September 15, 2022 YYYY. AA:1–46 https://doi.org/10.1146/((please add article doi)) Copyright © YYYY by Annual Reviews. All rights reserved Galactic Dynamos Axel Brandenburg1,2,3,4 & Evangelia Ntormousi5,6 1Nordita, KTH Royal Institute of Technology and Stockholm University, Hannes Alfvéns väg 12, SE-10691 Stockholm, Sweden 2The Oskar Klein Centre, Department of Astronomy, Stockholm University, AlbaNova, SE-10691 Stockholm, Sweden 3McWilliams Center for Cosmology & Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA 4School of Natural Sciences and Medicine, Ilia State University, 3-5 Cholokashvili Avenue, 0194 Tbilisi, Georgia 5Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy 6Institute of Astrophysics, Foundation for Research and Technology-Hellas, Vasilika Vouton, GR-70013 Heraklion, Greece Keywords Spiral galaxies, Milky Way, large-scale magnetic fields, circumgalactic medium, magnetic Reynolds number Abstract Spiral galaxies, including the Milky Way, have large-scale magnetic fields with significant energy densities. The dominant theory attributes these magnetic fields to a large-scale dynamo. We review the current status of dynamo theory and discuss various numerical simulations designed to explain either particular aspects of the problem or to reproduce galactic magnetic fields globally. Our main conclusions can be summarized as follows. • Idealized direct numerical simulations produce mean magnetic fields, whose saturation energy density tends to decline with increasing magnetic Reynolds number. This is a manifestation of catastrophic quenching, which is still an unsolved problem. • Small-scale dynamos, which create magnetic fields at the resistive scale, are important throughout a galaxy’s life, and probably provide strong seed fields at early stages. • Large-scale galactic magnetic fields of microGauss strengths can probably only be explained if helical magnetic fields of small or moderate length scales can rapidly be ejected or destroyed. • The circumgalactic medium (CGM) may play an important role in driving dynamo action at small and large length scales. These interactions between the galactic disk and the CGM may be a key aspect in understanding galactic dynamos. We expect future research in galactic dynamos to focus on the cosmological history of galaxies and the interaction with the CGM as means of replacing the idealized boundary conditions used in earlier work. 1 Contents 1. INTRODUCTION ................................................................................................ 2 2. DYNAMOS ....................................................................................................... 3 2.1. Historical remarks ........................................................................................... 3 2.2. The need for magnetic diffusivity: a clear case for steady flows......................................... 6 2.3. Turbulence and time-dependence .......................................................................... 8 2.4. Early examples of dynamos................................................................................. 9 2.5. Large-scale dynamos and averaging ....................................................................... 11 2.6. Types of large-scale dynamos .............................................................................. 11 2.7. Small-scale dynamos........................................................................................ 14 3. CATASTROPHIC QUENCHING AND MAGNETIC HELICITY FLUXES .................................... 15 3.1. Catastrophic quenching for uniform magnetic fields...................................................... 15 3.2. Catastrophically slow saturation in closed domains....................................................... 15 3.3. Alleviating catastrophic quenching by magnetic helicity fluxes .......................................... 16 4. TEST-FIELD METHOD AND NONLOCALITY IN SPACE AND TIME .................................... 17 4.1. Parameterization of the mean electromotive force........................................................ 17 4.2. Mean-field coefficients ...................................................................................... 18 4.3. Methods for measuring α and other effects ............................................................... 18 4.4. Using test fields ............................................................................................. 18 4.5. Nonlocality .................................................................................................. 20 5. SETTING THE SCENE FOR DYNAMO ACTION IN REAL GALAXIES .................................... 21 5.1. Possibilities for seed magnetic fields ....................................................................... 21 5.2. Gravitational collapse dynamos ............................................................................ 26 6. GALACTIC MEAN-FIELD DYNAMOS......................................................................... 27 6.1. Global magnetic field structure ............................................................................ 27 6.2. Dynamo models for specific galaxies ...................................................................... 28 6.3. Galactic models with magnetic helicity flux ............................................................... 29 6.4. Rotation Measure signature of mean-field models ........................................................ 30 6.5. Synchrotron emission from mean-field models ............................................................ 31 6.6. E and B polarizations....................................................................................... 31 7. TURBULENCE SIMULATIONS OF GALACTIC DYNAMOS................................................. 32 7.1. Physical parameters of the ISM............................................................................ 32 7.2. Numerical approaches ...................................................................................... 33 7.3. Local dynamo simulations of galaxy portions ............................................................. 35 7.4. Global isolated galaxy simulations ......................................................................... 36 8. INTERACTION WITH THE CGM.............................................................................. 38 8.1. Magnetization of the galaxy by inflows.................................................................... 38 8.2. Magnetization of the CGM by outflows ................................................................... 39 8.3. The impact of the environment on the galactic dynamo................................................. 40 9. CONCLUSIONS .................................................................................................. 40 1. INTRODUCTION Many spiral galaxies have microgauss magnetic fields, so that their magnetic energy densities are comparable to the thermal, kinetic, and cosmic ray energy densities; see Ruzmaikin et al. (1988) for an early textbook on the subject, and Shukurov & Subramanian (2022) for a recent one. Similar magnetic field strengths have also been detected in galaxies at larger redshifts (z '1) (Bernet et al. 2008, Mao et al. 2017). 2 Brandenburg & Ntormousi Galactic magnetic fields often also show large-scale coherence. The first evidence for a global Galactic magnetic field comes from optical polarization (Hiltner 1949, Hall 1949). The existence of magnetic fields was later confirmed for other galaxies using synchrotron emission (Segalovitz et al. 1976), which showed systematic large-scale magnetic fields roughly in the direction of the galactic spiral arms. There has long been a debate about the origin of such magnetic fields: are they primordial or dynamo-generated, or perhaps a combination of the two? Over the past few decades, there has been a shift of attention from a primordial to a dynamo-generated origin. In the meantime, however, there have also been significant developments in dynamo theory, and global numerical simulations are now becoming more realistic. They tend to show that large-scale magnetic fields can be generated by a dynamo, but the amplitudes may be insufficient or the time scales for their generation too long for the simulations presented so far. The broader problem of galactic magnetism that was addressed in Beck et al. (1996), will not be reviewed; we refer here to other excellent reviews by Beck (2001, 2015a) and Han (2017), as well as the recent textbook by Shukurov & Subramanian (2022). Regarding small-scale turbulent dynamos in partially ionized plasmas, we refer to the recent review by Xu & Lazarian (2021). For a recent review of observational magnetic field tracers, we recommend the article by Pattle et al. (2022). In the present review, we focus on galactic dynamos and highlight some of the main developments since the time of the review of Beck et al. (1996). At that time, there were results suggesting that the dynamo effect in mean-field theory is “catastrophically” quenched, i.e., it goes to zero as the magnetic Reynolds number (ReM) becomes large (Vainshtein & Cattaneo 1992, Cattaneo & Hughes 1996). Specifically, the mean-field effect in question is what is referred to as the α effect, which quantifies the component of the mean electromagnetic force in the direction of the mean magnetic field. “Catastrophic” can mean that the coefficient α goes to zero as ReM goes to infinity. More generally, however, it means that the resulting mean (or large-scale) magnetic field cannot be generated at the expected amplitudes or time scales. This led to a major crisis in dynamo theory, questioning the possibility of an α effect dynamo in the nonlinear regime beyond just infinitesimally weak kinematic dynamo-generated magnetic fields. Although there are still unresolved questions in nonlinear dynamo theory today, there have also been major developments in this field: the importance of magnetic helicity fluxes has been recognized, mean-field dynamo coefficients can now be determined from simulations without the restrictions imposed by analytic techniques, and new dynamo mechanisms beyond just the α effect have been identified. At the same time, there has been significant progress in performing realistic three-dimensional magnetohydrodynamic (MHD) simulations of galaxy formation, allowing a new theoretical view of the problem, where the circumgalactic medium (CGM) plays an integral part. All these developments motivate a new review on galactic dynamos. 2. DYNAMOS We begin by discussing historical and theoretical aspects of MHD and dynamos that are of particular importance in connection with the new developments in galactic dynamo research over the past few decades. The presence of a microphysical magnetic diffusivity plays here an important role. We begin with a historical perspective. 2.1. Historical remarks In Section 6 we will discuss the observational signatures of dynamo models in detail. Here, we will only briefly mention some observational results that were significant for the development of www.annualreviews.org • Galactic Dynamos (Sticky Note comment ostriker 14/10/2022, 6:21:02 PM It is helpful to readers also to include references to other relatively recent reviews on magnetic fields in the ISM, including Hennebelle & Inutsuka (2019), and Crutcher (2012)) (Sticky Note comment ostriker 16/10/2022, 3:02:15 AM It would be good to define magnetic helicity in the margin here. It does not seem to have been defined until p. 15, in the middle of a paragraph, but it would be helpful for readers to define it much earlier.) Catastrophic quenching: limits the large-scale field at large ReM. 3 Observational tracers of galactic magnetic fields Dust extinction / emission polarization Elongated dust grains in the interstellar medium tend to align their minor axes with the mean magnetic field direction (Davis & Greenstein 1951). As a result, the light they emit in infrared wavelengths is polarized, with the polarization direction perpendicular to the mean direction of the magnetic field. Only recently has it been possible to trace extragalactic magnetic fields with this method (e.g.,the recent survey by Lopez-Rodriguez et al. 2022). Since starlight emission is unpolarized, we can also measure dust polarization in absorption against stellar sources. If the distances to the stars are known, it is possible to map the magnetic field at different locations along the line-of-sight (e.g., the PASIPHAE survey, Tassis et al. 2018). Synchrotron emission Ultra-relativistic cosmic ray particles emit polarized synchrotron radiation in radio wavelengths as they gyrate around the galactic magnetic field. Synchrotron emission, which yields the plane-of-the-sky magnetic field component in the warm/hot ISM, has been the tracer of choice for studying extragalactic magnetic fields (see, e.g., Beck 2012, for a review). In general, polarization is measured through the Stokes parameters I, Q, and U. Then the angle of the magnetic field with respect to an arbitrarily chosen axis is calculated from the expression 1 U arctanχ = 1. 2 Q p and the polarization fraction p = Q2 ÅU2/I. Faraday Rotation When polarized radiation passes through an ionized magnetized medium, the plane of polarization is rotated by an angle ψ = RM¸2, where λ the observed wavelength and Z D RM = 0.81 neBÒdl 2. 0 2 ¡3 in units of radians/m , where D is the distance to the source in pc, ne is the electron density in cm , and BÒ the line-of-sight magnetic field of the medium in Gauss (Burn 1966, Brentjens & de Bruyn 2005). If we have an estimate of the electron density, then we can Faraday Rotation can yield the line-of-sight magnetic field. dynamo models. For later reference, we provide here a short overview of the existing galactic magnetic field tracers in the text box on “Observational tracers of galactic magnetic fields”. In the late 1970s, synchrotron radiation from spiral galaxies began to reveal the presence of large-scale ordered magnetic fields broadly aligned with the galactic spiral pattern. At the time, an obvious possibility was that these magnetic fields were the result of winding up a pre-existing, large-scale field. This idea goes back to Piddington (1964) and Ôki et al. (1964), and is now commonly referred to as primordial origin of the magnetic field. The resulting magnetic field is then expected to have the form of a bisymmetric spiral (BSS). The BSS form has a characteristic signature in the Faraday Rotation Measure (RM): when we observe an external galaxy almost, but not exactly, face-on, the line-of-sight magnetic field measured through RM probes the azimuthal component of the galactic field – see Figure 1 for a sketch. Using this technique, Tosa & Fujimoto (1978) 4 Brandenburg & Ntormousi Figure 1 Sketch of the rotation measure (RM) signature of a tilted galaxy with a ring, axisymmetric, or bisymmetric magnetic field. Here, i stands for the inclination angle of the observed galaxy with respect to a completely face-on view. Adapted from Tosa & Fujimoto (1978). found evidence for a BSS in M51. In an early review on galactic magnetic fields, Sofue et al. (1986) contrasted this with the axisymmetric spiral (ASS), expected from mean-field dynamo theory. Figure 1 also sketches the expected ASS signature. However, a purely primordial origin of galactic magnetic fields implies that a tremendous amount of winding has occurred over the past 14Gyr. For example, in the solar neighborhood, the angular velocity of the Galaxy is ≈ 30Gyr¡1, i.e., the rotation period is (2¼/30)Gyr ≈ 0.2Gyr. This yields 70 revolutions in 14Gyr, so we would expect the magnetic field to be strongly wound up. Figure 2 gives a quantitative illustration of this wind-up process. It shows color scale images of jB| together with field lines corresponding to the contours of the normal component of the magnetic vector potential, Az (x, y), so that the magnetic field in the plane is given by B = r× (zˆ Az ). To obtain the result shown in Figure 2, we solved the two-dimensional induction equation, which corresponds to an advection–diffusion equation of the form DAz (Sticky Note comment ostriker 14/10/2022, 6:31:26 PM In the caption is would be useful to indicate which is the projected major axis.) (Sticky Note comment ostriker 16/10/2022, 4:51:20 PM It would be helpful to explain the winding in term of *differential* rotation, since it is not rotation itself but shear that is relevant for winding ) = ´r2 Az , 3. Dt where D/Dt = @/@t + u· r. Here, we assumed that u = ­(x, y)$, where $ = (x, y,0) is the cylindrical position vector and Ω = ­0/[1+ ($/$0)2n]1/n is the angular velocity with $0 = 5kpc, n = 3, and ­0 = 40Gyr¡1. This experiment demonstrates the extreme winding of the magnetic field. The turbulent magnetic diffusivity is here 5× 10¡4 kpckms¡1, corresponding to 1.5 × 1023 cm2 s¡1, which is three orders of magnitude below the canonical estimates. We see about six windings at 1Gr with a thirty-fold increase of jBj. This amount of winding is not observed in any galaxy. Mean-field dynamo theory was originally developed in the solar context (Steenbeck & Krause 1969) and is able to predict both axisymmetric and nonaxisymmetric magnetic fields. For ASS and BSS magnetic fields, one sometimes talks about m = 0 and m = 1 magnetic fields, respectively, (Sticky Note comment ostriker 15/10/2022, 7:25:16 PM This choice for Omega may seem puzzling to readers since constant circular velocity at large radius (with Omega = V_c/\pomega) is a better approximation for realistic galaxies than velocity varying as 1/cylindrical radius (with Omega ~ \pomega^2). ) www.annualreviews.org • Galactic Dynamos Dynamos: convert kinetic energy into magnetic energy 6 Figure 2 Snapshots of field lines together with representations of jB| color-coded (in units of its original value) at 0.1 and 1Gyr for thewind-upproblemdescribed in the text. The inset inthe upper left corner shows the field lines at the time 0.01Gr. which means that they vary with azimuth φ like eimφ . Parker (1971) was the first to show that the most easily excited axisymmetric large-scale magnetic fields in oblate bodies such as galaxies have an azimuthal component that is symmetric about the midplane, i.e., the fields have quadrupolar symmetry – in contrast to the dipolar symmetry that is often found in spherical bodies. While dynamo theory can also produce BSS-type fields, it was shown that they are not the most easily excited modes (Elstner et al. 1990, Brandenburg et al. 1990), unless the turbulence is strongly anisotropic or the dynamo is controlled by strongly anisotropic flow structures (Moss et al. 1993). Today, the significance of primordial magnetic fields is still not resolved, and global three- dimensional numerical simulations suggest that both primordial and magnetic fields of astrophysical origin may be present in typical galaxies (e.g., Martin-Alvarez et al. 2021, see Figure 3). 