturbations at scales much larger than the resistive

scale, the magnetic spectrum will both grow in time and spread in wavenumber space

until it hits the resistive scale. This early evolution during which all k kη is called

the diffusion-free regime and is illustrated in Fig. 19. Assuming that we initially excite

a spectrum of magnetic modes M0(k), the evolution of the spectrum can be computed exactly using Green’s function of equation (3.48) with negligible magnetic diffusion,

M (k, t) = e3/4γt

∞ 0

dk k

M0(k

)

k k

3/2

5 4πγt

exp

−5

ln2 (k/k 4γt

)

.

(3.50)

On the large-scale side, the magnetic energy spectrum grows as k3/2. This result was first derived in Kazantsev’s (1967) paper and is therefore called the Kazantsev spectrum. Interestingly, the energy in each wavenumber grows at the rate 3γ/4, but the total energy integrated over wavenumber space grows at the rate 2γ because the number of excited Fourier modes also grows in time.
At the end of the diffusion-free regime, the spectrum hits k ∼ kη and we enter the diffusive regime, for which the long-time asymptotic form is

M (k, t) ∝

k kη

3/2
K0

k kη

eλγt ,

(3.51)

where K0 is a Macdonald function, kη = γ/(10η) ∼ P m1/2kν, and λ here is a nondimensional growth-rate prefactor. The spectrum at large scales is still k3/2, but now peaks at the resistive scale and falls off exponentially at even smaller scales. While the exact value of λ depends weakly on P m and on the boundary condition imposed on the viscous side at low k, the asymptotic total energy growth rate is now essentially 3γ/4, as the number of excited modes remains constant. The evolution of the spectrum in this regime is illustrated in Fig. 20.
Interestingly, this general spectral shape is reminiscent of the numerical results shown in Fig. 10. In fact, and despite all the assumptions behind the Kazantsev derivation, all

Dynamo theories

41

Spectral power

k 5/3

Forcing scales

Kinetic energy spectrum

Viscous scale stretching

Magnetic energy spectrum
k3/2
M0(k)

k0

k⌫ ⇠ Re3/4k0

Peak scale
k

Figure 19. Evolution of the magnetic energy spectrum in the kinematic, diffusion-free large-P m regime, starting from an initial magnetic spectrum M0(k). The magnetic energy in each wavenumber increases, and so does the the peak wavenumber as the spectrum spreads.

Spectral power

k 5/3

Forcing scales

Kinetic energy spectrum

Magnetic energy spectrum
k3/2

Viscous scale

stretching

k

k0

k⌫ ⇠ Re3/4k0

k⌘ ⇠ P m1/2k⌫

Figure 20. Evolution of the magnetic energy spectrum in the kinematic, diffusive large-P m regime. The shape of the magnetic spectrum is now fixed in time and peaks at the resistive scale η, but the magnetic energy continues to grow exponentially.

small-scale dynamo simulations at P m > 1 exhibit a spectrum with a positive slope at wavenumbers larger than but close to kν. Observing a clean k3/2 scaling, though, requires to go to fairly large P m to reach a proper scale separation between kν and kη.
Magnetic probability density function and moments. Probability density functions
(p.d.f.) and statistical moments of different orders can diagnose subtle dynamical features
in both experiments and numerical simulations and as such provide important statistical

42

F. Rincon

tools to tackle many turbulence problems. To complete our overview of the Kazantsev theory, we will therefore finally outline a mathematical procedure to calculate the magnetic p.d.f. and moments in the Kazantsev description in the diffusion-free regime. The spatial dependence of the field is ignored for simplicity in this particular derivation: this removes turbulent mixing effects from the model, but not inductive ones. The key point of the derivation (see, e.g., Schekochihin & Kulsrud 2001; Boldyrev 2001) is to introduce the characteristic function,

Z(µ, t) = exp iµiBi(t) = Z˜

(3.52)

where Z˜ = exp iµiBi(t) and the bracket average is over all the realisations of the velocity field. This function is interesting because it is the Fourier transform in B of the p.d.f. of B,

Z(µ, t) = P [B] exp iµiBi(t) d3B .

(3.53)

Using the simplified induction equation

∂Bi ∂t

=

σki Bk

,

(3.54)

where σki = ∂ui/∂xk is the velocity strain tensor, we obtain an evolution equation for Z,

∂Z ∂t

=

µi

∂ ∂µk

σki Z˜

.

(3.55)

This equation can be closed in the Kazantsev model using the same Gaussian integration trick as in §3.4.3. The result is

∂Z ∂t

=

κ2 2

Tkijl

µi

∂ ∂µk

µj

∂ ∂µl

Z

,

(3.56)

where

Tkijl

=

−

1 κ2

∂2κij ∂rk∂rl

= δij δkl −

1 4

δki δlj + δliδkj

(3.57)

is the strain correlator for a 3D, incompressible, isotropic velocity field. Performing an

inverse Fourier transform from µ to B variables and using the transformation

µi

∂ ∂µk

()

→

∂ ∂Bi

Bk ()

,

(3.58)

we obtain a Fokker-Planck equation for the p.d.f.

∂ ∂t

P

[B]

=

κ2 2

Tkijl

Bk

∂ ∂B

i

B

l

∂

∂ Bj

P

[B] .

(3.59)

Thanks to the isotropy assumption, this equation simplifies further as a 1D diffusion equation with a drift for the p.d.f. PB[B](t) of the field strength B,

∂ ∂t

PB

[B]

=

κ2 4

1 B2

∂ ∂B

B4

∂ ∂B

PB

[B]

,

(3.60)

which has the log-normal solution

PB [B ](t)

=

√1 πκ2t

∞ 0

dB B

PB [B

] (t

=

0) exp

− [ln(B/B

) + (3/4)κ2t]2 κ2t

, (3.61)

i.e. the statistics of the logarithm of the field strength are Gaussian. Therefore, the

Dynamo theories

43

magnetic field structure is strongly intermittent, despite the fact that the velocity field itself is Gaussian. This is actually expected from the central limit theorem on a general basis, not just in the Kazantsev formalism, as the induction equation is linear in B and involves the multiplicative random variable ∇u on the r.h.s. The magnetic moments of different order n, defined here as

grow as

∞

Bn(t) = 4π

dBB2+nPB [B] ,

0

(3.62)

Bn(t)

∝ exp

1 4

n(n

+

3)κ2t

.

(3.63)

There is currently no similar general result for the magnetic-field p.d.f. in the diffusive regime, although expressions for the n > 2 moments of the field have been derived in this regime by Chertkov et al. (1999) using a different mathematical technique. Their results show that the structure of the field remains strongly intermittent after the folds’ reversal scale hits the magnetic-diffusion scale.

3.4.9. Miscellaneous observations
Instability threshold subtleties. Equation (3.63) indicates that magnetic moments of different order have different ideal growth rates. Accordingly, we would find that each moment becomes unstable at a different Rm in the resistive case. This behaviour can be attributed to the multiplicative stochastic nature of the inductive term in the linear induction equation. Which moment, then, provides the correct prediction for the overall dynamo threshold ? It has been argued that a proper instability threshold in this context is only well-defined after taking into account nonlinear saturation terms that tend to suppress the large-deviations events responsible for the field intermittency. The dynamo onset of the full nonlinear problem then appears to be given by the linear threshold derived from the statistical average of the logarithm of the magnetic field, not of the magnetic energy (second-order moment) as calculated in the Kazantsev model (Seshasayanan & P´etr´elis 2018).
Connection with finite-correlation-time turbulence. Statistics of the magnetic field in the kinematic, large-P m regime have been extensively studied in numerical simulations of the MHD equations. While the log-normal shape of the magnetic p.d.f. appears to be a robust feature of the simulations, intermittency corrections imprinted in magnetic moments, such as predicted by equation (3.63), cannot be easily diagnosed due to subtle finite-size effects. An in-depth-discussion of this thorny aspect of the problem can be found in Schekochihin et al. (2004c). More generally, and despite its crude closure assumptions, the predictions of the Kazantsev model (dynamo growth rate and statistics) are in good overall agreement with the results of DNS at P m > 1. Nonzero correlation-time effects appear to be of relatively minor qualitative importance (see e.g. Vainshtein (1980); Chandran (1997); Bhat & Subramanian (2015) for theoretical arguments, and Mason et al. (2011) for a detailed numerical comparison with turbulence DNS), and intermittent statistics of the fluctuation dynamo fields are predicted by the model despite the gaussianity of the velocity field. This suggests that the kinematic smallscale fluctuation dynamo process is relatively insensitive to the details of the flow driving the dynamo, as long as the former is chaotic.

3.5. Dynamical theory
We have found that a small-scale magnetic field seed can grow exponentially on fast eddy-turnover timescales in a generic chaotic/turbulent flow in the kinematic dynamo

44

F. Rincon

regime. The dynamo field, however small initially, will therefore inevitably become “sufficiently large” (soon to be discussed) after a few turbulent turnover times for the backreaction of Lorentz force on the flow to become dynamically significant. Understanding the evolution of small-scale dynamo fields in this nonlinear dynamical regime is at least as important as understanding the kinematic regime, not least because all observed magnetic fields in the Universe are thought to be in a nonlinear dynamical state involving a strong small-scale component. Definitive quantitative theoretical results remain scarce though, in spite of the guidance now provided by high-resolution numerical simulations. We will therefore mostly discuss this problem from a qualitative, phenomenological perspective.

3.5.1. General phenomenology

The first and most natural question to ask about saturation is arguably that of the

dynamo efficiency: how much magnetic energy should we expect relative to kinetic energy

in the statistically steady saturated state of the dynamo ? Two answers were historically

given to this question, one by Batchelor (1950) and the other by Schlu¨ter & Biermann

(1950). Batchelor observed that the induction equation for the magnetic field has the

same form as the evolution equation for hydrodynamic vorticity. As the vorticity peaks

at the viscous scale in hydrodynamic turbulence, he then argued that the dynamo should

saturate when |B|2 ∼ u2ν. This argument predicts that very weak fields well below equipartition are sufficient to saturate the dynamo in Kolmogorov turbulence (uν ∼ Re−1/4u0) with Re 1,

|B|2 ∼ Re−1/2 |u|2 ,

(3.64)

admittedly not a very interesting astrophysical prospect ! Schlu¨ter & Biermann, on the other hand, argued that the outcome of saturation should be global equipartition between kinetic and magnetic energy,

|B|2 ∼ |u|2 ,

(3.65)

as a result of a gradual build-up of scale-by-scale equipartition from the smallest, least energetic scales. Which one (if either) is correct ? Batchelor’s analogy between vorticity and magnetic field was shown to be flawed a long time ago, and numerical simulations exhibit much higher saturation levels than predicted by his model. However, scaleby-scale equipartition does not appear to hold either, as can be seen for instance by inspecting the saturated spectra in Fig. 10.
In order to get a better handle on the issue, we must have a detailed look at how magnetic growth is quenched physically. Geometry, not just energetics, is very important in this problem, particularly in the large-P m case. First, not all velocity fields are created equal for the magnetic field, as equation (2.14) clearly illustrates. Only certain types of motions relative to the magnetic field can make the field change, and it is only by quenching such motions that the magnetic field can saturate. Second, the magnetic tension force B · ∇ B mediating the dynamical feedback (in incompressible flows) is proportional to the variation of the magnetic field along itself, i.e. to magnetic curvature. A quick look at the folded fields structure in Fig. 14 reveals the geometric complexity of the problem. Magnetic curvature is very small along the regions of straight magnetic field, and very large in regions where the magnetic field bends to reverse, and therefore so is the Lorentz force back-reaction.
These two observations are not independent though because the energy gained by the field through the anisotropic induction term must always be equal and opposite to that lost by the velocity field through the effects of the Lorentz force (§2.1.3). In practice, the Lorentz force effectively reduces Lagrangian chaos in the flow in comparison to the kinematic regime, thereby quenching the inductive stretching of the magnetic field. This

Dynamo theories

45

Figure 21. Spatial distribution of the finite-time Lyapunov stretching exponent in a dynamo simulation of the GP flow. Light shades correspond to integrable regions with little or no exponential stretching, dark shades to chaotic regions with strong stretching. Left: kinematic regime map exhibiting fractal regions of chaotic dynamics and stability islands. Right: dynamical regime. Strongly chaotic regions have almost disappeared (adapted from Cattaneo et al. 1996).
Figure 22. Magnetic energy in a simulation of small-scale dynamo action driven by turbulent thermal convection. Dashed-line: energy of the saturated magnetic field in the dynamical regime. Dash-dotted line: energy of an independent dummy magnetic field seeded in the course of the saturated phase. The dummy magnetic field grows exponentially, even though the true magnetic field has already saturated the velocity field (adapted from Cattaneo & Tobias 2009).
effect is illustrated in Fig. 21 by two maps of the stretching Lyapunov exponent of the GP flow in the kinematic and saturated regimes for P m = 4. The Lyapunov exponent reduction is strongly correlated spatially with the structure of the particular realisation of the dynamo magnetic field being saturated. The velocity field saturated by a given small-scale dynamo field realisation therefore remains a kinematic dynamo flow for other independent field realisations/dummy magnetic fields governed by an induction equation (Tilgner & Brandenburg 2008; Cattaneo & Tobias 2009, see also Brandenburg (2018) for a more detailed discussion). A numerical illustration of this effect is provided in Fig. 22.

46

F. Rincon

3.5.2. Nonlinear growth
Now that we have a better appreciation of the effects of the Lorentz force on the flow, let us come back to the dynamical history of saturation. Various numerical simulations (e.g. Schekochihin et al. 2004c; Ryu et al. 2008; Beresnyak 2012), suggest that the kinematic stage is followed by a nonlinear growth phase during which magnetic energy grows linearly in time instead of exponentially. A possible scenario for such nonlinear growth at large P m is as follows (Schekochihin et al. 2002b). In this parameter regime, the end of the kinematic phase occurs when the stretching by the most dynamo-efficient viscous scales gets suppressed,

B · ∇B ∼ u · ∇u ∼ u2ν / ν , ∼ B2/ ν (magnetic folds at viscous scale).

(3.66) (3.67)

This corresponds to the magnetic “Batchelor level” |B|2 ∼ Re−1/2 |u|2 . However, this need not be the final saturation level. Once the fastest viscous motions are quenched, the baton of magnetic amplification can be passed to increasingly slower but more energetic field-stretching motions at increasingly larger scales. To see how this process can lead to nonlinear growth, let us denote the scale of the smallest, weakest, but fastest as yet unsaturated motions at time t by ∗(t), and their stretching rate by γ∗(t) = u ∗ / ∗. These motions are the most efficient at amplifying the field at this particular time, but by definition their kinetic energy is also such that |B|2 (t) ∼ u2∗ . From the induction equation, we can then estimate that

d |B|2 dt

∼ γ∗(t) |B|2

,

∼ u3∗ / ∗ .

(3.68) (3.69)

Now, u3∗ / ∗ is proportional to the rate of turbulent transfer of kinetic energy ε at scale ∗, but note that this quantity is actually time- and scale-independent in Kolmogorov
turbulence. Equation (3.69) therefore shows that for this kind of turbulence, magnetic
energy should grow linearly in time at a rate corresponding to a fraction ζ of ε,

|B|2 ∝ ζεt .

(3.70)

In this regime, the magnetic field quenches stretching motions at increasingly large ∗(t) until its energy becomes comparable to the kinetic energy of the largest, most energetic shearing motions u0 at the integral scale of the flow 0, i.e. the dynamo saturates in a state of global equipartition

|B|2 ∼ |u|2 .

(3.71)

As the scale ∗(t) of the motions driving the dynamo slowly increases, there should be a
corresponding slow increase in the length of magnetic folds. Also, note that the gradual decrease of the maximum instantaneous stretching rate γ∗(t) leaves the diffusion of the
magnetic field unbalanced at increasingly larger scales in the course of the process. This
secular increase of magnetic energy should therefore be accompanied by a corresponding slow increase of the resistive scale as η ∼ (η/γ∗(t))1/2 ∼ (ηt)1/2 for Kolmogorov turbulence. Perpendicular magnetic reversals associated with the spectral peak at η do not play an obvious active role in the problem though, as they do not contribute to
the Lorentz force. Overall, we therefore expect that the folded structure of the field and
general shape of the large-P m magnetic spectrum should be preserved in this scenario.
The particular form of the Lorentz force implies that saturation is possible without the
need for scale-by-scale equipartition. Spectra derived from numerical simulations of the

Dynamo theories

47

saturated regime appear to support the latter conclusion qualitatively (Schekochihin et al. 2004c).
The previous argument is non-local in the sense that the magnetic field, which has reversals at the resistive scale, appears to strongly interact with the velocity field at scales much larger than that (that saturated isotropic MHD turbulence has a nonlocal character remains controversial though, see e.g. Aluie & Eyink 2010). A different nonlinear dynamo-growth scenario can be constructed if one instead assumes that nonlinear interactions between the velocity and magnetic field are predominantly local (Beresnyak 2012), as could for instance (but not necessarily exclusively) be the case for P m of order one or smaller. Imagine a situation in which the magnetic spectrum at time t now peaks at a scale (also denoted by ∗(t) for simplicity) where there is a local equipartition between kinetic and magnetic energy. The dynamics at scales larger than ∗(t) is hydrodynamic and the turbulent rate of energy transfer from these scales to ∗(t) is therefore the standard hydrodynamic transfer rate ε. At scales below ∗(t), the magnetic field becomes dynamically significant and the turbulent cascade turns into a joint MHD cascade of magnetic and kinetic energy characterised by a different constant turbulent transfer rate εMHD. Locality of interactions and energy conservation in scale space then lead to the prediction that the magnetic energy should grow as (ε−εMHD)t. For Kolmogorov turbulence, u ∼ 1/3, and therefore a linear in time increase of the magnetic energy is associated with an increase of the equipartition scale as ∗(t) ∼ t3/2. Numerical results obtained by Beresnyak (2012) for P m = 1 suggest that nonlinear magneticenergy growth in this scenario may be relatively inefficient, (ε − εMHD)/ε ∼ 0.05. It has been argued recently that this nonlinear growth regime may be relevant to galaxy clusters accreting through cosmic time (Miniati & Beresnyak 2015). As the turbulence injection scale 0 also grows in such systems, the relative inefficiency of the dynamo in this nonlinear regime would suggest that the ratio between ∗ and 0 remains small as both scales grow, so that a statistically steady saturated state is never achieved.

3.5.3. Saturation at large and low P m
It remains unclear which of the previous scenarios, if any, applies to the large-P m limit and, more generally, whether a unique, universal nonlinear growth scenario applies to different P m regimes. Numerical simulations of the P m > 1 regime (e.g. Haugen et al. 2004; Schekochihin et al. 2004c; Beresnyak 2012; Bhat & Subramanian 2013) do not as yet provide a definitive answer to these questions. As far as the P m 1 regime is concerned, it has so far proven impossible to explore numerically the astrophysically relevant asymptotic limit Rm Re 1 in which the viscous and integral scales of the dynamo flow are properly separated.
The physics of saturation at low P m also very much remains terra incognita at the time of writing of these notes. The only nonlinear results available in this regime are due to Sahoo et al. (2011) and Brandenburg (2011). One of the positive conclusions of the latter study is that dynamical simulations at low P m appear to be slightly less demanding in terms of numerical resolution than their kinematic counterpart because the saturated magnetic field diverts a significant fraction of the cascaded turbulent energy before it reaches the viscous scale. Brandenburg’s simulations at largest Rm also suggest that the efficiency of the dynamo should be quite large even at low P m, with magnetic energy of the order of at least 30% of the kinetic energy.

3.5.4. Reconnecting dynamo fields
Another potentially very important problem in the nonlinear regime is that of the stability of small-scale dynamo fields (in particular magnetic folds at large P m) to fast

48

F. Rincon

Figure 23. 2D snapshots of |u| (left) and |B| (right) in a nonlinear simulation of small-scale dynamo driven by turbulence forced at the box scale at Re = 290 Rm = 2900, P m = 10 (the magnetic energy spectra for this 5123 spectral simulation suggest that it is reasonably well resolved). At such large Rm, the dynamo field becomes weakly supercritical to a secondary fast-reconnection instability in regions of reversing field polarities associated with strong electrical currents. The instability generates magnetic plasmoids and outflows, leaving a small-scale dynamical imprint on the velocity field (unpublished figure courtesy of A. Iskakov and A. Schekochihin).
MHD reconnection for Lundquist numbers Lu = UAL/η of the order of a few thousands (or equivalently Rm of a few thousands in an equipartitioned nonlinear dynamo regime characterised by a typical Alfv´en velocity UA of the order of the r.m.s. flow velocity). This regime has only become accessible to numerical simulations in the last few years, but plasmoid chains typical of this process (Loureiro et al. 2007, 2012) have now been observed in different high-resolution simulations of small-scale dynamos by Andrey Beresnyak and by Alexei Iskakov and Alexander Schekochihin (Fig. 23), as well as in simulations of MHD turbulence driven by the magnetorotational instability (Kadowaki et al. 2018). The relevance and implications of fast, stochastic reconnection processes for the saturation of dynamos in general, and the small-scale dynamo in particular, are currently not well understood, although there is some nascent theoretical activity on the problem (Eyink, Lazarian & Vishniac 2011; Eyink 2011). A tentative phenomenological model of the reconnecting small-scale dynamo, inspired by similar results on MHD turbulence in a guide field, is due in an upcoming review by Schekochihin (2019).
3.5.5. Nonlinear extensions of the Kazantsev model*
Let us finally quickly review a few nonlinear extensions of the Kazantsev model that readers begging for more quantitative mathematical derivations may enjoy exploring further. The general idea of these models is to solve equation (3.25) for the magnetic correlator for a dynamical velocity field consisting of the original stochastic kinematic Kazantsev-Kraichnan field, plus a nonlinear magnetic-field-dependent dynamical correction uNL accounting for the effect of the Lorentz force on the flow. By massaging the new unclosed mixed correlators associated with uNL in equation (3.25), one picks up new quasilinear or nonlinear terms in Kazantsev’s equation (3.29). Stationary solutions of this modified equation for the saturated magnetic correlator are then sought. Of course, in the absence of a good analytical procedure to solve the Navier-Stokes equation with a magnetic back-reaction, the exact form of uNL must be postulated on phenomenological

Dynamo theories

49

grounds. Subramanian (1998), for instance, introduced an ambipolar drift proportional
to the Lorentz force, uNL = a J × B, where a is a dimensional constant. The closure procedure leads to a simple dynamical renormalisation of the magnetic diffusivity η →

ηNL = η + 2aHL(0, t), and therefore to a renormalisation of the magnetic Reynolds number, i.e. RmNL = u0 0/ηNL (HL is the longitudinal magnetic correlator introduced in equation (3.23)). The saturated state of the model is the neutral eigenmode of
the marginally-stable linear problem, scaled by an amplitude fixed by the condition

RmNL = Rmc. This model predicts a total saturated magnetic energy much smaller than the kinetic energy (by a factor Rmc) with locally strong equipartition fields organised into non-space-filling narrow rope structures.

