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) small- and 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 large-scale 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.
ContentsSubsections marked with asterisks contain some fairly advanced, technical or specialised material, and may be skipped on a first reading.
- 1 Introduction
2 Setting the stage for MHD dynamos
- 2.1 Magnetohydrodynamics
- 2.2 Important scales and dimensionless numbers
- 2.3 Dynamo fundamentals
3 Small-scale dynamo theory
- 3.1 Evidence for small-scale dynamos
- 3.2 Zel’dovich-Moffatt-Saffman phenomenology
- 3.3 Magnetic Prandtl number dependence of small-scale dynamos
3.4 Kinematic theory: the Kazantsev model
- 3.4.1 Kazantsev-Kraichnan assumptions on the velocity field
- 3.4.2 Equation for the magnetic field correlator
- 3.4.3 Closure procedure in a nutshell*
- 3.4.4 Closed equation for the magnetic correlator
- 3.4.5 Solutions
- 3.4.6 Different regimes
- 3.4.7 Critical
- 3.4.8 Selection of results at large *
- 3.4.9 Connection with finite-correlation-time turbulence
- 3.5 Dynamical theory
4 Fundamentals of large-scale dynamo theory
- 4.1 Evidence for large-scale dynamos
- 4.2 Some phenomenology
4.3 Kinematic theory: mean-field electrodynamics
- 4.3.1 Two-scale approach
- 4.3.2 Mean-field ansatz
- 4.3.3 Symmetry considerations
- 4.3.4 Mean-field equation for pseudo-isotropic homogeneous flows
- 4.3.5 , and dynamo solutions
- 4.3.6 Calculation of mean-field coefficients: First Order Smoothing
- 4.3.7 FOSA derivations of and for homogeneous helical turbulence
- 4.3.8 Third-order-moment closures: EDQNM and -approach*
- 4.4 Mean-field effects in stratified, rotating, and shearing flows
- 4.5 Difficulties with mean-field theory at large
- 4.6 Dynamical theory
- 4.7 Overview of mean-field dynamo theory applications
5 The diverse, challenging complexity of large-scale dynamos
- 5.1 Dynamos driven by rotating convection: the solar and geo- dynamos
- 5.2 Large-scale shear dynamos driven by turbulence with zero net helicity
5.3 Subcritical dynamos driven by MHD instabilities in shear flows
- 5.3.1 Numerical evidence, and a few puzzling observations
- 5.3.2 Self-sustaining nonlinear processes
- 5.3.3 Nonlinear dynamo cycles and subcritical bifurcations
- 5.3.4 -dependence of the MRI dynamo transition*
- 5.3.5 From the MRI dynamo to large-scale accretion-disc dynamos*
- 5.3.6 Other instability-driven and subcritical dynamos
- 5.4 The large- frontier
6 Dynamos in weakly-collisional plasmas
- 6.1 Kinetic versus fluid descriptions
- 6.2 Plasma dynamo regimes
- 6.3 Collisionless dynamo in the unmagnetised regime
- 6.4 Introduction to the dynamics of magnetised plasmas
- 6.5 Collisionless dynamo in the magnetised regime
- 6.6 Uncharted plasma physics
- 7 A subjective outlook for the future
- A Some good reads
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 entry-point 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 to make 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 suceeded 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 disk 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-to-day 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-years magnetic cycle timescale over which the large-scale solar magnetic field reverses. The solar cycle shows up in many different observational records, the most well-known of which is probably the number of sunspots as a function of time, see Fig. 1 (left) (note that the eleven-year cycle is also chaotically modulated on longer timescales). Large-scale solar magnetism exceeds kiloGauss strengths in sunspots, and has been monitored for many centuries (and for a long time without the knowledge that sunspots or northern lights were symptoms of solar magnetic activity). But there is also a lot of dynamical, small-scale, disordered magnetism in the solar surface photosphere and chromosphere, evolving on fast time and spatial scales comparable to those of thermal convective motions at the surface (from a minute to an hour, and from a few kilometers to a few thousands of kilometers). This so-called “network” and “internetwork” small-scale magnetism, depicted in Fig. 1 (right), was discovered much more recently (Livingston and 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 and Landstreet 2009), but only for the Sun do we have accurate, temporally and spatially-resolved 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 ( T). The dynamical evolution and structure of the field, including its many irregular reversals over a hundred-thousands 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 (left). 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 (right) shows auroral emissions on Jupiter, whose magnetic field has a typical strength of a few Gauss ( T). 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 Gauss. 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 (left). But recent high-resolution observations of polarised dust emission in our galaxy, displayed in Figure 3 (right), 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 Gauss are also measured in the hot intracluster medium (ICM) of galaxy clusters. 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 (left), suggest that there is a significant “small-scale”, turbulent ICM field component on scale comparables to or even smaller than a kiloparsec. Visible-light observations of the ICM, including in the H spectral line, also reveal the presence of colder gas structured into magnetised filaments, see Fig. 4 (right).