2.2. The need for magnetic diffusivity: a clear case for steady flows The evolution of the magnetic field B is governed by the usual induction equation, @B = r× (U × B − J/¾), 4. @t where U is the velocity and J = r× B/¹0 is the current density with ¹0 being the vacuum permeability. Equation 4 also includes an electric conductivity ¾, because the mean-free path of the electrons in the interstellar medium (of the order of a few thousand astronomical units) is much smaller that the scales under study here. The magnetic diffusivity is then given by η = 1/¾¹0. Dynamos convert kinetic energy into magnetic energy through what is called the dynamo instability. It occurs when the magnetic Reynolds number, ReM, exceeds a certain critical value (see the text box on “Characteristic nondimensional numbers” for a definition of ReM). Here, kf is the typical wavenumber of the flow. A rigorous definition of this instability is only possible for steady flows. Then an eigenvalue problem can be expressed through: B(x,t) = B¸(x)e¸t , where λ is the eigenvalue and B¸(x) the eigenfunction. For steady, mass conserving, compressible flows, Moffatt Brandenburg & Ntormousi Figure 3 Figure 1 from Martin-Alvarez et al. (2021), showing cosmological galaxy models evolved with different magnetic field seeds: primordial, or “injected” on small scales by stellar feedback. In this visualization, the first two rows show models including both primordial and injected fields, while the last two correspond to only primordial, or only injected fields. The last column color-codes the total magnetic energy according to its origin: green for the primordial, red for the injected, and blue for the cross-term field. www.annualreviews.org • Galactic Dynamos Characteristic nondimensional numbers Fluid and magnetic Reynolds numbers and their ratio, the magnetic Prandtl number, are defined as (Sticky Note comment ostriker 16/10/2022, 4:54:31 PM The viscosity nu has not yet been introduced; please define nu within the text box. Also, for the convenience of readers it would be good to state within the text box that (1) eta is the resistivity, and (2) k_f is a characteristic flow wavenumber) Re = urms/ºkf, ReM = urms/´kf, PrM = º/´. 5. Note that the Reynolds numbers are sometimes based on the length scale, 2¼/kf, which leads to about six times larger values. The present definition is commonly used in numerical simulations of turbulence. As a rule of thumb, the number of mesh points needed in a numerical simulation is similar to the value of the Reynolds number. In simulations with partial ionization, the ionization ratio ½i/½n, i.e., the density ratio of ions and neutrals, enters as another nondimensional number. & Proctor (1985) proved that dynamos (i.e., λ > 0) cannot exist for η = 0, i.e., in the strictly ideal case. This does not preclude dynamos in the astrophysically relevant limit η → 0, which are called fast dynamos (Soward 1987), but it is important to stress that the limit η → 0 is quite different from the case η = 0. The case η = 0 is arguably pathological, because without resistivity, there is no Joule heating and the field line topology cannot change. This case is therefore only of academic interest, because with zero dissipation, there can be no turbulence like that observed in real astrophysical systems. In principle, the case η = 0 can be described using Euler potentials, ©(x,t) and ª(x,t), such that B(x,t) = rΦ × rª, where ©(x,t) and ª(x,t) obey (e.g., Rosswog & Price 2007), DΦ DΨ @B Æ= 0 (⇒ = r× (U × B). 6. Dt Dt @t We see that in the special case of 2-D, this equation agrees with the advection–diffusion equation Equation 3, where Φ = Az and Ψ = z has been assumed. In this special case, it is possible to recover the induction equation in the presence of microphysical magnetic diffusion. Dynamos have not been found in this formulation – even for three-dimensional turbulent or Dynamos: have not other flows that allow for dynamo action in the limit η → 0 (Brandenburg 2010a). The method been found with Euler forbids even a very weakly diffusive advection of Φ and ª, which is needed in any numerical simu potentials lation to prevent the formation of infinitely sharp gradients. The tangling of a pre-existing magnetic field for η → 0 can convert kinetic energy into magnetic energy for some period of time, but it is then not through a dynamo instability, and can happen for a purely two-dimensional field, B = B(x, y), as we have seen in Figure 2. The magnetic field is amplified by perpetual stretching, so it continuously develops smaller scale structures. This continuous change in the field structure makes it impossible to describe the evolving field by an eigenfunction of the form B¸(x), which is independent of time. The actual solution B(x,t) to the case η = 0 would continuously develop smaller length scales as time goes on. Thus, even though the growth may still be exponential, the solution cannot be separated into a purely temporal and a purely spatial part. 2.3. Turbulence and time-dependence All astrophysically relevant flows are time-dependent. Turbulent flows can be statistically steady, so one can still determine an eigenvalue problem by averaging over the fluctuations; see Subramanian & Brandenburg (2014) for detailed studies of kinematic dynamos in helical and fractionally helical 8 Brandenburg & Ntormousi turbulence at large magnetic Reynolds numbers. Even in those turbulent time-dependent flows, when eigenvalues and statistical eigenfunctions with certain energy spectra are obtained empirically for finite η by suitable averaging, no dynamos have been found in the case η = 0, when Euler potentials can be used, as discussed above. In practice, we are often also interested in decaying or collapsing turbulent flows. Dynamos may occur in those cases, but they are hard to define rigorously. Nevertheless, amplification suggestive of dynamo action has been reported both for decaying turbulence (Brandenburg et al. 2019) and for turbulent gravitational collapse (Sur et al. 2012, Xu & Lazarian 2020), as will be discussed later in Section 5.2 of this review. (Sticky Note comment ostriker 16/10/2022, 4:57:46 PM Considering the central role of the alpha effect and the necessity for ARAA articles to be accessible to general readers, the article needs a brief section explaining what the alpha effect is, both mathematically and physically. The best place may be here, in between the current sections 2.3 and 2.4. Presently, the Roberts flows are introduced in the text box, and then eq. 9 states without explanation that the mean EMF contains a term linear in B. Novice/general readers will find themselves puzzled by this. ) 2.4. Early examples of dynamos Cowling (1933) formulated an anti-dynamo theorem, concluding that “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.” At that time, there was no hint that the solution to the problem could lie in the third dimension. It was only later that the use of a two-dimensional analysis in the work of Cowling (1933) was understood as not just a simplification, but as a crucial restriction precluding dynamo action. Even after Parker’s discovery (Parker 1955) of what is now called the α effect, it was not generally accepted that dynamos could work even in principle. For example, Chandrasekhar (1956) found that particular flow geometries could prolong the resistive decay time to half a billion years when using the magnetic diffusivity of the Earth’s outer core. He speculated that the Earth’s magnetic field could be explained in that way, rather than by a dynamo. This suggests that the existence of dynamos was far from being widely accepted at that time. The first rigorous examples of dynamos were presented by Herzenberg (1958) and Backus (1958). The former, consisting of two rotors (eddies) with an oblique angle between them, was also realized experimentally (Lowes & Wilkinson 1963). However, the length scale of those magnetic fields was only comparable to that of the rotors. This property could classify the Herzenberg result as a small-scale dynamo. During the kinematic phase, small-scale dynamos produce a field at the resistive scale and can later grow to the scale of turbulent eddies as the dynamo saturates. They do not possess a mean field. However, the Herzenberg dynamo is not a typical example of a small-scale dynamo, because it does not involve turbulence, and laminar dynamos cease to exist for large values of ReM. A typical example would be isotropic homogeneous turbulence, which can generate magnetic fields at the resistive scale, as will be described later. In the early 1970s, Roberts (1972) showed that several two-dimensional, spatially periodic steady flows can exhibit dynamo action. These flows are now called Roberts flows I–IV. They are large-scale dynamos and their properties have been investigated with modern tools (Rheinhardt et al. 2014). The expressions for the four Roberts flows are included in a dedicated text box. Flow I has maximum kinetic helicity with hω · ui= kfhu2i, where angle brackets denote volume averaging. Flow II to has !¢u = 0 pointwise, while flows III and IV have vanishing helicity only on average (hω · ui= 0), but not pointwise. We summarize the essential features of flows I–IV in Table 1. The resulting mean fields for flow I can be interpreted in terms of an α effect. The mean field for flow IV was identified to be due to a negative turbulent magnetic diffusivity (Devlen et al. 2013). The origin of the mean field for flows II and III involves two different effects: turbulent pumping, which acts like an advection velocity without actual material motion, and a memory effect, which means that the electromotive force also involves the mean magnetic field from earlier times. These classifications can be formalized once we define an averaged magnetic field B, which www.annualreviews.org • Galactic Dynamos Small-scale dynamos: generate magnetic fields at the resistive scale Kinetic helicity: hω · ui, where ω = r× u is the vorticity. 9 The four Roberts flows The four Roberts flows are classic examples of large-scale dynamos. They serve as simple benchmarks and highlight the existence of completely different mechanisms. Only the first one corresponds to the classical α effect, which is traditionally believed to operate in galaxies. All four flows have the following x and y components: ux = v0 sink0x cosk0y, uy Æ¡v0 cosk0x sink0 y, 7. but the z components are different for each flow: ⎧ > ⎨ uz = w0 > ⎩ sink0x sink0 y (for flow I), cosk0x cosk0 y (for flow II), (Sticky Note comment ostriker 16/10/2022, 4:59:15 PM Some words of explanation regarding *why* the mean EMF has this form for Flow I would be helpful) 8. (cos2k0x + cos2k0 y)/2 (for flow III), sink0x (for flow IV), where v0, w0, and k0 are constants. Particular solutions are obtained by specifying the magnetic Reynolds number ReM = v0/´k0 and the ratio w0/v0. The magnetic field must always be three-dimensional and varies in the z direction like eikz z , where kz is sometimes chosen such that it maximizes the growth rate. The evolution of B: is governed by the mean electromotive force E ≡ u× b Scale dependence: α and ´t decrease for small scales. Magnetic Reynolds number: ReM = urms/´kf, where kf is the wavenumber of velocity fluctuations 10 can here be an xy planar average, so B = B(z,t) depends just on time and on one spatial coordinate. This defines what we call the fluctuating field b ≡ B − B. For the Roberts flows, there is no mean flow, i.e., u = 0, so the evolution of B is only governed by the mean electromotive force E ≡ u× b, consisting of fluctuations only. In all cases, the mean magnetic field along the z axis, Bk≡ (0,0,Bz ), vanishes. The perpendicular components, B?≡ (Bx ,By ,0) are finite and we only need to focus on the components E?, B?, and J?. For flow I, we have E = ®B − ´t¹0J, 9. which is a prototype of an α effect dynamo. In Eq. 9, ´t is the turbulent magnetic diffusivity, because it adds to the microphysical magnetic diffusivity η to give the total magnetic diffusivity ´T = ´t + ´. For flows II–IV, the situation is more complicated in that α is now a tensor with vanishing diagonal components (flows II and III), or it is zero and ´t is negative and can thus lead to exponential growth (flow IV). For all those flows, it is important to realize that α and ´t are in general scale-dependent, and ´t becomes positive when B(z,t) has high spatial Fourier components, i.e., for mean fields of smaller scale in the z direction. The dependence of E on the mean magnetic field B and its associated mean current density, J = r× B/¹0, will be discussed below. To determine all components of the tensors ®ij and ´i jk in the representation E i = ®ij Bj + ´i jk B j,k with a rank-3 tensor ´i jk , one must solve the equation for the fluctuations in terms of the mean magnetic field. Here, a comma denotes partial differentiation. In Table 1, we indicate the form of E⊥ for each of the four flows. We also indicate the critical values of the magnetic Reynolds number Recrit, above which dynamo action occurs. Here, Recrit is defined with kf = k0, and we have MM fixed w0 = v0 and kz = k0/2 to ensure that dynamos are possible for all four flows. For flows II and III, for example, no dynamos are possible for kz = k0. Brandenburg & Ntormousi Table 1 Robert flows I–IV as simple benchmarks, and their dynamo properties. Flow helicity interpretation E⊥ Recrit M I II III IV yes, and constant pointwise zero α zero only on average zero only on average α effect off-diagonal α tensor with memory effect pumping effect with memory effect negative “turbulent” diffusion ®B⊥ − ´tJ⊥ µ¶ 0 a B⊥ − ´tJ⊥ a 0 µ¶ 0 γ B⊥ − ´tJ?¡γ 0 ¡´tJ⊥ with ´t < 0 on large length scales 1.99 6.86 3.92 4.55 In all cases, kz = k0/2 was used. The values of urms are 0.866 for flows I–III and unity for flow IV. 2.5. Large-scale dynamos and averaging As already alluded to above, an important feature of the four Roberts flow dynamos is that all of them are examples of large-scale dynamos, i.e., one can define an average (here an xy average) under which the magnetic field retains most of its energy and still captures its essential spatiotemporal evolution. The most suitable type of averaging depends on the type of the mean magnetic field that can emerge in certain geometries and in certain parameter regimes; see Gent et al. (2013) for a discussion. For example, in the context of disk galaxies, the azimuthally averaged magnetic field plays an important role. In cylindrical coordinates, ($,Á,z), such a field depends – not necessarily smoothly – on the cylindrical radius $ and the height z above the midplane. This variability can easily lead to a confusing terminology when we want to split the magnetic field into mean and fluctuations, B = BÅb. To avoid the temptation to refer to the non-smoothness of B as fluctuations, we sometimes refer to ordered and random fields instead (Shukurov & Subramanian 2022). An azimuthal average has obviously no azimuthal dependence and cannot describe nonaxisymmetric magnetic fields. On the other hand, in a mean-field model, one can always just assume that B does also depend on Á. A slight problem may then only occur if one wants to make direct comparisons with observations or simulations, because there is no spatial average that describes nonaxisymmetric magnetic fields under which self-consistent mean-field equations can be derived. For Fourier filtering, for example, the product of a mean and a fluctuation vanishes only approximately under averaging if the scales of mean fields and fluctuations are sufficiently different. Regarding the periodic flow patterns in Cartesian coordinates discussed in Section 2.4, it is important to stress that there can be examples where planar xy averaging is not suitable. An example is the Taylor–Green flow, where a one-dimensional average (here a z average) must be taken to demonstrate the existence of a large-scale dynamo due to a negative turbulent magnetic diffusivity (Andrievsky et al. 2015). In that case, the mean field depends on x, y, and t. 2.6. Types of large-scale dynamos Historically, the α effect was the first distinct dynamo effect that was discovered. It emerged in the derivation of mean-field effects in stratified rotating turbulence (Steenbeck et al. 1966), but in its essence, it was already obtained by Parker (1955) using a more phenomenological approach. It is Large-scale dynamos: Create coherent structures on large spatial and temporal scales. The type of averaging: depends on the magnetic field that can emerge in certain geometries and in certain parameter regimes www.annualreviews.org • Galactic Dynamos 11 Table 2 Summary of different types of dynamos. Flow main dynamo effect small-scale fast isotropic turbulence (nonhelical) isotropic turbulence (helical) Roberts flow-I (laminar, helical) Roberts flows-II and III (laminar, nonhelical) Roberts flows-IV Rädler effect with shear Shear–current effect Incoherent ®–shear effect small-scale dynamo ®2 dynamo X time delay neg turb diff Ω × J effect SJ effect fluctuating α effect X X — — — — — X X X — — — X X X Roberts flow I: this is a prototype of an ®2 dynamo 12 ‘Small-scale’ refers to possibility that a small-scale dynamo would operate. ‘Fast’ refers to the possibility that the dynamo works in the limit η → 0, as discussed in Section 2.2. intrinsically connected with the presence of kinetic helicity. Dynamos can work with an α effect alone, in which case one talks about an ®2 dynamo. Astrophysical dynamos often have strong shear, but shear alone cannot produce a dynamo. When shear is complemented by an α effect, one talks about an ®Ω dynamo, or even an ®2Ω dynamo if one wants to emphasize that both α and Ω effects play a role. At the moment, we do not know for sure whether galactic dynamos are of ®Ω type, as is generally believed, or of some other, possibly as yet unknown type of large-scale dynamo. To demonstrate the range of possibilities, we summarize in Table 2 the different large-scale dynamos known so far. Here we also indicate whether a small-scale dynamo might operate and whether the dynamo is expected to be fast, i.e., to grow even for very large values of ReM. This is usually not the case for laminar flows, unless the flow has chaotic streamlines.1 2.6.1. Helical dynamos. The first of the four Roberts flows is maximally helical. It is a prototype of an ®2 dynamo, whereby the two nonvanishing horizontally averaged mean-field components, Bx and By , are being amplified by the α effect. If shear is important, and we have an ®Ω dynamo, the dynamo is often oscillatory and can exhibit traveling wave solutions. In oblate bodies such as galaxies, however, ®Ω dynamos are usually non-oscillatory (Parker 1979, Stix 1975). 