AA YYYY

# Galactic Dynamos

Axel Brandenburg1,2,3,4 & Evangelia Ntormousi5,6 1Nordita, KTH Royal Institute
of Technology and Stockholm University, Hannes Alfvéns väg 12, SE-10691
Stockholm, Sweden  2The Oskar Klein Centre, Department of Astronomy, Stockholm
University, AlbaNova, SE-10691 Stockholm, Sweden  3McWilliams Center for
Cosmology & Department of Physics, Carnegie Mellon University, Pittsburgh, PA
15213, USA  4School of Natural Sciences and Medicine, Ilia State University,
3-5 Cholokashvili Avenue, 0194 Tbilisi, Georgia  5Scuola Normale Superiore,
Piazza dei Cavalieri 7, I-56126 Pisa, Italy  6Institute of Astrophysics,
Foundation for Research and Technology-Hellas, Vasilika Vouton, GR-70013
Heraklion, Greece

###### Abstract

Spiral galaxies, including the Milky Way, have large-scale magnetic fields
with significant energy densities. The dominant theory attributes these
magnetic fields to a large-scale dynamo. We review the current status of
dynamo theory and discuss various numerical simulations designed to explain
either particular aspects of the problem or to reproduce galactic magnetic
fields globally. Our main conclusions can be summarized as follows.  •
Idealized direct numerical simulations produce mean magnetic fields, whose
saturation energy density tends to decline with increasing magnetic Reynolds
number. This is still an unsolved problem. • 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. • Small-scale dynamos are important throughout a galaxy’s life, and
probably provide strong seed fields at early stages. • 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 provide important insights into our understanding of galactic dynamos. We
expect future research in galactic dynamos to focus on the cosmological
history of galaxies and the interaction with the CGM as means of replacing the
idealized boundary conditions used in earlier work.

###### doi:

10.1146/((please add article doi))

###### keywords:

Spiral galaxies, Milky Way, large-scale magnetic fields, circumgalactic
medium, magnetic Reynolds number

††journal:

###### Contents

  1. 1 INTRODUCTION
  2. 2 Observational tracers of galactic magnetic fields
  3. 3 DYNAMOS
    1. 3.1 Historical remarks
    2. 3.2 The need for magnetic diffusivity: the example of steady flows
  4. 4 Characteristic nondimensional numbers
    1. 4.1 Dynamos in turbulent and time-dependent flows
  5. 5 The $\alpha$ effect: example of a mean-field dynamo
    1. 5.1 Early examples of dynamos
  6. 6 The four Roberts flows
    1. 6.1 Large-scale dynamos and averaging
    2. 6.2 Types of large-scale dynamos
      1. 6.2.1 Helical dynamos
      2. 6.2.2 Nonhelical large-scale dynamos
      3. 6.2.3 Rädler and shear–current effects
      4. 6.2.4 Incoherent and shear dynamo effects
    3. 6.3 Small-scale dynamos
      1. 6.3.1 Early work on the subject
      2. 6.3.2 Effect of ambipolar diffusion
  7. 7 CATASTROPHIC QUENCHING AND MAGNETIC HELICITY FLUXES
    1. 7.1 Catastrophic quenching for uniform magnetic fields
    2. 7.2 Catastrophically slow saturation in closed domains
  8. 8 Derivation of Equation 5
    1. 8.1 Alleviating catastrophic quenching by magnetic helicity fluxes
  9. 9 MEAN-FIELD COEFFICIENTS AND NONLOCALITY
    1. 9.1 Parameterization of the mean electromotive force
    2. 9.2 Mean-field coefficients
    3. 9.3 Methods for measuring $\alpha$ and other effects
    4. 9.4 Using test fields
    5. 9.5 Nonlocality in space and time
  10. 10 Evolution equation for nonlocality in space and time
  11. 11 Dynamos from the memory effect
  12. 12 SETTING THE SCENE FOR DYNAMO ACTION IN REAL GALAXIES
    1. 12.1 Possibilities for seed magnetic fields
      1. 12.1.1 The need for sufficiently strong seed magnetic fields
  13. 13 Battery mechanisms
    1. 13.0.1 Small-scale dynamos as a seed for the large-scale dynamo
    2. 13.0.2 Battery versus plasma instabilities
    3. 13.0.3 Primordial seed magnetic fields
    4. 13.0.4 Primordial fields during structure formation
    5. 13.0.5 Possible importance of cluster mergers
    6. 13.1 Dynamos from gravitational collapse and other instabilities
      1. 13.1.1 Nature of collapse dynamos
      2. 13.1.2 Magneto-buoyancy and magneto-rotational instabilities
  14. 14 GALACTIC MEAN-FIELD DYNAMOS
    1. 14.1 Global magnetic field structure
      1. 14.1.1 Early analytic approaches
      2. 14.1.2 Boundary conditions
    2. 14.2 Dynamo models for specific galaxies
    3. 14.3 Galactic models with magnetic helicity flux
    4. 14.4 Galactic rotation measure signature
    5. 14.5 Synchrotron emission from mean-field models
    6. 14.6 $E$ and $B$ polarizations
  15. 15 TURBULENCE SIMULATIONS OF GALACTIC DYNAMOS
    1. 15.1 Physical parameters of the ISM
    2. 15.2 Numerical approaches
    3. 15.3 Local dynamo simulations of galaxy portions
    4. 15.4 Global isolated galaxy simulations
  16. 16 INTERACTION WITH THE CGM
    1. 16.1 Magnetization of the galaxy by inflows
    2. 16.2 Magnetization of the CGM by outflows
    3. 16.3 The impact of the environment on the galactic dynamo
  17. 17 CONCLUSIONS