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 thought to be of the order of, but no larger than Gauss. A detailed discussion of the current observational bounds on the scales and amplitudes of magnetic fields in the early Universe can for instance be found in the review by Durrer and Neronov (2013).
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. Absent such mechanisms, 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 and King (2013) for geomagnetic estimates. Besides, even in relatively high-conductivity environments such as stellar interiors, the fossil-field hypothesis can not 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 fluid/plasmas are electrically conducting. This raises the possibility that internal flows create a motional 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 sustainement 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 dynamo-generated 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 and Herrero-Bervera 2007), and the book by *molokov07 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“The theory proposed by Sir Joseph Larmor, that the magnetic field of a sunspot is 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.” and apparently even led Einstein to voice a pessimistic outlook on the subject (Krause 1993). The first significant positive developments only occurred after the second world war, when Elsasser (1946, 1947), followed by Bullard and Gellman (1954), set about formulating a spherical theory of magnetic field amplification by non-axisymmetric convective motions in the liquid core of the Earth. In the same period, Batchelor (1950) and Schlüter and 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, *steenbeck66 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 pionneering 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 and 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, *meneguzzi81 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 power-supply, dissipation and security 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 and Müller 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 only been reported in the Von Kármán Sodium experiment (VKS, Monchaux et al. 2007). The decisive role of soft-iron solid propellers 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 and Forest (2011).
1.5 An imperfect dichotomy
As we have seen, 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 scale as large-scale dynamo mechanisms. Global rotation and/or large-scale shear usually (though not always) plays an important role in this context. On the other hand, and while large-scale dynamos naturally produce a significant amount of small-scale magnetic field of their own, magnetic fields at scales comparable to or smaller than that of the flow can also be excited by independent small-scale dynamo mechanisms that can be active even in the absence of large-scale dynamo mechanisms.
This dichotomy between small- and large-scale dynamos has the merits of clarity and simplicity, and will therefore be used in this document 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 do 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 document.
1.6 Outline of the document
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 as the magnetorotational instability. Finally, §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
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
where is the gas density and is the fluid velocity field, and the momentum equation
where is the gas pressure, is the viscous stress tensor ( is the dynamical viscosity and is the kinematic viscosity), stands for any kind of external stirring (impellers, gravity, spoon, supernovae, meteors etc.), is the magnetic field, is the current density, and , the Lorentz force, describes the dynamical feedback exerted by the magnetic field on fluid motions. The evolution of is governed by the induction equation
supplemented with the solenoidality condition
Equation (2.1.1) is derived from the Maxwell-Faraday equation and a simple, isotropic Ohm’s law for collisional electrons,
where is the electrical conductivity of the fluid. The first term on the r.h.s. of equation (2.1.1) 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 term describes the diffusion of magnetic field in a non-ideal fluid of magnetic diffusivity . Both the Lorentz force and EMF terms in equations (2.1.1)-(2.1.1) 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
where is the gas temperature, is the entropy ( is the adiabatic index), and stand for the viscous and resistive dissipation, 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 like the perfect gas law for the thermodynamic variables is also required in order to close this system.