2.6.2. Nonhelical large-scale dynamos. There are various examples of large-scale dynamos that do not involve magnetic helicity. Three of the four Roberts flows have clearly demonstrated that large-scale dynamos do not have to be helical and they can even have pointwise zero helicity. Common to all three examples of Roberts flows II–IV is the fact that the two components, Bx and By , evolve independently of each other. In these examples, they have the same growth rate, but there are other flows, such as the Willis flow (Willis 2012), where the growth rates of Bx and By are different and only one of the two components grows. This is unusual and different from conventional dynamos of ®Ω or ®2 type, where the two components have strictly the same growth rate. Mathematically, the coupling of the two mean field components is caused by the cross product in the expression r× (®B) on the right-hand side of the evolution equation for B. In the presence of shear, for example by a mean flow with constant shear S = @Uy /@x, one has @By /@t = SBx + ..., where the ellipsis denotes further terms not relevant to the present discussion. 1The Galloway–Proctor flow is an example of a laminar flow that may be fast. It is a Roberts flow with time- dependent phases in the trigonometric functions, which causes its streamlines to be chaotic. Brandenburg & Ntormousi The reason for the decoupling of the two magnetic field components in some examples is the fact that the dynamo-active terms operate on each field component separately, i.e., @Bx /@t = ¡γ · rBx and @B y /@t = ¡γ · rB y for dynamos where the pumping velocity γ has a memory effect. In its simplest form, a memory effect has an exponential kernel proportional to e¡(t¡t0)/τ for t > t0 , and zero otherwise. Here, t is the current time and t0 the integration variable. In Fourier space, it leads to a factor 1/(1 − i!¿) in Fourier space, where ω is the frequency and τ is the turnover time. When °!τ > (η + ´t)k, dynamo action becomes possible. 2.6.3. Rädler and shear–current effects. Early in the history of mean-field dynamo theory, Rädler found a novel large-scale dynamo effect for rotating, but unstratified bodies, whereby E has an extra term proportional to Ω × J. Here, Ω is a pseudovector pointing along the rotation axis. The azimuthal velocity is then uφ = $ × ­. However, it is easy to see that the Ω × J term in E does 2 not contribute to the generation of mean-field energy proportional to B , because the dot product with J vanishes. Therefore, additional effects are needed to achieve an actual dynamo. Shear is one such effect, which can also generate another large-scale dynamo, similar to the Rädler dynamo: the shear–current effect. Most of the numerical evidence today shows that this effect does not have a favorable sign for dynamo action (Brandenburg et al. 2008a). There is the possibility that this finding would change when the shear-current effect is strongly controlled by the magnetic field from a small-scale dynamo (Squire & Bhattacharjee 2015). While it is true that large-scale magnetic fields can be generated, it is possible that the real reason behind this is actually the incoherent α effect, as will be discussed below; see also Zhou & Blackman (2021) for a detailed assessment of the different possibilities. 2.6.4. Incoherent and shear dynamo effects. There is another important class of large-scale dynamos that may explain the phenomenon of large-scale magnetic field generation in shear flows without helicity. Such nonhelical dynamo action was first found in a more complex shear flow geometry, relevant to the solar tachocline at mid to low latitudes (Brandenburg 2005). In this environment, it was found that large-scale fields can be generated both with and without helicity in the driving of the turbulence. Subsequent studies by Yousef et al. (2008) and Brandenburg et al. (2008a) produced such dynamos in simpler shearing box simulations, but gave different interpretations, which we discuss below. An incoherent α One interpretation involves helicity fluctuations, which lead to what is known as an incoherent α effect and, in conjunction with shear, to large-scale dynamo action (Vishniac & Brandenburg effect: implies negative turbulent magnetic diffusivity 1997). An incoherent α effect leads to a negative turbulent magnetic diffusivity (Kraichnan 1976). In that sense, the incoherent α effect is actually similar to the dynamo effect in Roberts flow IV. Another interpretation is what is sometimes called the shear dynamo. Attempts to interpret this as a mean-field effect amounts to invoking the shear-current effect. The magnetic sheer current effect, by contrast, is based on correlated fluctuations of the magnetic field from a small-scale dynamo, which is assumed to operate in the background. The role of the incoherent α effect in galactic dynamos is uncertain and may have been underestimated in the past. It might be important if the net kinetic helicity above and below the midplane is small. This may well be the case, especially when there is significant interaction with the CGM. Such interactions can generate strong fluctuations of opposite sign in the kinetic helicity, which would cancel out. www.annualreviews.org • Galactic Dynamos 13 Nonmagnetic Kolmogorov turbulence: hardly exists in astrophysics Figure 4 Magnetic (red lines) and kinetic (blue lines) energy spectra during the kinematic (dashed lines) and saturated (solid lines) phases. Adapted from Brandenburg et al. (2022). 2.7. Small-scale dynamos 2.7.1. Early work on the subject. Under fully isotropic conditions and without helicity, dynamo action is still possible—both for large and small values of the magnetic Prandtl number PrM. This was first shown by Kazantsev (1968). Earlier work suggested dynamo action only for PrM > 1, but this turns out to be incorrect; see the detailed discussion by Schekochihin et al. (2004). The existence of small-scale dynamos under isotropic conditions implies that the concept of nonmagnetic Kolmogorov turbulence hardly exists in astrophysics, where the medium is usually always highly conducting. In the early kinematic regime, when the magnetic field is still weak and exponentially growing, its energy spectrum increases with wavenumber k proportional to k3/2 and has a peak at the resistive wavenumber, provided PrM À 1. Kulsrud & Anderson (1992) found that the peak occurs at p a wavenumber kη that depends on the growth rate λ through kη = °/10´. This is demonstrated in Figure 4, where we plot magnetic and kinetic energy spectra during the kinematic and saturated phases. More recent work on small-scale dynamos is numerical, and will be covered in detail in the following sections. 2.7.2. Effect of ambipolar diffusion. Xu & Lazarian (2016) found a strong similarity between the regime of large magnetic Prandtl numbers and the regime of partial ionization. Their results have been confirmed in numerical simulations (Xu et al. 2019). In those simulations, the microphysical magnetic Prandtl number remained undetermined, because no explicit viscosity or magnetic diffusivity were used. Two-fluid direct numerical simulations (Brandenburg 2019) showed that at large magnetic Prandtl number, the kinetic energy spectra for neutrals and ions show different slopes. The energy spectra of ions and neutrals depart from each other only a small scales when k/kν > 1. For larger ambipolar diffusion coupling, the kinetic energy spectra of neutrals decrease further while those of the ions increase slightly. Brandenburg & Ntormousi 14 3. CATASTROPHIC QUENCHING AND MAGNETIC HELICITY FLUXES 3.1. Catastrophic quenching for uniform magnetic fields As long as the magnetic field is weak, the Lorenz force plays no significant role. Many dynamo effects, including those discussed in Section 2.4, can then be fully described by a given velocity field, because the induction equation 4 is linear in B. However, as soon as the velocity field is determined or modified by the magnetic field, the dynamo problem becomes nonlinear. Eventually, the growing effect of the Lorenz force on the flow can limit (or quench) the magnetic field growth. The term catastrophic quenching was coined by Blackman & Field (2000) to denote any type of detrimental ReM dependence of the nonlinear feedback. Ignoring the effect of turbulent diffusion, i.e., the term ¡´t¹0J in Equation 9, Cattaneo & Hughes (1996) found numerically that 2 α ∝ (1ÅReMB /B2 ). This dependence was originally anticipated by Vainshtein & Cattaneo (1992) (Sticky Note comment ostriker 16/10/2022, 5:03:06 PM this assumes EMF = alpha B_0; it would be good to mention this explictly and explain the motivation for this assumption) eq based on earlier analogous results by Cattaneo & Vainshtein (1991) for the suppression of just ´t in two dimensions. Gruzinov & Diamond (1996) explained these results as a consequence of the conservation of magnetic helicity hA· B〉 in 3-D and of hA2〉 in 2-D, respectively. However, the dependence of α on ReM is a peculiar property of the magnetic helicity equation in the presence of a mean magnetic field B0. In that case, the magnetic helicity corresponding to the departure from the imposed field, b, yields, in the steady state, 0 Æh(u × B0) · bi− 2´¹0hj · bi. Using h(u£B0)¢biÆ¡(hu£bi)¢B0 Æ¡®B02, we have α Æ¡´¹0hj¢bi/B02, so α → 0 as η → 0 or ReM !1. Much of the original work on catastrophic quenching adopted periodic domains. This is clearly only of limited value when thinking about galaxies. This result for α in the nonlinear regime was first obtained by Keinigs (1983). However, it is not really relevant in practice, because it assumes that the magnetic field can meaningfully be described by volume averages. This is not the case, because a volume-averaged magnetic field is always constant in a periodic domain. A relevant mean field for this kind of problem can be defined as planar averages, as discussed in Section 2.4. We denote that by overbars. The diffusion term ´tJ can then not be neglected and the relation of Keinigs (1983) can then be written in the form α − ´tkm Æ¡´¹0hj · bi/B02, where 2 km = J · B/B . 3.2. Catastrophically slow saturation in closed domains In reality, even if the restriction to closed or periodic domains is retained, neither α nor ´t are quenched in such a catastrophic fashion (Brandenburg et al. 2008b). Instead, the time scale for reaching ultimate saturation is “catastrophically” prolonged, i.e., the final saturation obeys (Brandenburg 2001) hi 2 ¡2´k2 B ¼hb2i(kf/k1)2 1− e 1(t¡tsat) (for t > tsat), 10. where kf is the typical wavenumber of the turbulence, k1 = 2¼/L is the lowest wavenumber of the cubic domain of size L3, and tsat marks the end of the early kinematic growth phase and the beginning of the slow saturation phase. Let us emphasize once again that in Equation 10, the value of η is the microphysical value, which is extremely small in galaxies. This motivates the characterization as “catastrophically slow”. Equation 10 has been derived based on just the magnetic helicity equation, i.e., no mean field theory was involved; see the derivation in the relevant text box. However, a phenomenological mean field theory can be formulated where the α effect has an extra magnetic contribution related to the magnetic helicity at small scales, which, in turn, is computed based on the large-scale magnetic helicity that is being being produced by the mean-field dynamo under the assumption that the total magnetic helicity is conserved. The α effect itself is then not catastrophically quenched www.annualreviews.org • Galactic Dynamos Catastrophic quenching means: α → 0 as ReM !∞ In a closed domain: final saturation is catastrophically slow 15 Derivation of Equation 10 The slow saturation behavior after t = tsat is governed by magnetic helicity conservation. The uncurled induction equation reads, @A/@t = U × B − ´¹0J − r', where ϕ is the electrostatic potential. The evolution of A· B is then given by ∂ (A· B) = (2(U(£(B() ¢ ( B − r· F , B − 2´¹0J · 11. @t (Sticky Note comment ostriker 15/10/2022, 7:22:42 PM Since it is not done elsewhere, in this box the definition of E in terms of the other variables needs to be written (including a few words explaining this physically)) where F = E × A+ 'B is the magnetic helicity flux density. (Note the analogy with the Poynting flux E × B/¹0 of magnetic energy density.) The equations involving A and F depend on the gauge, i.e., on the form of ', which can be chosen freely. One frequently adopts the Weyl gauge, ϕ = 0. Next, we consider spatial averages B = r£A and J = r£B/¹0, along with the resulting fluctuations, a = A¡A, b = B − B, and j = J − J, so, after averaging, Equation 11 becomes ³´ ∂ (A· B + a· b) Æ¡2´¹0 J · B + j · b − r· F. 12. @t Our analysis concerns only the phase when the small-scale dynamo has already saturated (for t > tsat), so a· b = const and j · b = const. Assuming the field to be helical with negative helicity at small scales and positive at large scales, we have ¹0j · b ¼¡kfb2 and A · B ≈ B2/k1 ≈ ¹0J · B/k12. Inserting this into Equation 15 and performing volume averaging over the periodic domain, indicated by angle brackets, so that the flux divergence term vanishes, one obtains d hB 2 iÆ¡2´k12hB 2 i+ 2´k1kfhb2i, 13. dt the solution of which for hb2i= const is given by Equation 10. (Blackman & Brandenburg 2002), so the magnetic field (in a periodic or closed domain) can still be strong, but only after a resistively long time. A detailed account of these developments is reviewed in Brandenburg & Subramanian (2005a). 3.3. Alleviating catastrophic quenching by magnetic helicity fluxes It has long been anticipated that the catastrophically or resistively slow saturation phase can be sped up by magnetic helicity fluxes. While the idea is simple and convincing, there is as yet no conclusive demonstration that this actually works. The decisive equation is the one for the magnetic helicity of the fluctuating field, a· b. The fluctuating field can be determined from the equation for the mean field, which, under the Weyl gauge, can be written as @A = U × B + E − ´¹0J, 14. @t This expression results in the following equation for the magnetic helicity of the mean magnetic field: ∂ (A· B) Æh2E · Bi− 2´¹0J · B − r· Fm. 15. @t The equation for a· b must also have corresponding E terms, but with h2E · B〉 being replaced by (Sticky Note comment ostriker 16/10/2022, 5:05:07 PM it would be good to remind the reader (right after eq. 14) that \bar \cal E \equiv \bar{u \cross b} ) ¡h2E · Bi, i.e., ∂ (a· b) = ¡2E · B − 2´¹0j · b− r· Ff. @t 16. 16 Brandenburg & Ntormousi Figure 5 ReM dependence of terms on the RHS of the small-scale magnetic helicity equation 16. Note that r· Ff becomes comparable to 2E · B and 2´hj · b〉 only for ReM > 1000. Adapted from Del Sordo et al. (2013). In the steady state, there are three terms on the right hand side, 2E · B,2´¹0j · b, and r· Ff. Simulations by Del Sordo et al. (2013) and Rincon (2021) showed that only at magnetic Reynolds numbers of the order of 1000, the helicity flux divergence begins to become more important than the resistive terms; see Figure 5. Thus, one could expect the saturation behavior to become inde (Sticky Note comment ostriker 15/10/2022, 8:29:13 PM I have recommended adding a short section on the alpha effect between the current sections 2.3 and 2.4. Presumably there the simpler form of eq. 17 (multiplication only) will be given, explaining where this comes from. Then, when the representation in eq. 17 is introduced here, it can be motivated as a generalization. ) 2 pendent of ReM. However, simulations still show that B declines with increasing ReM. This could mean that ReM needs to be much larger than 1000, but probing this regime requires larger simulations. It remains then to be seen if future simulations with different setups can result in situations where ´j · b does become clearly subdominant. (Sticky Note comment ostriker 16/10/2022, 3:45:57 AM It would help readers to provide more of a physical understanding of the role of mangetic helicity flux. Mathematically, it has been demonstrated that if the fluxes integrate to zero, saturation is very slow. But it would be helpful to provide some intuition about why removal of magnetic helicity might hasten the approach to saturation.) 