% template.tex, dated April 5 2013 % This is a template file for Annual Reviews 1 column Journals % % Compilation using ar-1col.cls' - version 1.0, Aptara Inc. % (c) 2013 AR % % Steps to compile: latex latex latex % % For tracking purposes => this is v1.0 - Apr. 2013 \documentclass{ar-1col} \usepackage[comma]{natbib} \usepackage{url,bm} \usepackage[utf8]{inputenc} \usepackage{fourier} \usepackage{array} \usepackage{makecell} \usepackage[makeroom]{cancel} \usepackage{amssymb} \usepackage[flushleft]{threeparttable} \renewcommand\theadalign{bc} \renewcommand\theadfont{\bfseries} \renewcommand\theadgape{\Gape[4pt]} \renewcommand\cellgape{\Gape[4pt]} \graphicspath{{./fig/}{./png/}} \input macros \def\red{\textcolor{red}} \def\blue{\textcolor{blue}} \setcounter{secnumdepth}{4} % Metadata Information %\jname{Xxxx. Xxx. Xxx. Xxx.} %put date (for now) \jname{\today} \jvol{AA} \jyear{YYYY} \doi{10.1146/((please add article doi))} % Document starts \begin{document} % Page header \markboth{Brandenburg \& Ntormousi}{Galactic Dynamos} % Title \title{Galactic Dynamos} \author{Axel Brandenburg$^{1,2,3,4}$ \& Evangelia Ntormousi$^{5,6}$ %[0000-0002-7304-021X]{Axel Brandenburg} \affil{$^1$Nordita, KTH Royal Institute of Technology and Stockholm University, Hannes Alfv\'ens v\"ag 12, SE-10691 Stockholm, Sweden} \affil{$^2$The Oskar Klein Centre, Department of Astronomy, Stockholm University, AlbaNova, SE-10691 Stockholm, Sweden} \affil{$^3$McWilliams Center for Cosmology \& Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA} \affil{$^4$School of Natural Sciences and Medicine, Ilia State University, 3-5 Cholokashvili Avenue, 0194 Tbilisi, Georgia} \affil{$^5$Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy} \affil{$^6$Institute of Astrophysics, Foundation for Research and Technology-Hellas, Vasilika Vouton, GR-70013 Heraklion, Greece} } \begin{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. \parbox{10cm}{ \begin{itemize} \item 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. \item 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. \item 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. \item 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. \end{itemize}} 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. \end{abstract} \begin{keywords} Spiral galaxies, Milky Way, large-scale magnetic fields, circumgalactic medium, magnetic Reynolds number \end{keywords} \maketitle \tableofcontents \section{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 \cite{RSS88} for an early textbook on the subject, and \cite{SS22} for a recent one. Similar magnetic field strengths have also been detected in galaxies at larger redshifts ($z\simeq$1) \citep{Bernet_2008,Mao2017}. Galactic magnetic fields often also show large-scale coherence. The first evidence for a global Galactic magnetic field comes from optical polarization \citep{Hiltner49, Hall49}. The existence of magnetic fields was later confirmed for other galaxies using synchrotron emission \citep{Segalovitz+76}, which showed systematic large-scale magnetic fields roughly in the direction of the galactic spiral arms. There has long been a debate about the origin of such magnetic fields: are they primordial or dynamo-generated, or perhaps a combination of the two? Over the past few decades, there has been a shift of attention from a primordial to a dynamo-generated origin. In the meantime, however, there have also been significant developments in dynamo theory, and global numerical simulations are now becoming more realistic. They tend to show that large-scale magnetic fields can be generated by a dynamo, but the amplitudes may be insufficient or the time scales for their generation too long for the simulations presented so far. The broader problem of galactic magnetism that was addressed in \cite{Beck+96}, will not be reviewed; we refer here to other excellent reviews by \cite{Beck01,Beck15} and \cite{Han17}, as well as the recent textbook by \cite{SS22}. Regarding small-scale turbulent dynamos in partially ionized plasmas, we refer to the recent review by \cite{XL21}. In the present review, we focus on galactic dynamos and highlight some of the main developments since the time of the review of \cite{Beck+96}. At that time, there were results suggesting that the dynamo effect in mean-field theory is ``catastrophically'' quenched, i.e., it goes to zero as the magnetic Reynolds number ($\Rm$) becomes large \citep{VC92,CH96}. Specifically, the mean-field effect in question is what is referred to as the $\alpha$ effect, which quantifies the component of the mean electromagnetic force in the direction of the mean magnetic field. ``Catastrophic'' can mean that the coefficient $\alpha$ goes to zero as $\Rm$ 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. %%%%%%%%%%%%%%% \begin{marginnote}[] \entry{Catastrophic quenching}{limits the large-scale field at large $\Rm$.} \end{marginnote} %%%%%%%%%%%%% This led to a major crisis in dynamo theory, questioning the possibility of an $\alpha$ 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 $\alpha$ 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. \begin{textbox}[t]\section{Observational tracers of galactic magnetic fields} \paragraph*{Dust extinction / emission polarization} Elongated dust grains in the interstellar medium tend to align their minor axes with the mean magnetic field direction \citep{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 \citep[e.g.,the recent survey by][]{LR2022}. 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 \citep[e.g., the PASIPHAE survey,][]{Tassis2018}. \paragraph*{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 \citep[see, e.g.,][for a review]{Beck_2012}. 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 \begin{equation} \frac{1}{2}\arctan{\chi} = \frac{U}{Q} \end{equation} and the polarization fraction $p=\sqrt{Q^2+U^2}/I$. \paragraph*{Faraday Rotation} When polarized radiation passes through an ionized magnetized medium, the plane of polarization is rotated by an angle $\psi = RM\lambda^2$, where $\lambda$ the observed wavelength and \begin{equation} RM=0.81 \int_0^D n_e B_{\parallel} dl \end{equation} in units of radians/$m^2$, where $D$ is the distance to the source in pc, $n_{\rm e}$ is the electron density in $\cm^{-3}$, and $B_{\parallel}$ the line-of-sight magnetic field of the medium in Gauss \citep{Burn_1966, Brentjens_deBruyn2005}. If we have an estimate of the electron density, then % EN removed %we can ` Faraday Rotation can yield the line-of-sight magnetic field. \end{textbox} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{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. \subsection{Historical remarks} \label{sec:history} In \Sec{sec:mean-field_models} 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 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 \cite{Pid64} and \cite{Oki+64}, 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 \Fig{sketch_aas_bss} for a sketch. Using this technique, \citet{TF78} found evidence for a BSS in M51. In an early review on galactic magnetic fields, \cite{SFW86} contrasted this with the axisymmetric spiral (ASS), expected from mean-field dynamo theory. \Fig{sketch_aas_bss} also sketches the expected ASS signature. \begin{figure}[t] \includegraphics[width=\textwidth]{sketch_aas_bss} \caption{ 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 \cite{TF78}. }\label{sketch_aas_bss} \end{figure} However, a purely primordial origin of galactic magnetic fields implies that a tremendous amount of winding has occurred over the past $14\Gyr$ For example, in the solar neighborhood, the angular velocity of the Galaxy is $\approx30\Gyr^{-1}$, i.e., the rotation period is $(2\pi/30)\Gyr\approx0.2\Gyr$. This yields 70 revolutions in $14\Gyr$, so we would expect the magnetic field to be strongly wound up. \FFig{psavss} gives a quantitative illustration of this wind-up process. It shows color scale images of $|\BB|$ together with field lines corresponding to the contours of the normal component of the magnetic vector potential, $A_z(x,y)$, so that the magnetic field in the plane is given by $\BB=\nab\times(\zzz A_z)$. To obtain the result shown in \Fig{psavss}, we solved the two-dimensional induction equation, which corresponds to an advection--diffusion equation of the form \begin{equation} \frac{\DD A_z}{\DD t}=\eta\nabla^2 A_z, \label{InductEq2D} \end{equation} where $\DD/\DD t=\partial/\partial t+\uu\cdot\nab$. Here, we assumed that $\uu=\Omega(x,y)\pom$, where $\pom=(x,y,0)$ is the cylindrical position vector and $\Omega=\Omega_0/[1+(\varpi/\varpi_0)^{2n}]^{1/n}$ is the angular velocity with $\varpi_0=5\kpc$, $n=3$, and $\Omega_0=40\Gyr^{-1}$. This experiment demonstrates the extreme winding of the magnetic field. The turbulent magnetic diffusivity is here $5\times10^{-4}\kpc\km\s^{-1}$, corresponding to $1.5\times10^{23}\cm^2\s^{-1}$, which is three orders of magnitude below the canonical estimates. We see about six windings at $1\Gr$ with a thirty-fold increase of $|\BB|$. This amount of winding is not observed in any galaxy. Mean-field dynamo theory was originally developed in the solar context \citep{SK69} 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, which means that they vary with azimuth $\phi$ like $e^{\ii m\phi}$. \cite{Par71} 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. \begin{figure}[t] \includegraphics[width=\textwidth]{psavss} \caption{ Snapshots of field lines together with representations of $|\BB|$ color-coded (in units of its original value) at $0.1$ and $1\Gyr$ for the wind-up problem described in the text. The inset in the upper left corner shows the field lines at the time $0.01\Gr$. }\label{psavss} \end{figure} While dynamo theory can also produce BSS-type fields, it was shown that they are not the most easily excited modes \citep{Elstner+90, BTK90}, unless the turbulence is strongly anisotropic or the dynamo is controlled by strongly anisotropic flow structures \citep{Moss+93}. 