Another model, put forward by Boldyrev (2001), postulates a velocity strain tensor of

the form

∂uiNL ∂xk

=

−

1 ν

BiBk

−

1 3

δik

B2

,

(3.72)

motivated by the dynamical feedback of the Lorentz force on viscous eddies expected in

the early phases of saturation in the large P m regime. This prescription neatly leads to a modified version of the Fokker-Planck equation (3.59) for the p.d.f. of magnetic-field

strength, whose stationary solution is a Gaussian. Simulations suggest that the magnetic

p.d.f. in the saturated regime of the dynamo is exponential, but the determination of the exact shape of the p.d.f. appears to bear some subtle dependence on rare but intense

stretching events. Phenomenological considerations can be used to fine-tune a very similar

model to generate exponential p.d.f.s (Schekochihin et al. 2002c). Yet another possibility,

explored by Schekochihin et al. (2004b), is to postulate a local magnetic-field-orientation dependent anisotropic correction κa to the correlation tensor κij of the Kazantsev velocity

field to model the effects of magnetic tension on the flow. The generalised correlator can

be expressed in spectral space as

κij (k) = κ(i)(k, |µ|) δij − kˆikˆj

+ κa(k, |µ|) BˆiBˆj + µ2kˆikˆj − µBˆikˆj − µkˆiBˆj ,

where µ = kˆ · Bˆ and kˆ = k/k. The Kazantsev equation derived from this model is more complicated and must be solved numerically. The results appear to reproduce the evolution of the magnetic spectra of simulations of the saturated large-P m regime of the small-scale dynamo reasonably well.
Overall, we see that multiple nonlinear extensions of the Kazantsev formalism are possible. The potential of this kind of semi-phenomenological approach to saturation has probably not yet been fully explored.

4. Fundamentals of large-scale dynamo theory
In the previous section, we studied the problem of small-scale dynamo action at scales comparable to, or smaller than the integral scale of the flow 0. We are now going to consider large-scale dynamo mechanisms by which the magnetic field is amplified at system scales L much larger than 0. At first glance, large-scale dynamos appear to be quite different from small-scale ones: while the latter rely on dynamical mechanisms that do not particularly care whether the system is globally isotropic or not, the former appear to generically require an element of large-scale symmetry-breaking such as rotation or shear. In particular, we will see that flow helicity is a major facilitator of largescale dynamos. In the absence of any such ingredient, it would seem extremely difficult

50

F. Rincon

(although not necessarily impossible) to maintain the spatial and temporal coherence of a large-scale magnetic-field component over many turbulent “eddies” and turnover times. We should therefore not be surprised to discover in this section that the classical theory of large-scale dynamos looks very different from what we have encountered so far. However, we will also find that the spectre of small-scale dynamos looms over largescale dynamos, and that both are in fact seemingly inextricable in large-Rm regimes of astrophysical interest. But, before we dive into such theoretical considerations, let us once again comfortably acquaint ourselves with the problem at hand by reviewing some straightforward numerical results and a bit of phenomenology.

4.1. Evidence for large-scale dynamos
As explained in §1.2, observational measurements of dynamical large-scale solar, stellar and planetary magnetic fields provide the best empirical evidence for large-scale MHD dynamo mechanisms. Reverse-engineering observations to understand the detailed underlying physics, however, is a very challenging task. Basic experimental evidence for the kind of physical mechanisms underlying large-scale dynamos that we will study in this section has also been incredibly long and difficult to obtain (Stieglitz & Mu¨ller 2001), and has so far unfortunately only provided relatively limited insights into the problem in generic turbulent flows, especially at high Rm. Therefore, despite their own flaws and limitations, numerical simulations have long been, and will for the foreseeable future, remain the most powerful tool available to theoreticians to make progress on this particular problem.
The first published numerical evidence for a growing large-scale dynamo is again due to Meneguzzi et al. (1981), and it was presented in the same paper as the first simulation of the small-scale dynamo. In the second simulation of their study, turbulent velocity fluctuations u with a net volume-averaged kinetic helicity u · ∇×u V are driven at an intermediate scale 0 < L of the (spatially periodic) numerical domain: magnetic energy grows exponentially and ends up being predominantly concentrated at the box scale L (Fig. 24). Comparing Fig. 24 to Fig. 10, we see that large-scale helical dynamo growth takes place at a much slower pace than the small-scale, non-helical dynamo. While Rm in these particular simulations is close to 40, the critical Rm for a large-scale, maximally helical dynamo in this configuration is actually O(1) (see e. g. Brandenburg 2001). This is significantly lower than Rmc,ssd for the small-scale dynamo, and explains why proofof-principle simulations of this problem could be done at relatively low resolution (323 in a spectral representation). What does the magnetic field of this dynamo look like ? Figure 25, reproduced from the work of Brandenburg (2001), illustrates the multiscale nature of the problem and the build-up of a large-scale magnetic-field component at scale L, on top of magnetic-field fluctuations at scales comparable to or smaller than 0. The simple conclusion of these selected controlled numerical experiments (and many others) is that there appears to be such a thing as large-scale statistical dynamo action. Now, how does this kind of dynamo work ? And why does flow helicity appear to be important in this context ?

4.2. Some phenomenology In order to answer these questions and to introduce the phenomenology of the problem in an intuitive way, let us travel back in time to the historical roots of dynamo theory.
4.2.1. Coherent large-scale shearing: the Ω effect The winding-up of a weak magnetic-field component parallel to the direction of the
velocity gradient into a stronger magnetic-field component parallel to the direction of

Dynamo theories

51

Figure 24. Numerical evidence for a kinematic large-scale dynamo driven by “small-scale” 3D homogeneous, pseudo-isotropic, helical turbulence forced at 0 corresponding to one fifth of the box size (Rm 40, P m = 4). Time-evolution of the kinetic energy (EV ), magnetic energy (EM ), and magnetic helicity (HM ). Time is measured in multiples of an O(1) fraction of the turnover time 0/u0 at the injection scale (Meneguzzi et al. 1981).
the velocity field itself, the Ω effect is undoubtedly a very important inductive process in any shearing or differentially rotating system (Cowling 1953). This effect is illustrated in Fig. 26 in spherical geometry. As discussed in §2.3.2, however, this effect is only good at producing toroidal field out of poloidal field, and cannot by itself sustain a dynamo†.
4.2.2. Helical turbulence: Parker’s mechanism and the α effect
After Cowling’s (1933) antidynamo work, it became clear that more complex, threedimensional, non-axisymmetric physical mechanisms were required to sustain the poloidal field against resistive decay. One such mechanism was first identified in a landmark publication by Parker (1955a), and is commonly referred to as the α effect after the theoretical work of Steenbeck et al. (1966); Steenbeck & Krause (1966) that we will introduce in §4.3. In his work, Parker considered the effects of helical fluid motions, typical of rotating convection in the Earth’s core or in the solar convection zone, on an initially straight magnetic field perpendicular to the axis of rotation (Fig. 27 top). Field lines rising with the hot convecting fluid also get twisted in the process, and thereby acquire a component perpendicular to the original field. A statistical version of this mechanism involving an ensemble of localised small-scale swirls, all with the same sign of helicity, would effectively couple the large-scale toroidal and poloidal field components, which could then lead to effective large-scale dynamo action. Parker’s mechanism can equally turn toroidal field into poloidal field, and poloidal field into toroidal field (the initial horizontal orientation of the magnetic field drawn in Fig. 27 (top) is arbitrary). It should therefore be sufficient to excite a dynamo even in the absence of large-scale shearing.
In practice, there are several issues with this simple picture, the most important of which will be discussed in §4.3.6 and in §4.6. Besides, as we will discover later in this section and in §5, the α effect is not the only statistical effect capable of exciting a large-scale dynamo. Parker’s idea, however, was seminal in the development of largescale dynamo theory and remains one of its central pillars, not least because it is directly
† A poloidal/toroidal and axisymmetric/non-axisymmetric decomposition and terminology is used here because the large-scale dynamo problem is most commonly discussed in the context of cylindrical or spherical systems such as accretion discs, “washing machines” filled with liquid sodium, or stars. The same phenomenology applies in Cartesian geometry though, as the physics discussed does not owe its existence to curvature effects or geometric constraints.

52

F. Rincon

Figure 25. Time-evolution of the x-component of the magnetic field in a plane in a 3D simulation of large-scale dynamo action driven by homogeneous, pseudo-isotropic helical turbulence forced at one fifth of the box size (Rm = 180, P m = 1). Time is measured in multiples of an O(1) fraction of the eddy turnover time (adapted from Brandenburg 2001).
connected to rotating dynamics, and rotation and large-scale magnetism in the Universe always seem to go hand-in-hand.
4.2.3. Writhe, twist, and magnetic helicity
An important aspect of the kinematic evolution of a magnetic field in a helical velocity field is the associated dynamics of magnetic helicity Hm subject to the constraint of total magnetic helicity conservation introduced in §2.1.2. To illustrate this, consider again the simple three-dimensional evolution of an initially straight magnetic flux tube in a steady right-handed helical swirl depicted in Fig. 27 (top). As the swirl acts upon the flux tube and makes it rise, a “Parker loop”, or large-scale writhe, is created. The current associated with this loop is anti-parallel to the large-scale field direction (and so is the vector potential in the Coulomb gauge), i.e. the process generates negative magnetic helicity on scales larger than that of the loop. But the same motion simultaneously

Dynamo theories

53

⌦

⌦

B

B

slow

fast

slow

Figure 26. A poor perspective drawing of the Ω effect in a spherical fluid system with latitudinal differential rotation Ω(θ)ez (maximum at the equator in this example): the latitudinal shear winds up an initial axisymmetric poloidal field into a stronger axisymmetric toroidal field in the regions of fastest rotation. In the absence of resistivity and any other dynamical effect, the growth of the toroidal field is linear, not exponential, in time. In the resistive case, and in the absence of further three-dimensional dynamical effects, the field as a whole is ultimately bound to decay (Cowling’s theorem).
twists the magnetic-field lines around the flux tube, thereby generating a local current within the tube with a positive projection along the local field, i.e. positive local magnetic helicity. If we ignore resistive effects and assume that there are no helicity losses out of the domain under consideration, equation (2.13) shows that the total magnetic helicity of the system is conserved, so that the negative helicity/left-handed writhe generated on large scales must be exactly balanced by the positive helicity/right-hand twist generated on small scales if Hm = 0 initially (for a left-handed helical swirl, the sign of large-scale and small-scale magnetic helicity is opposite). Overall, Parker’s kinematic mechanism therefore tends to generate a partition of large- and small-scale magnetic helicities of opposite signs.
This effect can be illustrated in a more quantitative way by computing for instance the magnetic helicity spectrum of a twisted magnetic flux tube (Fig. 27 bottom). Similar computations for fully turbulent helically-driven systems lead to the same results and conclusions (e.g. Mininni 2011). In the period preceding the very first simulations of helical MHD turbulence, it was suggested that such dynamics may be a consequence of the existence of an inverse “cascade” of magnetic helicity associated with the magnetic helicity conservation constraint (Frisch et al. 1975). This interpretation can be broadly justified in the framework of a three-scale interaction model, however numerical simulations suggest that the magnetic helicity transfer process in helical MHD turbulence is very different in nature from hydrodynamic cascades, as it does not proceed on turbulent dynamical timescales and is non-local in spectral space (Brandenburg, Dobler & Subramanian 2002; Brandenburg & Subramanian 2005a).
Finally, let us point out that the magnetic tension associated with the small-scale curvature of twisted magnetic-field lines should be an important source of back-reaction

54 B

F. Rincon j
J
u

Figure 27. Top: sketch of the dynamics of a magnetic flux tube in Parker’s mechanism for a right-handed helical velocity fluctuation u, showing a left-handed large-scale magnetic writhe associated with a large-scale current J, and a right-handed internal twist associated with a small-scale current j. This particular configuration is generally thought to be representative of the dynamics in the southern hemisphere of rotating stars with a strongly-stratified convection zone, where motions have a net cyclonic bias (§4.4.1). Bottom: computation of the Cauchy solution of an initially straight magnetic flux tube in a cyclonic velocity field (left), and corresponding magnetic-helicity spectrum (right) (adapted from Yousef & Brandenburg 2003).
of the field on the flow in the dynamical stages of the dynamo. Accordingly, we will discover in §4.6 that the dynamics of magnetic helicity is a key ingredient of nonlinear theories of large-scale helical dynamos driven by an α effect.
4.3. Kinematic theory: mean-field electrodynamics
How can we turn this kind of phenomenology into a mathematical theory of large-scale dynamos ? The classical approach to this problem is called mean-field dynamo theory, or mean-field electrodynamics (Steenbeck et al. 1966; Vainshtein 1970; Moffatt 1970a). Its mathematical machinery, and applications, have now been covered by a countless number of authors. We will therefore only provide a superficial and rather casual presentation of it in this tutorial, with the main objective being to emphasise the main underlying ideas and to frame the theory into a broader discussion. Readers are referred to the classic textbooks by Moffatt (1978) and Krause & Ra¨dler (1980), and to recent dedicated reviews by Hughes (2018) and Brandenburg (2018) for more exhaustive presentations.
4.3.1. Two-scale approach
The general idea behind mean-field theory is to split the magnetic field, velocity field and the corresponding MHD equations into mean and fluctuating parts in order to isolate the net r.m.s. effect of the fluctuations on the mean (“large-scale”) fields. Depending on the problem, the mean can be defined as an average over a statistical ensemble of realisations of the flow, as an average over one dimension (e.g., the toroidal direction for

Dynamo theories

55

problems in spherical or cylindrical geometry), over two dimensions, or, if there is a proper spatial and temporal dynamical scale-separation in the problem, as an average over the small-scale dynamics. In what follows, we will assume that any such averaging procedure satisfies the so-called Reynolds rules (for a more detailed discussion of averaging in the large-scale dynamo problem, see Brandenburg & Subramanian (2005a), Chap. 6.2, and Moffatt (1978), Chap. 7.1), and will essentially consider a formulation based on a two-scale decomposition of the dynamics. Namely, the magnetic and velocity fields are split into large-scale, mean-field parts ( 0, denoted by an overline) and small-scale, fluctuating parts ( 0, denoted by b and u),

U=U+u , B=B+b .

(4.1) (4.2)

We will also occasionally interpret equation (4.1) as a decomposition into axisymmetric and non-axisymmetric field components in systems with a rotation axis†. In order to minimise the physical complexity, we also restrict the analysis to the incompressible formulation of the kinematic dynamo problem with a uniform magnetic diffusivity, equation (2.18).
Let us start with the induction equation averaged over scales larger than 0,

∂B ∂t

+

U

·

∇

B

=

B

·

∇

U

+

∇×(u

×

b)

+

η∆B

.

(4.3)

To predict the evolution of B, we must determine the mean electromotive force (EMF) E = u × b driving the dynamo. This term involves the statistical cross-correlations between small-scale magnetic and velocity fluctuations of the kind that we encountered in Parker’s phenomenology. The big question, of course, is how do we calculate it ? Ideally, we would like to find an expression for E in terms of just B and the statistical properties of the fluctuating velocity field u. The logical next step is therefore to attempt to solve the induction equation for b in terms of u and B, and to substitute the solution into the above expression for E. This equation is obtained by subtracting equation (4.3) from the full induction equation (2.18),

∂b ∂t

=

∇×

u×B

+ U×b

+

u×b−u×b

+ η∆b .

(4.4)

The term in the first pair of parentheses on the r.h.s. describes the induction of smallscale magnetic fluctuations, and mixing of the field, due to the tangling and shearing of the mean field by small-scale velocity fluctuations. This is the term that we are most interested in in order to compute a mean-field dynamo effect. The term in the second pair of parentheses describes the effect of a large-scale velocity field on smallscale magnetic fluctuations. Such a velocity field usually has a much slower turnover time and amplitude compared to the fluctuations, and we will therefore consider that it has a subdominant role in the inductive dynamics of small-scale fluctuations for the purpose of this discussion. Finally, we have a combination of two terms in the third pair of parentheses, which will be henceforth referred to as the “tricky term” (also often referred to as the “pain in the neck” term). This term is quadratic in fluctuations, and there is no obvious way to simplify it without making further assumptions: so we meet again, old closure foe ! We will come back to this problem in §4.3.6.

† In the general case, we should formally distinguish between azimuthal averages and small-scale or ensemble averages. This distinction is not necessary in the context of this presentation, and is therefore ignored for the sake of mathematical and notational simplicity.

56

F. Rincon

4.3.2. Mean-field ansatz
Equation (4.4) is a linear relationship between b and B if U is independent of B (as is the case in the kinematic regime), i.e.

LU(b) = ∇× u × B ,

(4.5)

where LU is a linear operator functionally dependent on U (Hughes 2010). Despite its seemingly innocuous nature, there is actually quite a lot of complexity hiding in this equation due to the presence of the tricky term on the l.h.s., but for the time being we are going to ignore the presence of this term and simply postulate that equation (4.5) is indicative of a straightforward linear relationship between the fluctuations b and the mean field B. If this holds, then we may expand the mean EMF as

(u × b)i = aij Bj + bijk∇kBj + · · · ,

(4.6)

where the spatial derivative is with respect to the slow spatial variables over which B is non-uniform. This is called the mean-field expansion. Terms involving higher-order derivatives are discarded because the spatial derivative is slow. In the general case of inhomogeneous, stratified, anisotropic, differentially rotating flows, this expansion is usually recast in terms of a broader combination of greek-letter tensors and vectors that neatly isolate different symmetries (Krause & R¨adler 1980; R¨adler & Stepanov 2006):

Ei = αijBj + γ × B i − βij ∇×B j − δ × ∇×B

i

−

κijk 2

∇j Bk + ∇kBj

. (4.7)

Here, α and β are symmetric second-order tensors, γ and δ are vectors, and κ is a third-order tensor, namely

αij

=

1 2

(aij

+

aji)

,

βij

=

1 4

(εikl

bjkl

+

εjkl

bikl)

,

(4.8)

γi

=

−

1 2

εijk

ajk

,

δi

=

1 4

(bjji

−

bjij )

,

(4.9)

κijk

=

−

1 2

(bijk

+

bikj )

.

(4.10)

All these quantities formally depend on Rm, P m, and the statistics of the velocity field

(and magnetic field, in the dynamical regime). The two most emblematic mean-field

effects are those deriving from α and β. The former is related to Parker’s mechanism

and can drive a large-scale dynamo, while the latter is easily interpreted as a turbulent

magnetic diffusion (albeit not necessarily a simple one).

4.3.3. Symmetry considerations
Symmetry considerations can be used to determine which of the previous mean-field quantitities are in principle non-vanishing for a given problem. In order to illustrate at a basic level how this is generally done, and what kind of mean-field effects we might expect, we will essentially discuss three particular symmetries in the rest of this section: isotropy, parity, and homogeneity.
As mentioned in §3.4, there is no preferred direction or axis of symmetry in an isotropic three-dimensional system. Tensorial dynamical quantities can only be constructed from δij and the antisymmetric Levi-Cevita tensor εijk in this case. On the other hand, if a fluid system has one or several particular directions, as is the case in the presence of rotation, stratification, a non-uniform large-scale flow U, or a strong, dynamical large-scale magnetic field, then we are in principle also allowed to use a combination of quantities such as the rotation vector Ω, the direction of gravity (or inhomogeneity)

Dynamo theories

57

g, the mean-flow deformation tensor D = ∇U + (∇U)T /2, the mean-flow vorticity
W = ∇×U or even B in the dynamical case to construct mean-field tensors.
A second important class of symmetry is parity invariance. In three dimensions, a parity transformation, or point reflection, is a combination of a reflection through a plane (an improper rotation) and a proper rotation of π around an axis perpendicular to that plane. Parity symmetry is therefore connected to mirror symmetry, although the two must be distinguished in principle in three-dimensions (see discussion in Moffatt 1977, Chap. 7, footnote 4). We essentially ignore this distinction in what follows and will talk indiscriminately of mirror/parity/reflection symmetry breaking. Under a point reflection, the position vector transforms as r → −r, and the velocity field transforms as U → −U. Now, if we look at some local, homogeneous, isotropic hydrodynamic solution of the Navier-Stokes equation driven by a non-helical force and with no rotation, it is clear that the image of the velocity field under a parity transformation is itself a solution of the equations. In the MHD and/or rotating case, on the other hand, the image of a solution is only a physical solution itself if we keep B and / or Ω the same, i.e. B → B, and Ω → Ω. Accordingly, we say that the velocity field is a true vector, while the magnetic field and rotation vectors are pseudo-vectors†. Scalars and higher-order tensors can also be divided into pseudo and true quantities, depending on how they transform under reflections. Pseudo-scalars simply change sign under reflection. Let us finally introduce a few simple rules for the manipulation of vectors: vector products of pairs of true vectors or pairs of pseudo-vectors produce a pseudo-vector, while mixed vector products involving a true and a pseudo-vector produce a true vector. Curl operators turn a true vector into a pseudo-vector, and vice-versa (as illustrated by Ampere’s law, or Biot-Savart’s law of magnetostatics between the electric current, a true vector field constructed from the velocities of charged particles, and the magnetic field). Let us now come back to equation (4.7). The mean EMF on the l.h.s., being the vector product of a true vector and a pseudo-vector, is a true vector, while the mean magnetic field on the r.h.s. is a pseudo-vector. Thanks to the particular decomposition used in the equation, it is then straightforward to see that α and κ must be pseudo-tensors, while β is a true tensor. Similarly, δ must be a pseudo-vector, and γ is a true vector.
Why is this important ? If a flow is parity-invariant, as is for instance the case of standard homogeneous, isotropic non-helical turbulence, then its image under a parity transformation has the exact same statistical properties, and should produce the exact same physical statistical effects (with the same sign). However, we have identified several pseudo-quantities, most importantly α, which by construction must change sign under a parity transformation. These quantities must therefore vanish for a parity-invariant flow. In order for them to be non-zero, parity invariance must be broken one way or the other, and a particular way to achieve this is by making the flow helical. Parker’s big idea was to suggest that this kind of effect can in turn lead to large-scale dynamo action. We will find in the next paragraphs that this can indeed be proven rigorously in some particular regimes. Of course, it is also this argument that motivated the set-up of the dynamo simulations of Meneguzzi et al. (1981) discussed in the introduction of this section.
More generally, the previous discussion highlights that the different large-scale statistical effects encapsulated by the greek-letter tensors introduced in equation (4.7) are only non-zero when some particular symmetries are broken. For pedagogical purposes, we will

† The terminology “true” and “pseudo” stems from the transformation laws of these quantities under simpler mirror transformations. Mirror-invariance of MHD requires that we reflect all vectors (true and pseudo) under a mirror transformation, but further flip the sign of pseudo-vectors such as B.