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 accomodate a large range of dynamical magnetic phenomena including dynamos and (fluid) reconnection. 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 the ideal () MHD equations and of their main properties.
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
where is the Lagrangian derivative. Equation (2.1.2) for has the same form as the equation describing the evolution of the Lagrangian separation vector between two fluid particles,
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 Alfvén’s theorem. Using this equation, it is also possible to show that the magnetic flux through material surfaces (deformable surfaces moving with the fluid) is conserved in ideal MHD,
If a material surface is deformed under the effect of either shearing or compressive/expanding motions, the magnetic field threading it must change accordingly so that remains the same. Alfvén’s theorem enables us to apprehend 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 form an intuition of how small and large-scale dynamos processes work.
A second important conservation law in ideal MHD in the context of dynamo theory is the conservation of magnetic helicity , where is the magnetic vector potential. To derive it, we first write the Maxwell-Faraday equation for ,
is the total magnetic-helicity flux. In the ideal case, we see that equation (2.1.2) reduces to an explicitly conservative local evolution equation for ,
where equation (2.1.1) with has been used to express the magnetic-helicity flux. Note that both and 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 and 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 associated with a fluid particle in ideal MHD (),
where 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 . Equation (2.1.3) follows directly from equations (2.1.1)-(2.1.2), and shows that any increase of results from either a stretching of the magnetic field along itself by a flow, or from a compression, and that the rate at which changes is proportional to the local shearing or compression rate of the flow. Note that incompressible motions with no parallel component do not affect the field strength at linear order, and only generate magnetic curvature perturbations (these are shear Alfvén waves). Going back to full resistive MHD, the global evolution equation for the magnetic energy, derived for instance in the classic textbook of Roberts (1967), is
The first term on the r.h.s. is a volumic term equal to the opposite of the work done by the Lorentz force on the flow, the second term is the Poynting flux surface term associated with electromagnetic radiation energy-exchanges at the boundaries of the domain under consideration, and the last term quadratic in 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, 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 caseIn 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.. Rescaling and by and by (i.e. now stands for the Alfvén velocity ), and introducing the total pressure , we can write the incompressible momentum equation as
The induction equation is rewritten as
This form separates the physical effects of the electromotive force into two parts: advection/mixing represented by on the left, and induction/stretching represented by 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 (2.1.4)-(2.1.4) must be supplemented with
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).
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 and 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 in polar coordinates (think of an accretion disc or a galaxy). To study the dynamics around a particular cylindrical radius , we can move to a frame of reference rotating at the local angular velocity, , and solve the equations of rotating MHD locally (including Coriolis and centrifugal accelerations) in a Cartesian coordinate system () centered on , neglecting curvature effects (all of this can be derived rigorously). Here, corresponds to the direction of the local angular velocity gradient (the radial direction in an accretion disc), and corresponds to the azimuthal direction. In the rotating frame, the differential rotation around reduces to a simple a linear shear flow , where 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 , and of a pure shear, parametrised by , on dynamos. For instance, we can study dynamos in non-rotating shear flows by setting and varying the shearing rate 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 while setting . Cyclonic rotation regimes, whereby the vorticity of the shear flow is aligned with the rotation vector, have negative in the shearing sheet with our convention, while anticyclonic rotation regimes correspond to positive . In particular, anticyclonic Keplerian rotation typical of accretion discs orbiting around a central mass , , is characterised by in this model.