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 \citep[e.g.,][see \Fig{fig:martinalvarez}]{MartinAlvarez2021}. \begin{figure}\centering\includegraphics[width=\textwidth]{fig/martin_alvarez_galaxies.png} \caption{Figure~1 from \citet{MartinAlvarez2021}, 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.} \label{fig:martinalvarez} \end{figure} \subsection{The need for magnetic diffusivity: a clear case for steady flows} \label{NeedMagneticDiffusivity} The evolution of the magnetic field $\BB$ is governed by the usual induction equation, \begin{equation} \frac{\partial\BB}{\partial t}=\nab\times\left(\UU\times\BB-\JJ/\sigma\right), \label{InductEq} \end{equation} where $\UU$ is the velocity and $\JJ=\nab\times\BB/\mu_0$ is the current density with $\mu_0$ being the vacuum permeability. \EEq{InductEq} also includes an electric conductivity $\sigma$, 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 $\eta=1/\sigma\mu_0$. %%%%% Text box %%%%%%% \begin{textbox}[t]\section{Characteristic nondimensional numbers} Fluid and magnetic Reynolds numbers and their ratio, the magnetic Prandtl number, are defined as \begin{equation} \Rey=\urms/\nu\kf,\quad \Rm=\urms/\eta\kf,\quad \Pm=\nu/\eta. \label{Numbers} \end{equation} Note that the Reynolds numbers are sometimes based on the length scale, $2\pi/\kf$, 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 $\rho_{\rm i}/\rho_{\rm n}$, i.e., the density ratio of ions and neutrals, enters as another nondimensional number. \end{textbox} %%%%%%%%%%%%%%%% Dynamos convert kinetic energy into magnetic energy through what is called the dynamo instability. It occurs when the magnetic Reynolds number, $\Rm$, \begin{marginnote}[] \entry{Dynamos}{convert kinetic energy into magnetic energy} \end{marginnote} exceeds a certain critical value (see the text box on ``Characteristic nondimensional numbers'' for a definition of $\Rm$). 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: $\BB(\xx,t)=\BB_\lambda(\xx)\,e^{\lambda t}$, where $\lambda$ is the eigenvalue and $\BB_\lambda(\xx)$ the eigenfunction. For steady, mass conserving, compressible flows, \cite{MP85} proved that dynamos (i.e., $\lambda>0$) cannot exist for $\eta=0$, i.e., in the strictly ideal case. This does not preclude dynamos in the astrophysically relevant limit $\eta\to0$, which are called fast dynamos \citep{Sow87}, but it is important to stress that the {\em limit} $\eta\to0$ is quite different from the {\em case} $\eta=0$. The case $\eta=0$ 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 $\eta=0$ can be described using Euler potentials, $\Phi(\xx,t)$ and $\Psi(\xx,t)$, such that $\BB(\xx,t)=\nab\Phi\times\nab\Psi$, where $\Phi(\xx,t)$ and $\Psi(\xx,t)$ obey \citep[e.g.,][]{RP07}, \begin{equation} \frac{\DD\Phi}{\DD t}= \frac{\DD\Psi}{\DD t}=0\quad \Longleftrightarrow\quad \frac{\partial\BB}{\partial t}=\nab\times\left(\UU\times\BB\right). \end{equation} We see that in the special case of 2-D, this equation agrees with the advection--diffusion equation \Eq{InductEq2D}, where $\Phi=A_z$ and $\Psi=z$ has been assumed. In this special case, it is possible to recover the induction equation in the presence of microphysical magnetic diffusion. Dynamos have {\em not} been found in this formulation -- even for three-dimensional turbulent or other flows that allow for dynamo action in the limit $\eta\to0$ \citep{Bra10}. The method forbids even a very weakly diffusive advection of $\Phi$ and $\Psi$, which is needed in any numerical simulation to prevent the formation of infinitely sharp gradients. \begin{marginnote}[] \entry{Dynamos}{have not been found with Euler potentials} \end{marginnote} The tangling of a pre-existing magnetic field for $\eta\to0$ {\em 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, $\BB=\BB(x,y)$, as we have seen in \Fig{psavss}. 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 $\BB_\lambda(\xx)$, which is independent of time. The actual solution $\BB(\xx,t)$ to the case $\eta=0$ 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. \subsection{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 \cite{SB14} for detailed studies of kinematic dynamos in helical and fractionally helical 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 $\eta$ by suitable averaging, no dynamos have been found in the case $\eta=0$, 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 \citep{Bran+19} and for turbulent gravitational collapse \citep{Sur+12, XL20}, as will be discussed later in \Sec{sec:grav_collapse} of this review. \subsection{Early examples of dynamos} \label{SimpleDynamos} \citet{Cow33} 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 \cite{Cow33} was understood as not just a simplification, but as a crucial restriction precluding dynamo action. Even after Parker's discovery \citep{Par55} of what is now called the $\alpha$ effect, it was not generally accepted that dynamos could work even in principle. For example, \cite{Cha56} 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. \begin{marginnote}[] \entry{Small-scale dynamos}{generate magnetic fields at the resistive scale} \end{marginnote} The first rigorous examples of dynamos were presented by \cite{Herzenberg58} and \cite{Backus58}. The former, consisting of two rotors (eddies) with an oblique angle between them, was also realized experimentally \citep{Lowes+Wilkinson63}. 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 $\Rm$. 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, \citet{Rob72} 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 \citep{RDRB14}. The expressions for the four Roberts flows are included in a dedicated text box. Flow~I has maximum kinetic helicity with $\bra{\oo\cdot\uu}=\kf\bra{\uu^2}$, where angle brackets denote volume averaging. Flow~II to has $\oo\cdot\uu=0$ pointwise, while flows~III and IV have vanishing helicity only on average ($\bra{\oo\cdot\uu}=0$), but not pointwise. %%%%----Margin note----%%%%% \begin{marginnote}[] \entry{Kinetic helicity}{$\bra{\oo\cdot\uu}$, where $\oo=\nab\times\uu$ is the vorticity.} \end{marginnote} %%%%-------------------%%%%% %%%%%%%%----Textbox----%%%%%%%% \begin{textbox}[t]\section{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 $\alpha$ effect, which is traditionally believed to operate in galaxies. All four flows have the following $x$ and $y$ components: \begin{equation} u_x = v_0 \, \sin k_0x \, \cos k_0y, \quad u_y = -v_0 \, \cos k_0x \, \sin k_0y, \label{KH21} \end{equation} but the $z$ components are different for each flow: \begin{equation} u_z=w_0 \left\{ \begin{array}{ll} \sin k_0x \, \sin k_0y \quad\mbox{(for flow I)}, \\ \cos k_0x \, \cos k_0y \quad\mbox{(for flow II)}, \\ (\cos 2 k_0x + \cos 2 k_0y)/2 \quad\mbox{(for flow III)}, \\ \sin k_0x \quad\mbox{(for flow IV)}, \label{KH29} \end{array} \right. \end{equation} where $v_0$, $w_0$, and $k_0$ are constants. Particular solutions are obtained by specifying the magnetic Reynolds number $\Rm=v_0/\eta k_0$ and the ratio $w_0/v_0$. The magnetic field must always be three-dimensional and varies in the $z$ direction like $e^{\ii k_z z}$, where $k_z$ is sometimes chosen such that it maximizes the growth rate. \end{textbox} %%%%-------------------%%%%% We summarize the essential features of flows~I--IV in \Tab{TRob}. The resulting mean fields for flow~I can be interpreted in terms of an $\alpha$ effect. The mean field for flow~IV was identified to be due to a negative turbulent magnetic diffusivity \citep{DBM13}. 