58

F. Rincon

essentially concentrate on kinematic and dynamical problems involving simple isotropic, homogeneous α and β effects in this section. Important variants of these effects, as well as additional effects arising in stratified, rotating and shearing flows will however also be touched upon in §4.4.

4.3.4. Mean-field equation for pseudo-isotropic homogeneous flows
In what follows, we will more specifically be concerned with the Cartesian version of the problem of large-scale dynamo action driven by isotropic, homogeneous, statisticallysteady, but non-parity-invariant flows (also referred to as pseudo-isotropic flows), the archetype of which is our now familiar helically-forced isotropic turbulence. Although the helical nature of many flows in nature is a consequence of rotation, we will not explicitly try to capture this effect here, and instead assume that helical motions are forced externally (a discussion of the α effect in rotating, stratified flow will be provided in §4.4.1). On the other hand, we retain a large-scale shear flow typical of differentially rotating systems to accommodate the possibility of an Ω effect, but neglect possible anisotropic statistical effects associated with this shear for simplicity (the latter will be discussed in §4.4.3). Under all these assumptions, αij = α δij and βij = β δij (or bijk = β ijk) are both finite and independent of space and time, and all the other meanfield effects in equation (4.7) vanish. Substituting equation (4.6) into equation (4.3), we obtain a closed evolution equation for the mean field,

∂B ∂t

+

U

·

∇

B

=

B

·

∇

U

+

α∇×B

+

(η

+

β)∆B

.

(4.11)

The term on the r.h.s. proportional to the curl of B, the α effect, is the only one, apart from the shear, that can couple the different components of the mean field for the simple system under consideration. The last term on the r.h.s., the β effect, acts as an effective turbulent magnetic diffusivity operating on the mean field.
The presence of the α and β terms in the linear equation (4.11) implies that we can now in principle obtain exponentially growing mean-field dynamo solutions if the statistical properties of the flow allow it†. This conclusion may seem puzzling at first in view of Cowling’s theorem. If we interpret large-scale averages as azimuthal averages, how is it possible that equation (4.11), which only involves axisymmetric quantities, has growing solutions ? The key here is to realise that we have massaged the original problem quite a bit to arrive at this result. Equation (4.11) is not the pristine axisymmetric induction equation but a simpler model equation. The exact evolution equation for B, equation (4.3), has a dynamo-driving term E, which is quadratic in non-axisymmetric (or small-scale) fluctuations. In order to arrive at equation (4.11), we have simply expressed this term using a simple closure equation (4.6) motivated by the linear nature of equation (4.4). In other words, the three-dimensional nature of the dynamo is now hidden in the mean-field coefficients α and β. We will demonstrate in §4.3.6 that these coefficients are indeed functions of the statistics of small-scale fluctuations.

4.3.5. α2, αΩ and α2Ω dynamo solutions
In a Cartesian domain with periodic boundary conditions, we may seek simple solutions of equation (4.11) in the form

B = Bk exp [st + ik · x] + c.c. ,

(4.12)

† If α = 0, this requires a negative turbulent diffusivity (β < −η), which is of course not a feature of daily-life turbulence but is possible under certain circumstances. In standard turbulence conditions at large Rm, β η > 0.

Dynamo theories

59

↵ e↵ect

Poloidal field Bpol

Toroidal field Btor

⌦ e↵ect (B · r U)

(+↵ e↵ect)

Poloidal field Bpol

Toroidal field Btor

↵ e↵ect

↵ e↵ect

Figure 28. The α2 (left), αΩ, and α2Ω (right) mean-field dynamo loops.

subject to k · Bk = 0. We first consider the case with no mean flow or shear. Substituting equation (4.12) into equation (4.11) with U = 0, and solving the corresponding eigenvalue problem, we find a branch of growing dynamo modes with a purely real eigenvalue corresponding to a dynamo growth rate

γ = |α|k − (η + β) k2 .

(4.13)

For β > 0, the maximum growth rate and optimal wavenumber are

γmax

=

α2 4 (η +

β),

kmax

=

|α| 2 (η + β)

(4.14)

(for consistency, we should have kmax 0 < 1, but to check this we first need to learn how to calculate α and β). The dynamo is possible here because the α effect couples the two independent components of B in both ways through the curl operator (in particular, it also couples the toroidal field component to the poloidal field component in cylindrical or spherical geometry). For this reason, this mean-field dynamo model is usually called the α2 dynamo. It is this kind of coupling, illustrated in Fig. 28 (left), that drives the helical large-scale kinematic dynamo effect in the simulations shown in Fig. 24 and Fig. 25. A discussion of magnetic helicity and linkage dynamics in the α2 dynamo, complementary to the Parker mechanism picture shown earlier in Fig. 27, can be found in Blackman & Hubbard (2014).
What about the astrophysically relevant situation involving a mean shear flow or differential rotation, on top of some smaller-scale turbulence ? While this case is of course most directly connected to problems in global spherical and cylindrical geometries, it it most easily analysed in the Cartesian shearing sheet model introduced in §2.1.5. In the presence of linear shear, U = US = −Sx ey, the mean-field equation (4.11) reads

∂B ∂t

=

−SBxey

+ α∇×B

+ (η

+ β)∆B

.

(4.15)

The U · ∇ B term on the l.h.s. of equation (4.11) vanishes for a purely azimuthal mean flow because B is axisymmetric† (independent of y in our notation). The couplings in equation (4.15) allow different forms of large-scale dynamo action illustrated in Fig. 28 (right). In the most standard case, commonly referred to as the αΩ dynamo, toroidal/azimuthal (y) field is predominantly generated out of the poloidal field (of which Bx is one component) via the Ω effect (the first term on the r.h.s.), while the

† We would have to retain this term in order to describe the advection of the mean field in meridional planes by a poloidal mean flow. This effect does not amplify the magnetic field on its own, but it can redistribute it spatially in a way that alters the nature of a large-scale dynamo in comparison to a fully homogeneous process. Meridional circulation is for instance thought to be important for the global solar dynamo mechanism, see e.g. Charbonneau (2014).

60

F. Rincon

α effect associated with the small-scale helical turbulence turns the toroidal field back into poloidal field. In α2Ω dynamos, the α effect also significantly contributes to the
conversion of poloidal field into toroidal field. Let us briefly look at axisymmetric mean-field solutions† B(x, z, t) of equation (4.15)
in the αΩ case. We again seek plane-wave solutions in the form of equation (4.12), with a
wavenumber k = (kx, 0, kz). In the αΩ limit S αkz, the solution of the linear dispersion relation leads to a branch of growing dynamo solutions if the so-called dynamo number D ≡ αSkz/ 2(η + β)2k4 is larger than unity (Brandenburg & Subramanian 2005a). The unstable eigenvalue in this case has an imaginary part,

R(s)

≡

γ

=

1 2

|S αkz |1/2

−

(η

+

β)k2,

(4.16)

I (s)

≡

ω

=

1 2

|S αkz |1/2

,

(4.17)

i.e. the solutions take the form of growing oscillations. The interplay between the α effect

and the Ω effect can therefore produce dynamo waves involving periodic field reversals.

This conclusion, first obtained by Parker (1955a), was key to the realisation that differ-

entially rotating turbulent MHD systems can host large-scale oscillatory dynamos, and

sparked an enormous amount of research on planetary, solar, stellar and galactic dynamo

modelling.

4.3.6. Calculation of mean-field coefficients: First Order Smoothing
The mean-field formalism appears to be a very convenient and intuitive framework to model large-scale turbulent dynamos and to simplify their underlying nonlinear MHD complexity. However, we have so far ignored several difficult key questions: under which conditions is the mean-field ansatz justified ? And are there systematic ways to derive mean-field coefficients such as α and β from first principles, given a velocity field with prescribed statistical properties ?
The former question largely conditions the answer to the latter. To illustrate the nature of the problem, recall that we have carefully avoided to discuss how to handle the tricky term in equation (4.4) in our earlier discussion of the mean-field ansatz. However, if we want to calculate α, we need to do something about it ! The radical closure option is to simply neglect this term. This approximation is often called the First Order Smoothing Approximation (FOSA), Second Order Correlation Approximation (SOCA), Born approximation, or quasilinear approximation. It has the merit of simplicity, but can only be rigorously justified in two limiting cases:
• small velocity correlation times (either small Strouhal numbers St = τc/τNL 1, or random wave fields)
• low magnetic Reynolds numbers Rm = τη/τNL 1 . where τNL = 0/urms and τη = 20/η in the two-scale model (assuming P m = O(1)). To see this, we borrow directly from Moffatt (1978) and consider the ordering of equation (4.4),

∂b ∂t

= ∇× u × B

O(brms /τc )

O(B /τNL )

+ ∇× (u × b) − u × b
Tricky term: O(brms/τNL)

+ η∆b .
O(brms/τη )

(4.18)

† Earlier in the text, we opted not to make a distinction between azimuthal and small-scale or ensemble averages. When this distinction is made, it is formally possible to seek non-axisymmetric mean-field solutions of equation (4.15) with an azimuthal wave number ky such that ky 0 1. The analysis of such solutions is a bit more involved due to the presence of the shear (Brandenburg & Subramanian 2005a).

Dynamo theories

61

where U = 0 has been assumed for simplicity. The first important thing to notice here is that the typical rate of change of b on the l.h.s. is ordered as the inverse correlation time of the velocity field, because significant field-amplification requires coherent flow stretching episodes whose typical duration is O(τc). Another handwaving way to look at this is to formally integrate equation (4.18) up to a time t, and to approximate the integral of the tricky term by τc (the times during which u remains self-correlated) times the integrand. Either way, we see that the tricky term is negligible compared to the l.h.s. in the limit St 1. The problem, of course, is that St is usually O(1) for standard fluid turbulence such as Kolmogorov turbulence. In this case, the only limit in which the tricky term is formally negligible is the diffusion-dominated regime Rm 1 in which the resistive term dominates. Unfortunately, this regime is not very interesting from an astrophysical perspective either.

4.3.7. FOSA derivations of α and β for homogeneous helical turbulence
The calculation of mean-field electrodynamics coefficients under FOSA can be done with a variety of mathematical methods, most if not all of which have already been skillfully laid out in textbooks and reviews by the inventors and main practicioners of the theory (e.g. Moffatt 1978; Krause & R¨adler 1980; Brandenburg & Subramanian 2005a; R¨adler & Rheinhardt 2007). Many of these presentations are very detailed though, and can feel a bit intimidating or overwhelming to newcomers in the field. The aim of this paragraph is therefore to provide concise, low-algebra versions of the calculations of the α and β effects distilling the essence of these methods.
In order to illustrate in the most simple possible way how this kind of derivation works, we start with a physically intuitive calculation in the St 1, Rm 1 regime. In this limit, both the tricky term and the resistive term can be neglected in equation (4.18), but we cannot formally neglect the first term on the r.h.s. of the equation, because there is no particular reason to assume that b and B should be of the same order. In fact, they are not and, as mentioned in §4.3.2, it is precisely this term that makes a large-scale dynamo possible in the first place. In order to calculate α and β, we simply substitute the formal integral solution b(x, t) of equation (4.18) into the expression of the mean EMF,

t
E(x, t) ≡ u(x, t) × b(x, t) = u(t) × ∇× u(x, t ) × B(x, t ) dt .
t0

(4.19)

In writing this expression, we have implicitly assumed that t − t0 is much larger than the typical flow correlation and resistive timescales, so that the correlation u(t) × b(t0) can be neglected, and the integral is understood to be independent of the lower bound

t0. The spatial dependence of E is to be understood as a large-scale dependence on the scale of the mean field itself. The r.h.s. of equation (4.19) only contains correlations

that are quadratic in fluctuations, as a result of the neglect of the tricky term in

equation (4.18), hence the name Second Order Correlation Approximation (or First Order

Smoothing Approximation, if we take the equivalent view that it is the quadratic term

in equation (4.18) which has been neglected). Now, a bit of tensor algebra shows that

equation (4.19) in the isotropic case reduces to

where

t
E(x, t) = αˆ(t − t )B(x, t ) − βˆ(t − t )∇×B(x, t ) dt ,
0

αˆ

=

−

1 3

u(t)

·

ω(t

)

,

βˆ

=

1 3

u(t)

·

u(t

)

,

(4.20) (4.21)

62

F. Rincon

and ω = ∇×u. The coefficients α and β are uniform in this derivation as a result of the assumption of statistical homogeneity. Now, since the correlation time of the flow is assumed small compared to its typical turnover time, we can approximate the integrals in equation (4.20) by τc times the integrand at t = t, i.e.

E = α B − β ∇×B

(4.22)

with

α

=

−

1 3

τc(u

·

ω),

β

=

1 3

τc

|u|2

.

(4.23)

As expected by virtue of the FOSA, equation (4.22) has the form of the isotropic version

of the mean-field expansion (4.6) postulated in §4.3.2, but we have now also found explicit

expressions for α and β.

The previous calculation relies on a description of the statistical properties of the flow

in configuration (correlation) space, and is representative of the general method used by

Krause, Steenbeck and R¨adler in the 1960s to develop a much broader theory of mean-field

electrodynamical effects in differentially rotating stratified flows (their most important

papers, originally published in German, have been translated in English by Roberts &

Stix (1971), and form the core of the textbook by Krause & R¨adler 1980). However, mean-

field coefficients can also be derived by means of a spectral representation of the turbulent

dynamics. This formalism is perhaps slightly less intuitive physically than the procedure

outlined above, but it makes the calculations a bit more compact and straightforward

in the general case, and is also frequently encountered in the literature (the reference

textbook here is Moffatt 1978). In order to introduce this alternative in a nutshell, we

concentrate uniquely on the problem of an α effect generated by helical homogeneous

incompressible turbulence acting upon a uniform mean magnetic field. However, we do

not a priori restrict the calculation to one of the two possible FOSA limits in this case

and, in anticipation of a further discussion of rotational effects in §4.4.3, we do not assume

isotropy from the start either (a more detailed version of this derivation, also including a

determination of the β tensor for a non-uniform field, can be found in Moffatt & Proctor

1982). Denoting the space-time Fourier transforms of u and b over the small-scale, fast variables by uˆ(k, ω) and bˆ(k, ω), we first solve equation (4.18) in spectral space subject

to FOSA as

bˆ(k, ω)

=

i B·k (−iω + ηk2) uˆ(k, ω)

.

(4.24)

Substituting this expression into the general expression for the mean EMF,

Ei =

εilmuˆ∗l (k , ω )ˆbm(k, ω)ei[(k−k )·x−(ω−ω )t]dk dω dk dω ,

(4.25)

and introducing the spectrum tensor of the turbulence Φlm(k, ω), defined by

uˆ∗l (k, ω)uˆm(k , ω ) = Φlm(k, ω)δ(k − k )δ(ω − ω ) , we find after integration over k and ω that

Ei =

iεilm (−iω +

kj ηk2

)

Φlm(k,

ω)

dk

dω

Bj .

(4.26) (4.27)

Recalling the mean-field ansatz for a uniform mean field, Ei = αijBj, the term in parenthesis is easily identified with αij. In order to simplify this expression further, we introduce the helicity spectrum function

H(k, ω) = −iknεnlmΦlm(k, ω) ,

(4.28)

Dynamo theories

63

and remark that

εilmΦlm(k, ω)

=

ki k2

H

(k,

ω)

(4.29)

for an incompressible flow (i.e., the l.h.s., being a cross-product of two vectors uˆ(k, ω)

and uˆ∗(k, ω) both perpendicular to k, is oriented along k). Substituting equation (4.29)

into equation (4.27) and using the condition H(−k, −ω) = H(k, ω) deriving from the

reality of u to eliminate the imaginary part of the integral, we arrive at the result

αij = −η

ω2

kikj + η2

k4

H

(k,

ω)

dk

dω

(4.30)

showing that the α tensor is a weighted integral of the helicity spectrum of the flow under

generic anisotropic conditions. In the isotropic case, H(k, ω) = H(k, ω), and αij = α δij

with

α

=

αii 3

=

−

η 3

(ω2

k2 + η2

k4)

H

(k,

ω)

dk

dω

.

(4.31)

One of the main advantages of the spectral formalism, clearly, is that it makes it easy

in principle to investigate either analytically or numerically the dynamo properties of

flows with different prescribed spectral distributions, from simple “monochromatic”

helical wave fields to more complex random flows characterised by a broad range of

frequencies and wavenumbers. As stressed by Moffatt (1978) though, one must be wary

of the application of the formalism at large Rm if the flow under consideration has

some significant helicity and energy at frequencies ω < u0/ 0, in which case the FOSA
approximation cannot be justified. On the other hand, we can safely use equation (4.31) to derive α analytically in the St = O(1), Rm 1 limit. In this case, ω ηk2 and the weight function in the integral in equation (4.31) can be approximated by 1/(η2k2).

Using the property

(u · ω) = H(k, ω) dk dω ,

(4.32)

we obtain the low-Rm result (Moffatt 1970a)

with

α

=

−

1 3

τη(H)(u

·

ω)

τη(H )

=

1 η

k−2H(k, ω) dk dω .
H(k, ω) dk dω

(4.33) (4.34)

An extension of the calculation to a non-uniform mean field similarly leads to

β

=

1 3

τη(E)

|u|2

,

(4.35)

with τη(E) similarly defined as a weighted integral of the kinetic energy spectrum E(k, ω). The previous results all lead to the comforting conclusion that the α effect is directly
proportional to the net kinetic helicity of the flow, in line with Parker’s original intuition of large-scale dynamo action driven by cyclonic convection. The only difference between equation (4.23) and equation (4.33) derived in the low-St and low-Rm FOSA limits respectively is the characteristic time involved. In both cases, the results translate that the twisting of magnetic-field lines into Parker loops occurs on a short time compared to the dynamical turnover time. The amount of coherent twisting by each individual swirl is limited by magnetic diffusion in the Rm 1 case, and by the fast decorrelation of

64

F. Rincon

the flow in the St 1 case. The small but systematic effects of these “impulsive” twists simply add up statistically.
The problem, however, gets significantly more complicated if we consider the more realistic and astrophysically relevant regime St = O(1), Rm 1 for which the field is essentially frozen-in. For instance, when τc = O( 0/u0), a Parker loop can do a full 360◦ turn before the swirl decorrelates, leaving us with zero net magnetic field in the direction perpendicular to the original field orientation. While this scenario is extreme, the point is that the net effect of a statistical ensemble of generic turbulent swirls with St = O(1) at large Rm is much harder to assess than in the calculations above due to cancellation effects. This kind of complication notably creates some significant difficulties with the determination of the kinematic value of α at large Rm for some families of chaotic flows with long correlation times (Courvoisier, Hughes & Tobias 2006), although not necessarily for generic isotropically-forced helical turbulence with St = O(1) (Sur et al. 2008). On top of all that, we will also discover later in §4.5 that the neglect of the tricky term is actually generically problematic at large Rm, even when St 1, as a result of the excitation of small-scale dynamo fields.
Note finally that the FOSA result (4.33), combined with β η, ensures that the theory is consistent with the two-scale assumption in the regime of low Rm. In the α2 dynamo problem, for instance, the wavenumbers k at which growth occurs at low Rm are of the order k 0 ∼ α 0/η ∼ Rm2 1 (assuming |ω|rms ∼ urms/ 0). In the St 1, Rm 1 regime, on the other hand, equation (4.23) applies, β η, and k 0 ∼ α 0/β ∼ (u · ω) 0/u2. The theory is therefore formally only self-consistent in this regime if the flow has small fractional helicity, i.e. |ω|rms urms/ 0.

4.3.8. Third-order-moment closures: EDQNM and τ -approach*
While the FOSA closure has been central to the historical development of mean-field electrodynamics, its very limited formal domain of validity implies that its predictive power and practical applicability is also formally very limited. Other closure schemes, such as the eddy-damped quasi-normal Markovian (EDQNM) (Orszag 1970; Pouquet, Frisch & Leorat 1976) or minimal τ approximation (MTA) closures (Vainshtein & Kichatinov 1983; Kleeorin, Rogachevskii & Ruzmaikin 1990; Blackman & Field 2002) operating at the next order in the hierarchy of moments, have been implemented in the context of large-scale dynamo theory in order to attempt to remedy this problem. To illustrate the idea behind these closures, consider the time-derivative of the mean EMF,

∂E ∂t

≡

u×

∂b ∂t

+

∂u ∂t

×b

,

(4.36)

and substitute ∂b/∂t by its expression in equation (4.18), and ∂u/∂t by its expression given by the Navier-Stokes equation for u, assuming that the contribution of the Lorentz force is of the form B · ∇b (this is formally only valid for small |B| brms, see e.g. discussion in Proctor 2003). From this equation, it is easy to see that the tricky term in equation (4.18) and the inertial term in the Navier-Stokes equation introduce third-order correlations in the resulting evolution equation for the second-order moment E. In order to close this equation, one possibility is to model the effects of these triple correlations (fluxes) in terms of a simple relaxation of the EMF, −E/τ (τ , the relaxation time, is sometimes assumed to be scale dependent). This qualitatively amounts to modelling the effects of the turbulent fluxes as a diffusion-like/damping effect. The result of the simplest version of this kind of calculation after applying this new closure ansatz is (e.g. Blackman

Dynamo theories

65

& Field 2002; Brandenburg & Subramanian 2005a):

∂E ∂t

=

α

B−β

∇×B −

E τ

,

(4.37)

where now

α

=

−

1 3

u · ω − (∇×b) · b

,

β

=

1 3

|u|2

.