4. TEST-FIELD METHOD AND NONLOCALITY IN SPACE AND TIME 4.1. Parameterization of the mean electromotive force The mean electromotive force, E, in Equation 14 can be expressed in terms of the mean magnetic field as E i = ®ij ∗ Bj + ´i jk ∗ @Bj /@xk , 17. where the asterisks denote a convolution over space and time, and ®ij and ´i jk are integral kernels and xk is the kth component of the spatial coordinate, i.e., @Bj /@xk = Bj,k . For planar averages that depend on just one direction, we can write E i = ®ij Bj ¡´ij Jj , where ®ij and ´ij would each only have four components. For the rest of this review, we restrict ourselves to this simpler case, but refer to the work of Warnecke et al. (2018) for a study in the context of three-dimensional convection in a sphere. Most of the published literature ignores the fact that ®ij and ´ij are integral kernels, and approximate the convolution by a multiplication. This approximation then assumes a local connection between E and the mean fields. Replacing the convolution by a multiplication is often not a good approximation that can easily be avoided. The convolution models the effects of nonlocality in space and/or time can play important roles, as we will see below. www.annualreviews.org • Galactic Dynamos 17 4.2. Mean-field coefficients One of the major advances in mean-field dynamo theory is the development of numerical methods to avoid the limitations imposed by using analytic approaches. This concerns mainly the linearization of the evolution equations for the magnetic and velocity fluctuations in a turbulent flow. To obtain expressions for ®ij and ´i jk , one has to solve the equations for the fluctuations u and b. The evolution equation for b is the most important one and is obtained by subtracting the evolution equation for B from that for B. The equations are nonlinear in the fluctuations. In analytic approaches, those nonlinear terms are often ignored, which is referred to as the second order correlation approximation, but this restriction is no longer required in the numerical evaluations of E ≡ u× b. This approximation is only valid when ReM ¿ 1, or when the correlation time ¿cor is extremely short, i.e., when ¿corurmskf ¿ 1. Neither of the two is relevant to astrophysics, so we will focus here on the nonlinear approach, where no approximation is used. When the linearization is abandoned, most of the changes in the coefficients ®ij and ´i jk are of quantitative nature, especially when the mean field is weak. There are a few examples where new effects emerge (turbulent pumping in the Galloway–Proctor flow, or the effect of helicity on the turbulent magnetic diffusivity, although those effects remain mainly of academic interest; (see Brandenburg 2018, for a review). 4.3. Methods for measuring α and other effects One approach is to use a nonlinear simulation to obtain u and b in the presence of an additional imposed magnetic field. The resulting u× b can be related to B by ignoring ´ij and J. This is referred to as the imposed-field method, but it can only be used when J vanishes, for example when the averages are zero-dimensional, i.e., volume averages. Another approach is to relate u× b to the actual B and J by correlating them to each other and computing ®ij and ´ij as correlation coefficients. This approach has been applied both for the integral kernels in the nonlocal approach (Brandenburg & Sokoloff 2002, Bendre & Subramanian 2022) and the coefficients in the local version (Simard et al. 2016). The reliability of this approach is unclear and it has not yet been verified for the simple examples of the Roberts flow mean-field dynamos discussed in Section 2.4. This method is sometimes referred to as the correlation method. The occurrence of unphysical results with this method can sometimes be alleviated by using the singular value decomposition (Simard et al. 2016). The most reliable method for calculating ®ij and ´ij is the test-field method (TFM), where one solves the equations for the fluctuations numerically for a sufficiently big set of test fields. In the following, we only describe its essence in a few words. A more detailed description can be found in the review of Brandenburg et al. (2010). 4.4. Using test fields The TFM was originally applied by Schrinner et al. (2005, 2007) to determine the dependence of all transport coefficients in a sphere using longitudinal averages. In that case, one has 9 coefficients for ®ij and 18 nonvanishing coefficients for rank three tensor ´i jk in the representation E i = ®ij Bj + ´i jk B j,k . (The nine coefficients ´ij φ do not enter the problem, because φ derivatives of φ averages vanish.) For systems in Cartesian coordinates, planar xy averages are often the most suitable; see Brandenburg (2005) and Brandenburg et al. (2008a,c) for the first applications. The number of relevant coefficients is then four for ®ij , because only i, j = 1,2 are relevant, and four for the rank two tensor ´ij in the representation E i = ®ij Bj ¡´ij ¹0 Jj , because there are only 18 Brandenburg & Ntormousi Figure 6 Dynamo coefficients from supernova-driven turbulence. The top panel shows the diagonal components of ®, the middle panel shows the off-diagonal components, ®ÁR and ¡®RÁ, contributing to the pumping velocity °z = (®ÁR − ®RÁ)/2. The bottom panel shows the diagonal components of ´ij , which are seen to increase away from the midplane. Courtesy Gressel et al. (2008b) two nonvanishing components of Bj,k that can be expressed as the two components of the mean current density with ¹0 Jx Æ¡By,z and ¹0 Jy = Bx,z . In that case, one can use four sinusoidal test fields (sinkz,0,0), (coskz,0,0), aswell as(0,sinkz,0), (0,coskz,0). In Figure 6 we reproduce results from the work of Gressel et al. (2008b), who performed simulations of dynamos from supernova-driven turbulence in a stratified portion of a galactic disk. Using the TFM, they found that ´t increases away from the midplane and that this leads to turbulent pumping toward the midplane, which is given by γ ¼¡(¿/2)r´t. The pumping velocity γ corresponds to off-diagonal components of the α tensor, which they confirmed. In particular, the www.annualreviews.org • Galactic Dynamos 19 Evolution equation for nonlocality in space and time In Fourier space, the spatial and temporal nonlocalities can be combined to a single expression, which reads ˜ ®˜ ij (k,!)B˜j (k,!)− ´˜ ij (k,!)¹0 J˜j (k,!) (Sticky Note comment ostriker 15/10/2022, 8:52:51 PM Please add an explanation that eq. 18 is a model equation that assumes a certain form of alpha_ij and eta_ij, as motivated by the numerical simulations mentioned in section 4.5 (for the spatial nonlocality) ) Ei (k,!) = , 18. 1− i!τ + `2k2 where `2 = a/kf2. Moving the denominator to the left-hand side, the equation becomes ³´ 1 − i!τ + `2k2 E˜i (k,!) = ®˜ ij (k,!)B˜j (k,!)− ´˜ ij (k,!)¹0 J˜j (k,!), 19. which, back in real space, becomes a simple evolution equation with a diffusion term on the right inside: 2r2 τ ` 20. @E i = ®ij Bj + ´ij ¹0 Jj + E i − E i . @t This equation for the electromotive force is still only an approximation, because there are in general also larger powers of ω and k, but it provides a substantial improvement over the local formulations. pumping velocity in the z direction is given by °z = (®ÁR ¡®RÁ)/2, where R denotes the cylindrical radius. 4.5. Nonlocality It was soon realized that the results for ®ij and ´ij always depend on the wavenumber k. This is explained in the box “Dynamos from the memory effect”. For many turbulent flows, it was found that the components of both ®ij and ´ij decline with increasing values of k in a Laplacian fashion approximately like [1+ (ak/kf)2]¡1, where a is found to depend on details of the flow. In this relation, the value of the empirical coefficient a was found to vary between 0.1 and 0.5, depending on the nature of the turbulent flow (Brandenburg et al. 2008c, Rheinhardt & Brandenburg 2012). The significance of nonlocality is that the transport coefficients become effectively quenched when the mean field is of small scale, i.e., smaller than the integral scale of the turbulence. Especially near boundaries, where sharp boundary layers may occur in calculations that ignore nonlocality, the actual field would be smoother. In fact, sharp contrasting structures have been found in earlier galactic dynamo simulations (Moss 1996). Such results would need to be revisited in view of the importance of nonlocality effects. Even more important than spatial nonlocality is temporal nonlocality. It is also referred to as memory effect, because it implies that the electromotive force depends not just on the magnetic field at the current time, but also on the field at earlier times. To leading order, the Fourier- transformed kernel of temporal nonlocality is proportional to (1 − i!¿)¡1, where τ is the turbulent turnover time. Thus, the electromotive force diminishes with increasing frequency !, but there is also a new imaginary component that was absent otherwise. This can lead to new dynamo effects such as that responsible in the dynamos for the Roberts flows II and III. Whether those effects play a role in any turbulent dynamos is unclear, however. Although the memory effect may not be strong enough to produce new dynamo effects in turbulent flows, it can be strong enough to produce significant phase shifts between the generation of 20 Brandenburg & Ntormousi Dynamos from the memory effect We emphasize, again, that dynamos from the memory effect are so far only known to occur for the Roberts flow, so the effect may be special. At this point, however, we can not exclude that it plays a role in galaxies, for example in connection with the strong vertical stratification leading to a pumping effect toward the midplane. With the tools now at hand, it is not easy to explain this effect. The dispersion relation for a problem with turbulent pumping γ and turbulent magnetic diffusion ´t is given by λ Æ¡ik°¡´tk2. Since Reλ < 0, the solution can only decay, but it is oscillating with the frequency ω = Imλ = °. In the presence of a memory effect, γ is replaced by °/(1¡i!¿), where τ is the memory time. Then, λ ¼¡ikγ (1¡i!¿)¡´tk2, and Reλ can be positive if total. This is the case for the Roberts flow discussed in Section 2.4. magnetic field in galactic arm and interarm regions. This has been studied in detail by Shukurov (1998) and Chamandy et al. (2013). Including a memory effect in numerical simulations is, in general, very cumbersome, because it requires storing the full spatial form of the mean field for many earlier times in order to evaluate the convolution integral in Equation 17. In the present case, however, and to leading order, the convolution integral can be converted into an evolution equation for the electromotive force, which is computationally much easier to solve; see Equation 20. This approach was first proposed by Rheinhardt & Brandenburg (2012) and was applied to dynamos in spheres (Brandenburg & Chatterjee 2018). This formalism also reproduces the dynamo effect from time delay for the Roberts flows II and III; see Section 2.4, as was demonstrated by Rheinhardt et al. (2014). This is explained in the box “Dynamos from the memory effect”. 5. SETTING THE SCENE FOR DYNAMO ACTION IN REAL GALAXIES 5.1. Possibilities for seed magnetic fields 5.1.1. The need for sufficiently strong seed magnetic fields. In Section 2.1 we calculated an average of 70 revolutions in the Galaxy’s lifetime. This is not very much, so we have to be concerned about possible effects of the strength and shape of the initial magnetic field. Typical estimates for the growth rate of the Galactic dynamo are of the order of Γ ≈ 2Gyr¡1 (Beck et al. 1996). This means that the mean magnetic field could be amplified by up to 12 orders of magnitude in about 14Gyr. To reach the current level of the mean magnetic field of about 3¹G, we would need a seed mag-Cosmic battery: A netic field of about 10¡18 G. This is just about the level that can be expected from the Biermann mechanism that can generate magnetic field battery mechanism (Rees 1987). The text box “Battery mechanisms” provides some more informa even when there is tion about the Biermann battery and other mechanisms that can generate magnetic fields in an none initially. un-magnetized plasma. Even though the growth rate of a galactic large-scale dynamo may just be large enough for explaining the current level of the mean field of ≈ 3¹G at the present time, it would not be sufficient for explaining large-scale magnetic fields in very young galaxies. In fact, it is very likely that galaxies were hosting strong, large-scale magnetic fields already by redshift z = 1. Kronberg et al. (1992) found evidence of strong magnetic fields in a z = 0.395 galaxy. In a more systematic search, Bernet et al. (2008) found strong RMs in quasar sightlines passing from z ' 1 galaxy haloes. More recently, Mao et al. (2017) estimated a ¹G, kpc-coherent magnetic field in a lensing galaxy at z ' 0.46. To explain smaller-scale magnetic fields at equipartition levels of 3¹G would still require dynamo action, but that may just be a small-scale dynamo. Since the typical dynamo growth rates www.annualreviews.org • Galactic Dynamos 21 Battery mechanisms The Biermann battery. When the density and the temperature gradient in a plasma are misaligned, the electrons move down the pressure gradient, generating an electromotive force that gives rise to a magnetic field. The resulting time derivative of the magnetic vector potential is then @A c @t = rpe qne 21. where ne and pe are the electron number density and pressure, c the speed of light, and q the electron charge. The Durrive battery. Massive stars are surrounded by a region of ionized gas. Durrive & Langer (2015) proposed that an electromotive force should be created by the surplus momentum transferred to the electron after the ionization of an atom. Then, the uncurled induction equation for zero initial magnetic field becomes: @A cc = rpe − p˙e 22. @t qne qne where p˙e is the rate of momentum transfer to the electrons and Equation 22 also includes the Biermann battery. In the early Universe, the Biermann battery can appear around rippled shocks, while both battery mechanisms should operate around ionization fronts (Subramanian et al. 1994, Kulsrud et al. 1997, Gnedin et al. 2000). scale with the turbulence turnover time, which is shorter at small scales, a small-scale dynamo is a viable seed-field mechanism, as we discuss next. 5.1.2. Small-scale dynamos as a seed for the large-scale dynamo. All dynamos require seed magnetic fields — even the small-scale one. However, a small-scale dynamo grows much faster than a large-scale one. It would therefore be able to produce equipartition strength magnetic fields from much weaker magnetic fields. The idea has been discussed by Beck et al. (1994). Another related idea is to produce galactic seed magnetic fields in the first stars. The simplest form of this idea is that stars could pick up a Biermann seed, which would then be amplified though a stellar dynamo and get ejected with the supernova explosion at the end of the star’s life. This scenario has been explored in many numerical cosmological models (Beck et al. 2013, Katz et al. 2019, Vazza et al. 2017, Martin-Alvarez et al. 2021), and shows that it can, in fact, magnetize galaxies very efficiently. However, these simulations use unrealistically high values for the supernova-injected magnetic field. Those fields could then well be large-scale ones, i.e., on the scale of stars, but those would grow on an even shorter time scale, because stars are much smaller than the envisaged turbulent eddies in the interstellar medium. Young stars and also their surrounding accretion disks can host powerful dynamos that also drive magnetized winds; see the estimates in Brandenburg (2000) and corresponding mean-field simulations in von Rekowski et al. (2003). Those winds could magnetize the surrounding interstellar medium and could well produce much more efficient seeds for the galactic dynamo than any battery mechanism. The wind-based injection model would be a viable alternative to the rather uncertain supernova seeding often used in cosmological simulations. 22 Brandenburg & Ntormousi 5.1.3. Battery versus plasma instabilities. When the electron distribution is anisotropic and the magnetic field is not too weak, the Weibel instability can amplify the magnetic field, Typically, the Weibel instability generates very small-scale fields. Nevertheless, it could play a role in an intermediate regime when the Biermann battery has generated a sufficiently strong magnetic field. Interestingly, this is also in an agreement with recent laser plasma experiments that have been able to access a regime relevant to astrophysical dynamos (Schoeffler et al. 2016). 