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 much smaller than the typical radius of curvature of the system. In order to accomodate the linear shear in this numerical problem, the coordinate is usually taken shear-periodicA detailed description of a typical implementation of shear-periodicity in the popular pseudospectral numerical MHD code SNOOPY (Lesur and Longaretti 2007) can be found in App. A of Riols et al. (2013)., the coordinate is taken periodic, and the choice of the boundary conditions in depends on whether some stratification is incorporated in the modelling (if not, periodicity in 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 (2.1.4)-(2.1.4). First, we define the scale of the system under consideration as , and the integral scale of the turbulence, or the scale at which energy is injected into the flow, as . Depending on the problem under consideration, we will have either , or . Turbulent velocity field fluctuations at scale are denoted by . The kinematic Reynolds number
measures the relative magnitude of inertial effects compared to viscous effects on the flow. The Kolmogorov scale is the scale at which kinetic energy is dissipated in Kolmogorov turbulence, with the corresponding typical velocity at that scale. The magnetic Reynolds number
measures the relative magnitude of inductive (and mixing) effects compared to resistive effects in equation (2.1.4), 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
is also a key quantity in dynamo theory. Unlike and , 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 .
One reason for this wide distribution is that is very strongly dependent on both temperature and density. For instance, in a pure, collisional hydrogen plasma with equal ion and electron temperature,
where is in Kelvin, is the Coulomb logarithm and is the particle number density in . This collisional formula gives or larger for the very hot ICM of galaxy clusters (although it is probably not very accurate in this context given the weakly collisional nature of the ICM). The much denser and cooler plasmas in stellar interiors have much lower , for instance ranges approximately from at the base of the solar convection zone to below the photosphere. Accretion-disc plasmas can have all kinds of , 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 , typically or smaller. This has proven a major inconvenience for dynamo experiments, as achieving even moderate in a very low fluid requires a very large 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 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 can in principle be controlled and varied in the range by changing either the temperature or density of the plasma, as illustrated by equation (2.2.2). Finally, due to computing power limitations implying finite numerical resolutions, most virtual MHD fluids of computer simulations have (with a few exceptions at large ). 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 lowish regimes can provide glimpses of the asymptotic dynamics.
The large and small MHD regimes are seemingly very different. To see this, consider first the ordering of the resistive scale , i.e. the typical scale at which the magnetic field gets dissipated in MHD, with respect to the viscous scale .
Large magnetic Prandtl numbers.
For , the resistive cut-off scale 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. This is illustrated in spectral space in Fig. 7.
To estimate 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 . For this kind of turbulence, the smallest viscous eddies are therefore also the fastest at stretching the magnetic field. To estimate the resistive scale , we balance the stretching rate of these eddies with the ohmic diffusion rate at the resistive scale . This gives
Low magnetic Prandtl numbers.
For , we instead expect the resistive scale to fall in the inertial range of the turbulence. This is illustrated in spectral space in Fig. 8.
To estimate in this regime, we simply balance the turnover/stretching rate of the eddies at scale with the magnetic diffusion rate . Equivalently, this can be formulated as . The result is
Intuitively, the large- regime seems much more favourable to dynamos. In particular, the fact that the magnetic field “sees” a lot of turbulent activity at low could create many complications. However, and contrary to what was for instance argued in the early days of dynamo theory by Batchelor (1950), we will see in the next sections that dynamo action is possible at low . Besides, the large- 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
This number measures the ratio between the correlation time and the nonlinear turnover time of an eddy with a typical velocity at scale . 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: anti-dynamo 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 and Soward (2007).
2.3.1 Kinematic versus dynamical regimes
The question of the amplification and further sustainement 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 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 above which magnetic-field generation becomes possible.
In the presence of an externally prescribed velocity field (independent of ), the induction equation (2.1.4) is linear in . 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 of the bifurcation and growing eigenmodes of equation (2.1.4). The velocity field in the kinematic dynamo problem can be computed numerically from the forced Navier-Stokes equation with negligible Lorentz forceOr, in dynamo problems involving thermal convection, the Rayleigh-Benard or anelastic systems including equation (2.1.1)., or using simplified numerical flow models, or prescribed analytically. This linear problem is relevant to the early stages of magnetic-field 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 (2.1.1)-(2.1.1) (or the simpler equations (2.1.4)-(2.1.4) 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 smaller than the kinematic . There is also an important class of dynamical magnetic-field-sustaining MHD processes, referred to as instability-driven dynamos, which do not originate in a linear bifurcation at all, and have no well-defined . 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 (2.1.3), there are actually many generic cases in which magnetic fields can not be sustained by fluid motions in the limit of infinite times, even at large . Two of them are particularly important (and annoying) for the development of theoretical models and experiments: axisymmetric magnetic fields can not be sustained by dynamo action (Cowling’s theorem, 1933), and planar, two-dimensional motions can not 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 . Assume that is an axisymmetric vector field
where is a poloidal flux function and is a toroidal magnetic stream function. Similarly, assume that is axisymmetric with respect to the same axis of symmetry as , i.e.