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 $\meanBB$, which can here be an $xy$ planar average, so $\meanBB=\meanBB(z,t)$ depends just on time and on one spatial coordinate. This defines what we call the fluctuating field $\bb\equiv\BB-\meanBB$. For the Roberts flows, there is no mean flow, i.e., $\meanuu=0$, so the evolution of $\meanBB$ is only governed by the mean electromotive force $\meanEMF\equiv\overline{\uu\times\bb}$, consisting of fluctuations only. %%%%-----Margin note------%%%%% \begin{marginnote}[] \entry{The evolution of $\meanBB$}{is governed by the mean electromotive force $\meanEMF\equiv\overline{\uu\times\bb}$} \end{marginnote} %%%%-------------------%%%%% In all cases, the mean magnetic field along the $z$ axis, $\meanBB_\|\equiv (0,0,\meanB_z)$, vanishes. The perpendicular components, $\meanBB_\perp\equiv(\meanB_x,\meanB_y,0)$ are finite and we only need to focus on the components $\meanEMF_\perp$, $\meanBB_\perp$, and $\meanJJ_\perp$. For flow~I, we have \begin{equation} \meanEMF=\alpha\meanBB-\etat\mu_0\meanJJ, \label{meanEMF} \end{equation} which is a prototype of an $\alpha$ effect dynamo. In Eq. \ref{meanEMF}, $\etat$ is the turbulent magnetic diffusivity, because it adds to the microphysical magnetic diffusivity $\eta$ to give the total magnetic diffusivity $\etaT=\etat+\eta$. For flows~II--IV, the situation is more complicated in that $\alpha$ is now a tensor with vanishing diagonal components (flows~II and III), or it is zero and $\etat$ is negative and can thus lead to exponential growth (flow~IV). For all those flows, it is important to realize that $\alpha$ and $\etat$ are in general scale-dependent, and $\etat$ becomes positive when $\meanBB(z,t)$ has high spatial Fourier components, i.e., for mean fields of smaller scale in the $z$ direction. The dependence of $\meanEMF$ on the mean magnetic field $\meanBB$ and its associated mean current density, $\meanJJ=\nab\times\meanBB/\mu_0$, will be discussed below. %%%%-----Margin note------%%%%% \begin{marginnote}[] \entry{Scale dependence}{$\alpha$ and $\etat$ decrease for small scales.} \end{marginnote} %%%%----------------------%%%%% To determine all components of the tensors $\alpha_{ij}$ and $\eta_{ijk}$ in the representation $\meanemf_i=\alpha_{ij}\meanB_j+\eta_{ijk}\meanB_{j,k}$ with a rank-3 tensor $\eta_{ijk}$, one must solve the equation for the fluctuations in terms of the mean magnetic field. Here, a comma denotes partial differentiation. In \Tab{TRob}, we indicate the form of $\meanEMF_\perp$ for each of the four flows. %%%%-----Margin note------%%%%% \begin{marginnote}[] \entry{Magnetic Reynolds number}{$\Rm=\urms/\eta\kf$, where $\kf$ is the wavenumber of velocity fluctuations} \end{marginnote} %%%%----------------------%%%%% We also indicate the critical values of the magnetic Reynolds number $\Rmc$, above which dynamo action occurs. Here, $\Rmc$ is defined with $\kf=k_0$, and we have fixed $w_0=v_0$ and $k_z=k_0/2$ to ensure that dynamos are possible for all four flows. For flows~II and III, for example, no dynamos are possible for $k_z=k_0$. \begin{table}[t] \tabcolsep7.5pt \caption{ Robert flows~I--IV as simple benchmarks, and their dynamo properties. } \label{TRob} \begin{center} \begin{tabular}{@{}c|c|c|c|c@{}} \hline Flow & helicity & interpretation & $\meanEMF_\perp$ & $\Rmc$ \\ \hline I & yes, and constant & $\alpha$ effect & $\alpha\meanBB_\perp-\etat\meanJJ_\perp$ & 1.99 \\ II & pointwise zero $\alpha$ & \makecell{off-diagonal $\alpha$ tensor \\ with memory effect} & $\pmatrix{0&a\cr a&0}\meanBB_\perp-\etat\meanJJ_\perp$ & 6.86 \\ III & zero only on average & \makecell{pumping effect \\ with memory effect} & $\pmatrix{0&\gamma\cr-\gamma&0}\meanBB_\perp-\etat\meanJJ_\perp$ & 3.92 \\ IV & zero only on average & \makecell{negative ``turbulent'' \\ diffusion} & \makecell{$-\etat\meanJJ_\perp$ with $\etat<0$ \\ on large length scales} & 4.55 \\ \hline \end{tabular} \end{center} \begin{tabnote} In all cases, $k_z=k_0/2$ was used. The values of $\urms$ are 0.866 for flows~I--III and unity for flow~IV. \end{tabnote} \end{table} \subsection{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 spatio-temporal evolution. %%%%-----Margin note------%%%%% \begin{marginnote}[] \entry{Large-scale dynamos}{Create coherent structures on large spatial and temporal scales.} \end{marginnote} %%%%----------------------%%%%% 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 \cite{Gent2013} for a discussion. %%%%-----Margin note------%%%%% \begin{marginnote}[] \entry{The type of averaging}{depends on the magnetic field that can emerge in certain geometries and in certain parameter regimes} \end{marginnote} %%%%----------------------%%%%% For example, in the context of disk galaxies, the azimuthally averaged magnetic field plays an important role. In cylindrical coordinates, $(\varpi,\phi,z)$, such a field depends -- not necessarily smoothly -- on the cylindrical radius $\varpi$ 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, $\BB=\meanBB+\bb$. To avoid the temptation to refer to the non-smoothness of $\meanBB$ as fluctuations, we sometimes refer to ordered and random fields instead \citep{SS22}. 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 $\meanBB$ does also depend on $\phi$. 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 \Sec{SimpleDynamos}, 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 \citep{ABNZ15}. In that case, the mean field depends on $x$, $y$, and $t$. \subsection{Types of large-scale dynamos} Historically, the $\alpha$ effect was the first distinct dynamo effect that was discovered. It emerged in the derivation of mean-field effects in stratified rotating turbulence \citep{SKR66}, but in its essence, it was already obtained by \cite{Par55} using a more phenomenological approach. It is intrinsically connected with the presence of kinetic helicity. Dynamos can work with an $\alpha$ effect alone, in which case one talks about an $\alpha^2$ dynamo. Astrophysical dynamos often have strong shear, but shear alone cannot produce a dynamo. When shear is complemented by an $\alpha$ effect, one talks about an $\alpha\Omega$ dynamo, or even an $\alpha^2\Omega$ dynamo if one wants to emphasize that both $\alpha$ and $\Omega$ effects play a role. At the moment, we do not know for sure whether galactic dynamos are of $\alpha\Omega$ 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 \Tab{TDynType} 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 $\Rm$. This is usually not the case for laminar flows, unless the flow has chaotic streamlines.\footnote{The 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.} \begin{table} \tabcolsep7.5pt \caption{ Summary of different types of dynamos. } \label{TDynType} \begin{center} \begin{tabular}{@{}l|c|c|c@{}} \hline Flow & main dynamo effect & small-scale & fast \\ \hline isotropic turbulence (nonhelical) & small-scale dynamo & \checkmark & \checkmark \\ isotropic turbulence (helical) & $\alpha^2$ dynamo & \checkmark & \checkmark \\ Roberts flow-I (laminar, helical) & \checkmark & --- & --- \\ Roberts flows-II and III (laminar, nonhelical) & time delay & --- & --- \\ Roberts flows-IV &neg turb diff& --- & --- \\ R\"adler effect with shear &$\OO\times\meanJJ$ effect& --- & \checkmark \\ Shear--current effect &$\SSSS\meanJJ$ effect & --- & \checkmark \\ Incoherent $\alpha$--shear effect &fluctuating $\alpha$ effect& \checkmark & \checkmark \\ \hline \end{tabular} \end{center} \begin{tabnote} `Small-scale' refers to possibility that a small-scale dynamo would operate. `Fast' refers to the possibility that the dynamo works in the limit $\eta\to0$, as discussed in \Sec{NeedMagneticDiffusivity}. \end{tabnote} \end{table} \subsubsection{Helical dynamos} The first of the four Roberts flows is maximally helical. It is a prototype of an $\alpha^2$ dynamo, whereby the two nonvanishing horizontally averaged mean-field components, $\meanB_x$ and $\meanB_y$, are being amplified by the $\alpha$ effect. %%%%%%%----Margin note----%%%%%%% \begin{marginnote}[] \entry{Roberts flow I}{this is a prototype of an $\alpha^2$ dynamo} \end{marginnote} %%%%%%%-------------------%%%%%%% If shear is important, and we have an $\alpha\Omega$ dynamo, the dynamo is often oscillatory and can exhibit traveling wave solutions. In oblate bodies such as galaxies, however, $\alpha\Omega$ dynamos are usually non-oscillatory \citep{Par79,Stix75}. \subsubsection{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, $\meanB_x$ and $\meanB_y$, evolve independently of each other. In these examples, they have the same growth rate, but there are other flows, such as the Willis flow \citep{Wil12}, where the growth rates of $\meanB_x$ and $\meanB_y$ are different and only one of the two components grows. This is unusual and different from conventional dynamos of $\alpha\Omega$ or $\alpha^2$ 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 $\nab\times(\alpha\meanBB)$ on the right-hand side of the evolution equation for $\meanBB$. In the presence of shear, for example by a mean flow with constant shear $S=\partial\meanU_y/\partial x$, one has $\partial\meanB_y/\partial t=S\meanB_x+...$, where the ellipsis denotes further terms not relevant to the present discussion. 