5.1.4. Primordial seed magnetic fields. In the early Universe, inflation and phase transitions, such as the decoupling of the weak and electromagnetic forces, or the formation of hadrons from quarks may have produced hydromagnetic turbulence (Widrow 2002). Owing to the lack of further energy input, any magnetic field generated at that time would be slowly decaying. The dilution of the magnetic field due to the expansion of the Universe is always scaled out by talking about the comoving magnetic field, which is B˜ = a2B, where a is the scale factor of the Universe. When time is be- R ing replaced by conformal time, t˜ = dt/a(t), the MHD equations have their usual form without expansion factors (Brandenburg et al. 1996). In the following, the tildes are therefore dropped. Not much is known about the strength of the comoving magnetic field today. There are only constraints. Upper limits can be derived from Big Bang nucleosynthesis (BBN) constraints; see Grasso & Rubinstein (1995) and Kahniashvili et al. (2022) for recent work taking into account the decay of the magnetic field between the moment of generation and the time of BBN. A lower limit on the present-day magnetic field strength has been proposed on the grounds that magnetic fields would prevent the reconnection of pair-created electrons and positrons when the TeV photons from powerful blazars interact with the extragalactic background light (e.g., Neronov & Vovk 2010). It should be noted, however, that the validity of this technique may have a systematic uncertainty in that plasma instabilities could potentially also provide an explanation for the non-observation of GeV halos (Broderick et al. 2012). Simulations by Sironi & Giannios (2014), confirm that plasma instabilities do indeed operate, but they can only account for about 90% of the loss of GeV photons and the suppression of the remaining 10% would still need to be explained by the presence of magnetic fields. Similar conclusions were reached by Alves Batista et al. (2019), who performed more detailed simulations for individual blazars. The lower limits derived by Neronov & Vovk (2010), which become less stringent for larger length scales, provide an exciting motivation for primordial magnetogenesis scenarios. At the present time, those primordial magnetic fields may have strengths in the range of 10¡16 G to 10¡9G (see Subramanian 2016, for a review), and could act as seed magnetic fields for any subsequent dynamo processes – once sufficient kinetic energy becomes available. Figure 7 shows the expected magnetic field ranges as a function of the typical scale ¸B of the magnetic field. Here, the white solid lines describe the decay of a helical magnetic field (B2 ∼ t¡2/3) along with the increase of its typical length scale (¸B ∼ t2/3), so that B2¸B = const. For a nonhelical magnetic field, there are still magnetic helicity fluctuations, which constrain the decay such that the correlation integral of the local magnetic helicity is conserved (Hosking & Schekochihin 2022). This leads to a decay with B2 ∼ t¡10/9 and ¸B ∼ t4/9, so that B4¸5 B = const. The first stars are expected to form about 108 yr after the Big Bang, marking the beginning of the reionization epoch. The first galaxies should form at about 109 yr after the Big Bang. Since dynamo action is fastest at small length scales, the magnetic field generation during the formation of the first collapsing structures is potentially important and may have produced a stronger seed magnetic field for the subsequent global galactic dynamo. Strong magnetic fields may also affect galaxy and large-scale structure formation of the Universe (Kahniashvili et al. 2013). www.annualreviews.org • Galactic Dynamos Weibel instability: Occurs in a nearly homogeneous plasma when there is an anisotropy in velocity space (Weibel 1959). 23 Figure 7 Summary of lower and upper magnetic field limits as a function of correlation length. Only the narrowly hashed region indicates permissible a few strength. Courtesy Korochkin et al. (2021), in which we have added the prediction of Hosking & Schekochihin (2022) B ∝ ¸¡5/4. The asterisk shows the scale where Hosking & B Schekochihin (2022) stop the line in their work, since they assumed that the relevant time scale is determined by magnetic reconnection and not by the Alfvén time. 5.1.5. Primordial fields during structure formation. Modern numerical simulations of cosmological large-structure formation are taking into account the evolution of the magnetic field, seeded by the various mechanisms outlined above. A central question in these studies is whether the topology and strength of these primordial fields leave measurable signatures on the cluster or galaxy structures. Vazza et al. (2017) performed a comprehensive suite of cosmological simulations using different magnetogenesis mechanisms: a uniform seed, meant to simulate the magnetic field created by inflation, a seed that follows the distribution of density perturbations to approximate the magnetic field generation by a Biermann battery, a seed that approximates the turbulent dynamo amplification, and an astrophysical seed that simulates the injection of magnetic fields by stellar sources. They find that, at z = 0, all mechanisms agree on the cluster magnetization (which they were designed to reproduce). However, there are large differences in the magnetic field structure both on galaxy scales and in the voids. More recently, Mtchedlidze et al. (2022) explored different initial topologies for the primordial magnetic field, including uniform and scale-invariant inflationary 24 Brandenburg & Ntormousi (Sticky Note comment ostriker 15/10/2022, 9:24:49 PM a higher quality version of this figure is needed if it is going to appear in the article) Figure 8 Faraday rotation maps from four cosmological simulations of Mtchedlidze et al. (2022) using inflationary uniform and scale-invariant fields (upper and lower left), as well as phase-transitional helical and nonhelical initial fields (upper and lower right). Courtesy of Mtchedlidze et al. (2022). fields, as well as helical and non-helical fields from the radiation-dominated epoch. They also reported that the final magnetic field distribution retained a memory of the initial seed. This can be seen from Figure 8, where we show maps of the Faraday rotation at the present time starting at a redshift of z = 60 with any one of the four initial conditions discussed above. The above works use a uniform spatial resolution, which offers the advantage of an unbiased view of cosmological magnetic field evolution. However, models with adaptive resolution can give a more detailed view of the magnetic field on galaxy scales, while following their cosmological history. One recent example is the work of Garaldi et al. (2021), who explored the evolution of cosmological volumes and zoom-ins using four different mechanisms for magnetic field generation: primordial, Biermann battery, Durrive battery, and stellar seeds. Contrary to the findings of the uniform-resolution, large-volume works mentioned above, they report that the initial conditions are forgotten by redshift z ∼ 2. However, none of their initial conditions contained magnetic helicity, which should not have decayed. Marinacci & Vogelsberger (2016) and Martin-Alvarez et al. (2020) focused on the effects of cosmological magnetic fields on galaxy formation. They found that fewer, smaller galaxies form for stronger primordial fields. A very interesting approach was employed by Martin-Alvarez et al. (2021), who traced the evolution of the primordial field and the field injected by stellar sources (Sticky Note comment ostriker 15/10/2022, 9:28:04 PM should refer to Fig 3 here with the discussion of Martin-Alvarez et al) www.annualreviews.org • Galactic Dynamos 25 separately. They achieved this by adding two tracer induction equations to the code, one for each seed. These induction equations are not connected to the gas evolution, but only follow the evolution of the two fields as it would be if they were independent of each other. They found that their evolved galaxies contain a mixture of both: metal-poor gas at the galaxy’s outskirts containing mostly primordial fields with large-scale coherence and SN-enriched gas containing mostly fields of stellar origin, coherent on small scales. The cold, star-forming gas contains a mixture of the two. However, in agreement with the Garaldi et al. (2021) result, the origin of the galactic magnetic field becomes practically indistinguishable very early on without the tracers. All these simulations result in microgauss magnetic fields, but their length scales are typically too small to explain the fields seen in actual galaxies. The results of these comprehensive simulations point to a complex picture in which various seeding mechanisms combine to give the initial and boundary conditions for dynamos on different scales and different epochs. Interestingly, they also point to cluster scales – rather than galaxy scales for an answer regarding the origin of cosmic magnetic fields. 5.1.6. Possible importance of cluster mergers. Mergers of galaxy clusters could amplify large- scale magnetic fields quickly to near-equipartition strengths. The merger itself could stretch a preexisting field and amplify it in conjunction with the existing (possibly helical) background turbulence. One could then think of this as some kind of ®Ω dynamo, where the Ω effect is associated with the large-scale shear generated during the merger. Such simulations were produced by Roettiger et al. (1999). The study of the relevance of cluster mergers to dynamos has not been followed up in recent years. In the meantime, there have been many relevant advances in dynamo theory in connection with time dependence of the flow and in the context of measuring field transport coefficients. In view of these advances, this approach might deserve more detailed follow-up studies in the future. However, there are similarities with recent studies of gravitational collapse dynamos that will be discussed next. 5.2. Gravitational collapse dynamos By the time the first gravitationally bound structures (stars or galaxies) formed, any primordial turbulence from the processes we mention in Section 5.1.4 had probably already decayed. However, the assembly onto these first structures certainly generated large amounts of turbulent kinetic energy, which could have triggered dynamo action. A number of numerical works show that the formation of the first stars is an ideal laboratory for amplifying nG (Sur et al. 2010, 2012, Federrath et al. 2011), or even 10¡20 G (Schober et al. 2012) fields to equipartition values through a small-scale dynamo. Gravitational compression can amplify the field further, although in the presence of turbulence, the resulting dependence of the magnetic field on the density is weaker than the prediction of ideal flux freezing (Sur et al. 2012). While flux freezing predicts jBj∝ ½2/3, the theory of Xu & Lazarian (2020), which includes the turbulent dynamo, predicts that jBj/½2/3 ∝ ½¡2®/57¡1/6 ≈ ½¡0.13, where α Æ¡1 was assumed for the scaling coefficient of their small-scale dynamo. This remarkable agreement with the theory is taken to be suggestive of the importance of reconnection diffusion and the breakdown of flux-freezing (Xu & Lazarian 2020), which makes compressional amplification less efficient. However, the present results still leave some questions unanswered. For example, is it possi ble to explain the slow-down of compressional field amplification even in the absence of dynamo action? Although this may be an academic question, it could be answered by performing collapse 26 Brandenburg & Ntormousi simulations in two dimensions, when dynamo action is impossible. Also, it would be interesting to see the early magnetic field growth starting from a much smaller initial magnetic field. In fact, the theory of Xu & Lazarian (2020) does not really address the question of when the dynamo is excited, but discusses instead the nonlinear regime of a supercritical dynamo. 5.2.1. Nature of collapse dynamos. An important tool in characterizing dynamo action in time- dependent flows such as decaying turbulence or gravitational collapse, is to compare the work done against the Lorentz force with the Joule dissipation rate, and to look at different contributions to the Lorentz work term. These work and dissipation terms emerge when deriving the evolution equation for the magnetic energy density. Taking the dot product of Equation 4 with B, averaging, and ignoring surface terms, we obtain d hB2/2¹0iÆhJ · (U × B)i− ´¹0hJ2i. 23. dt Using J · (U × B) Æ¡U · (J × B), the first term on the right-hand side of Equation 23 can be written as the work against the Lorentz force. Two further refinements can then be employed (Brandenburg & Ntormousi 2022). First, one can decompose (U × B)i Æ¡Uj @j Ai + Uj @iAj to write WL = W 2D + W 3D , where W 2D ÆhJiUj @j Ai 〉 and W 3D = ¡hJiUj @iAj i. The second term, W 3D, LLLL L vanishes for two-dimensional magnetic fields oriented in the plane and describes therefore the work term associated with two-dimensional compression, stretching, compression, and bending. Second, one can decompose J × B into contributions from the magnetic pressure force, the tension force, and curvature force. The corresponding work terms are then referred to as WLc (for compression), W k (for tension force, i.e., along the field), and W ⊥ (for the curvature force, i.e., LL perpendicular to the field). To determine the reality and nature of dynamo action during a turbulent self-gravitational collapse more carefully, Brandenburg & Ntormousi (2022) computed the aforementioned terms that enter the magnetic energy balance. The basic conclusion is that there is indeed dynamo action during the early phase of the collapse while the initial turbulence is slowly decaying, but that dynamo action diminishes when the flow becomes dominated by three-dimensional compression toward the various collapsing potential minima, where only the irrotational flow component gains in strength, which, however, does not (or not much) contribute to dynamo action in their simulations. In Figure 9 we show a visualization of the collapsing magnetic field, the diminishing of the vorticity, expressed here as a wavenumber kω = !rms/urms, the gain of compressive motions, here expressed through kr¢u = (r· u)rms/urms, along with several other quantities, kpr¢u = ¡hpr· ui/p0urms 2 and k!¢u = jh!¢uij/u , characterizing the work done by compression and the amount of kinetic rms helicity, respectively. A potential problem with the simulations of Brandenburg & Ntormousi (2022) is the relatively short collapse time compared with the turnover time of the turbulence. 6. GALACTIC MEAN-FIELD DYNAMOS 6.1. Global magnetic field structure 6.1.1. Early analytic approaches. The idea that the large-scale magnetic field of galaxies could be explained through an ®Ω dynamo was formulated early on (Vainshtein & Ruzmaikin 1971, Parker 1971), just after the first successful mean-field models were proposed for the Sun and the Earth. An important early result was the finding that the most preferred magnetic field mode in flat geometries like galaxies is quadrupolar, i.e., the toroidal field is even about the midplane. (Here and elsewhere, quadrupolar means not just a quadrupole, but all modes of even symmetry about the www.annualreviews.org • Galactic Dynamos 27 (Sticky Note comment ostriker 16/10/2022, 2:24:57 AM A higher qualtity version of this figure is needed ) Dynamo number: Defined as D = Rω R®, where Rω and Rα are mean-field magnetic Reynolds numbers associated with differential rotation and ®-effect, respectively (Stix 1975). 28 Figure 9 Left: Visualization of the magnetic field. Right: Characteristic wavenumbers kω (blue), k!¢u (red), kr¢u (solid black), and kpr¢u (dotted black). midplane.) In view of many early claimed discoveries of BSS fields (for a review, see Sofue et al. 1986), an important question in those early days concerned the possibility of preferred nonaxisymmetric magnetic fields. Such models were never found. However, when the assumption of what is known as the pure ®Ω approximation was made, i.e., the toroidal field is only generated by differential rotation (Ω effect) and the α effect is neglected in the generation of the toroidal magnetic field, spurious nonaxisymmetric modes where found. This approximation turned out to be not permissible, when the magnetic field is nonaxisymmetric. 6.1.2. Boundary conditions. Standard dynamo problems are usually formulated with vacuum boundary conditions, i.e., the magnetic field is current-free and extends to infinity outside the domain (Krause & Rädler 1980). However, such boundary conditions can only be formulated for spheres or ellipsoids, but not for cylinders, for example. Stix (1975) employed ellipsoidal coordinates and obtained an axisymmetric solution. Contrary to Parker (1971), he found that oscillatory solutions occur only at substantially larger dynamo numbers. Unfortunately, the implementation of ellipsoidal coordinates in a numerical code is rather cumbersome. This led to the approach of embedding the galaxy in a sufficiently large poorly conducting halo, which itself is then contained either in a cylinder with perfectly conducting boundaries (Elstner et al. 1990), or in a sphere with vacuum boundaries (Brandenburg et al. 1990). These two alternatives are rather different from each other, but the hope is that those boundaries are far enough away from the physical ones that those differences are without consequence. 