where is an axisymmetric poloidal velocity field in the plane and is an axisymmetric toroidal differential rotation. The poloidal and toroidal components of equation (2.1.4) respectively read
Equation (2.3.2) has a source term, , which describes the stretching of poloidal field into toroidal field by the differential rotation, and is commonly referred to as the effect in the astrophysical dynamo community (more on this in §4.2.1). However, there is no similar back-coupling between and in equation (2.3.2), 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 in equation (2.3.2). The advection term on the l.h.s. describes the redistribution/mixing of the flux by the axisymmetric poloidal flow in the plane and can only amplify the field locally and transiently. The presence of resistivity on the r.h.s. then implies that must ultimately decay, and therefore so must the source term in equation (2.3.2), and . Overall, the constrained geometry of this system therefore makes it impossible for the magnetic field to be sustainedFrom 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 and Jackson (2004) for a discussion in the dynamo context)..
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 can not be sustained by a simple combination of axisymmetric differential rotation and meridional circulation. Note however that axisymmetric flows like the Dudley and James (1989) flow or von Kármán flows (Marié et al. 2003), on which the designs of several dynamo experiments are based, can excite non-axisymmetric dynamo fields with dominant equatorial dipole geometry ( 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), , can not excite a dynamo. A purely toroidal flow can not excite a dynamo either, and a magnetic field of the form alone can not 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. 2.5D (or 2D-3C) flows of the form with three non-vanishing components (i.e., including a 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 Galloway-Proctor flow (GP, Galloway and Proctor 1992). These flows have relatively simple kinematic dynamo eigenmodes of the form , where , the wavenumber of the magnetic perturbation along the direction, is a parameter of the problem and is the (a priori complex) growth rate of the dynamo for a given . 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- 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 and their growth rate tends to zero as , for instance (but not necessarily) as some inverse power of . For this reason, they are probably not relevant to astrophysical systems with very large and dynamical magnetic timescales much shorter than . A classic example is the Roberts (1970, 1972) dynamo at the core of the first experimental demonstrations of the dynamo effect. Fast dynamos, on the other hand, are dynamos whose growth rate remains finite and become independent of as . Although it is usually very hard to formally prove that a dynamo is fast, most dynamos processes discussed in the next sections are thought to be fast, and so is for instance the previously mentioned Galloway and 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 and Herrero-Bervera (2007).
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 is (inverse turnover times). Only a tiny magnetic diffusivity is required 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 and 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 and Gilbert (1995) and a chapter by Andrew Soward in a collective book of lectures on dynamos edited by Proctor and Gilbert (1994).
3 Small-scale dynamo theory
Dynamos processes exciting magnetic fields at scales smaller than the typical integral or forcing scale 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 driven randomly by an external artificial body force, or through natural hydrodynamic instabilities (e.g. Rayleigh-Bénard 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, 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 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 simulationsPragmatic 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..
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 (2.1.4) at 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 (2.1.4)-(2.1.4) are integrated from there on ( 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 , (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 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 randomly at regular time intervals, and by Moffatt and Saffman (1964) for the simpler case of time-independent linear shear, based on earlier hydrodynamic work by Pearson (1959).