##  1 INTRODUCTION

Many spiral galaxies have microgauss magnetic fields, so that their magnetic
energy densities are comparable to the thermal, kinetic, and cosmic ray energy
densities; see Ruzmaikin et al. (1988) for an early book on the subject, and
Shukurov & Subramanian (2022) for a recent one; hereafter SS22. Similar
magnetic field strengths have also been detected in galaxies at larger
redshifts up to $z\simeq 1$ (Bernet et al., 2008, Mao et al., 2017).

Galactic magnetic fields often also show large-scale coherence. The first
evidence for a global Galactic magnetic field comes from optical polarization
(Hiltner, 1949, Hall, 1949). The existence of magnetic fields for other
galaxies was later confirmed using synchrotron emission (Segalovitz et al.,
1976), which showed systematic large-scale magnetic fields roughly in the
direction of the galactic spiral arms. There has long been a debate about the
origin of such magnetic fields: are they primordial or dynamo-generated, or
perhaps a combination of the two? [] Dynamosconvert kinetic energy into
magnetic energy Over the past few decades, attention has shifted 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 timescales for their generation too long for the
simulations presented so far.

In this review, we focus on galactic dynamos and highlight the main
developments since the time of the review of Beck et al. (1996). The broader
problem of galactic magnetism that was addressed there will not be reviewed;
we refer readers to the reviews by Beck (2001, 2015a), Beck & Wielebinski
(2013), and Han (2017), and the book by SS22. A review of astrophysical
dynamos covering the era before 2005 is given by Brandenburg & Subramanian
(2005a). The mathematics of small-scale turbulent dynamos is explained in the
book by Zeldovich et al. (1990), and those in partially ionized plasmas are
discussed by Xu & Lazarian (2021). We also recommend the reviews on ISM
magnetic fields by Crutcher (2012), Hennebelle & Inutsuka (2019), and Pattle
et al. (2022).

At the time of the review by Beck et al. (1996), 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 ($\mbox{\rm Re}_{\rm
M}$) becomes large (Vainshtein & Cattaneo, 1992, Cattaneo & Hughes, 1996). []
Catastrophic quenching means$\alpha\to 0$ as $\mbox{\rm Re}_{\rm M}\to\infty$,
so it limits the large-scale field at large $\mbox{\rm Re}_{\rm M}$
Specifically, the mean-field effect in question has been termed the $\alpha$
effect, which quantifies the component of the mean electromagnetic force in
the direction of the mean magnetic field. More generally, however, it means
that the resulting mean (or large-scale) magnetic field cannot be generated at
the expected amplitudes or time scales. This led to a major crisis in dynamo
theory, questioning the possibility of an $\alpha$ 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 explored. [] Magnetic helicityVolume-integrated dot
product of magnetic vector potential $\bm{A}$ and magnetic field $\bm{B}$,
$\int\bm{A}\cdot\bm{B}dV$ At the same time, there has been significant
progress in performing realistic three-dimensional (3-D) 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.

[t]

##  2 Observational tracers of galactic magnetic fields

##### Dust extinction / emission polarization.

Elongated dust grains in the interstellar medium tend to align their minor
axes with the mean magnetic field direction (Davis & Greenstein, 1951). As a
result, the light they emit in infrared wavelengths is polarized, with the
polarization direction perpendicular to the mean direction of the magnetic
field. Only recently has it been possible to trace extragalactic magnetic
fields with this method (e.g.,the recent survey by Lopez-Rodriguez et al.,
2022).

Since starlight emission is unpolarized, we can also measure dust polarization
in absorption against stellar sources. If the distances to the stars are
known, it is possible to map the magnetic field at different locations along
the line-of-sight (e.g., the Polar-Areas Stellar-Imaging in Polarization High-
Accuracy Experiment (PASIPHAE) survey, Tassis et al., 2018).