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., $\partial\meanB_x/\partial t=-\ggamma\cdot\nab\meanB_x$ and $\partial\meanB_y/\partial t=-\ggamma\cdot\nab\meanB_y$ for dynamos where the pumping velocity $\ggamma$ has a memory effect. In its simplest form, a memory effect has an exponential kernel proportional to $e^{-(t-t')/\tau}$ for $t>t'$, and zero otherwise. Here, $t$ is the current time and $t'$ the integration variable. In Fourier space, it leads to a factor $1/(1-\ii\omega\tau)$ in Fourier space, where $\omega$ is the frequency and $\tau$ is the turnover time. When $\gamma\omega\tau>(\eta+\etat)k$, dynamo action becomes possible. \subsubsection{R\"adler and shear--current effects} Early in the history of mean-field dynamo theory, R\"adler found a novel large-scale dynamo effect for rotating, but unstratified bodies, whereby $\meanEMF$ has an extra term proportional to $\OO\times\meanJJ$. Here, $\OO$ is a pseudovector pointing along the rotation axis. The azimuthal velocity is then $\uu_\phi=\pom\times\OO$. However, it is easy to see that the $\OO\times\meanJJ$ term in $\EMF$ does not contribute to the generation of mean-field energy proportional to $\meanBB^2$, because the dot product with $\meanJJ$ 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 R\"adler dynamo: the shear--current effect. Most of the numerical evidence today shows that this effect does not have a favorable sign for dynamo action \citep{BRRK08}. 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 \citep{SB15c}. 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 $\alpha$ effect, as will be discussed below; see also \cite{ZB21} for a detailed assessment of the different possibilities. \subsubsection{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 \citep{Bra05}. 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 \cite{You+08} and \cite{BRRK08} produced such dynamos in simpler shearing box simulations, but gave different interpretations, which we discuss below. One interpretation involves helicity fluctuations, which lead to what is known as an incoherent $\alpha$ effect and, in conjunction with shear, to large-scale dynamo action \citep{VB97}. \begin{marginnote}[] \entry{An incoherent $\alpha$ effect}{implies negative turbulent magnetic diffusivity} \end{marginnote} An incoherent $\alpha$ effect leads to a negative turbulent magnetic diffusivity \citep{Kra76}. In that sense, the incoherent $\alpha$ 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 $\alpha$ 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. \subsection{Small-scale dynamos} \subsubsection{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 $\Pm$. This was first shown by \cite{Kaz68}. Earlier work suggested dynamo action only for $\Pm>1$, but this turns out to be incorrect; see the detailed discussion by \cite{Scheko+04}. %%% --- Margin note--- %%% \begin{marginnote}[] \entry{Nonmagnetic Kolmogorov turbulence}{hardly exists in astrophysics} \end{marginnote} 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 $k^{3/2}$ and has a peak at the resistive wavenumber, provided $\Pm\gg1$. \cite{KA92} found that the peak occurs at a wavenumber $k_\eta$ that depends on the growth rate $\lambda$ through $k_\eta=\sqrt{\gamma/10\eta}$. This is demonstrated in \Fig{pspec_comp_kinsat_Pm1_kf4_rev}, where we plot magnetic and kinetic energy spectra during the kinematic and saturated phases. \begin{figure}[t] \includegraphics[width=.7\textwidth]{pspec_comp_kinsat_Pm1_kf4_rev} \caption{ Magnetic (red lines) and kinetic (blue lines) energy spectra during the kinematic (dashed lines) and saturated (solid lines) phases. Adapted from \cite{Bra+22}. }\label{pspec_comp_kinsat_Pm1_kf4_rev} \end{figure} More recent work on small-scale dynamos is numerical, and will be covered in detail in the following sections. \subsubsection{Effect of ambipolar diffusion} \cite{XL16} 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 \citep{Xu+19}. In those simulations, the microphysical magnetic Prandtl number remained undetermined, because no explicit viscosity or magnetic diffusivity were used. Two-fluid direct numerical simulations \citep{Bra19} 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 $k/k_\nu>1$. For larger ambipolar diffusion coupling, the kinetic energy spectra of neutrals decrease further while those of the ions increase slightly. \section{CATASTROPHIC QUENCHING AND MAGNETIC HELICITY FLUXES} \subsection{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 \Sec{SimpleDynamos}, can then be fully described by a given velocity field, because the induction equation \eq{InductEq} is linear in $\BB$. 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. \begin{marginnote}[] \entry{Catastrophic quenching means}{$\alpha\to0$ as $\Rm\to\infty$} \end{marginnote} The term catastrophic quenching was coined by \cite{BF00} to denote any type of detrimental $\Rm$ dependence of the nonlinear feedback. Ignoring the effect of turbulent diffusion, i.e., the term $-\etat\mu_0\meanJJ$ in \Eq{meanEMF}, \cite{CH96} found numerically that $\alpha\propto(1+\Rm\meanBB^2/\Beq^2)$. This dependence was originally anticipated by \cite{VC92} based on earlier analogous results by \cite{CV91} for the suppression of just $\etat$ in two dimensions. \cite{GD96} explained these results as a consequence of the conservation of magnetic helicity $\bra{\AAA\cdot\BB}$ in 3-D and of $\bra{\AAA^2}$ in 2-D, respectively. However, the dependence of $\alpha$ on $\Rm$ is a peculiar property of the magnetic helicity equation in the presence of a mean magnetic field $\BB_0$. In that case, the magnetic helicity corresponding to the departure from the imposed field, $\bb$, yields, in the steady state, $0=\bra{(\uu\times\BB_0)\cdot\bb}- 2\eta\mu_0\bra{\jj\cdot\bb}$. Using $\bra{(\uu\times\BB_0)\cdot\bb} =-(\bra{\uu\times\bb})\cdot\BB_0=-\alpha\BB_0^2$, we have $\alpha=-\eta\mu_0\bra{\jj\cdot\bb}/\BB_0^2$, so $\alpha\to0$ as $\eta\to0$ or $\Rm\to\infty$. 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 $\alpha$ in the nonlinear regime was first obtained by \cite{Keinigs+83}. 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 \Sec{SimpleDynamos}. We denote that by overbars. The diffusion term $\etat\meanJJ$ can then not be neglected and the relation of \cite{Keinigs+83} can then be written in the form $\alpha-\etat k_{\rm m}=-\eta\mu_0\bra{\jj\cdot\bb}/\BB_0^2$, where $k_{\rm m}=\meanJJ\cdot\meanBB/\meanBB^2$. \subsection{Catastrophically slow saturation in closed domains} In reality, even if the restriction to closed or periodic domains is retained, neither $\alpha$ nor $\etat$ are quenched in such a catastrophic fashion \citep{BRRS08}. %%%%----Margin note----%%% \begin{marginnote}[] \entry{In a closed domain}{final saturation is catastrophically slow} \end{marginnote} %%%%%%--------%%%%%% Instead, the time scale for reaching ultimate saturation is ``catastrophically'' prolonged, i.e., the final saturation obeys \citep{Bra01} \begin{equation} \meanBB^2\approx\bra{\bb^2}\,(\kf/k_1)^2\, \left[1-e^{-2\eta k_1^2(t-t_{\rm sat})}\right] \quad\mbox{(for $t>t_{\rm sat}$)}, \label{FinalSat} \end{equation} where $\kf$ is the typical wavenumber of the turbulence, $k_1=2\pi/L$ is the lowest wavenumber of the cubic domain of size $L^3$, and $t_{\rm sat}$ marks the end of the early kinematic growth phase and the beginning of the slow saturation phase. Let us emphasize once again that in \Eq{FinalSat}, the value of $\eta$ is the microphysical value, which is extremely small in galaxies. This motivates the characterization as ``catastrophically slow''. \begin{textbox}[t]\section{Derivation of \Eq{FinalSat}} The slow saturation behavior after $t=t_{\rm sat}$ is governed by magnetic helicity conservation. The uncurled induction equation reads, $\partial\AAA/\partial t=\UU\times\BB -\eta\mu_0\JJ-\nab\varphi$, where $\varphi$ is the electrostatic potential. The evolution of $\AAA\cdot\BB$ is then given by \begin{equation} \frac{\partial}{\partial t}(\AAA\cdot\BB)=\cancel{2(\UU\times\BB)\cdot\BB} -2\eta\mu_0\JJ\cdot\BB-\nab\cdot\FFF, \label{dABdt} \end{equation} where $\FFF=\EE\times\AAA+\varphi\BB$ is the magnetic helicity flux density. (Note the analogy with the Poynting flux $\EE\times\BB/\mu_0$ of magnetic energy density.) The equations involving $\AAA$ and $\FFF$ depend on the gauge, i.e., on the form of $\varphi$, which can be chosen freely. One frequently adopts the Weyl gauge, $\varphi=0$. Next, we consider spatial averages $\meanBB=\nab\times\meanAA$ and $\meanJJ=\nab\times\meanBB/\mu_0$, along with the resulting fluctuations, $\aaaa=\AAA-\meanAA$, $\bb=\BB-\meanBB$, and $\jj=\JJ-\meanJJ$, so, after averaging, \Eq{dABdt} becomes \begin{equation} \frac{\partial}{\partial t}(\meanAA\cdot\meanBB+\overline{\aaaa\cdot\bb}) =-2\eta\mu_0\left(\meanJJ\cdot\meanBB+\overline{\jj\cdot\bb}\right) -\nab\cdot\meanFFFF. \label{dAmBmdt} \end{equation} Our analysis concerns only the phase when the small-scale dynamo has already saturated (for $t>t_{\rm sat}$), so $\overline{\aaaa\cdot\bb}=\const$ and $\overline{\jj\cdot\bb}=\const$. Assuming the field to be helical with negative helicity at small scales and positive at large scales, we have $\mu_0\overline{\jj\cdot\bb}\approx-\kf\overline{\bb^2}$ and $\meanAA\cdot\meanBB\approx\overline{\BB^2}/k_1 \approx\mu_0\meanJJ\cdot\meanBB/k_1^2$. Inserting this into \Eq{dAmBmdt} and performing volume averaging over the periodic domain, indicated by angle brackets, so that the flux divergence term vanishes, one obtains \begin{equation} \frac{\dd}{\dd t}\bra{\meanBB^2} =-2\eta k_1^2\bra{\meanBB^2}+2\eta k_1\kf\bra{\bb^2}, \label{dBm2dt} \end{equation} the solution of which for $\bra{\bb^2}=\const$ is given by \Eq{FinalSat}. \end{textbox} \EEq{FinalSat} 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 $\alpha$ 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 $\alpha$ effect itself is then not catastrophically quenched \citep{BB02}, 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 \cite{BS05}. \subsection{Alleviating catastrophic quenching by magnetic helicity fluxes} \label{sec:helicityflux} 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, $\overline{\aaaa\cdot\bb}$. The fluctuating field can be determined from the equation for the mean field, which, under the Weyl gauge, can be written as \begin{equation} \frac{\partial\meanAA}{\partial t}=\meanUU\times\meanBB+\meanEMF -\eta\mu_0\meanJJ, \label{dAmdt} \end{equation} This expression results in the following equation for the magnetic helicity of the mean magnetic field: \begin{equation} \frac{\partial}{\partial t}(\meanAA\cdot\meanBB)=\bra{2\meanEMF\cdot\BB} -2\eta\mu_0\meanJJ\cdot\meanBB-\nab\cdot\meanFFFFm. \label{dAmBmdt} \end{equation} The equation for $\overline{\aaaa\cdot\bb}$ must also have corresponding $\meanEMF$ terms, but with $\bra{2\meanEMF\cdot\BB}$ being replaced by $-\bra{2\meanEMF\cdot\BB}$, i.e., \begin{equation} \frac{\partial}{\partial t}(\aaaa\cdot\bb)=-2\meanEMF\cdot\meanBB -2\eta\mu_0\overline{\jj\cdot\bb}-\nab\cdot\meanFFFFf. \label{dabdt} \end{equation} \begin{figure}[t] \includegraphics[width=.8\textwidth]{presults2b} \caption{ $\Rm$ dependence of terms on the RHS of the small-scale magnetic helicity equation \eq{dabdt}. Note that $\nab\cdot\meanFFFFf$ becomes comparable to $2\meanEMF\cdot\meanBB$ and $2\eta\bra{\jj\cdot\bb}$ only for $\Rm>1000$. Adapted from \cite{DSGB13}. }\label{presults2b} \end{figure} In the steady state, there are three terms on the right hand side, $2\meanEMF\cdot\meanBB$, $2\eta\mu_0\overline{\jj\cdot\bb}$, and $\nab\cdot\meanFFFFf$. Simulations by \cite{DSGB13} and \cite{Rin21} 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 \Fig{presults2b}. Thus, one could expect the saturation behavior to become independent of $\Rm$. However, simulations still show that $\meanBB^2$ declines with increasing $\Rm$. This could mean that $\Rm$ 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 $\eta\overline{\jj\cdot\bb}$ does become clearly subdominant. \section{TEST-FIELD METHOD AND NONLOCALITY IN SPACE AND TIME} \subsection{Parameterization of the mean electromotive force} The mean electromotive force, $\meanEMF$, in \Eq{dAmdt} can be expressed in terms of the mean magnetic field as \begin{equation} \meanemf_i=\alpha_{ij}\ast\meanB_j+\eta_{ijk}\ast\partial\meanB_j/\partial x_k, \label{EMFi} \end{equation} where the asterisks denote a convolution over space and time, and $\alpha_{ij}$ and $\eta_{ijk}$ are integral kernels and $x_k$ is the $k$th component of the spatial coordinate, i.e., $\partial\meanB_j/\partial x_k=\meanB_{j,k}$. For planar averages that depend on just one direction, we can write $\meanemf_i=\alpha_{ij}\meanB_j-\eta_{ij}\meanJ_j$, where $\alpha_{ij}$ and $\eta_{ij}$ 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 \cite{Warnecke+18} for a study in the context of three-dimensional convection in a sphere. Most of the published literature ignores the fact that $\alpha_{ij}$ and $\eta_{ij}$ are integral kernels, and approximate the convolution by a multiplication. This approximation then assumes a local connection between $\meanEMF$ 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. \subsection{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 $\alpha_{ij}$ and $\eta_{ijk}$, one has to solve the equations for the fluctuations $\uu$ and $\bb$. The evolution equation for $\bb$ is the most important one and is obtained by subtracting the evolution equation for $\meanBB$ from that for $\BB$. 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 $\meanEMF\equiv\overline{\uu\times\bb}$. This approximation is only valid when $\Rm\ll1$, or when the correlation time $\tau_{\rm cor}$ is extremely short, i.e., when $\tau_{\rm cor}\urms\kf\ll1$. 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 $\alpha_{ij}$ and $\eta_{ijk}$ 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; \citep[see][for a review]{Bran18}. \subsection{Methods for measuring $\alpha$ and other effects} One approach is to use a nonlinear simulation to obtain $\uu$ and $\bb$ in the presence of an additional imposed magnetic field. The resulting $\overline{\uu\times\bb}$ can be related to $\meanBB$ by ignoring $\eta_{ij}$ and $\meanJJ$. This is referred to as the imposed-field method, but it can only be used when $\meanJJ$ vanishes, for example when the averages are zero-dimensional, i.e., volume averages. Another approach is to relate $\overline{\uu\times\bb}$ to the actual $\meanBB$ and $\meanJJ$ by correlating them to each other and computing $\alpha_{ij}$ and $\eta_{ij}$ as correlation coefficients. This approach has been applied both for the integral kernels in the nonlocal approach \citep{BS02,Bendre+22} and the coefficients in the local version \citep{Simard+16}. 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 \Sec{SimpleDynamos}. 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 \citep{Simard+16}. The most reliable method for calculating $\alpha_{ij}$ and $\eta_{ij}$ 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 \cite{Bran+10}. \subsection{Using test fields} The TFM was originally applied by \cite{Sch05,Sch07} to determine the dependence of all transport coefficients in a sphere using longitudinal averages. In that case, one has 9 coefficients for $\alpha_{ij}$ and 18 nonvanishing coefficients for rank three tensor $\eta_{ijk}$ in the representation $\meanemf_i=\alpha_{ij}\meanB_j+\eta_{ijk}\meanB_{j,k}$. (The nine coefficients $\eta_{ij\phi}$ do not enter the problem, because $\phi$ derivatives of $\phi$ averages vanish.) For systems in Cartesian coordinates, planar $xy$ averages are often the most suitable; see \cite{Bra05} and \cite{BRRK08,BRS08} for the first applications. The number of relevant coefficients is then four for $\alpha_{ij}$, because only $i,j=1,2$ are relevant, and four for the rank two tensor $\eta_{ij}$ in the representation $\meanemf_i=\alpha_{ij}\meanB_j-\eta_{ij}\mu_0\meanJ_j$, because there are only two nonvanishing components of $\meanB_{j,k}$ that can be expressed as the two components of the mean current density with $\mu_0\meanJ_x=-\meanB_{y,z}$ and $\mu_0\meanJ_y=\meanB_{x,z}$. In that case, one can use four sinusoidal test fields $(\sin kz,0,0)$, $(\cos kz,0,0)$, as well as $(0,\sin kz,0)$, $(0,\cos kz,0)$. \begin{figure}[t] \includegraphics[width=.7\textwidth]{Gressel} \caption{ Dynamo coefficients from supernova-driven turbulence. The top panel shows the diagonal components of $\alpha$, the middle panel shows the off-diagonal components, $\alpha_{\phi R}$ and $-\alpha_{R\phi}$, contributing to the pumping velocity $\gamma_z=(\alpha_{\phi R}-\alpha_{R\phi})/2$. The bottom panel shows the diagonal components of $\eta_{ij}$, which are seen to increase away from the midplane. Courtesy \cite{Gressel+08AN} }\label{Gressel} \end{figure} In \Fig{Gressel} we reproduce results from the work of \cite{Gressel+08AN}, who performed simulations of dynamos from supernova-driven turbulence in a stratified portion of a galactic disk. Using the TFM, they found that $\etat$ increases away from the midplane and that this leads to turbulent pumping toward the midplane, which is given by $\ggamma\approx-(\tau/2)\nab\etat$. The pumping velocity $\ggamma$ corresponds to off-diagonal components of the $\alpha$ tensor, which they confirmed. In particular, the pumping velocity in the $z$ direction is given by $\gamma_z=(\alpha_{\phi R}-\alpha_{R\phi})/2$, where $R$ denotes the cylindrical radius. \subsection{Nonlocality} It was soon realized that the results for $\alpha_{ij}$ and $\eta_{ij}$ 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 $\alpha_{ij}$ and $\eta_{ij}$ 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 \citep{BRS08, RB12}. 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 \citep{Moss96}. 