6.2. Dynamo models for specific galaxies Various attempts have been made to produce dynamo models of individual galaxies. One such example is M31, i.e., the Andromeda nebula. Its magnetic field is often described as a ring field. It is also often regarded as an analog of the Milky Way. Corresponding models have been presented by Poezd et al. (1993) using nonlinear α quenching. An important challenge here is to reproduce Brandenburg & Ntormousi Figure 10 Left: Evolution of the energy of the mean magnetic field in a model with a shear-induced magnetic helicity flux for different values of ReM (here denoted by Rm). Adapted from Brandenburg & Subramanian (2005b). Right: Magnetic field evolution in models with advective magnetic helicity fluxes for ReM = 105 and different values strength of advection. The advection velocity is characterized by the parameters CU . Courtesy of Shukurov et al. (2006). the right pitch angle of the magnetic field. Another interesting case is M81, whose magnetic field is known to be predominantly nonaxisymmetric. This was difficult to explain. Moss et al. (1993) showed however that such a field could result from an initial magnetic field that might have survived for long enough times, at least in the outer parts of the galaxy. Yet another very different case is NGC 6946, whose field may consist of what is called magnetic arms. Those are structures that are interlaced with the stellar arms, but could also be phase shifted relative to them; see Shukurov (1998) and Chamandy et al. (2015) for a more detailed discussion and references therein. 6.3. Galactic models with magnetic helicity flux In Section 3.3 we have discussed the potential importance of magnetic helicity fluxes. It is often believed that they would be required to explain strong magnetic fields in galaxies. Here, we demonstrate the effect of magnetic helicity fluxes in specific models. Shukurov et al. (2006) have presented nonlinear models in a one-dimensional geometry using the flux associated with a galactic fountain flow. The main nonlinearity was here given through the dynamical quenching formalism with advective magnetic helicity fluxes included. The authors found that a magnetic helicity flux does indeed lead to larger magnetic field amplitudes provided the magnetic helicity flux is strong enough. In their model, the helicity flux was accomplished through a galactic fountain flow with a speed of atleast 300ms¡1. More detailed studies have been performed by Prasad & Mangalam (2016). There is some uncertainty regarding the main contributors to the magnetic helicity flux; see the more recent study by Vishniac & Shapovalov (2014). In addition to advection, shear could modify the turbulent correlations in such a way as to transport magnetic helicity efficiently outward. This was first proposed by Vishniac & Cho (2001) and can lead to episodic magnetic field amplification, especially as ReM is increased; see Brandenburg & Subramanian (2005b) and the left-hand side of Figure 10. On the right-hand side of Figure 10, we reproduce the simulation result of Shukurov et al. (2006) with an advective magnetic helicity flux. In the models with insufficient advective flux, the magnetic energy decreases to very small values. The earlier simulations (by Brandenburg & Subramanian 2005b, their Figure 7) with somewhat smaller values of the magnetic Reynolds number www.annualreviews.org • Galactic Dynamos 29 Figure 11 Full-sky view of the Galactic magnetic field in (A) RM of extragalactic sources (Hutschenreuter et al. 2022) and (B) synchrotron emission (Bennett et al. 2013). showed that the magnetic field can recover after some time, but then, again, it begins to fall off, just like what is seen in Figure 10. 6.4. Rotation Measure signature of mean-field models In Section 2.1 we mentioned the historical importance of RM studies for distinguishing between an ASS, characteristic of dynamo models, and a BSS, characteristic of a wound-up primordial fields. The varied findings of RM studies indicate that galactic magnetic field evolution might be more complex than this simple dichotomy. At the time of the review of Sofue et al. (1986), the majority of galaxies were thought to be of BSS type; the authors listed seven out of 11 galaxies as having a BSS field. However, more accurate subsequent surveys confirmed a predominantly BSS type structure only for M81 (Krause et al. 1989). In a later review, Beck et al. (1996) listed the field structures for 33 galaxies. The picture became more complicated, with four examples primarily of ASS fields (albeit two where marked as uncertain). The dominance of ASS over BSS continues to persist even today. For M33, Tabatabaei et al. (2008) found an axisymmetric field in the inner regions and a superposition of axisymmetric and bisymmetric fields in the outer regions. Beck (2015b) observed a weak (0.5¹G) axisymmetric field in IC 342, and Beck et al. (2020) found a dominating ASS field in M31 combined with a six-times weaker BSS component. Classifying the Milky Way’s magnetic field structure is much harder. The existing parametric models of the Galactic magnetic field are largely based on full-sky RM maps of extragalactic sources (e.g., Oppermann et al. 2015, Hutschenreuter et al. 2022) and synchrotron emission maps of the Milky Way (mainly from WMAP; Page et al. 2007, Bennett et al. 2013, – see Figure 11). The model of Sun et al. (2008) assumes an axisymmetric spiral with a reversal in the inner 5 kpc. Jansson & Farrar (2012) include magnetic spiral arms and an X-shaped field in the halo (see also the following section for the observational motivation). Jaffe et al. (2010) also fit magnetic spiral arms to the disk data, also including the random magnetic field component. Terral & Ferrière (2017), using analytic forms for the three-dimensional field, conclude that a bisymmetric (m1) halo field best fits the RM data. However, West et al. (2020) found evidence for an axisymmetric (m = 0) quadrupolar magnetic field with a small net vertical component in RM. A newer analysis (Dickey 2022) shows that a combination of an axisymmetric and a bisymmetric mode, based on analytical galactic dynamo models from Henriksen et al. (2018), best explains the large-scale morphology of the Galactic RM 30 Brandenburg & Ntormousi data. As the quality of the observational data improves, a complex picture of galactic magnetic field morphology emerges, including that of the Milky Way. 6.5. Synchrotron emission from mean-field models Synchrotron emission provides an important means of measuring the magnetic field in galaxies and comparing models with simulations (see Beck et al. 2019, for a review on bridging dynamo models and observations). Early attempts of computing the polarized synchrotron emission from models were presented by Donner & Brandenburg (1990), who computed the linearly polarized emission from a galactic mean-field models, which contained both axisymmetric and nonaxisymmetric magnetic fields. They confirmed the idea of distinguishing these modes by measuring the RM along a ring around the galaxy. Another example was the computation of linear polarization from the magnetic field on both sides of the midplane in edge-on galaxies. In galaxies seen edge-on, synchrotron emission reveals X-shaped halo magnetic fields (e.g., Golla & Hummel 1994, Tüllmann et al. 2000, Krause et al. 2006, Stein et al. 2019, Krause et al. 2020). Although the three-dimensional morphology of these fields is unknown, the X-shaped signature can be reproduced by dynamo models that include an outflow (Brandenburg et al. 1993, Elstner et al. 1995). The resulting magnetic fields have a quadrupole symmetry both in the disk and the halo. An alternative would be a dynamo in the halo itself, which could produce dominant dipole modes (Brandenburg et al. 1992, Moss & Sokoloff 2008). The synchrotron intensity gives an indication about the magnetic field strength. It is proportional to the product of the density of relativistic cosmic ray electrons and a power close to 2 of the local magnetic field component perpendicular to the line of sight. However, the relativistic cosmic ray electron density may itself depend on the local magnetic energy density, because cosmic rays and magnetic fields have supernova explosions as a common source of energy. These arguments have been reviewed by Seta & Beck (2019), who also compare with numerical simulations of cosmic ray confinement by a local dynamo-generated magnetic field, similar to what was done earlier by Snodin et al. (2006). Seta & Beck (2019) conclude that the commonly made assumption of an equipartition between cosmic ray and magnetic energy densities is not valid on scales smaller than at least 100pc. They argue that earlier estimates based on the equipartition assumption may have overestimated the actual magnetic field by a factor of about 1.5. An interesting comparison between radio synchrotron and dust polarization in emission can be found in Borlaff et al. (2021) for the M51 galaxy. They find that the magnetic pitch angles of the two tracers differ, with the dust polarization showing a more tightly wound spiral than the radio. In light of this and forthcoming comparisons, predictions from dynamo models should take into account the multi-phase nature of the ISM. 6.6. E and B polarizations The linear polarization described by Stokes Q and U can also be expressed in terms of the rotationally invariant parity-even E polarization and the parity-odd B polarization, as is commonly done in cosmology (Seljak & Zaldarriaga 1997, Kamionkowski et al. 1997). Here, the symbols E and B have nothing to do with electric and magnetic fields, except that both can qualitatively be described by as gradient-like and curl-like fields. It is important to stress, however, that E and B are only defined on a two-dimensional surface. Therefore, the parity-odd B polarization has no immediate correspondence with the helicity of the underlying magnetic field. www.annualreviews.org • Galactic Dynamos 31 Figure 12 Left: Galactic B mode polarization. Right: longitudinally averaged B mode polarization. Here, θ and φ are Galactic colatitude (= 90±− latitude) and longitude. Mathematically, E and B are obtained as the real and imaginary parts of a quantity R(µ,Á) with N`` XX R ≡ E + iB = R˜ `mY`m (µ,Á). 24. `Æ2 mÆ¡` and R˜ `m are coefficients that have been computed as Z R˜ `m = (Q + iU)2Y ∗ (µ,Á) sinθ dθ dÁ, 25. `m 4π with 2Y`m(µ,Á) being the spin-2 spherical harmonics and the asterisk denoting the complex conjugate. Brandenburg & Brüggen (2020) found that the B polarization averaged over Galactic longitude is very small owing to longitudinal cancelation, but there is a small hemispheric antisymmetry. This is shown in Figure 12, where we plot the Galactic B mode polarization together with the longitudinally averaged B mode polarization. It may be tempting to associate this hemispheric dependence with that anticipated for the α effect, which is also a parity-odd quality with hemispheric same change. However, it turned out that the observed hemispheric antisymmetry is actually explained by the spiral nature of the magnetic field. Looking toward northern and southern galactic latitudes yields mirror images of each other, which explains the observed hemispheric antisymmetry of the mean B. 7. TURBULENCE SIMULATIONS OF GALACTIC DYNAMOS 7.1. Physical parameters of the ISM The interstellar gas can be found in various phases, characterized by different temperatures, densities, and degrees of ionization. The neutral, atomic gas is found in a cold (50K < T < 100K) and a warm (103 K < T < 104 K) phase, usually referred to as the cold and warm neutral media (CNM and WNM). The ionized gas is also found in a hot (T ' 106K) and a warm (T ' 104 K) phase (HIM and WIM,respectively). Finally,thedensestandcoldest(10K < T < 20K) gas is mostly molecular (MM), with a very low ionization fraction. Dynamos can easily be excited in the interstellar medium, since all phases (apart from the MM) are characterized by large values of Re, ReM, and PrM, although these vary greatly between phases; see Table 3, with order-of-magnitude values taken from Ferrière (2020) and Draine (2011). This vast range of parameters also poses a challenge for accurately modeling interstellar turbulence. 32 Brandenburg & Ntormousi Table 3 Parameters of the interstellar medium. Phase Re Rem Pm HIM 102 1023 1021 WIM 107 1019 1011 WIM 107 1018 1011 CNM 1010 1011 104 WNM 107 1018 1011 MM 107 104 105 Table 4 Overview of the numerical codes mentioned in this review. Code name Approach Other properties AREPO1 r· B-clean Finite volume Unstructured moving mesh ENZO2 CT or r· B-clean AMR, Riemann, split and unsplit schemes FLASH3 CT or r· B-clean AMR, Riemann, split and unsplit schemes Gadget4 r· B-clean SPH Nirvana5 CT AMR, Godunov (Riemann) PENCIL CODE6 B = r× A centered finite diff., sixth order RAMSES 7 CT AMR, Godunov (Riemann) 1 Pakmor et al. (2017), van de Voort et al. (2021), Whittingham et al. (2021) 2 Vazza et al. (2017), Mtchedlidze et al. (2022) 3 Sur et al. (2010), Federrath et al. (2011), Sur et al. (2012) 4 Steinwandel et al. (2019) 5 Gressel et al. (2008a) 6 Brandenburg (2001, 2005, 2010a), Brandenburg et al. (2010), Brandenburg (2017, 2019), Brandenburg et al. (2017), Brandenburg & Boldyrev (2020), Gent et al. (2021), Brandenburg & Ntormousi (2022) 7 Rieder & Teyssier (2016, 2017a,b), Ntormousi et al. (2020a) Most of the numerical work on ISM turbulent dynamos so far has been isothermal (e.g., Schekochihin et al. 2004, Seta et al. 2020). In an interesting extension, Seta & Federrath (2022) modelled a small-scale dynamo in driven turbulence simulations of a two-phase ISM and identified the processes responsible for vorticity generation in each phase. They found that the magnetic to turbulent kinetic energy ratio is lower in the cold phase. This will be discussed in more detail further below. 7.2. Numerical approaches Including magnetic fields in simulations of astrophysical systems is a non-trivial task because the chosen discretization must obey the zero divergence constraint. In Eulerian codes, a commonly used approach is the Constrained Transport (CT) scheme (Evans & Hawley 1988), which ensures r· B = 0 by defining the magnetic field components on cell faces. However, no similar scheme is applicable to Lagrangian codes such as Smoothed Particle Hydrodynamics (SPH), which rely on divergence-cleaning algorithms (e.g., Brackbill & Barnes 1980, Powell et al. 1999, Dedner et al. 2002). www.annualreviews.org • Galactic Dynamos 33 Figure 13 Figure 2 from Balsara & Kim (2004), where different divergence-cleaning algorithms: (a) Scalar divergence cleaning scheme from Kim et al. (1999); (b) scalar divergence cleaning scheme from Balsara (1998); (c) vector divergence cleaning scheme from Balsara (1998) were compared to a divergence-conserving scheme, (d). Lagrangian codes are particularly well-suited for MHD modeling of galaxies and cosmological volumes due to the natural adaptation of the resolution to areas of interest. They are also naturally Galilean-invariant. However, their dependence on divergence cleaning poses a significant drawback when modeling astrophysical dynamos. This was clearly demonstrated by Mocz et al. (2016), who compared CT and divergence cleaning approaches, both implemented on the moving-mesh code AREPO. They found that divergence cleaning systematically creates artifacts that mimic physical effects. Some of these artifacts are illustrated in Figures 13 and 14. Out of these effects, particularly notorious for dynamo studies is an artificial increase of the magnetic energy when using scalar divergence-cleaning schemes, as pointed out by Balsara & Kim (2004) for supernova-driven turbulence. Another very relevant example of an artifact caused by divergence cleaning is the spontaneous production of magnetic helicity, which has been found by Brandenburg & Scannapieco (2020), who compared simulations that employed a divergence 34 Brandenburg & Ntormousi Figure 14 Figure 7 from Mocz et al. (2016), where the same disk is evolved with CT and a Powell-like divergence cleaning scheme. The winding pattern of the magnetic field is better captured with CT, while an artifact is formed at the center with divergence cleaning. cleaning algorithm with one that advances instead the magnetic vector potential, A, so that B = r× A is always divergence-free. They found that, for a helically driven flow in a periodic domain, spurious net magnetic helicity is generated on dynamical time scales. An interesting experiment by Tricco et al. (2016) compared an SPH code to the FLASH grid code, both using divergence cleaning, in simulations of turbulent dynamos. They found very good agreement between the codes, both in the growth rates and the saturation level of the dynamo. Yet another method of dealing with the r· B = 0 constraint is to employ the Euler or Clebsch potentials. However, this method only works in the strictly ideal case when the microphysical magnetic diffusivity vanishes; see Section 2.2. As we have stressed above, even the addition of an almost negligibly small diffusivity in the evolution equations for the Euler potentials does not correspond to any physical magnetic diffusivity and leads to wrong results where no dynamo is possible – even for flows that are known to be either fast or slow dynamos (Brandenburg 2010b). Table 4 outlines some characteristics of the codes mentioned in this review. Next to each code we have mentioned the works cited in this review that use it. The fact that many of them have to rely on divergence cleaning methods is evidence of the difficulty in dealing with the divergence problem, but also a sign to use caution when interpreting the results in the context of dynamo action. 