(4.38)

One seeming advantage of this scheme, compared to FOSA, is that τ is not formally

required to be small in comparison to the typical eddy turnover time. However, it is not

a panacea either, because there is no guarantee that the net effect of triple correlations

is to relax the mean EMF in the form prescribed by an MTA-like closure. In fact, it is

all but certain that this is not strictly true for most turbulence problems, and that even

the qualitative validity of this kind of prescription is problem- and regime-dependent. In

the mean-field dynamo context, it has for instance notably been found that analytical

predictions that come out of the MTA are not always in agreement with the FOSA

predictions in the (admittedly rather peculiar) regimes in which the latter is formally

valid (e.g. R¨adler & Rheinhardt 2007, see §4.4.3 below for a discussion of a specific

example). On the other hand, analyses of numerical simulations of turbulent transport

and large-scale dynamo problems suggest that an MTA closure with τ /τNL = O(1) is not entirely unfounded either in some common fluid turbulence regimes (Brandenburg,

K¨apyl¨a & Mohammed 2004; Brandenburg & Subramanian 2005b) (of course, that τ in

this model should be of the order of τNL is expected from a simple order of magnitude

analysis of the different correlation functions involved in the closure). Overall, it seems

like we have no choice at the moment but to accept the theoretical uncertainties and

confusion that come with this kind of problem. Closures are notoriously difficult in all

areas of turbulence research (Krommes 2002), and dynamo theory is no exception to this.

Keeping these caveats in mind, it is nevertheless interesting to note that the MTA

closure applied to the helical dynamo problem gives rise in equation (4.38) to a magnetic

contribution to the α effect associated with small-scale helical magnetic fluctuations

interacting with the turbulent velocity field. The questions of the exact meaning, validity

and applicability of this result are beyond the realm of pure kinematic theory, and will

be critically discussed in §4.6 in the context of saturation of large-scale dynamos.

4.4. Mean-field effects in stratified, rotating, and shearing flows
While it is eminently instructive and captures the essence of the α effect, the problem of large-scale dynamo in a homogeneous, isotropic helical flow discussed so far in this section is quite idealised, and does not capture the full physical complexity of largescale dynamos in astrophysical, planetary and even experimental MHD flows involving a combination of rotation, shear, stratification and thermal physics. Intuitively, we may for instance expect that statistical magnetic-field generation effects and turbulent magnetic diffusion become anisotropic in the presence of a large-scale stratification, inhomogeneity, or rotation, or in the presence of a large-scale magnetic field in the dynamical regime of the dynamo. This, in itself, gives rise to significant technical complications. The problem, however, is not just one of generalising some already familiar effects to less symmetric cases. Inhomogeneity, stratification, rotation, and shear all break some symmetry and can therefore be expected to give rise to a zoo of additional large-scale statistical effects already at the kinematic level.
Before supercomputing became everyday routine, mean-field electrodynamics provided the only available, albeit not necessarily always very physically transparent, means to explore this seemingly outstanding statistical complexity in a somewhat systematic way.

66

F. Rincon

Despite all their limitations, theoretical calculations of this kind have therefore played an essential role in the development of large-scale astrophysical dynamo theory and have had a profound, long-lasting impact on how the community speaks and thinks of largescale statistical effects. The classic reference on this subject is the book of Krause & R¨adler (1980). The aim of the next few paragraphs is to single out several such effects that appear to be most generically relevant to large-scale astrophysical, planetary or experimental dynamos, and to explain how they arise within the framework of meanfield electrodynamics.

4.4.1. α effect in a stratified, rotating flow

As we discovered in §4.3.3, an α effect is only possible in flows that are not parityinvariant, a particular example of which is helically-forced, pseudo-isotropic homogeneous turbulence. The kinematic derivations presented in §4.3.7 showed that α in this problem is directly proportional (in the FOSA regimes) to the net average kinetic helicity of the flow. Here, we will see that the α effect is also non-zero in a rotating, stratified flow involving a gradient of density and/or kinetic energy, such as stratified rotating thermal convection typical of many stellar interiors (see e.g. Brandenburg & Subramanian (2005a) for a similar discussion). Beyond its obvious astrophysical relevance, this particular problem provides a good illustration of how one can use symmetry arguments to simplify matters. Zooming in on a patch of stratified, rotating turbulence, we can locally identify two preferred directions in the system, that of gravity denoted by the unit vector gˆ = g/g, and that of rotation, denoted by the unit vector Ωˆ = Ω/Ω. The pseudo-tensor α can only be constructed from these two vectors in the following way (to lowest order in g and Ω):

αij = α0 gˆ · Ωˆ δij + α1gˆj Ωˆi + α2gˆiΩˆj .

(4.39)

Of course, we still have to calculate the three mean-field coefficients. This is difficult to achieve in the general case, but explicit expressions can be obtained for a few analyticallyprescribed flows. One of the most well-known results in this context is the expression

α0

gˆ

·

Ωˆ

=

−

16 15

τc2|u|2

Ω

·

∇

ln

ρ

|u|2

,

α1

=

α2

=

−

α0 4

(4.40)

derived in the St 1 FOSA regime for an anelastic flow model (characterised by ∇ · (ρu) = 0) encapsulating the effects of stratification through a simple exponential vertical-dependence parametrisation of the background density and turbulence intensity (Krause 1967; Steenbeck & Krause 1969). Equation (4.40) obviously provides a more explicit mathematical validation of Parker’s idea that rotating convection can generate an α effect than equation (4.23), however it also suggests that some stratification along the rotation axis, not just rotation, is actually required in a rotating flow to obtain a nonzero α tensor. The physical reason underlying this result was identified by Steenbeck et al. (1966). In a stratified environment, rising fluid expands horizontally, while sinking fluid is compressed. Thereby, and upon the action of the Coriolis force, upflows in the northern hemisphere are made to rotate clockwise, while downflows are made to rotate counterclockwise. In both cases, the flow acquires negative helicity. The effect is opposite in the southern hemisphere. The net result, expressed by equation (4.40), is an antisymmetric distribution of α with respect to the equator. This argument and equation (4.40) also predict that αϕϕ (or αyy in the Cartesian model with a vertical rotation axis), the key coefficient that couples back the toroidal field to the poloidal field in an αΩ dynamo, is positive in the northern hemisphere. We will discover later in §5.1 that this result is actually problematic in the solar dynamo context.

Dynamo theories

67

The stratification effect described above formally vanishes in the incompressible Boussinesq regime of convection rotating about a vertical axis, because in this limit all motions are assumed to take place at a vertical scale much smaller than the typical density scaleheight. Note however that the presence of vertical boundaries in the system also generates converging and diverging motions that get acted upon by the Coriolis force. It is actually this type of horizontal motions, not expansions or compressions in a stratified atmosphere, that Parker apparently had in mind when he introduced the idea of an α effect generated by cyclonic convection (he used the words “influxes” and “effluxes”). In his work, Parker focused on helical motions of a given sign, but in Boussinesq convection the dynamics of overturning rotating convective motions actually leads to the generation of flow helicity of opposite signs at the top and bottom. In plane-parallel Boussinesq convection between two plates and rotating about the vertical axis (the rotating Rayleigh-B´enard problem) in particular, the existence of an “up-down” dynamical mirror symmetry with respect to the mid-plane of the convection layer (Chandrasekhar 1961) implies that the profile of (horizontally-averaged) flow helicity induced by the Coriolis force is antisymmetric in z. Accordingly, the volume-averaged flow helicity of the system is zero in this case†. In the density-stratified case, this up-down symmetry is broken (e.g. Graham 1975; Gough et al. 1976; Graham & Moore 1978; Massaguer & Zahn 1980) and the vertical profile of horizontally-averaged helicity has no particular symmetry (K¨apyl¨a et al. 2009). The volume-averaged helicity does not therefore in general vanish in this case, illustrating further that it is stratification effects that generate a helicity imbalance in this problem.

4.4.2. Turbulent pumping*

In an inhomogeneous fluid flow, the statistics of the velocity field depend on the

coordinate along the inhomogeneous direction gˆ. If we thread such a flow with a large-

scale magnetic field, then the latter will be expelled from the regions of higher turbulence

intensity to the regions of lower turbulent intensity. In a stratified turbulent fluid layer in

particular, urms usually decreases along g as density increases, and we therefore expect the field to be brought downwards. This is usually referred to as diamagnetic pumping

(Zel’dovich 1956; R¨adler 1968; Moffatt 1983). While this effect is not inducing magnetic

field on its own, it can contribute to the large-scale transport of the field, and may notably

lead to its accumulation deep into stellar interiors.

From a mean-field theory perspective, inhomogeneity implies the presence of a non-

vanishing γ × B term in equation (4.7), as we now have a preferred direction to construct

a true vector γ = γ g . Remark that γ × B has the same form as U × B, so that γ

is easily interpreted as an effective large-scale velocity advecting the field. A detailed

FOSA derivation in the St 1 limit, for a simple flow model encapsulating the effects

of inhomogeneity through an exponential dependence of the turbulence intensity, gives

(Krause 1967)

γ

=

−

1 6

τc∇|u|2

,

(4.41)

confirming the diamagnetic character of the effect. Note finally that the further presence

of rotation or large-scale flows in the problem raises additional contributions to γ, not

all of which are oriented along g (see e.g. R¨adler & Stepanov 2006). In spherical geom-

etry, this notably makes for the possibility of large-scale pumping along the azimuthal

direction.

† This, however, does not formally rule out the existence in rotating Rayleigh-B´enard convection of helical large-scale kinematic dynamo eigenmodes consistent with the vertical boundary conditions and with the symmetry of the vertical profile of α(z), see e.g. Soward (1974) and Hughes & Cattaneo (2008).

68

F. Rincon

4.4.3. R¨adler and shear-current effects*
Additional statistical effects distinct from the standard α effect formally arise if the turbulence is rendered anisotropic by the presence of large-scale rotation or shear, even in the absence of stratification. These effects are not necessarily very intuitive physically but their mathematical form can be captured easily within the framework of mean-field electrodynamics using some of the symmetry arguments introduced in §4.3.3. Namely, in the presence of rotation, we can now form a non-vanishing δ × ∇×B term in the expression (4.7) for the mean EMF by making δ proportional to the rotation vector, δ ≡ δΩΩ. The associated statistical effect is generally referred to as the R¨adler effect after its original derivation by R¨adler (1969a,b) or, more explicitly, as the “Ω × J” effect. Similarly, we may expect that δ is formally non-vanishing if the turbulence is sheared by a large-scale flow U with associated vorticity W = ∇×U, in which case δ ≡ δW W. This effect is generally referred to as the shear-current effect, or “W × J” effect (Rogachevskii & Kleeorin 2003).
The R¨adler effect has a subtle connection to parity invariance and flow helicity. To see this, let us consider a flow whose statistics are rendered anisotropic (more precisely axisymmetric) by the presence of rotation. In particular, the energy and helicity spectra are such that E(k, ω) = E(k, µ, ω) and H(k, ω) = H(k, µ, ω), where µ = kˆ ·Ωˆ . A spectral FOSA derivation similar to that outlined in §4.3.7 shows that (Moffatt & Proctor 1982)

δΩ Ω

=

2η2 5

k3µ ωH(k, µ, ω) (ω2 + η2k4)2

dk dω

,

(4.42)

and

αij = α δij + α1 δij − 3 ΩˆiΩˆj ,

(4.43)

with

α

=

−

η 3

(ω2

k2 + η2k4)

H (k,

µ,

ω)

dk

dω

(4.44)

and

α1

=

η 6

k2 3 µ2 − 1 (ω2 + η2k4) H(k, µ, ω) dk dω .

(4.45)

These expressions show that both δΩ and αij are weighted integrals of the helicity spectrum of the flow, and therefore vanish if the flow has no helicity at all. However, notice
that the weight function in the δΩ integral is odd in µ, and even in the αij integrals. This implies that δΩ, unlike the α effect, can be non-zero even if there is an equal statistical amount of right- and left-handed velocity fluctuations, and the flow as a whole has zero net-helicity. In particular, δΩ = 0 and α = 0 if H(k, −µ, ω) = −H(k, µ, ω). This situation corresponds to a flow with an up-down symmetry with respect to planes perpendicular to

the rotation axis, such as the incompressible Rayleigh-B´enard convection rotating about the vertical axis discussed in §4.4.1. A corollary of this result is that the R¨adler effect is formally present in unstratified rotating flows. The derivation of Moffatt & Proctor

(1982) also makes it clear that the effect survives even if the forcing of the flow is itself

non-helical, and all the flow helicity is induced by the effects of the Coriolis force. In other words, we expect the R¨adler effect to be present even in incompressible, homogeneous

turbulence forced isotropically and non-helically, as long as the system rotates. In a

generic stratified, rotating, turbulent system, both α and R¨adler effects are expected

to coexist, although it is sometimes argued that the latter should be smaller due to the presence of an extra slow spatial derivative (∇×B) in the expression of its EMF.

Analyses of simulations of stratified rotating convection in which both effects are present

Dynamo theories

69

suggest that the R¨adler effect is smaller than the α effect, but not altogether negligible, with δΩ being comparable in magnitude to the isotropic turbulent diffusivity coefficient β (K¨apyl¨a et al. 2009).
To understand how the R¨adler and shear-current effects may affect large-scale dynamos in rotating shear flows, let us have a look at the mean-field dynamo problem for smallscale homogeneous turbulence forced isotropically and non-helically in the simplest possible unstratified, rotating shearing sheet configuration, Ω = Ω ez, US = −Sx ey, for which W = −S ez and the only non-zero components of the deformation tensor are Dxy = Dyx = −S/2. Under these assumptions, there are no mean α and γ effects and the kinematic evolution equations for the x and y components of a z-dependent mean magnetic field (defined as the average over x and y of the total magnetic field) can be cast in the simple form

∂Bx ∂t

=

−ηyx

∂2By ∂z2

+

(η

+

ηyy

)

∂2Bx ∂z2

,

∂By ∂t

=

−SBx

−

ηxy

∂2Bx ∂z2

+

(η

+

ηxx)

∂2By ∂z2

,

(4.46) (4.47)

where we have introduced a contracted generalised anisotropic turbulent diffusion tensor ηij appropriate to the configuration of the problem, namely

bijz = ηilεjzl .

(4.48)

Using equations (4.8)-(4.10), it can be shown that

ηxx = ηyy = β ,

(4.49)

ηyx = Ω

δΩ

−

κΩ 2

−S

δW

−

1 2

(κW

−

βD

+

κD )

,

(4.50)

ηxy = −Ω

δΩ

−

κΩ 2

+S

δW

−

1 2

(κW

+

βD

−

κD )

,

(4.51)

where β is the usual isotropic turbulent diffusion coefficient, βD is an anisotropic contribution to the β tensor arising from the presence of the large-scale strain D associated with the shear flow, and κΩ, κW and κD are contributions to the mean-field κ tensor arising (similarly to δΩ and δW ) from the presence of rotation, large-scale vorticity, and strain associated with the shear flow (for a detailed derivation, see R¨adler & Stepanov 2006; Squire & Bhattacharjee 2015a).
We see that all these effects amount to a special kind of off-diagonal diffusion that couples the components of the magnetic field perpendicular to the new special direction introduced in the problem. Can these couplings, most importantly that associated with the ηyx coefficient that couples back the toroidal field to the poloidal field, drive a dynamo of their own ? To discover this, we can derive from equations (4.46)-(4.47) the complex growth rate of a horizontal mean-field mode with a simple exp(st + ikzz) dependence,

s = kz ηyx (−S + ηxykz2) − (η + β)kz2 .

(4.52)

Further assuming |S| ηxykz2, similarly to what we did when we derived the αΩ dispersion relation in §4.3.5, we find that a necessary condition for dynamo growth is

Sηyx < 0 ,

(4.53)

i.e. S and ηyx must have opposite sign. If the dynamo is realisable, the maximum growth

70

F. Rincon

rate and optimal wavenumber are

γmax

=

|Sηyx| 4 (η + β)

,

kz,max =

(Sηyx)1/2 2 (η + β)

.

(4.54)

But under which circumstance(s), if any, is the condition (4.53) satisfied ? To answer this question, we need to calculate ηyx using a closure assumption such as the FOSA or MTA. At this point, things become tricky. Indeed, FOSA calculations (Krause & R¨adler 1980; Moffatt & Proctor 1982) show that the R¨adler effect alone can promote a dynamo (ηyxS < 0) when the rotation is anticyclonic (corresponding to positive S/Ω > 0 with our convention), but not the shear-current effect (Sηyx 0, R¨adler & Stepanov 2006; Ru¨diger & Kitchatinov 2006; Sridhar & Subramanian 2009; Sridhar & Singh 2010; Squire & Bhattacharjee 2015a). Calculations based on the MTA closure agree with FOSA calculations as far as the R¨adler effect is concerned, but find dynamo growth (Sηyx < 0) for the shear-current effect (Rogachevskii & Kleeorin 2003, 2004) ! Which one, if any, is right in practice ? We know that the results derived with FOSA at least hold rigorously if Rm 1 or St 1, which establishes the absence of a shear-current-driven dynamo and the existence of a R¨adler-effect-driven dynamo in these limits (at least for the kind of sheared or rotating random flows considered in the FOSA derivations). However, there is no guarantee that these conclusions extend to standard MHD turbulence regimes. As discussed in §4.3.8, the MTA is an empirical closure model that seems to be in reasonable agreement with numerical results for a few simple turbulent-transport problems with St = O(1), but it cannot be rigorously shown to be valid in any particular regime either, leaving us in the dark for the time being.
Looking at the bigger picture, we see that the previous discussion of a shear-currenteffect dynamo (or lack thereof) raises the more general, very intriguing question of the possibility of large-scale dynamo excitation in non-rotating sheared turbulence with zero mean helicity. This problem, including the sign of Sηyx in non-rotating sheared turbulence, will be discussed much more extensively in §5.2 in the light of several recent numerical developments.

4.5. Difficulties with mean-field theory at large Rm
4.5.1. The overwhelming growth of small-scale dynamo fields
In the previous paragraphs, we discussed the very limited formal asymptotic mathematical range of parameters under which kinematic mean-field electrodynamics can be rigorously derived, but there is actually an even bigger potential physical problem looming over the theory at large Rm. Indeed, we have learned in §3 that homogeneous, isotropic turbulence independently generates small-scale magnetic fluctuations b through a small-scale dynamo effect, provided that Rm exceeds a value Rmc,ssd of a few tens to a few hundreds, depending on P m (Fig. 15). This dynamo produces equipartition field strengths over a dynamical (turbulent) turnover timescale, which is usually much smaller than the rotation or shearing timescales over which kinematic mean-field dynamo modes grow. This strongly suggests that the derivation of a meaningful solution of the generic problem of large-scale magnetic-field growth at large Rm should start from a state of saturated small-scale MHD turbulence characterised by |b|2 |B|2 and ρ|u|2 ∼ |b|2, rather than from a state of hydrodynamic turbulence. This was recognised as a major issue for large-scale astrophysical dynamo theory after the work of Kulsrud & Anderson (1992) described in §3.4.8 (see also the review by Kulsrud et al. 1997), although obviously the tricky questions of the interactions of small-scale MHD turbulence and large-scale dynamos already pervaded several earlier calculations, including the Pouquet et al. (1976)

Dynamo theories

71

paper mentioned earlier (see also Ponty & Plunian (2011) for a specific numerical example of a classic large-scale helical dynamo being overwhelmed by a small-scale dynamo as Rm increases).
We are unfortunately not in a very good place to address this problem at this stage of exposition of the theory. A key observation is that small-scale dynamo modes were purely and simply discarded in the FOSA treatment as a result of the neglect of the tricky term in equation (4.4). Mathematically, we can think of these modes as fastgrowing homogeneous solutions of equation (4.5), which have nothing to do with B. In other words, our earlier assumption that magnetic fluctuations b are small and uniquely the product of the stretching and tangling of the mean field B is incorrect at large Rm. This in turn begs the question of the interpretation and practical applicability of the linear mean-field ansatz (4.6) in this regime (Cattaneo & Hughes 2009; Hughes 2010; Cameron & Alexakis 2016). In order to make progress on a unified theory of large- and small-scale dynamos, our first priority should therefore be to derive a linear theory that at the very minimum accommodates both types of dynamos. Such a theory is available in the form of a helical extension of the Kazantsev model discussed in §3.4.

4.5.2. Kazantsev model for helical turbulence*
A generalisation of the Kazantsev model to helical flows was first derived by Vainshtein (1970) shortly after the introduction of mean-field electrodynamics. Vainshtein used a Fourier-space representation (see also Kulsrud & Anderson 1992; Berger & Rosner 1995), but in the following we will stay in the correlation-vector space as a matter of continuity with §3.4. Helicity can be accommodated in the model by adding a new term to the correlation tensor of the flow,

κij(r) = κN (r)

δij

−

rirj r2

+

κL

(r)

rirj r2

+ g(r)εijkrk

,

(4.55)

where g is a scalar function that vanishes for parity-invariant flows. A corresponding helical term is added to the magnetic correlation tensor,

Hij(r, t) = HN (r, t)

δij

−

rirj r2

+

HL(r,

t)

rirj r2

+ K(r)εijkrk

.

(4.56)

Proceeding as in §3.4 to close the problem, we obtain a system of two coupled equations for HL(r) and K(r) (Vainshtein & Kichatinov 1986; Subramanian 1999; Boldyrev, Cattaneo & Rosner 2005),

∂HL ∂t

=

1 r4

∂ ∂r

∂K ∂t

=

1 r4

∂ ∂r

r4

κ

∂HL ∂r

+ GHL − 4hK ,

r4

∂ ∂r

(κK

+

hHL)

,

(4.57) (4.58)

where κ(r) is given by equation (3.30), h(r) = g(0) − g(r), and G(r) = κ + 4κ /r. When g = 0 (no helicity in the flow), HL decouples from K, equation (4.57) reduces to equation (3.29), and equation (4.58) for the evolution of the magnetic helicity correlator K reduces to a diffusion equation with no source term. From there, it can be shown (Vainshtein 1970; Subramanian 1999; Boldyrev 2001; Boldyrev et al. 2005) that taking the r → ∞ limit of this problem leads to a mean-field α2 equation reminiscent of equation (4.11),

∂B ∂t

= α∇× B + (η + β)∆ B

,

(4.59)

72

F. Rincon

where α ≡ g(0), β ≡ κL(0)/2, and the mean field here is to be interpreted as an average

of the full magnetic field over the Kraichnan velocity-field ensemble. Because the velocity

is correlated on scales 0, this averaging procedure is equivalent to a spatial average over

scales much larger than 0.