Let us consider the incompressible, kinematic dynamo problem (2.1.4) paired with the simplest possible model of time-evolving, non-uniform and spatially “smooth” incompressible velocity field, a random linear shear with , and further assume that the magnetic field at , , has finite total energy, no singularity and . The evolution of the separation vector connecting two fluid particles is given by
We first analyse the situation considered by Moffatt and Saffman (1964) where C is constant. In an appropriate basis , with . There are two possible ways to stretch and squeeze the magnetic field, namely we can form magnetic ropes if , or magnetic pancakes if . 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 to lead to magnetic amplification as in ideal MHD. However, the perpendicular squeezing implies that even a tiny magnetic diffusion matters. Is growth still possible in that case ? To answer this question, we decompose into shearing Fourier modes
where , the initial lagrangian wavenumbers, can be thought of as labelling each evolving mode, and the different are time-evolving wavenumbers with ( in this context should not be confused with the unit vector, introduced in §2.1.3, defining the orientation of ). Replacing by this expression in the induction equation, we have
for each , and
with at all times. The diffusive part of the evolution goes as
and leads to the super-exponential decay of most Fourier modes because . At any given time , the “surviving” wavenumbers live in an exponentially narrower cone of Fourier space such that
In the rope case, the initial wavenumber of the modes still surviving at time is given by
Accordingly, the initial magnetic field of these surviving modes is
(from the solenoidality condition for ). As the field is stretched along , we then find that the amplitude of the surviving rope modes at time goes as
We can now estimate from equation (3.2) the total magnetic field in physical space. The first term in the integral goes as from equation (3.2), and the wavevector space element as from equation (3.2), so that overall
Hence the magnetic field is stretched and squeezed into a rope that decays pointwise asymptotically. But what about the total magnetic energy
in the volume of fluid ? , but the volume occupied by the field goes as . 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. Regrouping everything, we obtain
The second twiddle equality only applies in 3D. Similar conclusions hold for the pancake case, except that . Overall, we see that the total magnetic energy of magnetic ropes decays in 2D, because 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 and the volume occupied by the magnetic field grows faster than the pointwise decay rate of the field itself (Moffatt and Saffman 1964).
This generates a succession of random area-preserving stretches and squeezes, which can be described in multiplicative matrix form. More precisely, the matrix such that is put in Volterra multiplicative integral form
where I is the unit tensor. From there, the formal solution of the linear induction equation is
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 corresponding to a stretching direction
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 cristallises exponentially fast in time, while the exponents converge as (Goldhirsch, Sulem, and 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 separation of initially nearby fluid trajectories is critical to the dynamo process. The linear shear assumption can be relaxed to accomodate 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 (3.2) through wavenumber couplings/scattering associated with the 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 in 3D. In 2D, and the field must be squeezed as much along as it is stretched along . In that case, decays ensues according to the first twiddle inequality in equation (3.2). 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 planes perpendicular to the most compressive direction , with reversals occurring along at the resistive scale . This is illustrated in Fig. 13.
3.3 Magnetic Prandtl number dependence of small-scale dynamos
3.3.1 Small-scale dynamo fields at
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 , which of course becomes very small at large . The problem described above is therefore implicitly typical of the large- MHD regime introduced in §2.2.2. The fastest shearing eddies at large 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 (random Stokes flow). On the other hand, must be large enough for stretching to win over diffusion. There is therefore 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 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 regime. Snapshots of the smooth velocity field and particularly clean folded magnetic field structures in the relatively asymptotic large- regime , , 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 , including the Meneguzzi et al. experiment, essentially produce a dynamo of the kind described above. Figure 15 provides a map in the plane of the dynamo growth rate of the small-scale dynamo, and a plot of the critical magnetic Reynolds number above which the dynamo is excited ( here and in Fig. 14 and Fig. 16 is defined as ). is found to be almost independent of and approximately equal to 60 for . As 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 in simulations at and .
3.3.2 Small-scale dynamo fields at
What about the 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 (a concise account of Batchelor’s arguments on the small-scale dynamo can be found in Davidson (2013b), § 18.3). As explained in §2.2.2, the magnetic field sees a very different, and much more irregular velocity field in the low- regime, and we would naturally expect this to have a negative impact on the dynamo. Whether the dynamo survived in this regime remained an open and somewhat controversial theoretical and numerical question for many years (Vainshtein 1982; Novikov et al. 1983; Vainshtein and Kichatinov 1986; Rogachevskii and Kleeorin 1997; Vincenzi 2002; Haugen et al. 2004; Schekochihin et al. 2004a; Boldyrev and Cattaneo 2004; Schekochihin et al. 2005b).