7.3. Local dynamo simulations of galaxy portions Simulating the magnetic field evolution over an entire galactic disk is another challenging task, due to the vast range of dynamical scales of the problem and the large shearing velocities involved. One approach to this challenge that can successfully capture many aspects of the problem is simulating galaxy portions. The local approach has been rather successful in the context of accretion disks, where simulations have been performed in what is known as shearing boxes. This means that the radial boundary is “shearing-periodic”, i.e., it is periodic with respect to an azimuthal position that shifts in time following the background shear flow. www.annualreviews.org • Galactic Dynamos 35 Figure 15 2D PDFs of magnetic field and density for (a) solenoidal and (b) compressive forcing. The dashed black lines show various b–ρ relations for simple gas compressions. Especially the solenoidal shows very little similarity with any of the simple relations. Using a shearing box, Gressel et al. (2008a) performed the first simulation of a galactic dynamo, including supernova-induced turbulence. It was similar to earlier multiphase simulations of supernova-driven turbulence of Korpi et al. (1999), where the magnetic Reynolds number was still too low to permit dynamo action. Gressel et al. (2008a) found that the rotation frequency of the considered galaxy portion is the dominant factor in determining the dynamo efficiency, while the supernova rate did not significantly affect the efficiency of the dynamo. This finding suggests that the simulations were able to capture large-scale dynamo action, but not the small-scale dynamo. UsingthePENCIL CODE andasimilarsetup,Gentetal. (2013)showedthatthemeanandfluctuating fields have different growth rates, indicating a co-existence of small-and large-scale dynamos. Theoretically, however, the possibility of large-scale and small-scale dynamos having different growth rates in one and the same system is not well understood (Subramanian & Brandenburg 2014). Recently, Gent et al. (2021) sought to derive criteria for the appearance of a small-scale dynamo in simulations of interstellar turbulence. By not employing a shearing-box setup or stratification, they focused only on the effects of the supernova-driven turbulence. They confirm that a small-scale dynamo is easily excited by ISM turbulence at sufficiently high resolution and magnetic Reynolds number. Seta & Federrath (2022) have shown that the multiphase aspect of the ISM tends to have a detrimental effect on the small-scale dynamo. This is mostly because of the stronger Lorentz force in the cold regions. Their simulations show that with solenoidal forcing, the magnetic field is mostly decoupled from the density behavior; see Figure 15. Simple Compression along magnetic field lines (½0), perpendicular to to magnetic field lines (b ∝ ½1/2 for cylindrical/filamentary geometry and b ∝ ½1 for disc-like/slab geometry), and spherical compression (b ∝ ½2/3) hardly occur. One might argue, however, that in the compressive case, the cold phase shows a higher slope than the warm phase. 7.4. Global isolated galaxy simulations The increasing efficiency and complexity of numerical codes and the availability of resources have led to a number of works studying the magnetic field evolution in global galaxy models. In contrast to the shearing-box approach, such models more naturally allow for the study of the large- scale dynamo. However, the limited resolution is still problematic for simultaneously capturing the 36 Brandenburg & Ntormousi small-scale dynamo, as outlined above. Wang & Abel (2009) performed disk galaxy simulations including nG ordered seeds with a code similar to Enzo, using Dedner et al. (2002) divergence-cleaning. They found that the tiny seed was amplified to ¹G levels over 500 Myr. They also noticed that the magnetic field in the cold gas saturates first. Their setup did not include stellar feedback, so the amplification process was driven by differential rotation only. In a series of papers, Rieder & Teyssier (2016, 2017a,b) performed multi-component simulations of the magnetic field evolution in dwarf and Milky-Way-like galaxies using the RAMSES code and including supernova feedback. Their setup includes dark matter and stars as collisionless particles, coupled to an Adaptive Mesh Refinement (AMR) grid on which the MHD equations are solved. In the first paper of the series, the authors found signatures of small-scale dynamo amplification during intense feedback epochs, followed by a large-scale dynamo at later, more quiescent evolution times. In the second paper, they examined the saturation of the dynamo, which occurs at only a small fraction of the turbulent kinetic energy. They observed that, if the feedback efficiency is artificially lowered after saturation, the turbulence decays and the galaxy settles in a thin disk with an equipartition field. Finally, in Rieder & Teyssier (2017b) they studied the magnetic field evolution in a cosmological context. Using a similar setup and separating the mean from the fluctuating component using a median filter, Ntormousi et al. (2020b) found large-scale dynamo action in a model of a massive spiral. However, their result was insensitive to the inclusion of supernova feedback. Since supernova feedback is considered an important driver of small-scale turbulence, this could mean that a small-scale dynamo was never captured in their models. This result is consistent with the Rieder & Teyssier (2016) results for the quiescent phase of galaxy evolution. However, as shown by Gent et al. (2021), the limited resolution of the simulation could be preventing the formation of a small-scale dynamo action in this quiescent phase. Pakmor et al. (2017) performed a suite of zoom-in cosmological simulations that includes 30 galaxies using the AREPO code (called the Auriga simulations). Similarly to Rieder & Teyssier (2017b), they reported an early exponential growth of the magnetic field, saturating at redshift z ' 2 − 3 at a few percent of the turbulent kinetic energy. Steinwandel et al. (2019) also claimed significant small-scale dynamo action in isolated galaxy models using an MHD version of the Gadget code. However, these simulations rely on the divergence-cleaning scheme that could suffer from the problems summarized in Section 7.2. A very different approach was adopted by Rodrigues et al. (2019), who modeled galactic magnetic fields by post-processing cosmological simulation data. Specifically, they inserted galaxy parameters such as shear rate and turbulence into a parameter-fitting package that returns a suitable dynamo solution (Shukurov et al. 2019). Although this approach cannot capture the back-reaction of the magnetic field on the gas, it can give estimates on the cosmological conditions that favor mean-field dynamo action, which the authors find sets in around redshift z < 3. While each numerical approach presents certain limitations, the tentative picture painted by global numerical simulations is that the small-scale and large-scale dynamos co-exist during galaxy evolution. Taking into account the large-scale gravitational collapse of the halo, as well as internal galactic processes such as star formation, appears to be fundamental for reconstructing the observed magnetic field evolution. (Stellar) Feedback: The energy and momentum deposited to the ISM by stars through radiation, winds, and supernova explosions. (Sticky Note comment ostriker 16/10/2022, 5:09:58 PM In numerical MHD simulations that employ finite volume methods, it is frequently the case that no explicit resistivity or viscosity is included. However, there is inevitably effective numerical resistivity and viscosity, and only moderate Re and Re_M are achievable. In discussing the simulations, it would be valuable to comment on this, in light of (1) the argument (section 3.1 and 3.2) that the rate at which saturation is reached is reduced at higher Re_M, corresponding to higher resolution, and (2) the hypothesis (section 3.3) that magnetic helicity flux -- which is present in both the shearing-box and global simulations -- enhances mean field growth. In particular, are there resolution studies from the simulations discussed in Section 7.3 and 7.4 that either show slower saturation at higher resolution (indicating a dominance of #1 at the achievable resolution) or convergence in saturation at higher resolution (possibly in combination with measurements of large magnetic helicity flux), supporting #2?) www.annualreviews.org • Galactic Dynamos 37 8. INTERACTION WITH THE CGM It was already suggested in the previous sections that the galactic environment must play an important role in the behavior of the dynamo, because it defines the boundary conditions for its operation. The immediate environment of a galaxy is its halo, which contains large amounts of diffuse gas, and is usually referred to as the CGM. The CGM is a powerful probe of galaxy evolution processes, because it contains traces of the cold (T < 105 K) and hot (T > 105 K) galactic inflows, as well as the hot (T ' 106 K), metal-enriched outflows from feedback events (Putman et al. 2012). The CGM also contains colder gas (T < 104 K) that can co-exist with these hotter phases for long periods of time. This observation has led to theories involving magnetic fields and cosmic rays in the dynamics of the CGM. This cold gas was studied in the context of cosmological simulations by Nelson et al. (2020), who found small-scale cool (T < 104 K) structures in massive (M ' 1013 − 1013.5M¯) galaxy haloes. In these simulations, the cloudlets are created by thermal instability, seeded by tidally stripped gas from in-falling haloes. In many cases, these structures are dominated by magnetic pressure. However, it is not clear whether these properties would persist at higher resolution. In spite of the dominant magnetic pressure, an otherwise identical simulation of a massive halo with the magnetic field set to zero showed essentially no change in the distribution and morphology of these cloudlets. 8.1. Magnetization of the galaxy by inflows The classical picture of gas accretion onto a galaxy halo predicts that the gas should shock and heat up to high temperatures (e.g. White & Rees 1978). However, cosmological simulations of galaxy formation (e.g., Dekel et al. 2009) showed that a high fraction of the in-flowing gas in high-redshift galaxies and present-day dwarfs is organized in cool (T ' 105 K) streams. RM observations suggest that this gas carries a nG-level magnetic field (e.g., Carretti et al. 2022). These estimates are compatible with the predictions of cosmological magnetic field evolution models, which predict detectable intergalactic magnetic fields from the evolution of primordial seeds (e.g., Vazza et al. 2015, 2017, –see also Section 5.1.5). Then a primordial galaxy might receive a strong seed for its own magnetic field. However, at the time of this review, the possible effect of magnetized inflowing gas on the galactic dynamo remains unexplored. As we have seen in Section 6, some models invoke a large-scale dynamo in galactic halos to explain the observed X-shaped magnetic fields therein. If there is really large-scale dynamo action in the halo, it may be predominantly of dipolar parity (Sokoloff & Shukurov 1990). This can lead to an interaction and competition with the quadrupolar magnetic field in the disk. Brandenburg et al. (1992) found that, during certain time intervals, the RM of these models shows a doubly peaked azimuthal variation, which could be falsely interpreted as an indication of a bisymmetric field structure. The galactic halo may also act as a buffer for the dynamo in the disk to dispose of excess magnetic helicity. Galaxy mergers are a particular form of inflow, which can influence the entire structure of the galaxy. As the galaxies approach each other, their star formation rate is enhanced and shocks form in their interstellar media. Both shocks and small-scale flows associated with feedback from young stars can strongly amplify the magnetic field. This highly non-linear type of interaction requires numerical modeling. Most numerical models of galaxy mergers with magnetization so far were done using La grangian codes, which have an obvious advantage in adapting their resolution in this setup. The first numerical simulation of a galaxy merger with magnetic fields was performed by Kotarba et al. 38 Brandenburg & Ntormousi (Sticky Note comment ostriker 16/10/2022, 4:29:04 AM need higher quality figure) Figure 16 Left: Figure 2 from van de Voort et al. (2021), showing a magnetized galactic outflow. The figure includes a slice through the middle of the central galaxy in magnetic field strength (top) and plasma β (bottom). Right: Adapted Figure 3 from Whittingham et al. (2021), showing a galaxy re-arranging after a merger event. The top two rows show density and the bottom two magnetic field strength. Both sets of simulations are part of the Auriga project (Grand et al. 2017). (2010), who modeled the Antennae using an MHD version of the Gadget code that subtracts the Lorenz force associated with magnetic divergence. They confirmed the expected amplification of the magnetic field during the galaxy encounters. However, the amplification was also accompanied by a surge in numerical magnetic divergence. The subsequent cosmological merger models of Beck et al. (2012) suffered from the same issue. Whittingham et al. (2021) showed more sophisticated merger models, modeled in a cosmological context using the AREPO code. They found a significant impact of the magnetic field on the morphology of the remnant galaxy. Specifically, a comparison between MHD and hydro models showed the presence of extended disks and spiral structure in the magnetized mergers, as opposed to compact remnants with a ring morphology in unmagnetized mergers. The right panel of Figure 16, taken from their work, shows the evolution of the galaxy post-merger. One exception to the Lagrangian models is the work of Rodenbeck & Schleicher (2016), who performed a grid simulation of a galaxy merger, using a simplified model without stellar feedback or a collisionless component. They found that the enhancement of the magnetic field is particularly pronounced in the central regions of the galaxy. 8.2. Magnetization of the CGM by outflows Galactic outflows are fundamental in any theory of galaxy evolution. In starburst and dwarf galaxies they are powered by stellar feedback (e.g., Zhang 2018), while a few galaxies host AGN-powered winds (Fabian 2012, Kormendy & Ho 2013, Martin 1998, Veilleux et al. 2005). Recent work has indicated that, in some cases, galactic winds can be driven by cosmic ray (CR) pressure (Hanasz et al. 2013, Girichidis et al. 2016). The material carried by these outflows can magnetize the CGM. In a recent study, van de Voort et al. (2021) performed direct galaxy evolution simulations while resolving the magnetized CGM, as part of the Auriga project. They found that, while the CGM re www.annualreviews.org • Galactic Dynamos 39 mained a high-beta plasma, magnetic fields can noticeably change its structure around galaxies, indirectly affecting numerous galactic processes (see left panel of Figure 16). For instance, galactic outflows become more collimated, resulting in less efficient mixing between enriched and unenriched gas. Outflow speeds are also reduced in the presence of magnetization, which means that more metals remain in the halo with respect to the un-magnetized situation. The overall structure of the CGM is smoother, due to the additional magnetic pressure. Arámburo-García et al. (2021) studied the magnetization of haloes by AGN and SN-driven out flows in the Illustris-TNG simulations. They found that both types of outflows contributed to the creation of over-magnetized bubbles, with the AGN-driven bubbles playing the dominant role. 8.3. The impact of the environment on the galactic dynamo The interaction of the galaxy with the CGM can have a crucial impact on the development of a dynamo. The loss of helicity flux through a galactic wind or fountain can help avoid catastrophic quenching and sustain a dynamo for longer (see Section 3.3), eventually reaching higher values of the saturated field. However, winds can also interfere with the dynamo itself if they are acting within the dynamo-active region. On the other hand, a strong magnetic field can suppress the galactic outflow. Whether we are considering inflows, outflows, or a galactic fountain, the galaxy is always embedded in a current system that will have an impact on the evolution of the dynamo. This is a non-trivial complication of the effective boundary conditions because the level of magnetization of the inflowing or outflowing gas is unknown, and the extent of these flows can be much larger than the virial radius of the galaxy. 