This seems promising, but let us now look at the full solutions of equations (4.57)-(4.58)

to gain a better understanding of the situation. Using the transformation

√

√

HL =

2 r2

Wh

,

K

=

−

2 r4

∂ ∂r

r2Wk

,

W=

Wh Wk

,

(4.60)

Boldyrev et al. (2005) showed that equations (4.57)-(4.58) can be cast into a self-adjoint spinorial form,

where

∂W ∂t

=

−R˜T ˜JR˜W

,

√

2/r

0

R˜ =

0

−

1 r2

∂ ∂r

r2

,

(4.61) (4.62)

˜J =

Eˆ C CB

,

(4.63)

Eˆ

=

−

1 2

r

∂ ∂r

B

∂ ∂r

r

+

√1 2

(A

−

rA

)

,

(4.64)

and

√

A(r) = 2 [2η + κN (0) − κN (r)] ,

(4.65)

√

B(r) = 2 [2η + κL(0) − κL(r)] ,

(4.66)

√

C(r) = 2 [g(0) − g(r)] r .

(4.67)

Therefore, the generalised helical case too can be diagonalised and has orthogonal eigenfunctions. It turns out that there are now two kinds of growing eigenmodes, discrete bound modes (“trapped particles” in the quantum-like description) with growth rates γn, and a continuous spectrum of free modes (“travelling particles”) with growth rates γfree. The growth rates are such that γn > γ0 > γfree > 0, where

γ0

=

g2(0) 2η + κL(0)

=

2α2 4(η + β)

,

(4.68)

is twice the maximum standard mean-field α2 dynamo growth rate for the magnetic field (see e.g. Malyshkin & Boldyrev 2007, remember that we are looking at the growth rates of the quadratic magnetic correlator here). In other words, not only do we have trapped growing modes reminiscent of the small-scale dynamo modes derived in §3.4, there are now also free “large-scale” modes asymptotic to helical mean-field dynamo modes in the limit r → ∞. Of course, this convergence is not totally surprising considering that the α2 mean-field theory can be derived rigorously in the asymptotic two-scale approach in the limit of short correlation times. The family of free large-scale growing modes can be envisioned in the quantum-like description as caused by a helicity-induced modification of the potential at large scales. When r/ 0 1, the effective helical potential is

Veff (r)

2 r2

−

(η

α2 + β)2

(4.69)

Ve↵ (r)

Dynamo theories

`0 `⌘

73 r

Figure 29. Helical Kazantsev potential as a function of correlation length r.
and therefore tends to a strictly negative constant value as r → ∞, allowing largescale eigenfunctions with negative energy (positive growth rates). The shape of the full potential is shown in Fig. 29, to be compared with Fig. 17 for the non-helical case.
We see that the smaller the helicity of the flow (as represented by α), the smaller the growth rate of the free large-scale modes must be, due to the constraint γ0 > γfree. The growth rate of the bound modes, on the other hand, is not bounded from above by the amount of helicity in the flow. In all cases, the model predicts that the bound modes grow faster than the free modes, and are therefore expected to dynamically saturate the turbulence before the free modes saturate.
Another major conclusion of this analysis, however, is that the kinematic helical dynamo problem at large Rm is not reducible to a simple dichotomy between fast, small-scale dynamo modes and slow, large-scale dynamo ones. Solving equations (4.57)(4.58) numerically for a helical velocity field with a Kolmogorov spectrum, Malyshkin & Boldyrev (2007, 2009) found that the faster-growing bound modes that reduce to “smallscale dynamo” modes when h = 0 in equation (4.58) themselves develop correlations on scales r > 0 in the presence of helicity, and are therefore expected to contribute to the growth of the large-scale magnetic field. A comparison between some of the bound eigenfunctions and the fastest-growing unbound eigenfunction is shown in Fig. 30 for two different regimes. The results show that the fastest kinematic modes in helical turbulence at large Rm are not pure mean-field or small-scale dynamo modes, but hybrid modes energetically dominated by small-scale fields, with a significant large-scale magnetic tail.
4.5.3. P m-dependence of kinematic helical dynamos
It is finally worth pointing out in relation to our discussion in §3.3.2 of small-scale dynamos at low P m that the helical Kazantsev model predicts that the presence of flow helicity at the resistive scale η ∼ Rm−3/4 has the effect of decreasing Rmc for bound modes at low P m in comparison to the non-helical case, down to a value comparable to or even smaller than in the large-P m case (Malyshkin & Boldyrev 2010). In other words, helicity/rotation facilitates the low-P m growth of modes identified as small-scale dynamo modes in the non-helical case. While this problem deserves further numerical scrutiny,

74

F. Rincon

Figure 30. Fourier spectra of selected dynamo eigenfunctions in the helical Kazantsev model (black thin lines: bound modes, red spiky solid line: most unstable unbound mode). Left: P m = 150 case. Right: P m = 6.7 × 10−4 case. The kinetic helicity of the flow is maximal in both cases (adapted from Malyshkin & Boldyrev 2009).

a rotationally-induced decrease of Rmc has for instance been reported in simulations of dynamos driven by turbulent convection at P m = 1 (Favier & Bushby 2012), and in simulations of kinematic dynamos driven by rotating 2.5D flows, but forced non-helically (Seshasayanan, Dallas & Alexakis 2017). More generally, numerical simulations suggest that large-scale dynamos in rotating/helical turbulent flows are much less dependent on P m than small-scale ones (see e.g. Mininni 2007; Brandenburg 2009b). A handwaving physical explanation for this is that the large-scale field always feels the whole sea of small-scale turbulent velocity and magnetic fluctuations. Pure “non-helical” small-scale dynamo fields, on the other hand, appear to be much more dependent on the details of the flow and dissipation.

4.6. Dynamical theory
Having completed our tour of linear theory, we are now in a slightly better position to discuss the dynamical evolution and saturation of large-scale dynamos, in particular helical dynamos driven by an α effect. As with small-scale dynamos, we would like to understand by which dynamical mechanisms saturation occurs, at which strength the magnetic field saturates (especially its large-scale component), and on which timescale this happens. Due notably to the problems described in §4.5, formulating a consistent mathematical theory of nonlinear large-scale dynamos at large Rm has kept a lot of theoreticians busy for many years, and still remains one of the most formidable problems in the field today.

4.6.1. Phenomenological considerations

The simplest possible outcome of large-scale dynamo saturation, which is also arguably the most desirable one to explain the characteristics of magnetic fields in astrophysical systems such as our Sun or galaxy, is that the large-scale magnetic field reaches equipartition with the underlying turbulence,

|B|2 ∼ ρ|u|2 ≡ Be2q .

(4.70)

Whether and how such a state can be achieved, however, is quite an enigma. Note in particular that the two dynamical fields involved in equation (4.70), B and u, are at very different scales. How can we get these two fields to communicate and equilibrate in the dynamical regime, considering that the part of the Lorentz force J × B due solely

Dynamo theories

75

to B acts on a much larger scale than that of the dynamo-driving flow ? Independently of whether equation (4.70) holds, it seems clear that the saturation process must involve dynamical interactions between velocity fluctuations u and magnetic field fluctuations b at scales similar to or below 0, the forcing scale of the turbulence. Besides, for the helical large-scale dynamo problem, we have actually found in §4.5.2 that small-scale magnetic fluctuations are in all likelihood already much more energetic than the large-scale field in the kinematic stage of the dynamo, and should therefore reach equipartition with the flow well before the large-scale field does. Once the Lorentz force associated with the small-scale fluctuations starts to affect the flow, it is not obvious that B itself can keep growing up to equipartition on dynamically relevant timescales. This problem should arise even in the absence of fast-growing small-scale dynamo modes. Indeed, according to Fig. 30, even unbound mean-field modes are characterised by |b|2 |B|2.
A classic argument illustrating the nature of the problem in a rather dramatic way is as follows. Consider a hydrodynamic velocity fluctuation u0 at scale 0, with a typical shearing rate ω0 = τN−L1 ∼ u0/ 0, threaded by a weak large-scale field B well below equipartition. In the initial stages of the evolution, the stretching of field lines will induce magnetic fluctuations |b|(t) ∼ |B| ω0t with increasingly smaller-scale gradients characterised by a typical wavenumber k(t) ∼ k0 ω0t. These increasingly thinner magnetic structures will hit the resistive scale at a time tres defined by ηk2(tres) ∼ ω0, at which point the typical energy of the fluctuations will be of the order |b|2 ∼ Rmp|B|2, with p = O(1) (the exact exponent of this relation, usually referred to as the Zel’dovich relation, formally depends on the number of dimensions and type of magnetic structures, see Zel’dovich (1956); Moffatt (1978); Zel’dovich, Ruzmaikin & Sokolov (1983), but appears to be close to unity in practice). At large Rm typical of astrophysical regimes, this formula predicts |b|2 |B|2, suggesting that dynamical effects become significant when |b|2 ∼ Be2q and |B|2 ∼ Be2q/Rmp Be2q (Vainshtein & Rosner 1991; Cattaneo & Vainshtein 1991; Vainshtein & Cattaneo 1992). If this line of reasoning is correct, it would suggest that large-scale astrophysical dynamos first start to saturate through the dynamical feedback of small-scale fluctuations at dramatically (asymptotically) low level of large-scale fields, and may therefore have a very hard time powering the nearequipartition large-scale magnetic fields that we observe in the Universe on timescales shorter than the cosmological Hubble time. Such a strong potential dependence of large-scale helical dynamos on Rm in the nonlinear regime is commonly referred to as catastrophic quenching.
It is important to stress that this particular argument is not universally accepted and does not constitute a proof that catastrophic quenching occurs, notably because it is based on a description of a transient evolution of magnetic fluctuations, starting from a large-scale field, rather than on a statistically steady 3D dynamo eigenfunction such as shown in Fig. 30 (see e.g. Blackman & Field (2005) for a critical discussion of the applicability of Zel’dovich relations in this context). Nevertheless, it provides a clear illustration of the broader problem that the generation of dynamically strong, predominantly small-scale fields can pose to large-scale magnetic field growth at large Rm, and also suggests that the microphysics of dissipation at the resistive scale can play a subtle but important role in this problem.

4.6.2. Numerical results
Let us now discover what our trusty brute-force simulations of large-scale helical dynamos driven by pseudo-isotropic turbulence in a periodic box have to say about this problem. First, numerical results confirm that the kinematic dynamo stage is dominated by small-scale magnetic fluctuations, and that saturation at small-scales does indeed

76

F. Rincon

occur well before the large-scale component reaches equipartition with the flow. This appears to be true at Rm both smaller (e.g. Brandenburg 2001) and larger (e.g. Mininni, G´omez & Mahajan 2005; Subramanian & Brandenburg 2014; Bhat, Subramanian & Brandenburg 2016b) than the critical Rm for small-scale dynamo action (Bhat, Subramanian & Brandenburg (2019) have recently been argued that a secondary, transient, “quasi-linear” growth phase of the large-scale mean field may occur once the smallscale dynamo saturates). Second, all simulations of this kind to date are ultimately plagued by catastrophic quenching, albeit not quite of the form discussed above. In the absence of shear (α2 case), the large-scale field does ultimately saturate at significant dynamical levels actually exceeding equipartition (as noted earlier, the dynamo in this configuration generates a large-scale force-free field, i.e. the mean-field itself has no backreaction on the flow), but it only does so on a timescale of the order of the prohibitively large resistive time at the scale of the mean field, not on a dynamical timescale (see e.g. Fig. 8.6 of Brandenburg & Subramanian (2005a) derived from the Brandenburg (2001) simulation set). A “catastrophic” timescale of saturation of the dynamo is also manifest in simulations of helical dynamos in the presence of large-scale shear (αΩ case) at Rm = O(100) (Brandenburg, Bigazzi & Subramanian 2001).
To illustrate these results, we show in Fig. 31 the time-evolution of the magnetic energy spectrum in one of the highest-resolution helical dynamo simulations to date with P m = 0.1 and Rm = 330 (Bhat et al. 2016b), and the evolution of the ratio |B|/Brms in three simulations with the same P m but different Rm, taken from the same study (the turbulence is forced at k0 = 4kL in these simulations). The spectrum of the kinematic eigenfunction clearly peaks at small scales k/kL ∼ 30 comparable to the resistive scale, and at first the whole eigenfunction grows exponentially as expected in a linear regime, until saturation occurs around t ∼ 400 (k0urms)−1. The larger-scale components of the field continue to grow nonlinearly after that, with the peak of the spectrum progressively shifting to larger scales. The energy of the “mean-field” (defined as the energy in the k/kL = 1 − 2 modes in the simulation) grows very slowly in the nonlinear regime, and only becomes comparable to that of the total saturated r.m.s. magnetic energy after 103 turnover times (Brms is itself only of the order 10% of Beq in this particular low-P m simulation). Note also that the mean field never reaches saturation in these already very long simulations, and that its typical time of nonlinear evolution seemingly increases with Rm. These results are therefore strongly suggestive of a catastrophic resistive scaling of the timescale of evolution of the dynamo in the nonlinear regime.

4.6.3. Magnetic helicity perspective on helical dynamo quenching

Is it possible to make sense of these results ? Let us start from the basics and ask what kind of dynamical effects may be important in the problem. Leaving aside the hard question of the interplay between small and large-scale dynamos for a moment, we have actually already identified a possible channel of back-reaction of the magnetic field on helical motions in our discussion in §4.2.3 of magnetic helicity dynamics in Parker’s mechanism, and that is through the magnetic tension associated with the small-scale curvature of the twisted field. This somewhat intuitive idea was first translated into a mathematical model in a paper by Pouquet et al. (1976), who used an EDQNM closure to derive the expression

α

=

−

1 3

τ

u · ω − (∇×b) · b

.

(4.71)

This result can also be derived from the MTA closure equations (4.37)-(4.38) presented in §4.3.8, assuming a steady state for the EMF. The second term on the r.h.s. of this

Dynamo theories

77

Figure 31. Left: Magnetic (full black lines) and kinetic energy (dotted blue lines) spectra in a simulation of large-scale helical dynamo action at P m = 0.1 and Rm ≡ urms 0/(2πη) = 330. Each spectrum corresponds to a time separation of a hundred turbulent turnover times. Right: time-evolution of |B|/Brms in three simulations with P m = 0.1 and different Rm (adapted from Bhat et al. 2016b).

equation can be traced back to the effect of the tension force term B · ∇b on u, and

does therefore capture a back-reaction of a helical, twisted magnetic field on the flow. An

easy way to see this is by noticing that α in equation (4.71) vanishes for torsional Alfv´en

waves. As will be discussed shortly, there are many subtleties and caveats attached to

the interpretation of equation (4.71), but let us also temporarily ignore them and simply

acknowledge for the time being that this result gives us an incentive to look at the

problem of saturation from a magnetic helicity dynamics perspective.

Magnetic helicity conservation in the two-scale approach. Arguably, the conservation

equation (2.11) for the total magnetic helicity is a particularly attractive feature in

this context and seemingly represents a good starting point to develop some multiscale

theory, as it provides a direct coupling between the small and large-scale components of

the field. In order to illustrate this, we will again use the simple two-scale decomposition.

Manipulating either the small and large-scale components of the induction equation or equation (2.11) directly, and using E · B = 0 (by definition of E), it is straightforward to

show that

∂ ∂t

(A

·

B)

+

∇

·

FHm,mean

=

2E

·

B

−

2η

∇×B

·B ,

(4.72)

∂ ∂t

(a

·

b)

+

∇ · FHm,fluct

=

−2 E

·

B

−

2η

(∇×b)

·

b

,

where we have introduced the mean and fluctuating helicity fluxes

(4.73)

FHm,mean = c ϕB + E × A ,

(4.74)

F = c Hm,fluct ϕb + e × a = FHm − FHm,mean ,

(4.75)

the total magnetic-helicity flux FHm is given by equation (2.12), a is the fluctuating smallscale magnetic vector potential, and e in the context of this equation is the small-scale

fluctuating electric field (not a unit vector). Averaging over a periodic domain, or over

a domain bounded by perfectly conducting boundaries, the surface integrals associated

with the flux terms vanish, leading to

d dt

A·B V =2

E · B V − 2η

∇×B · B V ,

(4.76)

78

F. Rincon

d dt

a · b V = −2 E · B V − 2η

(∇×b) · b
V

,

(4.77)

where · V denotes a volume average. Let us stress at this stage that the equations above are independent of any dynamical closure for E, such as equation (4.71). Besides, all

these equations can be derived from the induction equation alone and therefore provide

no information of their own regarding the dynamical effects of the Lorentz force. Finally,

note that the production terms of large and small-scale helicities, ±2 E · B, are equal in

magnitude but opposite in sign. This is the mathematical translation of the conservation

of the total magnetic helicity, and of our earlier observation in §4.2.3 that large-scale

helical dynamos generate magnetic helicity of one sign at large scales, and the same

amount of helicity of opposite sign at small scales.

It is clear at this stage that we need a closure prescription for E of the kind provided

by equation (4.71) in order to solve for the full dynamical evolution of the system in a

way that consistently factors in the effects of the Lorentz force. Before we go down this

path, however, let us see whether we can learn something about the long-time evolution

of the system from equations (4.76)-(4.77) alone. To this end, imagine a situation in which |b|2 > |B|2 in the kinematic regime, so that the small-scale fluctuating field

component attains dynamical levels first. We then assume that the small-scale field
reaches a statistical steady state with |b|2 of the order of Be2q at a time tsat,fluct, and that this also corresponds to a steady state for the small-scale magnetic helicity. How

does the large-scale field evolve for t > tsat,fluct ? In this regime, the time-derivative in equation (4.77) becomes negligible, leaving us with

E · B V = −η (∇×b) · b V .

(4.78)

This simply tells us that u and b (and therefore E) have dynamically co-evolved in such a way that a balance between the production and dissipation of small-scale magnetic helicity is established. But we also know that the small-scale helicity production term is equal in magnitude to that of large-scale magnetic helicity as a result of total helicity conservation. Equation (4.78) therefore suggests that the growth of the large-scale field in this “partially-saturated” regime is tied to the microscopic ohmic dissipation of the (steady) small-scale magnetic helicity (Gruzinov & Diamond 1994).
In the simplest α2 dynamo case, the kinematic mean-field mode is an eigenvector of the curl operator, so that

(∇×B) · B = kL2 A · B = ∓kL|B|2

(4.79)

in the Coulomb gauge, ∇·A = 0 (here we have assumed for simplicity that the large-scale field has a scale comparable to the system size L ≡ 1/kL). Similarly,

(∇×b) · b = k02 a · b = ±k0|b|2 .

(4.80)

Using these results and substituting equation (4.78) in equation (4.76), we then find that

d dt

|B|2 V

The solution of this equation is

2ηk0kLBe2q − 2ηkL2 |B|2 V .