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 $(1-\ii\omega\tau)^{-1}$, where $\tau$ is the turbulent turnover time. Thus, the electromotive force diminishes with increasing frequency $\omega$, 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. \begin{textbox}[t]\section{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 \begin{equation} \tilde{\cal E}_i(k,\omega)=\frac {\tilde{\alpha}_{ij}(k,\omega)\tilde{B}_j(k,\omega) -\tilde{\eta}_{ij}(k,\omega)\mu_0\tilde{J}_j(k,\omega)} {1-\ii\omega\tau+\ell^2k^2}, \label{EMFifrac} \end{equation} where $\ell^2=a/\kf^2$. Moving the denominator to the left-hand side, the equation becomes \begin{equation} \left(1-\ii\omega\tau+\ell^2k^2\right)\tilde{\cal E}_i(k,\omega)= \tilde{\alpha}_{ij}(k,\omega)\tilde{B}_j(k,\omega) -\tilde{\eta}_{ij}(k,\omega)\mu_0\tilde{J}_j(k,\omega), \label{EMFifrac2} \end{equation} which, back in real space, becomes a simple evolution equation with a diffusion term on the right inside: \begin{equation} \tau\frac{\partial\meanemf_i}{\partial t} =\alpha_{ij}\meanB_j+\eta_{ij}\mu_0\meanJ_j +\ell^2\nabla^2\meanemf_i-\meanemf_i. \label{EMFi_nonloc} \end{equation} This equation for the electromotive force is still only an approximation, because there are in general also larger powers of $\omega$ and $k$, but it provides a substantial improvement over the local formulations. \end{textbox} 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 magnetic field in galactic arm and interarm regions. This has been studied in detail by \cite{Shukurov98} and \cite{Chamandy+13}. 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 \Eq{EMFi}. 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 \Eq{EMFi_nonloc}. This approach was first proposed by \cite{RB12} and was applied to dynamos in spheres \citep{BC18}. This formalism also reproduces the dynamo effect from time delay for the Roberts flows~II and III; see \Sec{SimpleDynamos}, as was demonstrated by \cite{RDRB14}. This is explained in the box ``Dynamos from the memory effect''. \begin{textbox}[t]\section{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 $\gamma$ and turbulent magnetic diffusion $\etat$ is given by $\lambda=-\ii k\gamma-\etat k^2$. Since $\Rey\lambda<0$, the solution can only decay, but it is oscillating with the frequency $\omega=\Imag\lambda=\gamma$. In the presence of a memory effect, $\gamma$ is replaced by $\gamma/(1-\ii\omega\tau)$, where $\tau$ is the memory time. Then, $\lambda\approx-\ii k\gamma\,(1-\ii\omega\tau)-\etat k^2$, and $\Rey\lambda$ can be positive if total. This is the case for the Roberts flow discussed in \Sec{SimpleDynamos}. \end{textbox} %boundary condition \section{SETTING THE SCENE FOR DYNAMO ACTION IN REAL GALAXIES} \subsection{Possibilities for seed magnetic fields} \subsubsection{The need for sufficiently strong seed magnetic fields} In \Sec{sec:history} 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 $\Gamma\approx2\Gyr^{-1}$ \citep{Beck+96}. This means that the mean magnetic field could be amplified by up to 12 orders of magnitude in about $14\Gyr$. To reach the current level of the mean magnetic field of about $3\uG$, we would need a seed magnetic field of about $10^{-18}\G$. This is just about the level that can be expected from the Biermann battery mechanism \citep{Rees87}. The text box ``Battery mechanisms'' provides some more information about the Biermann battery and other mechanisms that can generate magnetic fields in an un-magnetized plasma. %%%%---- Margin note----%%%% \begin{marginnote}[] \entry{Cosmic battery}{A mechanism that can generate magnetic field even when there is none initially.} \end{marginnote} %%%%--------%%%% 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 $\approx3\uG$ 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 $z=1$. \cite{Kronberg+92} found evidence of strong magnetic fields in a $z=0.395$ galaxy. In a more systematic search, \citet{Bernet_2008} found strong RMs in quasar sightlines passing from $z\simeq 1$ galaxy haloes. More recently, \citet{Mao2017} estimated a $\mu$G, kpc-coherent magnetic field in a lensing galaxy at $z\simeq0.46$. To explain smaller-scale magnetic fields at equipartition levels of $3\uG$ would still require dynamo action, but that may just be a small-scale dynamo. Since the typical dynamo growth rates 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. \begin{textbox}[t]\section{Battery mechanisms} \paragraph*{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 \begin{equation} \frac{\partial\AAA}{\partial t} = \frac{c}{q n_{\rm e}}\nab p_{\rm e} \end{equation} where $n_e$ and $p_e$ are the electron number density and pressure, $c$ the speed of light, and $q$ the electron charge. \paragraph*{The Durrive battery.} Massive stars are surrounded by a region of ionized gas. \citet{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: \begin{equation} \frac{\partial\AAA}{\partial t} = \frac{c}{q n_{\rm e}}\nab p_{\rm e} -\frac{c}{qn_{\rm e}}\dot{\bf{p}}_{\rm e} \label{eq:durrive} \end{equation} where $\dot{\bf{p_e}}$ is the rate of momentum transfer to the electrons and \Eq{eq:durrive} 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 \citep{Subramanian_1994,Kulsrud+97,Gnedin2000}. \end{textbox} \subsubsection{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 \cite{Beck+94}. 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 \citep{beck2013,Katz2019,Vazza2017,MartinAlvarez2021}, 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 \cite{Bra00} and corresponding mean-field simulations in \cite{vonRekowski+03}. 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. \subsubsection{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 \citep{Schoeffler+16}. %%%%---- Margin note----%%%% \begin{marginnote}[] \entry{Weibel instability}{Occurs in a nearly homogeneous plasma when there is an anisotropy in velocity space \citep{weibel1959}.} \end{marginnote} %%%%--------%%%% \subsubsection{Primordial seed magnetic fields} \label{subs:bigbangturb} 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 \citep{Widrow02}. 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 $\tilde{\BB}=a^2\BB$, where $a$ is the scale factor of the Universe. When time is being replaced by conformal time, $\tilde{t}=\int\dd t/a(t)$, the MHD equations have their usual form without expansion factors \citep{BEO96}. 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 \cite{Grasso+Rubinstein95} and \cite{Kahniashvili+22} 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 \citep[e.g.,][]{neronov}. 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 \citep{Broderick+12}. Simulations by \cite{Sironi14}, 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 \cite{AlvesBatista+19}, who performed more detailed simulations for individual blazars. The lower limits derived by \cite{neronov}, 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 $10^{-9}\G$ \citep[see][for a review]{subramanian2016}, and could act as seed magnetic fields for any subsequent dynamo processes -- once sufficient kinetic energy becomes available. \Fig{IGMF_hints} shows the expected magnetic field ranges as a function of the typical scale $\lambda_B$ of the magnetic field. Here, the white solid lines describe the decay of a helical magnetic field ($\BB^2\sim t^{-2/3}$) along with the increase of its typical length scale ($\lambda_B\sim t^{2/3}$), so that $\BB^2\lambda_B=\const$. 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 \citep{Hosking_Schekochihin_2022}. This leads to a decay with $\BB^2\sim t^{-10/9}$ and $\lambda_B\sim t^{4/9}$, so that $\BB^4\lambda_B^5=\const$. The first stars are expected to form about $10^{8}\yr$ after the Big Bang, marking the beginning of the reionization epoch. The first galaxies should form at about $10^{9}\yr$ 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 \citep{Kahniashvili+13}. \begin{figure}[t] \includegraphics[width=\textwidth]{fig/IGMF_hints_addedhosking2.png} \caption{ 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 \cite{Korochkin+21}, in which we have added the prediction of \citet{Hosking_Schekochihin_2022} $B\propto\lambda_{B}^{-5/4}$. The asterisk shows the scale where \citet{Hosking_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 Alfv\'en time. }\label{IGMF_hints} \end{figure} \subsubsection{Primordial fields during structure formation} \label{sec:cosmosims} 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. \citet{Vazza2017} 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, \citet{Mtchedlidze+22} explored different initial topologies for the primordial magnetic field, including uniform and scale-invariant inflationary 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 \Fig{Salome}, 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. \begin{figure}[t] \includegraphics[width=\textwidth]{Salome} \caption{ Faraday rotation maps from four cosmological simulations of \cite{Mtchedlidze+22} 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 \cite{Mtchedlidze+22}. }\label{Salome} \end{figure} 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 \citet{Garaldi2021}, 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 $z\sim2$. However, none of their initial conditions contained magnetic helicity, which should not have decayed. \citet{Marinacci2016} and \citet{MartinAlvarez2020} 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 \citet{MartinAlvarez2021}, who traced the evolution of the primordial field and the field injected by stellar sources 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 \citet{Garaldi2021} 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. \subsubsection{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 pre-existing field and amplify it in conjunction with the existing (possibly helical) background turbulence. One could then think of this as some kind of $\alpha\Omega$ dynamo, where the $\Omega$ effect is associated with the large-scale shear generated during the merger. Such simulations were produced by \cite{Roettiger+99}. 