9. CONCLUSIONS After 70 years of inquiry into the possibility of dynamos in galaxies, several important questions can now be answered. First of all, in virtually all astrophysical settings, there is turbulence and this turbulence is always magnetized because of small-scale dynamo action. Dynamos also work in decaying and otherwise nonstationary turbulence and can produce equipartition-strength magnetic fields exponentially on a turbulent turnover time scale. This realization makes the question of cosmological seed magnetic fields for galaxies and galaxy clusters almost obsolete, because it would always be overpowered by small-scale dynamos that can operate very rapidly when the scales are sufficiently small. While primordial magnetic fields may still be present and interesting in their own rights, the simple idea of them being wound up to explain the bisymmetric spiral of nonaxisymmetric magnetic fields in some galaxies is essentially ruled out. While global numerical simulations are now beginning to show the production of magnetic fields in galaxies, there remains a big uncertainty regarding the question of what actually produces large-scale dynamo action. Is it really the α effect or some other mechanisms at play? In this review, we have outlined several known mechanisms that could produce large-scale magnetic fields, but this question remains a major research topic in the years to come. One reason behind this is that the problem of catastrophic quenching is still not fully resolved. Low resolution simulations of relatively diffusive dynamos may have been promising in terms of field strength and structure, but have so far not survived the test of higher resolution. It will remain important to continue investigating this. At the same time, it is important to open ones mind and think about dynamos beyond just the immediate proximity of a galaxy. The interaction with the CGM may be of crucial importance, and not all of the different processes, which are important, may qualify as a dynamo. 40 Brandenburg & Ntormousi SUMMARY POINTS 1. Small-scale dynamos work in all turbulent astrophysical environments. 2. Simulations suggest that the efficiency of large-scale dynamos decreases with increasing resolution. 3. The relevance of an α effect dynamo in galaxies remains unclear. 4. Modern numerical simulations of galactic magnetic fields tend to take the past evolution of and the interaction with the environment into account. FUTURE ISSUES 1. The problem of catastrophic quenching remains relevant and will need to be addressed in high-resolution models with realistic boundary conditions. 2. Numerical codes of galaxy evolution should take special care of the accurate treatment of magnetic fields, especially on the solenoidality constraint, a problem that is accentuated through the sub-grid modeling of star formation and feedback. 3. The next generation of numerical models should make an effort to identify observables related to the galactic dynamo that go beyond the BSS/ASS signature in the RM of galaxies. 4. Future numerical models will have to quantify the importance of dynamo action in different stages of a galaxy’s evolution. DISCLOSURE STATEMENT The authors are not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review. ACKNOWLEDGMENTS We acknowledge support for the Nordita program on “Magnetic field evolution in low density or strongly stratified plasmas” in May 2022, when part of this work was done. EN also acknowledges Interstellar Institute’s program "With Two Eyes" and the Paris-Saclay University’s Institut Pascal for hosting fruitful discussions that nourished some ideas included in this work. We acknowledge very useful discussions with Fabio Del Sordo, Frederick Gent, Sergio Martin-Alvarez, and Jennifer West. EN acknowledges funding from the ERC Grant ”Interstellar” (Grant agreement 740120) and the Hellenic Foundation for Research and Innovation (Project number 224). This work was supported by the Swedish Research Council (Vetenskapsrådet, 2019-04234). Nordita is sponsored by Nordforsk. We acknowledge the allocation of computing resources provided by the Swedish National Allocations Committee at the Center for Parallel Computers at the Royal Institute of Technology in Stockholm and Linköping. LITERATURE CITED Alves Batista R, Saveliev A, de Gouveia Dal Pino EM. 2019. MNRAS 489(3):3836–3849 Andrievsky A, Brandenburg A, Noullez A, Zheligovsky V. 2015. ApJ 811:135 www.annualreviews.org • Galactic Dynamos 41 Arámburo-García A, Bondarenko K, Boyarsky A, Nelson D, Pillepich A, Sokolenko A. 2021. MNRAS 505(4):5038– 5057 Backus G. 1958. Ann. Phys. 4(4):372–447 Balsara DS. 1998. ApJS 116(1):119–131 Balsara DS, Kim J. 2004. ApJ 602(2):1079–1090 Beck AM, Dolag K, Lesch H, Kronberg PP. 2013. MNRAS 435(4):3575–3586 Beck AM, Lesch H, Dolag K, Kotarba H, Geng A, Stasyszyn FA. 2012. MNRAS 422(3):2152–2163 Beck R. 2001. SSR 99:243–260 Beck R. 2012. SSR 166(1-4):215–230 Beck R. 2015a. AAPR 24:4 Beck R. 2015b. A&A 578:A93 Beck R, Berkhuijsen EM, Gießübel R, Mulcahy DD. 2020. A&A 633:A5 Beck R, Brandenburg A, Moss D, Shukurov A, Sokoloff D. 1996. ARA&A 34:155–206 Beck R, Chamandy L, Elson E, Blackman EG. 2019. Galaxies 8(1):4 Beck R, Poezd AD, Shukurov A, Sokoloff DD. 1994. A&A 289:94–100 Bendre AB, Subramanian K. 2022. MNRAS 511(3):4454–4463 Bennett CL, Larson D, Weiland JL, Jarosik N, Hinshaw G, et al. 2013. ApJS 208(2):20 Bernet ML, Miniati F, Lilly SJ, Kronberg PP, Dessauges-Zavadsky M. 2008. NATUR 454(7202):302–304 Blackman EG, Brandenburg A. 2002. ApJ 579(1):359–373 Blackman EG, Field GB. 2000. ApJ 534(2):984–988 Borlaff AS, Lopez-Rodriguez E, Beck R, Stepanov R, Ntormousi E, et al. 2021. ApJ 921(2):128 Brackbill JU, Barnes DC. 1980. J. Comp. Phys. 35(3):426–430 Brandenburg A. 2000. Phil. Trans. Roy. Soc. Lond. A 358:759–774 Brandenburg A. 2001. ApJ 550(2):824–840 Brandenburg A. 2005. ApJ 625:539–547 Brandenburg A. 2010a. MNRAS 401(1):347–354 Brandenburg A. 2010b. MNRAS 401(1):347–354 Brandenburg A. 2017. A&A 598:A117 Brandenburg A. 2018. J. Plasma Phys. 84:735840404 Brandenburg A. 2019. MNRAS 487:2673–2684 Brandenburg A, Boldyrev S. 2020. ApJ 892:80 Brandenburg A, Brüggen M. 2020. ApJL 896:L14 Brandenburg A, Chatterjee P. 2018. Astron. Nachr. 339:118–126 Brandenburg A, Chatterjee P, Del Sordo F, Hubbard A, Käpylä PJ, Rheinhardt M. 2010. PhysS 142:014028 Brandenburg A, Donner KJ, Moss D, Shukurov A, Sokoloff DD, Tuominen I. 1993. A&A 271:36–50 Brandenburg A, Donner KJ, Moss D, Shukurov A, Sokolov DD, Tuominen I. 1992. A&A 259:453–461 Brandenburg A, Enqvist K, Olesen P. 1996. PhRvD 54(2):1291–1300 Brandenburg A, Kahniashvili T, Mandal S, Roper Pol A, Tevzadze AG, Vachaspati T. 2019. PhRvF 4:024608 Brandenburg A, Ntormousi E. 2022. MNRAS 513(2):2136–2151 Brandenburg A, Rädler KH, Rheinhardt M, Käpylä PJ. 2008a. ApJ 676:740–751 Brandenburg A, Rädler KH, Rheinhardt M, Subramanian K. 2008b. ApJL 687:L49–L52 Brandenburg A, Rädler KH, Schrinner M. 2008c. A&A 482:739–746 Brandenburg A, Scannapieco E. 2020. ApJ 889(1):55 Brandenburg A, Schober J, Rogachevskii I, Kahniashvili T, Boyarsky A, et al. 2017. ApJ 845:L21 Brandenburg A, Sokoloff D. 2002. Geophys. Astrophys. Fluid Dynam. 96(4):319–344 Brandenburg A, Subramanian K. 2005a. PHYSREP 417:1–209 Brandenburg A, Subramanian K. 2005b. Astronomische Nachrichten 326(6):400–408 Brandenburg A, Tuominen I, Krause F. 1990. Geophys. Astrophys. Fluid Dynam. 50(1):95–112 Brandenburg A, Zhou H, Sharma R. 2022. arXiv e-prints :arXiv:2207.09414 Brentjens MA, de Bruyn AG. 2005. A&A 441(3):1217–1228 Broderick AE, Chang P, Pfrommer C. 2012. ApJ 752(1):22 42 Brandenburg & Ntormousi Burn BJ. 1966. MNRAS 133:67 Carretti E, Vacca V, O’Sullivan SP, Heald GH, Horellou C, et al. 2022. MNRAS 512(1):945–959 Cattaneo F, Hughes DW. 1996. PhRvE 54(5):R4532–R4535 Cattaneo F, Vainshtein SI. 1991. ApJL 376:L21 Chamandy L, Shukurov A, Subramanian K. 2015. MNRAS 446:L6–L10 Chamandy L, Subramanian K, Shukurov A. 2013. MNRAS 428:3569–3589 Chandrasekhar S. 1956. ApJ 124:244 Cowling TG. 1933. MNRAS 94:39–48 Davis Leverett J, Greenstein JL. 1951. ApJ 114:206 Dedner A, Kemm F, Kröner D, Munz CD, Schnitzer T, Wesenberg M. 2002. J. Comp. Phys. 175(2):645–673 Dekel A, Birnboim Y, Engel G, Freundlich J, Goerdt T, et al. 2009. NATUR 457(7228):451–454 Del Sordo F, Guerrero G, Brandenburg A. 2013. MNRAS 429:1686–1694 Devlen E, Brandenburg A, Mitra D. 2013. MNRAS 432:1651–1657 Dickey JMa. 2022. ApJ, submitted Donner KJ, Brandenburg A. 1990. A&A 240(2):289–298 Draine BT. 2011. Physics of the Interstellar and Intergalactic Medium Durrive JB, Langer M. 2015. MNRAS 453(1):345–356 Elstner D, Golla G, Rudiger G, Wielebinski R. 1995. A&A 297:77–82 Elstner D, Meinel R, Rüdiger G. 1990. Geophys. Astrophys. Fluid Dynam. 50(1):85–94 Evans CR, Hawley JF. 1988. ApJ 332:659 Fabian AC. 2012. ARA&A 50:455–489 Federrath C, Sur S, Schleicher DRG, Banerjee R, Klessen RS. 2011. ApJ 731(1):62 Ferrière K. 2020. Plasma Physics and Controlled Fusion 62(1):014014 Garaldi E, Pakmor R, Springel V. 2021. MNRAS 502(4):5726–5744 Gent FA, Mac Low MM, Käpylä MJ, Singh NK. 2021. ApJL 910(2):L15 Gent FA, Shukurov A, Sarson GR, Fletcher A, Mantere MJ. 2013. MNRAS 430:L40–L44 Girichidis P, Naab T, Walch S, Hanasz M, Mac Low MM, et al. 2016. ApJL 816(2):L19 Gnedin NY, Ferrara A, Zweibel EG. 2000. ApJ 539(2):505–516 Golla G, Hummel E. 1994. A&A 284:777–792 Grand RJJ, Gómez FA, Marinacci F, Pakmor R, Springel V, et al. 2017. MNRAS 467(1):179–207 Grasso D, Rubinstein HR. 1995. Astroparticle Physics 3(1):95–102 Gressel O, Elstner D, Ziegler U, Rüdiger G. 2008a. A&A 486(3):L35–L38 Gressel O, Ziegler U, Elstner D, Rüdiger G. 2008b. Astron. Nachr. 329(6):619 Gruzinov AV, Diamond PH. 1996. Physics of Plasmas 3(5):1853–1857 Hall JS. 1949. Science 109(2825):166–167 Han JL. 2017. ARA&A 55(1):111–157 Hanasz M,LeschH, Naab T, Gawryszczak A, Kowalik K, Wólta´ nski D. 2013. ApJL 777(2):L38 Henriksen RN, Woodfinden A, Irwin JA. 2018. MNRAS 476(1):635–645 Herzenberg A. 1958. Phil. Trans. Roy. Soc. Lond. Ser. A 250(986):543–583 Hiltner WA. 1949. Science 109(2825):165 Hosking DN, Schekochihin AA. 2022. arXiv e-prints :arXiv:2203.03573 Hutschenreuter S, Anderson CS, Betti S, Bower GC, Brown JA, et al. 2022. A&A 657:A43 Jaffe TR, Leahy JP, Banday AJ, Leach SM, Lowe SR, Wilkinson A. 2010. MNRAS 401(2):1013–1028 Jansson R, Farrar GR. 2012. ApJ 757(1):14 Kahniashvili T, Clarke E, Stepp J, Brandenburg A. 2022. PhRvL 128(22):221301 Kahniashvili T, Maravin Y, Natarajan A, Battaglia N, Tevzadze AG. 2013. ApJ 770(1):47 Kamionkowski M, Kosowsky A, Stebbins A. 1997. PhRvL 78(11):2058–2061 Katz H, Martin-Alvarez S, Devriendt J, Slyz A, Kimm T. 2019. MNRAS 484(2):2620–2631 Kazantsev AP. 1968. Sov. J. Exp. Theor. Phys. 26:1031 Keinigs RK. 1983. Physics of Fluids 26(9):2558–2560 Kim J, Ryu D, Jones TW, Hong SS. 1999. ApJ 514(1):506–519 www.annualreviews.org • Galactic Dynamos 43 Kormendy J, Ho LC. 2013. ARA&A 51(1):511–653 Korochkin A, Kalashev O, Neronov A, Semikoz D. 2021. ApJ 906(2):116 Korpi MJ, Brandenburg A, Shukurov A, Tuominen I, Nordlund Å. 1999. ApJL 514(2):L99–L102 Kotarba H, Karl SJ, Naab T, Johansson PH, Dolag K, et al. 2010. ApJ 716(2):1438–1452 Kraichnan RH. 1976. JFM 75:657–676 Krause F, Rädler KH. 1980. Mean-Field Magnetohydrodynamics and Dynamo Theory. Oxford: Pergamon Press (also Akademie-Verlag: Berlin) Krause M, Beck R, Hummel E. 1989. A&A 217:17–30 Krause M, Irwin J, Schmidt P, Stein Y, Miskolczi A, et al. 2020. A&A 639:A112 Krause M, Wielebinski R, Dumke M. 2006. A&A 448(1):133–142 Kronberg PP, Perry JJ, Zukowski ELH. 1992. ApJ 387:528 Kulsrud RM, Anderson SW. 1992. ApJ 396:606 Kulsrud RM, Cen R, Ostriker JP, Ryu D. 1997. ApJ 480(2):481–491 Lopez-Rodriguez E, Clarke M, Shenoy S, Vacca W, Coude S, et al. 2022. ApJ 936(1):65 Lowes FJ, Wilkinson I. 1963. NATUR 198(4886):1158–1160 Mao SA, Carilli C, Gaensler BM, Wucknitz O, Keeton C, et al. 2017. Nature Astron. 1:621–626 Marinacci F, Vogelsberger M. 2016. MNRAS 456(1):L69–L73 Martin CL. 1998. ApJ 506(1):222–252 Martin-Alvarez S, Katz H, Sijacki D, Devriendt J, Slyz A. 2021. MNRAS 504(2):2517–2534 Martin-Alvarez S, Slyz A, Devriendt J, Gómez-Guijarro C. 2020. MNRAS 495(4):4475–4495 Mocz P, Pakmor R, Springel V, Vogelsberger M, Marinacci F, Hernquist L. 2016. MNRAS 463(1):477–488 Moffatt HK, Proctor MRE. 1985. JFM 154:493–507 Moss D. 1996. A&A 315:63–70 Moss D, Brandenburg A, Donner KJ, Thomasson M. 1993. ApJ 409:179 Moss D, Sokoloff D. 2008. A&A 487(1):197–203 Mtchedlidze S, Domínguez-Fernández P, Du X, Brandenburg A, Kahniashvili T, et al. 2022. ApJ 929(2):127 Nelson D, Sharma P, Pillepich A, Springel V, Pakmor R, et al. 2020. MNRAS 498(2):2391–2414 Neronov A, Vovk I. 2010. Science 328:73 Ntormousi E, Tassis K, Del Sordo F, Fragkoudi F, Pakmor R. 2020a. A&A 641:A165 Ntormousi E, Tassis K, Del Sordo F, Fragkoudi F, Pakmor R. 2020b. A&A 641:A165 Ôki T, Fujimoto M, Hitotuyanagi Z. 1964. Progress of Theoretical Physics Supplement 31:77–115 Oppermann N, Junklewitz H, Greiner M, Enßlin TA, Akahori T, et al. 2015. A&A 575:A118 Page L, Hinshaw G, Komatsu E, Nolta MR, Spergel DN, et al. 2007. ApJS 170(2):335–376 Pakmor R, Gómez FA, Grand RJJ, Marinacci F, Simpson CM, et al. 2017. MNRAS 469(3):3185–3199 Parker EN. 1955. ApJ 122:293 Parker EN. 1971. ApJ 163:255–278 Parker EN. 1979. Cosmical Magnetic Fields: Their Origin and Their Activity. Oxford: Clarendon Press Pattle K, Fissel L, Tahani M, Liu T, Ntormousi E. 2022. arXiv e-prints :arXiv:2203.11179 Piddington JH. 1964. MNRAS 128:345 Poezd A, Shukurov A, Sokoloff D. 1993. MNRAS 264:285–297 Powell KG, Roe PL, Linde TJ, Gombosi TI, De Zeeuw DL. 1999. J. Comp. Phys. 154(2):284–309 Prasad A, Mangalam A. 2016. ApJ 817(1):12 Putman ME, Peek JEG, Joung MR. 2012. ARA&A 50:491–529 Rees MJ. 1987. QJRAS 28:197–206 Rheinhardt M, Brandenburg A. 2012. Astron. Nachr. 333:71–77 Rheinhardt M, Devlen E, Rädler KH, Brandenburg A. 2014. MNRAS 441:116–126 Rieder M, Teyssier R. 2016. MNRAS 457(2):1722–1738 Rieder M, Teyssier R. 2017a. MNRAS 471(3):2674–2686 Rieder M, Teyssier R. 2017b. MNRAS 472(4):4368–4373 Rincon F. 2021. Physical Review Fluids 6(12):L121701 Roberts GO. 1972. Phil. Trans. Roy. Soc. Lond. Ser. A 271(1216):411–454 44 Brandenburg & Ntormousi Rodenbeck K, Schleicher DRG. 2016. A&A 593:A89 Rodrigues LFS, Chamandy L, Shukurov A, Baugh CM, Taylor AR. 2019. MNRAS 483(2):2424–2440 Roettiger K, Stone JM, Burns JO. 1999. ApJ 518(2):594–602 Rosswog S, Price D. 2007. MNRAS 379(3):915–931 Ruzmaikin A, Shukurov A, Sokoloff D. 1988. Magnetic Fields of Galaxies. Dordrecht: Kluwer Schekochihin AA, Cowley SC, Taylor SF, Maron JL, McWilliams JC. 2004. ApJ 612(1):276–307 Schober J, Schleicher D, Federrath C, Glover S, Klessen RS, Banerjee R. 2012. ApJ 754(2):99 Schoeffler KM, Loureiro NF, Fonseca RA, Silva LO. 2016. Physics of Plasmas 23(5):056304 Schrinner M, Rädler KH, Schmitt D, Rheinhardt M, Christensen U. 2005. Astron. Nachr. 326:245–249 Schrinner M, Rädler KH, Schmitt D, Rheinhardt M, Christensen UR. 2007. Geophys. Astrophys. Fluid Dynam. 101:81–116 Segalovitz A, Shane WW, de Bruyn AG. 1976. NATUR 264:222–226 Seljak U, Zaldarriaga M. 1997. PhRvL 78(11):2054–2057 Seta A, Beck R. 2019. Galaxies 7:45 Seta A, Bushby PJ, Shukurov A, Wood TS. 2020. Physical Review Fluids 5(4):043702 Seta A, Federrath C. 2022. MNRAS 514(1):957–976 Shukurov A. 1998. MNRAS 299(1):L21–l24 Shukurov A, Rodrigues LFS, Bushby PJ, Hollins J, Rachen JP. 2019. A&A 623:A113 Shukurov A, Sokoloff D, Subramanian K, Brandenburg A. 2006. A&A 448:L33–L36 Shukurov A, Subramanian K. 2022. Astrophysical Magnetic Fields: From Galaxies to the Early Universe. Cambridge: Cambridge University Press Simard C, Charbonneau P, Dubé C. 2016. Adv. Spa. Res. 58(8):1522–1537 Sironi L, Giannios D. 2014. ApJ 787(1):49 Snodin AP, Brandenburg A, Mee AJ, Shukurov A. 2006. MNRAS 373:643–652 Sofue Y, Fujimoto M, Wielebinski R. 1986. ARA&A 24:459–497 Sokoloff D, Shukurov A. 1990. NATUR 347(6288):51–53 Soward AM. 1987. JFM 180:267–295 Squire J, Bhattacharjee A. 2015. PhRvL 115(17):175003 Steenbeck M, Krause F. 1969. Astron. Nachr. 291:49–84 Steenbeck M, Krause F, Rädler KH. 1966. Zeitschr. Naturforsch. A 21:369 Stein Y, Dettmar RJ, We˙ zgowiec M, Irwin J, Beck R, et al. 2019. A&A 632:A13 Steinwandel UP, Beck MC, Arth A, Dolag K, Moster BP, Nielaba P. 2019. MNRAS 483(1):1008–1028 Stix M. 1975. A&A 42(1):85–89 Subramanian K. 2016. RPPh 79(7):076901 Subramanian K, Brandenburg A. 2014. MNRAS 445:2930–2940 Subramanian K, Narasimha D, Chitre SM. 1994. MNRAS 271:L15 Sun XH, Reich W, Waelkens A, Enßlin TA. 2008. A&A 477(2):573–592 Sur S, Federrath C, Schleicher DRG, Banerjee R, Klessen RS. 2012. MNRAS 423(4):3148–3162 Sur S, Schleicher DRG, Banerjee R, Federrath C, Klessen RS. 2010. ApJL 721(2):L134–L138 Tabatabaei FS, Krause M, Fletcher A, Beck R. 2008. A&A 490(3):1005–1017 Tassis K, Ramaprakash AN, Readhead ACS, Potter SB, Wehus IK, et al. 2018. arXiv e-prints :arXiv:1810.05652 Terral P, Ferrière K. 2017. A&A 600:A29 Tosa M, Fujimoto M. 1978. PASJ 30:315–326 Tricco TS, Price DJ, Federrath C. 2016. MNRAS 461(2):1260–1275 Tüllmann R, Dettmar RJ, Soida M, Urbanik M, Rossa J. 2000. A&A 364:L36–L41 Vainshtein SI, Cattaneo F. 1992. ApJ 393:165 Vainshtein SI, Ruzmaikin AA. 1971. AZh 48:902 van de Voort F, Bieri R, Pakmor R, Gómez FA, Grand RJJ, Marinacci F. 2021. MNRAS 501(4):4888–4902 Vazza F, Brüggen M, Gheller C, Hackstein S, Wittor D, Hinz PM. 2017. Classical and Quantum Gravity 34(23):234001 Vazza F, Ferrari C, Brüggen M, Bonafede A, Gheller C, Wang P. 2015. A&A 580:A119 www.annualreviews.org • Galactic Dynamos 45 Veilleux S, Cecil G, Bland-Hawthorn J. 2005. ARA&A 43(1):769–826 Vishniac ET, Brandenburg A. 1997. ApJ 475:263–274 Vishniac ET, Cho J. 2001. ApJ 550(2):752–760 Vishniac ET, Shapovalov D. 2014. ApJ 780(2):144 von Rekowski B, Brandenburg A, Dobler W, Dobler W, Shukurov A. 2003. A&A 398:825–844 Wang P, Abel T. 2009. ApJ 696(1):96–109 Warnecke J, Rheinhardt M, Tuomisto S, Käpylä PJ, Käpylä MJ, Brandenburg A. 2018. A&A 609:A51 Weibel ES. 1959. PhRvL 2(3):83–84 West JL, Henriksen RN, Ferrière K, Woodfinden A, Jaffe T, et al. 2020. MNRAS 499(3):3673–3689 White SDM, Rees MJ. 1978. MNRAS 183:341–358 Whittingham J, Sparre M, Pfrommer C, Pakmor R. 2021. MNRAS 506(1):229–255 Widrow LM. 2002. Reviews of Modern Physics 74(3):775–823 Willis AP. 2012. PhRvL 109(25):251101 Xu S, Garain SK, Balsara DS, Lazarian A. 2019. ApJ 872(1):62 Xu S, Lazarian A. 2016. ApJ 833(2):215 Xu S, Lazarian A. 2020. ApJ 899(2):115 Xu S, Lazarian A. 2021. Rev. Mod. Plasma Phys. 5(1):2 Yousef TA, Heinemann T, Schekochihin AA, Kleeorin N, Rogachevskii I, et al. 2008. PhRvL 100:184501 Zhang D. 2018. Galaxies 6(4):114 Zhou H, Blackman EG. 2021. MNRAS 507(4):5732–5746 46 Brandenburg & Ntormousi