(4.81)

|B|2 V

k0 kL

Be2q

1 − e−2ηkL2 (t−tsat,fluct)

,

(4.82)

i.e. the mean-field saturates at a super-equipartition value, but only on the catastrophically slow mean-field resistive timescale tsat,mean = tη,mean = (ηkL2 )−1. In the transient

Dynamo theories

79

phase (t − tsat,fluct)/tη,mean 1, the mean field in this model grows linearly with time,

|B|2 V

|B|2 V (t = tsat,fluct) + Be2qγsat(t − tsat,fluct) ,

(4.83)

where γsat ≡ 2ηk0kL, so that the timescale to reach equipartition can actually be much shorter than tη,mean for significant scale separations. Interestingly, this model, originally derived by Brandenburg (2001) (see Brandenburg et al. (2001) for a similar analysis of the αΩ dynamo), matches quite well simulation results of the nonlinear evolution of helical dynamos in periodic boxes for Rm below or comparable to Rmc,ssd. The result does not appear to be explicitly dependent on a closure as we usually envision them, but note that it is nevertheless heavily dependent on the assumption of a preliminary saturation of the small-scale field. This assumption is not unreasonable at all, but it is clearly additional information on the dynamics (as opposed to the kinematics) of the field that we inject by hand in the induction equation to understand the evolution of the system in the nonlinear regime. Overall though, it seems that we can already learn something on saturation without having to bother too much about the tricky details of small-scale dynamical closures such as that suggested by equation (4.71).
α-quenching models. As mentioned earlier, solving the full dynamical time-evolution requires closing equations (4.76)-(4.77), and it is very tempting to make use of the meanfield ansatz with α given by equation (4.71) to do this. This closure has some seemingly nice features: first, it suggests that saturation starts to take place when the small-scale field reaches equipartition, which is consistent with our earlier arguments and numerical results. Second, it seems to capture the effects of the magnetic tension associated with the small-scale helical twisting of field lines. Finally, it directly involves a small-scale magnetic current helicity, raising the prospect of a relatively straightforward mathematical closure of equations (4.76)-(4.77) when used in combination with equations (4.79)-(4.80). As emphasised by several authors though, one has to be very careful with the interpretation and definition of b in equation (4.71) in the context of the dynamo problem (e.g. Blackman & Field 1999, 2000; Proctor 2003). The reason is that in the original study of Pouquet et al. (1976), b is not tied to the large-scale field B through equation (4.5), but stands for the magnetic component of some underlying small-scale background turbulent helical magnetic field predating a large-scale dynamo. In §4.3.8, the problem was also linearised around such a state to derive equation (4.38). In other words, the magnetic current helicity term in these derivations cannot be easily formally associated with a dynamical quenching of the α effect resulting from the exponential growth of the smallscale helical component of the dynamo field.
Overall, we should be mindful that equation (4.71) probably at best only qualitatively captures some, but not all the possibly relevant dynamical effects that affect the largescale growth of helical dynamos. It is clear from the algebra that a magnetic current helicity term associated with small-scale magnetic fluctuations specifically generated by the helical tangling and stretching of a growing large-scale dynamo field B should also contribute to α and reduce the kinematic helical dynamo effect when these fluctuations reach equipartition with the flow, but there is no guarantee that this effect is either dominant or essential to saturation. Remember for instance that we have so far swept all the small-scale dynamo fields present at large Rm under the carpet, and those are not obviously accounted for in the closure procedures leading to equation (4.71).
Keeping all these caveats in mind, let us nevertheless quickly review some simple flavours of so-called α-quenching models that result from using equation (4.71) as a closure. Roughly speaking, static quenching models (e.g. Gruzinov & Diamond 1994; Bhattacharjee & Yuan 1995; Vainshtein 1998; Field, Blackman & Chou 1999) are relevant to the partially-saturated regime in which the small-scale field has already saturated, and

80

F. Rincon

essentially provide closed expressions for α(Rm, |B|) that only depend on time through the time-dependence of |B| itself. For instance, the expression

α(|B|) = 1+

αkin |B|/Beq

2

,

(4.84)

describes a regular “quasi-linear” quenching of the dynamo as the large-scale field grows

towards equipartition (see e.g. Gilbert & Sulem (1990); Fauve & Petrelis (2003) for

derivations of expressions of this kind for a variety of flows, using weakly nonlinear

asymptotic multiscale analyses close to the dynamo threshold). On the other hand, the

formula

α(|B|) =

αkin 1 + Rm |B|/Beq

2

,

(4.85)

describes a form of catastrophic quenching. Here αkin refers to the (unquenched) value of α in the kinematic regime. The catastrophic quenching expression (4.85) actually follows directly from combining equation (4.71) and equation (4.78) (Gruzinov & Diamond 1994). A castastrophic dependence of the α effect on Rm of this kind was notably observed in the first numerical study of α-quenching (Cattaneo & Hughes 1996). In this experiment, a uniform, externally imposed mean field was used to probe the EMF response to the introduction of a large-scale field of an underlying, fully developed turbulent helical MHD turbulence in a periodic domain. This set-up, while not a full-on helical dynamo experiment, has the advantage of keeping things as close as possible to the conditions of the derivation of equation (4.71). This, in hindsight, probably made the attainment of the result (4.85) quite inevitable in these simulations, considering that equation (4.78) is also necessarily satisfied if the measurements of the EMF are carried out in a steady state. The possible limitations and flaws of static quenching models and of the aforementioned numerical experiments exhibiting catastrophic quenching have been debated at length in the literature (e.g. Field et al. 1999; Field & Blackman 2002; Proctor 2003; Brandenburg & Subramanian 2005a; Hughes 2018). In the end, equation (4.85) appears to be no more than a simple translation in mean-field terms of the earlier finding that the EMF along the mean field (expressed as a difference between the small-scale kinetic and current helicity in the MTA closure framework) in the partially-saturated regime must have dynamically decreased down to a level at which it matches the resistive dissipation of helicity.
Equation (4.85) should be regarded with utmost caution, as it is not generically valid and should not be blindly applied or trusted in all possible situations. First and foremost, remember that it is the result of using both the mean-field ansatz and a dynamical closure formula whose interpretation remains disturbingly fragile, and whose domain of applicability is clearly limited. Second, equation (4.85) was derived in a regime in which the small-scale field has already fully saturated. While the first limitations are very hard to overcome, the second can be easily circumvented. Using equation (4.71) in combination with equations (4.79)-(4.80), it is easy to see that equations (4.76)-(4.77) describing the time-dependent dynamics of magnetic helicity can be turned into a relatively simple nonlinear dynamical quenching model in which α itself becomes time-dependent (e.g. Kleeorin & Ruzmaikin 1982; Zel’dovich et al. 1983; Field & Blackman 2002). These models predict for instance that the catastrophic quenching of the α2 dynamo is not engaged when |B|2 ∼ Be2q/Rm, as equation (4.85) would suggest, but when |B|2 ∼ (kL/k0)Be2q, which is admittedly a bit less catastrophic (Field & Blackman 2002). Overall, however, the long-time asymptotics of two-scale dynamical quenching models of large-scale helical dynamos in homogeneous periodic domains reduce to the Brandenburg (2001) model described earlier, and therefore also predict that such dynamos are ultimately resistively

Dynamo theories

81

limited. The main difference with static models is that they provide a closed description of the dynamical transition into resistive saturation.
Is there a way out of catastrophic quenching ? Our discussion so far suggests that the catastrophic quenching of large-scale helical dynamo fields observed in numerical simulations in periodic domains is an inevitable outcome of the back-reaction of smallscale fields on the flow. However, all these models and numerical simulations are limited in some way, either because they rely on over-simplifying assumptions, are too idealised, or are not asymptotic in Rm. Their conclusions should therefore probably not be taken as the final word on catastrophic quenching, and we may ask what kind of physics overlooked until this point could potentially improve the efficiency of such dynamos.
One possibility would be for the dissipation of magnetic helicity to behave in unforeseen ways at very large Rm. In the two-scale model of the saturated α2 dynamo, the r.h.s. of equation (4.78) is directly proportional to η (assuming k0 is independent of η and |b|2 ∼ Be2q) and therefore quickly goes to zero as η → 0 (Rm → ∞). Accordingly, so must the large-scale saturated EMF, resulting in the resistively limited growth of the largescale field. But a different outcome may be possible if the actual dissipation of magnetic helicity in the full MHD problem remains finite or asymptotes to zero as a relatively small power of η as η → 0 (definitely < 1, for the whole dynamo process to become astrophysically relevant). It is often argued, though, that the dissipation of magnetic helicity is generically less efficient than that of magnetic energy for physically reasonable magnetic energy and helicity spectra, with the implication that the resistively limited growth of the dynamo is quite inevitable (e.g. Brandenburg et al. 2002; Brandenburg & Subramanian 2005a; Blackman 2015). A common argument made in support of this claim is that if the volume-averaged small-scale magnetic energy dissipation Dη,fluct V = η |∇b|2 V in the saturated state is finite and independent of η as η → 0, then η ∼ η1/2 and η (∇×b) × b V ∼ η1/2 should still asymptote to zero as η → 0, just not quite as drastically as in the two-scale model (Brandenburg 2001; Brandenburg et al. 2002).
Another possible way out of catastrophic quenching put forward by Blackman & Field (2000) (see also Kleeorin et al. 2000; Ji & Prager 2002; Brandenburg et al. 2002) may be through the removal of small-scale magnetic helicity from the system via magnetichelicity flux losses through its boundaries. To understand this proposition, let us again consider equation (4.73), but this time without assuming any spatial periodicity of perfectly conducting boundaries. The spatial average of the dynamical evolution equation for the small-scale magnetic helicity now reads

d dt

a·b

V

=

−

1 V

FHm,fluct · dS − 2 E · B V − 2η
∂V

(∇×b) · b .
V

(4.86)

The main idea is that a dynamically significant large-scale EMF may survive in the regime of small-scale saturation if a dominant balance

E ·B

V

∼

−

1 2V

FHm,fluct · dS
∂V

(4.87)

between the EMF term and a non-resistive boundary flux term is established, rather than equation (4.78) (the full time-dependent equation (4.86) can also be used as a basis for dynamical quenching models, see e.g. discussion in Brandenburg 2018). There have been a variety of phenomenological proposals as to how the removal of smallscale magnetic helicity may occur in astrophysical conditions, notably with the help of differential rotation and/or large-scale winds and magnetic coronae (e.g. Vishniac & Cho 2001; Blackman & Brandenburg 2003; Brandenburg & Subramanian 2005a). Numerical efforts to make large-scale helical dynamos work in the presence of open boundaries

82

F. Rincon

remain work in progress and have so far only given mixed results. Non-zero helicity fluxes (Brandenburg & Dobler 2001; K¨apyl¨a, Korpi & Brandenburg 2010; Hubbard & Brandenburg 2011, 2012; Del Sordo, Guerrero & Brandenburg 2013) have been measured in simulations, but they do not seem to reach the balance (4.87) in the range of Rm investigated so far, and do not unambiguously produce fast large-scale dynamo fields on dynamical timescales. Some recent simulations even suggest that helicity flux losses preferentially occur at large scales (Brandenburg 2019). For a more detailed presentation of this line of research, we refer to the recent review of Brandenburg (2018), in which it is notably conjectured that helicity fluxes should become dominant dynamically at asymptotically large Rm (see for instance Del Sordo et al. (2013) for numerical results up to Rm = O(103) suggestive of this trend).
Whether any of the previous arguments survives the test of numerical simulations at asympotically large Rm and Re remains to be found. The discussion above, at the very least, suggests that significant further progress on the question may require a much deeper understanding of the dynamics of magnetic helicity dissipation, reconnection and magnetic relaxation in high-Rm nonlinear MHD than we currently have, and this problem also appears to be very difficult to solve in a general way on its own (Moffatt (2015, 2016), see also our ealier discussion in §3.5.4). Let us finally point out that the whole magnetic helicity dynamics approach to saturation is not universally accepted, notably because magnetic helicity conservation itself is a by-product of the ideal induction equation, and because non-helical small-scale dynamo fields may also play a role in saturation. Two very different takes on this problem can be found in the recent reviews of Brandenburg (2018) and Hughes (2018) published in this journal. A selection of analytical and numerical results offering different perspectives on saturation at large Rm is presented in the next paragraph, and in §5.4.

4.6.4. Dynamical saturation in the helical Kazantsev model*

In the previous discussion of dynamical quenching, we did not really pay attention to the origin of the small-scale fields saturating the large-scale dynamo. Some of the nonlinear numerical simulations of the α2 dynamo we mentioned had Rm smaller than Rmc,ssd, while others had Rm larger than that. Catastrophic quenching appeared to be the norm for all larger-scale helical dynamo simulations in periodic domains, up to Rm = O(103). Overall, it is therefore not clear at this stage whether small-scale dynamo fields have a specific dynamical impact on the evolution of large-scale fields at large Rm.
Perhaps not entirely surprisingly considering its complexity, analytical results on this problem are scarce. An instructive calculation possibly linking the dynamical evolution of the large-scale field to the saturation of small-scale dynamos can be found in the work of Boldyrev (2001), who considered the extension to the helical case of the derivation of the small-scale dynamo magnetic p.d.f. discussed in §3.4.8. In the derivation of the non-helical small-scale dynamo problem, there was no helical contribution to the velocity correlation function, and we could also invoke the statistical isotropy of the magnetic field to simplify the problem into a 1D Fokker-Planck evolution equation (3.60) for the p.d.f. of the magnetic-field strength, from which the log-normal statistics of the field strength were inferred. In the helical case, however, this isotropy is broken by the growth of the mean field B (x, t), and it becomes necessary to solve a more general Fokker-Planck equation for the full field p.d.f.

P [B](t) = PB[B](t)G[Bˆ ](x, t) ,

(4.88)

including the p.d.f. of magnetic-field orientations G[Bˆ ](x, t). In the non-resistive, largeP m limit in which the flow correlator can be expanded according to equation (3.46), it

Dynamo theories

83

can be shown using the characteristic function technique introduced in §3.4.8 that

∂G ∂t

=

κ0 2

∆G

+

κ2 2

(δik

−

Bˆ

i Bˆ k

)

∂

∂2 Bˆ i ∂ Bˆ k

−

κ2 Bˆ i

∂ ∂Bˆ

i

+

g εikl Bˆ i

∂

∂ Bˆ k

∇l

G

(4.89)

in 3D, in the presence of flow helicity†. We seek solutions of this equation in the form of a series in powers of Bˆi, with terms of increasing order describing the angle distribution

of the magnetic field on an increasingly fine level. Equation (4.89) then suggests that we

write

G[Bˆ ](x, t) = 1 + BˆiBi(x, t)e−κ2t + O(Bˆ Bˆ ) terms

(4.90)

with the higher order terms decaying faster than the first order term. Keeping only the latter is sufficient to study the behaviour of Bi. Substituting equation (4.90) in equation (4.89), we then simply recover the evolution equation for the large-scale field in mean-field dynamo theory

∂B ∂t

=

κ0 2

∆B

+ g∇×B

.

(4.91)

This result is of course not very surprising, considering that the calculation is essentially

another way to derive the helical Kazantsev model discussed in §4.5.2. However, it brings

the growth of the mean field under a slightly different light. Equation (4.90) shows that

the anisotropic part of the magnetic-field distribution, to which the mean field is directly

tied, decays over time due to the isotropisation of the field by the turbulence. Taken

at face value, this would suggest that the mean field should decay. However, in the

kinematic regime, the average field strength B still grows exponentially according to

equation (3.63), at a rate κ2 that compensates exactly the decay rate of the anisotropic

part of G. Integrating the first moment of the full probability distribution function (4.88)

over all possible magnetic-field strengths and orientations, we then find that the actual

mean field B (now understood as an average over the Kraichnan ensemble) is simply

proportional to B, whose evolution is governed by equation (4.91). And, of course, we

know that this equation has unstable α2 solutions.

The interesting aspect of this calculation in comparison to §4.5.2 becomes clear if we

consider a regime in which the small-scale field is saturated. We already briefly discussed

in §3.5.5 different ways in which the Kazantsev model may be “patched” to describe such

a regime, but for the purpose of this discussion it is enough to assume that the moments

of the magnetic-field strength stop growing when the small-scale field saturates. In this

situation, the relaxation of the p.d.f. of magnetic orientations is not balanced anymore

by the growth of magnetic-field strength, and it cannot be balanced by the growth of

B through equation (4.91) either as a result of a flow helicity realisability condition

g2 < (5/8)κ0κ2. The calculation therefore predicts a quenching of the growth of the

large-scale field as a result of the saturation of the magnetic-field strength. Boldyrev

(2001) argues that this quenching should be independent of Rm (non-catastrophic) if the

saturation of the field strength occurs while the small-scale field is still in the diffusion-free

regime described in §3.4.8. This conclusion only reinforces the feeling that the question

of quenching is intimately related to the physics of magnetic dissipation.

This theory of saturation, unlike that presented in §4.6.3, does not explicitly invoke

helicity conservation. It is not a priori obvious whether and how the two approaches can

be bridged, although of course one of the (kinematic) helical Kazantsev model equations is

the evolution equation (4.58) for the magnetic helicity correlator. Overall, this discussion

† There is a factor 1/2 difference between this expression and that given by Boldyrev (2001) due to his different treatment of the integration of the δ(t − t ) function (Stanislav Boldyrev, private communication).

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and §4.5.2 nevertheless make it clear that the helical Kazantsev model represents one of the most promising (and well-posed) analytical frameworks for future research on nonlinear large-scale helical dynamos at large Rm.

4.6.5. Quenching of turbulent diffusion
In the previous subsections, we discussed the phenomenology of large-scale helical mean-field generation in a turbulent MHD fluid, and derived this effect mathematically rigorously within the limits of the FOSA in the form of an α dynamo coupling. We then discovered the many uncertainties surrounding the question of how the α effect changes as magnetic fields of dynamical strength develop. Furthermore, in the process of our simple kinematic mean-field derivation, we found another effect, the β effect, which has a clear interpretation as a turbulent magnetic diffusion. The question therefore naturally also arises as to how this effect changes as dynamical saturation takes place. Considering that turbulent magnetic diffusion also significantly affects the growth of large-scale magnetic fields (just ask your favourite colleague doing liquid-metal dynamo experiments at very large Re for confirmation !), it is important to at least mention this question here. Its resolution, however, unsurprisingly turns out to be just as difficult and technical as that of the saturation of the α effect and, in order to keep the matters at a simple introductory level, we will therefore not dive much into it here. A thorough presentation of the fundamental theoretical aspects of the problem (and of some of the main arguments stirring the debate in the community) can be found in Diamond et al. (2005).
In a nutshell, turbulent diffusion is strongly quenched in two dimensions (with an in-plane magnetic field). This result can be simply traced back to the conservation of the out-of-plane mean-square magnetic vector potential (e.g. Cattaneo & Vainshtein 1991; Vainshtein & Cattaneo 1992) in 2D. The three-dimensional case, however, turns out to be significantly more complex due to the extra freedom of motion that tangled magnetic-field lines enjoy in 3D. Analytical calculations similar to that of Gruzinov & Diamond (1994) leading to the catastrophic α-quenching equation (4.85) for instance suggest that there is no quenching of the kinematic turbulent diffusion at all in three dimensions (see also Avinash (1991) for a derivation in the FOSA limit of the vanishing of a “magnetic” β effect associated with magnetically-driven fluctuations), while other analytical studies predict a full range of possible outcomes, including regular (Kitchatinov et al. 1994; Rogachevskii & Kleeorin 2000) and catastrophic quenching (Vainshtein & Cattaneo 1992). In practice, numerical experiments, in their typical way of not revealing anything particularly definitive or extreme, suggest that a significant, albeit not necessarily catastrophic quenching of turbulent magnetic diffusion occurs in dynamical regimes (Brandenburg 2001; Blackman & Brandenburg 2002; Yousef et al. 2003; Brandenburg et al. 2008b; K¨apyl¨a et al. 2009; Gressel et al. 2013; Karak et al. 2014; Simard et al. 2016). As with α quenching though, it should be kept in mind that no such simulation is truly asymptotic in Rm and that the effects of a small-scale dynamo on the results are still not well understood. Also, some particular kinds of numerical experiments which at first glance appeared well-suited to study investigate dynamical quenching in a simple, systematic way, have turned out to be far from ideal in this respect in practice. This is for instance the case of our good old simulations of the α2 turbulent dynamo which, besides having the conservation of magnetic helicity hard-wired into them, tend to generate largescale force-free Beltrami fields, resulting in a degeneracy between the α and β effects (see e.g. Blackman & Brandenburg 2002; Brandenburg et al. 2008b).

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4.7. Overview of mean-field dynamo theory applications
Having discussed some of the most fundamental aspects of linear and nonlinear largescale dynamos, we will now go through an overview of the most common applications of their popular mean-field theory. These are essentially of two kinds: as low-dimensional nonlinear mathematical models aiming at describing in relatively simple effective ways the complex nonlinear behaviour of large-scale astrophysical and planetary MHD dynamos, and as a numerical analysis tool to fit and extract effective dynamical information from multidimensional nonlinear MHD simulations.

4.7.1. Low-dimensional nonlinear mean-field models

Low-dimensional dynamical systems derived from simple dynamical and symmetry considerations are well-suited to study bifurcations, weakly-nonlinear solutions and chaotic behaviour in fluid dynamos that operate in the vicinity of their critical Rm, and are for instance commonly used to shed some light on the dynamics and reversals of system-scale magnetic fields in experimental dynamos close to threshold (Ravelet et al. 2008; Berhanu et al. 2009; P´etr´elis & Fauve 2008, 2010). But what about hugely supercritical systems, such as those encountered in astrophysics ? In this context, it has long been argued on the basis of generic dynamical and symmetry arguments that a low-dimensional, weakly-nonlinear approach still makes sense, provided that we think of the various transport and coupling coefficients involved in the equations, such as the viscosity or magnetic diffusivity, as being dominated by turbulent processes rather than microscopic processes.
Mean-field electrodynamics (or rather, mean-field magnetohydrodynamics in the nonlinear regime) can be viewed as a particular application of these principles in the dynamo context, and has therefore long been used to devise low-dimensional dynamical representations of large-scale astrophysical and planetary magnetism at highly-supercritical Rm (and large Re), see e.g. Meinel & Brandenburg (1990); Stefani et al. (2006a). At their core, astrophysical mean-field dynamo models are relatively simple systems of nonlinear partial differential equations governing the evolution of large-scale magnetic and velocity fields coupled through mean-field coefficients, such as in equation (4.3.5), and nonlinear terms. Different large-scale dynamical transport processes, such as the advection of the largescale field by meridional circulations, or its rise under the effect of magnetic buoyancy, can be included with varying degrees of mathematical rigour on the basis of phenomenological considerations and observational incentives. Some nonlinear terms usually included in these models, such as the effect of the large-scale Lorentz force J × B on mean flows for instance, correspond to pristine dynamical nonlinearities that may also contribute to the equilibration of large-scale dynamos (e.g. Malkus & Proctor 1975). Other nonlinearities stem from introducing more or less empirical magnetic-field dependences in the meanfield coefficients, such as the dynamical quenching effects discussed in §4.6.
Many variations and phenomenological prescriptions are possible within this framework. To illustrate in the simplest possible way this popular modelling approach and how it makes use of the theory discussed so far, we will consider the following system of equations, which aims at describing axisymmetric mean-field dynamo action in the convection zone of a differentially rotating star such as the Sun through a minimal nonlinear extension of the local Cartesian αΩ dynamo model introduced in §4.3.5 (Jones (1983); Weiss, Cattaneo & Jones (1984); Jennings & Weiss (1991), see Fauve et al. (2007) for a similar discussion):

∂A ∂t

=

Cα cos z 1 + τB2

B

+

∂2A ∂z2

,

(4.92)

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∂B ∂t

=

CS sin z 1 + κB2

∂A ∂z

+

∂2B ∂z2

− λB3

,

(4.93)

A = B = 0 at z = 0, π .