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. \subsection{Gravitational collapse dynamos} \label{sec:grav_collapse} By the time the first gravitationally bound structures (stars or galaxies) formed, any primordial turbulence from the processes we mention in \Sec{subs:bigbangturb} 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 \citep{Sur+10,Sur+12,Federrath+11b}, or even $10^{-20}\G$ \citep{Schober+12} 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 \citep{Sur+12}. While flux freezing predicts $|\BB|\propto\rho^{2/3}$, the theory of \citet{XL20}, which includes the turbulent dynamo, predicts that $|\BB|/\rho^{2/3} \propto \rho^{-2\alpha/57-1/6}\approx\rho^{-0.13}$, where $\alpha=-1$ 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 \citep{XL20}, which makes compressional amplification less efficient. However, the present results still leave some questions unanswered. For example, is it possible 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 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 \cite{XL20} does not really address the question of when the dynamo is excited, but discusses instead the nonlinear regime of a supercritical dynamo. \subsubsection{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 \Eq{InductEq} with $\BB$, averaging, and ignoring surface terms, we obtain \begin{equation} \frac{\dd}{\dd t}\bra{\BB^2/2\mu_0}= \bra{\JJ\cdot(\UU\times\BB)}-\eta\mu_0\bra{\JJ^2}. \label{dB2dt} \end{equation} Using $\JJ\cdot(\UU\times\BB)=-\UU\cdot(\JJ\times\BB)$, the first term on the right-hand side of \Eq{dB2dt} can be written as the work against the Lorentz force. Two further refinements can then be employed \citep{BN22}. First, one can decompose $(\UU\times\BB)_i=-U_j\partial_j A_i+U_j\partial_i A_j$ to write $W_{\rm L}=W_{\rm L}^{\rm2D}+W_{\rm L}^{\rm3D}$, where $W_{\rm L}^{\rm2D}=\bra{J_i U_j\partial_j A_i}$ and $W_{\rm L}^{\rm3D}=-\bra{J_i U_j\partial_i A_j}$. The second term, $W_{\rm L}^{\rm3D}$, 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 $\JJ\times\BB$ into contributions from the magnetic pressure force, the tension force, and curvature force. The corresponding work terms are then referred to as $W_{\rm L}^{\rm c}$ (for compression), $W_{\rm L}^\|$ (for tension force, i.e., along the field), and $W_{\rm L}^\perp$ (for the curvature force, i.e., perpendicular to the field). To determine the reality and nature of dynamo action during a turbulent self-gravitational collapse more carefully, \cite{BN22} 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 \Fig{bb3_0020} we show a visualization of the collapsing magnetic field, the diminishing of the vorticity, expressed here as a wavenumber $k_{\oo}=\orms/\urms$, the gain of compressive motions, here expressed through $k_{\,\nab\cdot\uu}=(\nab\cdot\uu)_{\rm rms}/\urms$, along with several other quantities, $k_{p\nab\cdot\uu}=-\bra{p\nab\cdot\uu}/p_0\urms$ and $k_{\oo\cdot\uu}=|\bra{\oo\cdot\uu}|/\urms^2$, characterizing the work done by compression and the amount of kinetic helicity, respectively. A potential problem with the simulations of \cite{BN22} is the relatively short collapse time compared with the turnover time of the turbulence. \begin{figure*}\begin{center} \includegraphics[width=\textwidth]{fig/magfield_wavenumbers_combined.jpeg} \end{center}\caption[]{ Left: Visualization of the magnetic field. Right: Characteristic wavenumbers $k_\omega$ (blue), $k_{\oo\cdot\uu}$ (red), $k_{\,\nab\cdot\uu}$ (solid black), and $k_{p\nab\cdot\uu}$ (dotted black). }\label{bb3_0020}\end{figure*} \section{GALACTIC MEAN-FIELD DYNAMOS} \label{sec:mean-field_models} \subsection{Global magnetic field structure} \subsubsection{Early analytic approaches} The idea that the large-scale magnetic field of galaxies could be explained through an $\alpha\Omega$ dynamo was formulated early on \citep{VR71,Par71}, 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 midplane.) In view of many early claimed discoveries of BSS fields \citep[for a review, see][]{SFW86}, 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 $\alpha\Omega$ approximation was made, i.e., the toroidal field is only generated by differential rotation ($\Omega$ effect) and the $\alpha$ 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. \subsubsection{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 \citep{KR80}. However, such boundary conditions can only be formulated for spheres or ellipsoids, but not for cylinders, for example. \cite{Stix75} employed ellipsoidal coordinates and obtained an axisymmetric solution. Contrary to \cite{Par71}, he found that oscillatory solutions occur only at substantially larger dynamo numbers. \begin{marginnote} \entry{Dynamo number}{Defined as $D=R_\omega~R_\alpha$, where $R_\omega$ and $R_\alpha$ are mean-field magnetic Reynolds numbers associated with differential rotation and $\alpha$-effect, respectively \citep{Stix75}.} \end{marginnote} 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 \citep{Elstner+90}, or in a sphere with vacuum boundaries \citep{BTK90}. 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. \subsection{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 \cite{Poezd+93} using nonlinear $\alpha$ quenching. An important challenge here is to reproduce 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. \cite{Moss+93} 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 \cite{Shukurov98} and \cite{Chamandy+15} for a more detailed discussion and references therein. \subsection{Galactic models with magnetic helicity flux} In \Sec{sec:helicityflux} 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. \cite{SSSB06} 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 at least $300\m\s^{-1}$. More detailed studies have been performed by \cite{Prasad+16}. \begin{figure*}\begin{center} \includegraphics[width=\textwidth]{fig/pcompsat2_Hl232_f1_combined.jpeg} \end{center}\caption[]{ Left: Evolution of the energy of the mean magnetic field in a model with a shear-induced magnetic helicity flux for different values of $\Rm$ (here denoted by $R_{\rm m}$). Adapted from \cite{Bra+05}. Right: Magnetic field evolution in models with advective magnetic helicity fluxes for $\Rm=10^5$ and different values strength of advection. The advection velocity is characterized by the parameters $C_U$. Courtesy of \cite{SSSB06}. }\label{Hl232_f1}\end{figure*} There is some uncertainty regarding the main contributors to the magnetic helicity flux; see the more recent study by \cite{Vishniac+Shapovalov14}. 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 \cite{Vishniac+Cho01} and can lead to episodic magnetic field amplification, especially as $\Rm$ is increased; see \cite{Bra+05} and the left-hand side of \Fig{Hl232_f1}. On the right-hand side of \Fig{Hl232_f1}, we reproduce the simulation result of \cite{SSSB06} with an advective magnetic helicity flux. In the models with insufficient advective flux, the magnetic energy decreases to very small values. The earlier simulations \cite[by][their Figure~7]{Bra+05} with somewhat smaller values of the magnetic Reynolds number showed that the magnetic field can recover after some time, but then, again, it begins to fall off, just like what is seen in \Fig{Hl232_f1}. \subsection{Rotation Measure signature of mean-field models} In \Sec{sec:history} 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 \cite{SFW86}, 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 \citep{KBH89}. In a later review, \cite{Beck+96} 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, \citet{Tabatabaei+2008} found an axisymmetric field in the inner regions and a superposition of axisymmetric and bisymmetric fields in the outer regions. \citet{Beck_2015} observed a weak ($0.5\uG$) axisymmetric field in IC~342, and \citet{Beck_2020} found a dominating ASS field in M31 combined with a six-times weaker BSS component. \begin{figure} \centering \includegraphics[width=\textwidth]{fig/RM_synchrotron_sky.png} \caption{Full-sky view of the Galactic magnetic field in (A) RM of extragalactic sources \citep{Hutschenreuter2022} and (B) synchrotron emission \citep{Bennett2013}.