The first conclusive numerical demonstrations of kinematic dynamos at low in non-helical 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 the low- dynamo regime is not entirely settled, the main results of these Meneguzzi-like simulations of homogeneous, isotropic turbulence are that the critical of the dynamo increases significantly as becomes smaller than one (see Fig. 15), and that the nature of the low- dynamo is quite different from its large- counterpart. The rightmost snapshot of Fig. 16 shows for instance that the structure of the magnetic field at is radically altered in comparison to even the 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 . Unfortunately, a clear physical understanding of the low- small-scale kinematic dynamo process comparable to that of the large case is still lacking. As we will see in the next paragraph, though, the increase in at low 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- regime is that one must simultaneously ensure that is large enough to trigger the dynamo ( at low appears to be at least a factor two larger than at large depending on how the problem is set-up), and that is significantly larger than ! In practical terms, a resolution of is required to simulate such high 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 and 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 and Plunian 2006; Buchlin 2011) and DNS and large-eddy 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 simulations of turbulent thermal convection, Boussinesq and stratified alike (Nordlund et al. 1992; Cattaneo 1999; Vögler and Schüssler 2007; Pietarila Graham et al. 2010; Moll et al. 2011; Bushby and Favier 2014). Low- 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 significantly smaller than one have still not been conducted.
3.4 Kinematic theory: the Kazantsev model
Would not it 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 -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 non-Gaussian and has 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 turbulent closure problems, and offers a different perspective on the physics of small-scale 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 and Anderson (1992), Vincenzi (2002), *schekochihin02, Boldyrev and Cattaneo (2004), and *tobias11b, 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
We assume that has Gaussian statistics,
where is a normalisation factor and the covariance matrix is related to by
We further assume that is -correlated in time,
where 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 helicity, we are only allowed to use and to construct If the turbulence is made helical (but remains isotropic), equation (3.4.1) must be complemented by an extra term proportional to the fully anti-symmetric Levi-Cevita tensor (see §4.5.2)., and the most general expression that we can form is
where and and are the tangential and longitudinal velocity correlation functions. For an incompressible/solenoidal vector field, we have
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, the magnetic energy spectrum)
For the same reasons as above, we can write
Taking the -th component of the induction equation at point and multiplying it by , then taking the -th component at point and multiplying it by we find, after adding the two results, the evolution equation for ,
where because of statistical homogeneity. Equation (3.4.2) is exact, but we are now faced with an important difficulty: the time-derivative of the second-order magnetic correlation function depends on mixed third-order 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, -correlated-in-time Gaussian velocity field come into play. The assumption of Gaussian statistics implies that th-order mixed correlation functions involving can be expressed in terms of th-order correlation functions using the Furutsu-Novikov (Gaussian integration) formula:
where stands for any functional of , and the are functional derivatives. We can use this formula in equation (3.4.2) 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,
Applying the Gaussian integration formula is a critical first step, but more work is needed. In particular, equation (3.0) 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 (3.4.1) in equation (3.0) removes the time-integral and leaves us with the task of calculating the equal-time functional derivative . This expression can be explicitly calculated using the expression of the formal solution of the induction equation,
Functionally differentiating this equation (and that for ) with respect to introduces and , which makes the space-integral in equation (3.0) 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 (3.4.2) reduces to the much simpler closed scalar equation for ,
can be interpreted as (twice) the “turbulent diffusivity”. If we now perform the change of variables
we find that equation (3.0) reduces to a Schrödinger equation with imaginary time
which describes the evolution of the wave function of a quantum particle of variable mass
in the potential
We can then look for solutions of equation (3.0) of the form
Keeping in mind that stands for the magnetic correlation function, we see that exponentially growing dynamo modes correspond to discrete 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 . The variational result for is