(4.94)

Distances and times in these equations are expressed in units of stellar radius R and turbulent magnetic diffusion time R2/β respectively (it is assumed that β η), x corresponds to the radial direction in spherical geometry, y to the azimuthal direction, z to the colatitude†. All quantities are assumed to depend on time and z only, and B(z, t) ≡ (−∂A/∂z, B, 0) is expressed in equipartition units (A is the poloidal flux function). This particular model also assumes that α ≡ α0 cos z in the kinematic regime is antisymmetric with respect to the equator, as expected in the solar convection zone and consistent with equation (4.40), while the background shear flow U = −S(z)x ey associated with the differential rotation (S ≡ −rdΩ/dr) in that same regime vanishes at the poles, S ≡ S0 sin z. The control parameter of the linear dynamo instability is again the dynamo number, defined here as D = CαCS = α0S0R3/β2, where Cα = α0R/β and CS = S0R/β. The model also includes non-catastrophic static dynamical quenching of both α and Ω effects in the presence of a dominant azimuthal field B (Stix 1972; Kleeorin & Ruzmaikin 1981), of the form given in equation (4.84), and an empirical cubic nonlinearity mimicking losses of magnetic flux through magnetic buoyancy. These effects are parametrised by three free parameters τ , κ and λ respectively.
As simple as it looks, this model already exhibits some interesting dynamical phenomena, including some symmetry breaking from pure oscillatory dipole and quadrupole solutions. An example bifurcation diagram of this system as a function of D is shown in Fig. 32. This diagram, which describes changes in the amplitude of nonlinear solutions and their branching into new solutions as the control parameter of the system is increased, is typical of the solutions of many astrophysical mean-field dynamo models. It is of course possible, and in fact very common in practical applications such as solar dynamo cycle prediction, to devise similar models in 2D or even 3D (if non-axisymmetric solutions are sought), and in more astrophysically realistic cylindrical or spherical geometries (see e.g. Jouve et al. 2008). Obviously, the results of any such model depends on the values of its free parameters, on the mean-field and nonlinear terms included, and on their particular mathematical form. Larger MHD dynamical systems including nonlinear evolution equations for the large-scale velocity field U and more advanced parametrisations such as time-delays, spatial non-localities or localised couplings usually exhibit an even larger dynamical complexity and chaotic behaviour including dynamocycle modulations (e.g. Ruzmaikin 1981; Weiss et al. 1984; Jones et al. 1985; Tobias 1996, see Weiss (2005) for a detailed discussion and further references). Models of this kind notably include the popular interface “flux-transport” models of the solar dynamo inspired by the early work of Babcock (1961) and Leighton (1969), see also Parker (1993) and reviews by Charbonneau (2010, 2014).
With time, applied astrophysical dynamo modelling of this kind has turned into an industry. The popularity of these models is of course related to their minimal formal mathematical complexity, to their dynamical phenomenological simplicity, and to the practical convenience and flexibility that low-dimensional dynamical systems with theoretically unconstrained parameters offer. Historically, this approach has been instrumen-

† For the sake of notation consistency, we have used the same coordinate system as in §4.3.5 in equations (4.92)-(4.94). These coordinates differ from those used by Weiss et al. (1984) and Jennings & Weiss (1991), for whom x refers to the colatitude and z to the radial direction. The explicit presence of D in their equations instead of Cα and CΩ here stems from a rescaling of B relative to A.

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Figure 32. Example bifurcation diagram of the nonlinear αΩ stellar mean-field dynamo equations (4.92)-(4.94) as a function of −D, computed for κ = λ and τ = 0 (adapted from Jennings & Weiss 1991).
tal in providing fundamental qualitative insights into the nonlinear dynamical essence of large-scale astrophysical and planetary magnetism at a time when three-dimensional MHD simulations were not available (for a perspective on this school of thinking, see again Weiss 2005). On the other hand, the impossibility to derive any such model rigorously from the pristine MHD equations under realistic assumptions, their degeneracies, and the arbitrary amounts of fine-tuning and ad hoc refinement that they are amenable to are considered by many, this author included, as severe structural theoretical weaknesses. It notably requires a big leap of faith to believe that such models can lead to reliable quantitative predictions on the magnetism and dynamics of just about any known astrophysical system. To be fair to this approach, the intrinsically chaotic nature of most known dynamos implies that prediction using any kind of nonlinear model, not just mean-field ones, is a particularly tricky business. A critical discussion and illustration of these problems can notably be found in two papers by Tobias, Hughes & Weiss (2006) and Bushby & Tobias (2007). Charbonneau (2014) provides a good overview of the various possible applications of mean-field modelling in the solar dynamo context, ranging from cycle-prediction activities to the phenomenological interpretation of global simulations discussed in §5.1.2 below.
4.7.2. Mean-field electrodynamics as a numerical analysis tool
Once an essentially theoretical and observational interpretation tool, mean-field electrodynamics has gradually morphed into a practical simulation analysis technique and computer-assisted astrophysical modelling tool in the supercomputing era. A systematic reduction of the challenging dynamical complexity of high-resolution 3D simulations into lower-dimensional mean-field models is of course appealing, and is now indeed routinely performed using tools that measure in situ (in the simulation data) the local EMF responses of the flow to the introduction of large-scale neutral test fields (the socalled test-field method of Schrinner et al. (2005)), or the actual E(B) relationship in the simulation (Brandenburg & Sokoloff 2002; Racine et al. 2011; Tobias & Cattaneo

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2013a; Charbonneau 2014; Squire & Bhattacharjee 2016), and project the results onto a tensorial mean-field relationship (or a generalised convoluted version of it that factors in time-delays and spatial non-locality, Brandenburg 2018). The test-field method, notably, solves equation (4.4) for the fluctuations without any approximation, making it possible to measure the exact total EMF acting on the mean field even in the presence of small-scale magnetic fluctuations induced by a small-scale dynamo. These various approaches are considered by many as a pragmatic, convenient way of simplifying the overall dynamical picture and modelling of otherwise theoretically intractable turbulent dynamos problems (Brandenburg et al. 2010), in that it provides the convenience and comfort of a data-driven interpretation of complex simulation results in relatively simple dynamical terms (e. g. “this simulated dynamo behaves as an αΩ dynamo”). It is for instance widely exploited to distill effective low-dimensional nonlinear mean-field dynamo models from simulations of large-scale planetary and stellar dynamos driven by rotating convection (Schrinner et al. 2007; Charbonneau 2010; Schrinner et al. 2011; Schrinner 2011; Schrinner et al. 2012; Miesch 2012; Charbonneau 2014; Brun & Browning 2017) and accretion-disc dynamos, (Brandenburg et al. 1995; Gressel 2010; Blackman 2012; Gressel & Pessah 2015; Brandenburg 2018), all of which will be further discussed in §5.
While it is undoubtedly practical and may be sufficient to understand some important dynamical aspects of natural dynamos at the phenomenological level, it is important to stress that this reductionist approach does not in itself vindicate the existing meanfield theory, and should not distract us from seeking a better, self-consistent theory of large-scale turbulent dynamos. In particular, this kind of analysis cannot easily escape its main criticism, that mean-field electrodynamics cannot be generically rigorously justified for nonlinear MHD at large Rm in the current state of our understanding. Finally, it should be pointed out that choosing to view all the dynamics through the meanfield prism by decomposing the effective dynamics into many seemingly independent statistical mean-field effects (related to shear, rotation, helicity, etc.) creates a significant risk of disconnection between the analysis and the underlying three-dimensional nonlinear physical dynamics, which often involves several physical effects working together. A clear illustration of this issue will be provided in our discussion of accretion-disc dynamos and the magnetorotational instability in §5.3.5.

5. The diverse, challenging complexity of large-scale dynamos
The observation that large-scale dynamos in nature are almost invariably associated with small-scale turbulence, rotation and shear has been one of the main drivers of the development of a seemingly universal statistical theoretical framework, the mean-field electrodynamics theory presented in §4, at the centre of which lies the α effect and the paradigm of helical α2 and αΩ dynamos. Astrophysical and planetary fluid flows, however, occur in diverse geometries, in diverse thermal, rotation, Reynolds and Prandtl number regimes (Fig. 6), and can accordingly be excited by very different means and instabilities in principle. We should therefore perhaps expect that all large-scale dynamos do not operate in the same way, and that their physical and dynamical peculiarities may not necessarily be easy or even possible to capture with a single universal formalism, not least one that is not formally valid in commonly encountered turbulent MHD regimes.
Whether or not we have confidence in the idea that actual large-scale dynamos at large Rm can be described by a low-dimensional statistical theory with some degree of universality (and the present author does so to some extent), it is fundamental that we keep exploring in parallel their full three-dimensional complexity without any theoretical preconception. Only by doing this can we hope to gain an in-depth understanding of the

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diverse dynamics at work in these systems, to relate them to descriptive mathematical theories in physically meaningful ways, and to identify and fix the possible flaws of existing theoretical paradigms. Over the last twenty years or so, supercomputing has made it possible to start investigating many aspects of large-scale dynamos from a variety of new angles, ranging from the exploration of asymptotic regimes of astrophysical and planetary dynamos using sheer numerical power, to numerical experiments focused on some very fundamental dynamical questions pertaining to the general instability and statistical problems. Very often, these efforts lead to new insights or results that do not seem to fit easily with the existing theory, or require that we use, extend, revise or reinterpret it in previously unforeseen ways.
The aim of this section is to provide an overview of the challenging dynamical and physical complexity that is continously emerging from a combination of old and new theoretical ideas, numerical simulations, and in a few cases also observations of a variety of large-scale dynamos driven by rotating convection, sheared turbulence, or even MHD instabilities such as the magnetorotational instability. As with many other aspects of this review, it is however almost impossible (and counterproductive pedagogically) to aim for exhaustivity on such a vast subject. The next paragraphs therefore concentrate on a selection of problems that the present author feels at least moderately qualified to write about, and believes are quite representative of the exceptional dynamical diversity and outstanding theoretical challenges that large-scale natural dynamos currently present us with.

5.1. Dynamos driven by rotating convection: the solar and geo- dynamos
The solar and geo- dynamos are, for human beings, the epitomes of large-scale MHD dynamos in the Universe, and are undoubtedly the most thoroughly studied and bestdocumented natural processes of this kind both observationally and numerically. These two dynamos belong to the same family of low-P m large-scale stellar and planetary MHD dynamos driven by helical thermal convection in a differentially rotating spherical shell, however they are very different from each other in many respects, and their study is strongly illustrative of the generic difficulties that we have to face as we attempt to understand almost any laboratory or natural dynamo. Although we are going to discuss these two problems in some detail, our main objective in this paragraph is not to provide a specialised review of any of them either, but rather to highlight some important trends and phenomenological aspects of these problems that relate to what we have discussed so far, and to the broader study of large-scale dynamos. Readers interested in specialised reviews are encouraged to consult Brandenburg & Subramanian (2005a); Charbonneau (2010); Miesch (2012); Charbonneau (2014); Brun & Browning (2017); Brandenburg (2018) on solar and stellar dynamos and Christensen (2010); Jones (2011); Roberts & King (2013) on geo- and planetary dynamos.
5.1.1. A closer look at the dynamo regimes of the Sun and the Earth
Stellar and planetary dynamos driven by rotating convection, of which the solar and geo- dynamos are the most nearby instances, share many important dynamical features: they excite a broad spectrum of magnetic fluctuations due to the large Rm involved, sustain a significant large-scale field component, and in some (but not all) cases display some dynamical variability, including large-scale field reversals. The liquidmetal alloy in the Earth’s core and the hydrogen gas in the Sun are both low-P m MHD fluids, meaning that turbulence in these systems extends down to scales well below the magnetic dissipation scale. The dynamics of rotating stellar and planetary interiors is

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also generically characterised by extremely low Ekman numbers E = ν/(ΩL2). In both the Earth’s core and the solar convection zone, E = O 10−15 .
The Sun and the Earth, however, are not in the same rotation regime. The ratio between inertial and Coriolis forces at the turbulence forcing scale, the Rossby number Ro = u0/(Ω 0), is O(10−6) in the Earth’s core and O(0.1) in the solar convection zone. In other words, flows in the Earth’s core down to very small scales are very strongly affected by rotation. In comparison, solar surface convection at scales smaller than that of the rather large-scale supergranulation flows (30 000 km, see Rincon & Rieutord 2018) is essentially unaffected by rotation. Rapidly-rotating convection in the Earth’s core is generally thought to be statistically organised into columns aligned with the rotation axis (at least if we think of the hydrodynamic regime rather than the dynamo-saturated regime), a natural consequence of the Taylor-Proudman theorem (Greenspan 1968). In contrast, we know from helioseismic inversions that the solar differential rotation in the bulk of the convection zone is not organised along axial cylinders, but rather in a spokelike pattern (see e.g. Sekii (2003) for a review). This difference is not only a matter of inertial effects, though, as thermal winds driven by latitudinal entropy gradients are thought to be a major determinant of the Sun’s internal rotation profile (Miesch 2005; Miesch et al. 2006; Balbus et al. 2009).
A second major difference between these two systems is the level at which their dynamo fields saturate. The magnetic energy in the Earth’s core is estimated to be four orders of magnitude larger than the kinetic energy (e.g. Gillet et al. (2010, 2011), see also remarks in Aubert et al. 2017). Measurements of dynamical kG-strength magnetic fields at the solar surface (Solanki et al. 2006) and the detection of large-scale torsional oscillations (e.g. Vorontsov et al. 2002) point to a solar dynamo magnetic field in rough equipartition with flows at both large and small scales.
Altogether, these key observations suggest that the solar and geo- dynamos do not operate in the same dynamical regime, despite both being low-P m and convectiondriven. The strong rotational effects in the Earth’s liquid convective core and its strong magnetic field point to a dynamical balance between Magnetic, Archimedean and Coriolis forces (the so-called MAC or magnetostrophic balance), with subdominant turbulent inertial and viscous effects. This notably leads to the prediction that the asymptotic state in which the nonlinear geodynamo resides should be essentially characterised by a vanishing of the azimuthal Lorentz force averaged over axial cylinders, the so-called Taylor constraint (Taylor 1963, not to be confused with the Taylor-Proudman theorem). The weaker magnetic-field and rotational influence in the Sun, on the other hand, suggest that the dominant dynamical balance of the solar dynamo involves turbulent inertia, with the Coriolis force having a real, but more subtle influence on the dynamo. Accordingly, there is no equivalent to the Taylor constraint in solar dynamo theory. This also seemingly implies that magnetic saturation proceeds differently in the two systems. The dynamical feedback of the magnetic field on the turbulence is thought to be real but energetically subdominant in the strong-field magnetostrophic dynamical regime (Roberts 1988; Roberts & Soward 1992; Davidson 2013; Dormy 2016; Hughes & Cattaneo 2016) but it is usually deemed essential to the saturation of the solar dynamo. In this respect, our earlier theoretical discussion of dynamical saturation in §4.6 appears to be more directly relevant to the solar dynamo than to the geodynamo.
Figure 33 shows snapshots of the magnetic field in two of the highest-resolution geo- and solar dynamo simulations to date. The magnetic field in the geodynamo simulation notably appears to have a much stronger latitudinal dependence and geometric connection to the cylinder tangent to the inner core and oriented along the rotation axis (the so-called tangent cylinder) than that in the solar dynamo simulation, although there

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Figure 33. Two very different simulations of large-scale dynamos driven by rotating turbulent convection at large Rm. Top (adapted from Schaeffer et al. 2017): rendering of the magnetic-field strength, and radial magnetic field at the outer boundary, in a nonlinear geodynamo simulation at Rm = 514, P m = 0.1, Ro = 2.7 × 10−3 and E = 10−7 (in the liquid iron Earth’s core, Rm = O(103), P m = O(10−6), Ro = O(10−6) and E = O(10−15)). Bottom (adapted from Hotta et al. 2016): horizontal projection of the magnetic field-strength in high-resolution simulations of the solar dynamo with implicit numerical diffusion, at an estimated P m = O(1), Rm 2000, Ro = 10−1 − 1, and E 10−5 (in the strongly stratified gaseous hydrogen solar convection zone, P m = 10−6 − 10−2, Rm = 106 − 1010, Ro = O(10−1) and E = O(10−15)). In both cases, Rm, Ro and E are defined on the thickness of the convective layer and typical (r.m.s. or mixing length) convective flow velocity.

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is also some clear latitudinal dependence in the latter. The solar dynamo simulation is also characterised by much more statistically homogeneous, smaller-scale magnetic fluctuations in strong dynamical interaction with the turbulence.

5.1.2. Global simulations of dynamos driven by rotating convection
MHD simulations in spherical geometry such as those shown in Fig. 33 have become a valuable tool to probe the rather extraordinary geometric and thermodynamic complexity of stellar and planetary dynamos. Their main strength is their capacity to factor in many potentially relevant dynamical phenomena (e. g. rotating convection, magnetic buoyancy, thermal winds, meridional circulations and other large-scale flows, strong shear layers) as well as important geometric constraints (e. g. tangent cylinders), whose individual or combined effects are harder or even sometimes impossible to foresee in more idealised local Cartesian settings, and are difficult to capture with statistical theories. A selection of work reflecting the historical evolution and increasing massive popularity of global simulations or large-scale dynamos driven by convection in rotating spherical shells includes Zhang & Busse (1989); Glatzmaier & Roberts (1995); Kuang & Bloxham (1997); Christensen, Olson & Glatzmaier (1999); Christensen & Aubert (2006); Kutzner & Christensen (2002); Kageyama, Miyagoshi & Sato (2008); Takahashi, Matsushima & Honkura (2008); Sakuraba & Roberts (2009); Soderlund, King & Aurnou (2012); Dormy (2016); Sheyko, Finlay & Jackson (2016); Yadav, Gastine, Christensen, Wolk & Poppenhaeger (2016); Aubert, Gastine & Fournier (2017); Schaeffer, Jault, Nataf & Fournier (2017); Sheyko, Finlay, Favre & Jackson (2018) in the geo- and planetary dynamos contexts, and Gilman & Miller (1981); Gilman (1983); Valdettaro & Meneguzzi (1991); Brun, Miesch & Toomre (2004); Dobler, Stix & Brandenburg (2006); Browning, Miesch, Brun & Toomre (2006); Browning (2008); Ghizaru, Charbonneau & Smolarkiewicz (2010); Brown, Miesch, Browning, Brun & Toomre (2011); K¨apyl¨a, Mantere & Brandenburg (2012); Nelson, Brown, Brun, Miesch & Toomre (2013); Fan & Fang (2014); Augustson, Brun, Miesch & Toomre (2015); Yadav, Christensen, Morin, Gastine, Reiners, Poppenhaeger & Wolk (2015); Yadav, Christensen, Wolk & Poppenhaeger (2016); Hotta, Rempel & Yokoyama (2016); Strugarek, Beaudoin, Charbonneau, Brun & do Nascimento (2017); Warnecke (2018) in the solar and stellar dynamo contexts.
Large-scale fields. Similarly to the real systems that they try to emulate, many highresolution global simulations of convection-driven dynamos now show organised, timedependent (and sometimes reversing) dynamically-strong large-scale fields emerging from a much more disordered field component (see e.g. Stefani et al. (2006b); Wicht et al. (2009); Amit et al. (2010); P´etr´elis & Fauve (2010) for descriptions of possible dynamoreversal mechanisms in the geodynamo context, and Charbonneau (2014) for a theoretical perspective on solar dynamo reversals). It remains unclear how these findings can be articulated with the results of the homogeneous simulations in periodic Cartesian domains presented in §4.6.2. One possibility is that helicity fluxes discussed at the end of §4.6.3 are indeed important in global simulations with open magnetic boundary conditions and somehow alleviate catastrophic quenching. Another possibility is that the growth, dynamical saturation and evolution timescales of the large-scale dynamo modes observed in global simulations are indeed subtly tied to the microscopic diffusion time at system scales, but that much higher Rm must be achieved in simulations to reveal the full extent of catastrophic quenching. Some global simulation results at relatively low Rm (of a few tens) are indicative of a strong quenching (Schrinner et al. 2012; Simard et al. 2016). Interestingly, a breadth of recent high-resolution local simulations also now suggest that cyclic large-scale dynamo fields can be generated by compressible rotating convection in Cartesian geometry, albeit with a period seemingly controlled by the resistive time at

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Figure 34. The quest for asymptotic geodynamo regimes (adapted from Schaeffer et al. 2017). Each circle represents a published direct or large-eddy numerical simulation, and the area of the discs represent Rm (again defined on the thickness of the convective layer and r.m.s. flow velocity). The pale-blue discs down to E = 10−8 at P m < 1 and Rm = O(103) correspond to the LES simulations of Aubert et al. (2017), the red ones to the DNS simulations of Schaeffer et al. (2017). A2 is the ratio between kinetic and magnetic energy. Many simulations have super-equipartition saturated fields, although such states appear more difficult to achieve computationally at low P m (Rm = O(103), P m = O(10−6), E = O(10−15), and A = O(10−2) in the liquid iron core of the Earth).
the system scale (Bushby et al. 2018). Some authors have argued that the solar cycle itself may be close to resistively limited (see for instance Brandenburg & Subramanian 2005a, Sect. 9.5). Note finally that the quenching issue may be of lesser importance in the context of the geodynamo, as the latter is characterised by a smaller Rm.
Parametric explorations. Unfortunately, global simulations require even more important sacrifices than local Cartesian simulations in terms of scale separation in order to encompass global system scales, turbulent forcing scales, inertial scales and bulk and boundary layer dissipation scales. The numbers given in the caption of Fig. 33 show that there remains a wide parameter gap between even the most massive simulations to date and the Sun or the Earth. A recurring question with any new generation of simulations is therefore to what extent their results are representative of the asymptotic dynamical regimes in which the systems that they try to emulate operate. A long-time strategy of the geodynamo community to address this question has been to carry out a methodical parametric numerical exploration of dynamical force balances, scaling laws, and nonlinear dynamo states along parameter paths consistent with the natural ordering of scales in the problem (Christensen et al. 1999; Olson & Christensen 2006; Christensen & Aubert 2006; Takahashi et al. 2008; Soderlund et al. 2012; Schrinner et al. 2012; Stelzer & Jackson 2013; Dormy 2016; Sheyko et al. 2016; Yadav et al. 2016; Aubert et al. 2017; Schaeffer et al. 2017; Sheyko et al. 2018). This approach, illustrated in Fig. 34, is now also gaining traction in astrophysics, notably on the problem of dynamos in rapidlyrotating fully convective stars (Christensen et al. 2009; Morin et al. 2011; Gastine et al. 2012, 2013; Raynaud et al. 2015) or convective stellar cores (Augustson et al. 2016), see also Augustson et al. (2019) for a recent discussion and meta-analysis of several sets of convective dynamo simulations.
Geodynamo simulations. Much analytical and numerical work has been done to understand the roles that the many kinds of flows relevant to fast-rotating planetary cores (Fearn 1998) may have on their internal dynamos. These include fast-rotating thermal