} \label{fig:MWobservations} \end{figure} 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 \citep[e.g.,][]{oppermann2015,Hutschenreuter2022} and synchrotron emission maps of the Milky Way \citep[mainly from WMAP;][-- see \Fig{fig:MWobservations}]{Page2007,Bennett2013}. The model of \citet{sun2008} assumes an axisymmetric spiral with a reversal in the inner 5~kpc. \citet{Jansson_Farrar_2012} include magnetic spiral arms and an X-shaped field in the halo (see also the following section for the observational motivation). \citet{Jaffe2010} also fit magnetic spiral arms to the disk data, also including the random magnetic field component. \citet{Terral_Ferriere_2017}, using analytic forms for the three-dimensional field, conclude that a bisymmetric (m1) halo field best fits the RM data. However, \citet{West2020} found evidence for an axisymmetric ($m=0$) quadrupolar magnetic field with a small net vertical component in RM. A newer analysis \citep{Dickey+22} shows that a combination of an axisymmetric and a bisymmetric mode, based on analytical galactic dynamo models from \citet{Henriksen_2018}, best explains the large-scale morphology of the Galactic RM data. As the quality of the observational data improves, a complex picture of galactic magnetic field morphology emerges, including that of the Milky Way. \subsection{Synchrotron emission from mean-field models} Synchrotron emission provides an important means of measuring the magnetic field in galaxies and comparing models with simulations \citep[see][for a review on bridging dynamo models and observations]{Beck_2019}. Early attempts of computing the polarized synchrotron emission from models were presented by \cite{DB90}, 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 \citep[e.g.,][]{Golla_hummel_1994,Tullmann2000,Krause2006,stein2019,krause2020}. Although the three-dimensional morphology of these fields is unknown, the X-shaped signature can be reproduced by dynamo models that include an outflow \citep{BDMSST93,Elstner+95}. 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 \citep{BDMSST92,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 \cite{Seta+Beck19}, who also compare with numerical simulations of cosmic ray confinement by a local dynamo-generated magnetic field, similar to what was done earlier by \cite{Snodin+06}. \cite{Seta+Beck19} 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 $100\pc$. 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 \citet{Borlaff2021} 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. \subsection{$E$ 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 \citep{Seljak+97,Kamionkowski+97}. 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. Mathematically, $E$ and $B$ are obtained as the real and imaginary parts of a quantity $R(\theta,\phi)$ with \begin{equation} R\equiv E+\ii B=\sum_{\ell=2}^{N_\ell}\sum_{m=-\ell}^{\ell} \tilde{R}_{\ell m} Y_{\ell m}(\theta,\phi). \end{equation} and $\tilde{R}_{\ell m}$ are coefficients that have been computed as \begin{equation} \tilde{R}_{\ell m}=\int_{4\pi} (Q+\ii U)\,_2 Y_{\ell m}^\ast(\theta,\phi)\, \sin\theta\,\dd\theta\,\dd\phi, \label{EBfromQU} \end{equation} with $_2 Y_{\ell m}(\theta,\phi)$ being the spin-2 spherical harmonics and the asterisk denoting the complex conjugate. \begin{figure}[t] \begin{center} \includegraphics[width=\textwidth]{RfromP_shift_last} \end{center} \caption{Left: Galactic $B$ mode polarization. Right: longitudinally averaged $B$ mode polarization. Here, $\theta$ and $\phi$ are Galactic colatitude ($=90\degr-\mbox{latitude}$) and longitude. }\label{RfromP_shift_last} \end{figure} \cite{BB20b} 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 \Fig{RfromP_shift_last}, 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 $\alpha$ 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 $B$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{TURBULENCE SIMULATIONS OF GALACTIC DYNAMOS} \subsection{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 ($50\K10^5\K$) galactic inflows, as well as the hot ($T\simeq10^6\K$), metal-enriched outflows from feedback events \citep{Putman2012}. The CGM also contains colder gas ($T<10^4\K$) 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 \citet{Nelson2020}, who found small-scale cool ($T<10^4$ K) structures in massive ($M\simeq10^{13}-10^{13.5}M_\odot$) 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. \subsection{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 \citep[e.g.][]{White_rees1978}. However, cosmological simulations of galaxy formation \citep[e.g.,][]{dekel2009} showed that a high fraction of the in-flowing gas in high-redshift galaxies and present-day dwarfs is organized in cool ($T\simeq10^5\K$) streams. RM observations suggest that this gas carries a $nG$-level magnetic field \citep[e.g.,][]{Caretti2022}. 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 \citep[e.g.,][--see also \Sec{sec:cosmosims}]{vazza2015,Vazza2017}. 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 \Sec{sec:mean-field_models}, 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 \citep{SS90}. This can lead to an interaction and competition with the quadrupolar magnetic field in the disk. \cite{BDMSST92} 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 Lagrangian 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 \citet{Kotarba2010}, 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 \citet{Beck+2012} suffered from the same issue. \citet{Whittingham2021} 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 \Fig{fig:magnetized_outflow}, taken from their work, shows the evolution of the galaxy post-merger. \begin{figure} \includegraphics[width=\linewidth]{fig/auriga_cgm_mergers_combined.jpeg} \caption{ Left: Figure~2 from \citet{vdv2021}, showing a magnetized galactic outflow. The figure includes a slice through the middle of the central galaxy in magnetic field strength (top) and plasma $\beta$ (bottom). Right: Adapted Figure~3 from \citet{Whittingham2021}, 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 \citep{grand2017}. }\label{fig:magnetized_outflow} \end{figure} One exception to the Lagrangian models is the work of \citet{Rodenbeck2016}, 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. \subsection{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 \citep[e.g.,][]{Zhang2018}, while a few galaxies host AGN-powered winds \citep{Fabian2012,Kormendy_Ho2013,Martin1998,Veilleux2005}. Recent work has indicated that, in some cases, galactic winds can be driven by cosmic ray (CR) pressure \citep{Hanasz2013,girichidis2016}. The material carried by these outflows can magnetize the CGM. In a recent study, \citet{vdv2021} performed direct galaxy evolution simulations while resolving the magnetized CGM, as part of the Auriga project. They found that, while the CGM remained a high-beta plasma, magnetic fields can noticeably change its structure around galaxies, indirectly affecting numerous galactic processes (see left panel of \Fig{fig:magnetized_outflow}). For instance, galactic outflows become more collimated, resulting in less efficient mixing between enriched and un-enriched 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. \citet{Aramburo-Garcia2021} studied the magnetization of haloes by AGN and SN-driven outflows 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. \subsection{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 \Sec{sec:helicityflux}), 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{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 {\em 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 $\alpha$ 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. \begin{summary}[SUMMARY POINTS] \begin{enumerate} \item Small-scale dynamos work in all turbulent astrophysical environments. \item Simulations suggest that the efficiency of large-scale dynamos decreases with increasing resolution. \item The relevance of an $\alpha$ effect dynamo in galaxies remains unclear. \item Modern numerical simulations of galactic magnetic fields tend to take the past evolution of and the interaction with the environment into account. \end{enumerate} \end{summary} \begin{issues}[FUTURE ISSUES] \begin{enumerate} \item The problem of catastrophic quenching remains relevant and will need to be addressed in high-resolution models with realistic boundary conditions. \item 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. \item 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. \item Future numerical models will have to quantify the importance of dynamo action in different stages of a galaxy's evolution. \end{enumerate} \end{issues} \section*{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. \section*{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 (Vetenskapsr{\aa}det, 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 Link\"oping. \bibliography{ref}{} \bibliographystyle{ar-style2} \end{document}