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convection and vortices (Childress & Soward 1972; Busse 1975, 1976; Roberts & Soward 1978; Roberts & Soward 1992; Kageyama & Sato 1997; Sarson & Busse 1998; Olson et al. 1999; Stellmach & Hansen 2004; Sreenivasan & Jones 2011; Jones 2011; Guervilly et al. 2015; Calkins et al. 2016), helical waves (Braginskii 1964a,b; Moffatt 1970b; Schaeffer & Cardin 2006; Davidson 2014; Davidson & Ranjan 2018) Ekman boundary layers (Ponty et al. 2001; Schaeffer & Cardin 2006), and zonal flows and internal shear layers (Simitev & Busse 2009; Sheyko et al. 2016) (this list of references is not meant to be exhaustive but to provide relevant entry points in the literature). Various analyses have recently shown that some aspects of the dynamics of geodynamo simulations conducted even just a few years ago down to E = O(10−5), P m = O(1) and Rm = O(100) were still significantly affected by viscous effects (Soderlund et al. (2012); King & Buffett (2013); Davidson (2013); Oruba & Dormy (2014); Dormy (2016), see also see also theoretical arguments in P´etr´elis & Fauve 2001; Fauve et al. 2007). The weak-field (equipartition or sub-equipartition) dynamo branch solutions obtained in these “classical” simulations have been interpreted as the product of an α2-type Parker mechanism driven by helical convection columns involving an axial flow component generated through the viscous coupling of the bulk convection with the boundary layer (Roberts & King 2013). The latest generation of high-resolution simulations of the geodynamo extending down to E = O(10−7), P m = 0.05 and Rm = O(103), on the other hand, now seems on the verge of convergence towards asymptotic strong-field magnetostrophic dynamo states (Yadav et al. 2016; Aubert et al. 2017; Schaeffer et al. 2017; Sheyko et al. 2018). These simulations paint a very complex, inhomogeneous, and multiscale dynamical picture that is not easy to reconcile with a simple statistical theoretical description (although interestingly it appears to be possible to construct magnetostrophic mean-field dynamo solutions, see Wu & Roberts 2015; Roberts & Wu 2018). The highly non-perturbative effects of the super-equipartition magnetic field on the dynamics of the fluid are a major theoretical complication, with the detailed force balance and flow properties depending both on the region (i.e. inside or outside the tangent cylinder, or in the boundary layers) and scale considered (Aubert et al. 2017).
Solar dynamo simulations. As complex as the geodynamo is to simulate realistically and to interpret theoretically, modelling the solar dynamo appears to be even more of a quagmire, so much so that a credible “numerical” dynamo solution (in the sense of being both in a credible parameter regime with the right scale-orderings, and showing convincing dynamical similarities with the observational characteristics of the solar cycle) still does not appear to be in sight at the moment. One of the main difficulties is that the solar dynamo involves many different processes that all appear to be of the same order. As explained earlier, the rotation regime of the Sun is not quite as asymptotic as that of the Earth’s core, and as a result turbulent inertial effects are likely to play a much more important dynamical role in the problem of its magnetic-field sustainment, as indicated by the near-equipartition levels of saturation. Turbulent transport of angular momentum through Reynolds stresses is also thought to significantly contribute to the maintenance of the Sun’s differential rotation, whose exact distribution has a strong impact on the generation of toroidal fields. The solar differential rotation profile is itself notoriously difficult to reproduce even in hydrodynamic numerical simulations, with the effect of thermal winds driven by dynamically established latitudinal entropy gradients being another key factor to consider in this context.
The thermodynamics and internal structure of the Sun adds yet another layer of complexity to the problem: the strong density stratification of the bulk of the solar convection zone, for instance, is fertile to many additional MHD phenomena relevant to the dynamo, including magnetic-buoyancy instabilities (more on this in §5.3) and the

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turbulent pumping of magnetic fields (see §4.4.2 and Nordlund et al. (1992); Brandenburg et al. (1996); Tobias et al. (1998, 2001); Browning et al. (2006) for simulations). The inner radiative zone of the Sun and its outer convection zone are also coupled through a thermodynamically and dynamically complex shear layer, the tachocline (Stix 2004; Hughes, Rosner & Weiss 2012). From a dynamo theory perspective, this region is a mixed blessing. On the one hand, it is an obvious locus for the generation of strong toroidal magnetic fields through the Ω effect (Parker 1993), potentially providing us with half of a solar dynamo mechanism without having to think too much. On the other hand, this layer turns out to be extremely difficult to model consistently both analytically and numerically, and it creates all sorts of dynamical complications ranging from being prone to its own large-scale MHD instabilities (more on this in §5.3.6) to being a complex processing and storage unit of angular momentum and thermal, potential, kinetic and magnetic energy. In other words, the lower boundary condition in the solar dynamo problem is arguably much more tricky and dynamically active than in the geodynamo problem.

One of the major problems of early global simulations of the solar dynamo is that they showed very few signs of strong large-scale field organisation, and instead essentially looked like standard Cartesian simulations of turbulent small-scale fluctuation dynamos (§3) mapped on a sphere (e.g. Brun et al. 2004; Dobler et al. 2006). The consistent obtention of more organised large-scale and often cyclic fields in simulations of increasing structural and dynamical complexity over the last ten years (e.g. Ghizaru et al. 2010; Brown et al. 2010, 2011; K¨apyl¨a et al. 2012; Nelson et al. 2013; Fan & Fang 2014; Augustson et al. 2015; Hotta et al. 2016; Strugarek et al. 2017; Warnecke 2018; Strugarek et al. 2018) therefore marks an important milestone in the field. While the α(2)Ω dynamo paradigm remains a pillar of the phenomenological interpretation of many such simulations (Racine et al. 2011; Charbonneau 2014; Brun & Browning 2017; Brandenburg 2018), many important dynamical effects such as small-scale dynamo activity, tachoclinic dynamics, turbulent pumping, magnetic buoyancy, and large-scale transport of magnetic fields by meridional circulations appear to be significant in many instances. All these effects work together in the simulations to produce (or impede) the generation of largescale field and are extremely difficult to disentangle in practice, making it very hard to assess with confidence what exactly drives and controls the solutions (not even mentioning the actual solar dynamo). Note that large-scale effects distinct from an α effect excited by rotating stratified convection must be important in the solar dynamo problem, because the theoretical prediction (also observed in simulations) of a positive α effect in the northern hemisphere (§4.4.1), combined with helioseismic measurements of the differential rotation profile in the Sun, leads to the prediction of a polewards migration of pure αΩ dynamo waves according to the so-called Yoshimura (1975) rule, opposite to the actual record of sunspot patterns.

Another significant complication to the analysis and comparison of the current generation of numerical solutions is the use of (different) subgrid scale models. For instance, all other things being close, different codes tend to produce cyclic dynamo solutions with significantly different cycle periods, and these differences have been attributed to the details of the numerical implementation (Charbonneau 2014). This observation once again raises the problem of the subtle but seemingly critical role of small-scale dynamics and dissipative processes in controlling large-scale dynamos.

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Figure 35. Space-time diagrams showing the evolution of a large-scale dynamo magnetic field By(z, t) in local Cartesian shearing box numerical simulations of dynamo action in the presence of forced, non-helical small-scale turbulence, shear US = −Sx ey, with or without global rotation Ω = Ω ez. Top row: simulations at Rm lower than Rmc,ssd for small-scale dynamo action (adapted from Squire & Bhattacharjee 2015c). Bottom row: simulations at Rm larger than Rmc,ssd. Hatches indicate the phase of small-scale dynamo growth (adapted from Squire & Bhattacharjee 2015b, 2016). Left: simulations with shear, but no rotation. Right: simulations with Keplerian rotation, Ω = (2/3) S. The box size (Lx, Ly, Lz) is (1, 1, 16) in the top-row calculations with no small-scale dynamo, and (1, 4, 2) in the more computationally demanding bottom-row calculations with a small-scale dynamo. All four simulations are in a regime of Re and Rm that is linearly and nonlinearly stable to hydrodynamic shear instabilities and MHD instabilities such as the magnetorotational instability. In all cases, the turbulence is forced at a scale 0 = Lx/3 much smaller than that at which the large-scale dynamo fields develop, and has a turnover rate urms/ 0 comparable to the shearing rate S.
5.2. Large-scale shear dynamos driven by turbulence with zero net helicity
5.2.1. Numerical simulations
A very different, but equally interesting development in large-scale dynamo theory and modelling over the past ten years has been the explicit numerical demonstration (by several independent groups using different methods and studying different flows) that small-scale turbulence with zero net helicity, but embedded in a large-scale shear flow, can drive a large-scale dynamo (Yousef et al. 2008b; Brandenburg et al. 2008a; K¨apyl¨a et al. 2008; Hughes & Proctor 2009; Singh & Jingade 2015). In the simplest possible Cartesian case, a shearing-box extension of the Meneguzzi et al. (1981) set-up, turbulence is forced at a small-scale 0 by an external, δ-correlated-in-time non-helical body force, and is embedded in a linear shear flow US = −Sx ey. This results in the generation of a large-scale horizontal magnetic field Bx(z, t), By(z, t) with a typical wavenumber kz 0 1 (the overline here denotes an average in the xy-plane). Magnetic field generation of this kind appears to be possible both with and without overall rotation Ω = Ω ez, and survives even in the presence of a small-scale dynamo at Rm > Rmc,ssd (Yousef et al. 2008a; Squire & Bhattacharjee 2015b, 2016), see Fig. 35. This mechanism, now commonly referred to as the shear dynamo, can also produce large-scale field reversals typical of dynamo cycles (Teed & Proctor 2017).
5.2.2. Shear dynamo driven by a kinematic stochastic α effect
Non-rotating case. That non-rotating turbulence with zero net helicity can drive a large-scale dynamo may seem quite surprising at first in the light of the discussion in

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§4.3.3 of the seemingly important role of reflection symmetry in the problem, and it has indeed proven quite difficult to make sense of these results. In view of its potential broad range of application in astrophysical dynamo theory and of the interest that it has drawn in recent years, this problem is definitely worthy of a brief review here. It is however unfortunately also among the most technical matters in dynamo theory, and will therefore only be treated at a rather superficial technical level. Readers are referred to the work of Heinemann, McWilliams & Schekochihin (2011), Sridhar & Singh (2014) and Squire & Bhattacharjee (2015c) for more thorough and quantitative, yet very clear presentations of the many analytical facets of the problem.
One of the first effects put forward as an explanation of the results of simulations of the non-helical shear dynamo has been the kinematic shear-current effect introduced in §4.4.3. In fact, the original analytical study of this effect provided one of the main motivations for the first numerical simulations of the shear dynamo by Yousef et al. (2008b). Its relevance as a dynamo-driving mechanism, however, remained quite controversial for a few years, as different closure calculations appeared to lead to conflicting conclusions (see discussion in §4.4.3). Measurements of mean-field responses (using the test-field method discussed in §4.7.2) in several independent numerical simulations of non-rotating, sheared, nonhelical turbulence with St = O(1), now seem to have confirmed that the contribution of the shear-current effect to ηyx has the wrong sign for a large-scale dynamo, both in the FOSA regime of low Rm (as it should be) and for moderate Rm up to O(100) (Brandenburg et al. 2008a; Squire & Bhattacharjee 2015c; Singh & Jingade 2015).
A further indication that the shear dynamo is not driven by the shear-current effect, at least in the non-rotating case, is that the phase of the large-scale field B in the simulations evolves randomly in time. This can be seen for instance in the time-distance representations of non-rotating shear dynamo simulations shown in Fig. 35 (left). Accordingly, if we were to perform a statistical ensemble average (denoted by · ) over independent realisations of the forcing (or over long-enough times), we would find that |B|2 grows,
but not B . If the dynamo was driven by a systematic, coherent statistical effect such as the shear-current effect, on the other hand, we would expect this phase to follow a deterministic pattern and B to grow, in accordance with equations (4.46)-(4.47). In that sense, the shear dynamo can be said to be “incoherent” (Heinemann et al. 2011; Squire & Bhattacharjee 2015c). This peculiarity is now considered as one of the key features of the shear dynamo, but was not immediately recognised at the time of its first numerical simulations.
An alternative possibility that may be squared more easily with this numerical observation is that the non-rotating shear dynamo below the small-scale dynamo threshold Rmc,ssd is driven by a stochastic α effect associated with fluctuations of kinetic helicity with zero mean, assisted by the Ω effect. The general idea that helicity fluctuations with zero mean may be capable of driving a large-scale dynamo actually also largely predates the numerical discovery of the shear dynamo, and has evolved along several distinct lines through the history of dynamo research. Its origins can be traced back to the work of Kraichnan (1976), who inferred from a simple stochastic model of short-correlated† fluctuations of the diagonal terms of the α tensor that the large-scale field in the presence of helicity fluctuations could be prone to an instability of negative-eddy-diffusivity type. Moffatt (1978) further showed that this effect is supplemented by an effective drift of the

† “Short-correlated” here must be understood in a restricted sense. Namely, α is allowed to fluctuate on a timescale τα much longer than the typical correlation time of the small-scale flow (otherwise there is no scale separation), but much shorter than α/αrms, where αrms is the typical r.m.s. value of the fluctuating α and α is a typical (large) spatial scale of the fluctuations of α. The latter approximation is necessary to solve the problem with quasilinear theory.

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magnetic field when the fluctuations of α are also allowed to fluctuate in space (for zero α-correlation-time τα, this effect does not induce any field). The full evolution equation for the magnetic field B “super-ensemble”-averaged over both statistical realisations of α and small-scales is

∂B ∂t = ∇× UM × B + (η + β + ηK ) ∆ B ,

(5.1)

where

∞
ηK = − dτ α(x, τ )α(x, 0)
0

∞
and UM = dτ α(x, τ )∇α(x, 0)
0

(5.2)

are called the Kraichnan diffusivity and Moffatt drift. The spatial and time dependence of α here must be understood as a slow dependence on scales much larger than that of the turbulence itself. Note finally that possible off-diagonal contributions to the fluctuating α tensor are discarded in this derivation and that α = 0 is assumed. A significant issue with the mean-field equation (5.1) is that it promotes growth at the smallest scales available (as the instability growth rate is proportional to k2), thereby compromising the scale-separation between the mean field and fluctuations on which the theory is constructed. The Kraichnan-Moffatt (KM) mechanism, at least in its basic form, is therefore in all likelihood not what drives the shear dynamo observed in simulations either. The latter does not appear to be of negative-eddy-diffusivity type (the field does not grow at the smallest-scales of the simulations at all), is not excited in the absence of shear, and as we mentioned earlier does not generate a coherent mean field B . Besides, the KM mechanism requires strong fluctuations of the diagonal elements of α in order for their r.m.s. effect to overcome the regular positive turbulent diffusivity β, and it has recently been pointed out that, even if this condition holds, an instability is only possible if the fluctuations of the off-diagonal elements of α are comparatively much smaller (Squire & Bhattacharjee 2015c). While such conditions may not be impossible to achieve in a non-shearing system, they do not appear to be typical of the sheared turbulence forced in the simulations of the shear dynamo. A generalised in-depth treatment of the KM mechanism in a variety of regimes, including the interesting case of small but non-zero Sτα in shearing regimes, can be found in the work of Sridhar & Singh (2014).
More general manifestations of a stochastic α effect have been explored in the context of shearing systems such as galaxies or accretion discs. However, most studies of this problem have been conducted at a semi-phenomenological level within the framework of meanfield electrodynamics, usually by plugging “by hand” a stochastic model for α into αΩ dynamo models (Sokolov 1997; Vishniac & Brandenburg 1997; Silant’ev 2000; Fedotov et al. 2006; Proctor 2007; Kleeorin & Rogachevskii 2008; Rogachevskii & Kleeorin 2008; Richardson & Proctor 2012). While such models generically produce growing dynamo solutions, their detailed results and conclusions appear to be strongly dependent on an ad hoc prescription for the stochastic α model and on the procedure used to find a closed expression for the super-ensemble-averaged correlator α · B appearing in the derivation. A rigorous first-principles calculation demonstrating (in some analytically tractable limit) the existence of an incoherent shear dynamo driven by fluctuations of helicity in the simplest case of sheared, forced non-helical turbulence only appeared a few years ago in the work of Heinemann et al. (2011) (see also McWilliams 2012; Mitra & Brandenburg 2012; Sridhar & Singh 2014). Although it is perturbative at its core too, this calculation is technically more involved than the FOSA derivation of the standard α effect presented in §4.3.6. We will therefore only explain its key features here. The main interest of the derivation lies in that it rigorously captures the effect of the shear on the

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turbulent motions underlying helicity fluctuations. To achieve this, the authors introduce a representation of the three-dimensional forced turbulent velocity field as an ensemble of plane “shearing waves” labelled by Lagrangian wavevectors k0 (we already encountered this mathematical representation in §3.2). Under the effect of large-scale shearing, the Eulerian wavevectors of plane waves evolve according to†

k(t) = k0 + Skyt ex .

(5.3)

Using this representation, a closed evolution equation for the the Fourier-transformed
mean-field B(kz, t) covariance four-vector, averaged over a statistical ensemble of realisations of a body force driving the turbulence, can be derived in the perturbative limits
of low Rm and Sτc 1 (the ensemble average is again denoted by · below, while the overline denotes spatial averages in the xy-plane). Crucially, this equation couples the Bx(kz, t)Bx(kz, t)∗ component of this vector to its By(kz, t)By(kz, t)∗ component through an off-diagonal term proportional to the coefficient

∞
D12(t) = 2 dt
0

αyy(t)αyy(t − t ) = 4

ky4

|Φk0 (t)|2 |uzk0 (t)|2 νη2k6(t)

k0

,

(5.4)

where Φk0 is the stream function associated with the horizontal velocity component of a shearing wave with Lagrangian wavevector k0. The usual Ω effect ensures the amplification of an azimuthal (y) mean-field component from the poloidal (x) component. Note that D12 is always positive, which appears to be a necessary condition for the dynamo. That equation (5.4) involves the product between the energies of the horizontal vortical and vertical components of the velocity field of the ensemble of shearing waves makes it explicit that this dynamo owes its existence to the r.m.s. fluctuations of the kinetic helicity of the sheared turbulent velocity field (the net average helicity k2(t) uzk0 (t)Φk0 (t)∗ for each shearing wavenumber is taken to be zero in the derivation, so that there is no systematic coherent α effect). Despite being formally only valid at low Rm and perturbative in the shear parameter, the Heinemann et al. theory appears to predict a maximal dynamo growth rate γ ∼ |S| at a wavenumber kz ∼ |S|1/2 consistent with the numerical results of Yousef et al. (2008b) obtained in the regime S 0/urms = O(1) and Rm > 1. In view of our earlier remarks on the evolution of the phase of the mean field in the shear dynamo, it is also important to emphasise that the product of this derivation is a dynamo equation for the super-ensemble-averaged mean energy of the magnetic field in the xy-plane, |B|2 , not for the super-ensemble-averaged
mean field B itself. The latter does not grow in the theory, to lowest-order in the expansion parameters.
That the non-rotating shear dynamo observed in simulations is primarily driven by an incoherent α effect as captured by the theory of Heinemann et al. (2011) is now further supported by parametric numerical explorations showing that the growth rate of the instability decreases if the horizontal dimensions Lx and Ly of the numerical domain are increased while keeping the scale 0 of the turbulent forcing fixed (Squire & Bhattacharjee 2015c). This behaviour is exactly what one expects from a purely random effect by averaging over a larger number of fluctuations, and implies that such a dynamo formally disappears in a system of infinite size, contrary to a coherent dynamo relying on a systematic α effect in a helical flow. This is not a fundamentally existential problem for the shear dynamo though, as for all practical purposes we are only interested in single realisations of dynamos in finite systems. However, it would seem to imply that this

† We once again define US = −Sx ey for internal consistency. Heinemann et al. (2011) use the opposite sign convention.

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dynamo effect could be much weaker than other possible effects in systems with very large-scale separations.
Rotating case. Another interesting question seemingly relevant to dynamos in many differentially rotating astrophysical bodies is what happens when we add global rotation to the shear dynamo. The anticyclonic Keplerian rotation regime characterised by Ω = (2/3)S in the local shearing sheet approximation has been the most studied in this context due to its relevance to accretion-disc dynamics. In this regime of rotation and in the presence of a magnetic field, it is well-known that an MHD fluid can become unstable to the magnetorotational instability (MRI, Velikhov 1959; Chandrasekhar 1960; Balbus & Hawley 1991) at wavenumbers such that k · UA ∼ Ω, where UA is the Alfv´en velocity (2.16), provided that Rm is larger than some critical Rm whose exact value depends on the magnetic-field configuration (and P m). This is important because the MRI can drive its own MHD turbulence, and can even sustain the very magnetic field that mediates it through a nonlinear dynamo process. We will study the latter problem in detail in §5.3, but for the time being we will keep Rm below the threshold of any kind of manifestation of the MRI in order to study possible large-scale dynamo effects excited solely by smallscale turbulence driven by a non-helical forcing. As shown in Fig. 35 (right) then, the shear dynamo studied earlier appears to survive in the Keplerian regime in a range of Rm where the MRI is not excited, which suggests that this mechanism may be relevant to the excitation of a dynamo field in accretion discs independently of an MRI-driven dynamo. This however would require that a robust turbulence-driving mechanism unrelated to the MRI, such as a hydrodynamic instability, is excited in Keplerian discs. Such a mechanism, if any, remains elusive (Fromang & Lesur 2017).
It has recently been observed that the nature of the shear dynamo in the Keplerian regime may be quite different from that in the non-rotating regime though (Squire & Bhattacharjee 2015c). An incoherent stochastic α effect is still present in the Keplerian case, however it appears to be supplemented, and overwhelmed, by a more coherent statistical effect reminiscent of the Ω ×J R¨adler effect discussed in §4.4.3. This conclusion comes from the observation that the coherent ηyx coefficient measured in the Keplerian simulations, unlike in the non-rotating simulations, is of opposite sign to that of the shear S. This result is also consistent with the observation made in §4.4.3 that a R¨adler effectdriven dynamo is only possible in anticyclonic regimes. Finally, Squire & Bhattacharjee (2015c) also argue that the phase of the mean field in Keplerian simulations is more stable than in non-rotating simulations, indicating that the dynamo is driven by a more coherent statistical effect in the former case.

5.2.3. Shear dynamo driven by nonlinear MHD fluctuations*
The previous results clearly show that kinematic large-scale dynamos can be excited by a much wider variety of statistical effects than the standard α effect on which much of our interpretation of so many dynamos has historically relied. Accordingly, the shear dynamo is now considered by many in the community as a promising way to circumvent (some would say to avoid having to deal with) helical dynamo quenching (§4.6). However, there is currently no guarantee either that the shear dynamo is not itself affected by some strong form of dynamical quenching at large Rm. Simulations of the shear dynamo problem above the critical Rmc,ssd for the small-scale dynamo (bottom row of Fig. 35) show that a large-scale dynamo can coexist with a small-scale dynamo in this problem (Yousef et al. 2008a; Squire & Bhattacharjee 2015b, 2016), but this in itself does not prove that large-scale growth is not quenched in some way in this regime.
An interesting recent twist on this problem has been the realisation that the presence of dynamical small-scale magnetic fluctuations (characterised by brms ∼ urms), driven