##### Synchrotron emission.

Ultra-relativistic cosmic ray particles emit polarized synchrotron radiation
in radio wavelengths as they gyrate around the galactic magnetic field.
Synchrotron emission, which yields the plane-of-the-sky magnetic field
component in the warm/hot ISM, has been the tracer of choice for studying
extragalactic magnetic fields (see, e.g., Beck, 2012, for a review).

In general, linear polarization is measured through the Stokes parameters $I$,
$Q$, and $U$. Then the observed polarization angle $\chi$ is calculated from
the expression $\chi=\frac{1}{2}\arctan(U/Q)$ and the polarization fraction is
$p=\sqrt{Q^{2}+U^{2}}/I$.

##### Faraday Rotation.

When polarized radiation passes through an ionized magnetized medium, the
plane of polarization is rotated by an angle $\Delta\chi=RM\lambda^{2}$, where
$\lambda$ is the observed wavelength and $RM=0.81\int_{0}^{D}n_{\rm
e}B_{\parallel}{\rm d}{}l$ is in units of radians/$m^{2}$. Here, $D$ is the
distance to the source in pc, $n_{\rm e}$ is the electron density in $\,{\rm
cm}^{-3}$, and $B_{\parallel}$ is the line-of-sight magnetic field of the
medium in microgauss (Burn, 1966, Brentjens & de Bruyn, 2005). If we have an
estimate of the electron density, then Faraday rotation can yield the line-of-
sight magnetic field.

##  3 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.

###  3.1 Historical remarks

In Section 14 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 external spiral galaxies began
to reveal the presence of large-scale ordered magnetic fields broadly aligned
with the galactic spiral pattern. At the time, an obvious possibility was that
these magnetic fields were the result of winding up a pre-existing, large-
scale field. This idea goes back to Piddington (1964) and Ôki et al. (1964),
and is now commonly termed the primordial origin of the magnetic field. The
resulting magnetic field is then expected to have the form of a bisymmetric
spiral (BSS). The BSS form has a characteristic signature in the Faraday
Rotation Measure (RM): when we observe an external galaxy almost, but not
exactly, face-on, the line-of-sight magnetic field measured through RM probes
the azimuthal component of the galactic field – see Figure 1 for a sketch.
Using this technique, Tosa & Fujimoto (1978), who used a sketch similar to
ours in Figure 1, found evidence for a BSS in M51. In an early review on
galactic magnetic fields, Sofue et al. (1986) contrasted BSS with an
axisymmetric spiral (ASS), expected from mean-field dynamo theory. Figure 1
also sketches the expected ASS signature.

Figure 1:  Sketch of the rotation measure (RM) signature of a tilted galaxy
with a ring, axisymmetric, or bisymmetric magnetic field. The inclination $i$
is the angle between the $z$ axis, indicated in the top left panel, and the
line of sight. Only when $i\neq 0$ can one see the RM signature as sketched in
the bottom panels.

However, a purely primordial origin of galactic magnetic fields implies that a
tremendous amount of winding has occurred over the past $14\,{\rm Gyr}$ due to
the shear induced by the differential rotation of the galaxy. For example, in
the solar neighborhood, the angular velocity of the Galaxy is $\approx
30\,{\rm Gyr}^{-1}$, i.e., the rotation period is $(2\pi/30)\,{\rm Gyr}\approx
0.2\,{\rm Gyr}$. This yields 70 revolutions in $14\,{\rm Gyr}$, so we would
expect the magnetic field to be strongly wound up. Figure 2 gives a
quantitative illustration of this wind-up process. It shows color scale images
of $|\bm{B}|$ 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
$\bm{B}={\bm{\nabla}}\times(\hat{\mbox{\boldmath$z$}}{}A_{z})$. To obtain the
result shown in Figure 2, we solved the two-dimensional (2-D) induction
equation, which corresponds to an advection–diffusion equation of the form

$\frac{{\rm D}{}A_{z}}{{\rm D}{}t}=\eta\nabla^{2}A_{z},$ (1)

where ${\rm D}{}/{\rm D}{}t=\partial/\partial t+\bm{U}\cdot{\bm{\nabla}}$.
Here, we assumed that $\bm{U}=\Omega(x,y)\mbox{\boldmath$\varpi$}{}$, where
$\mbox{\boldmath$\varpi$}{}=(x,y,0)$ is the cylindrical position vector and
$\Omega=\Omega_{0}/[1+(\varpi/\varpi_{0})^{n}]^{1/n}$ is the angular velocity
with $\varpi_{0}=5\,{\rm kpc}$, $n=3$, and $\Omega_{0}=40\,{\rm Gyr}^{-1}$.
This experiment demonstrates the extreme winding of the magnetic field. The
turbulent magnetic diffusivity is here $5\times 10^{-4}\,{\rm kpc}\,\,{\rm
km}\,{\rm s}^{-1}$, corresponding to $1.5\times 10^{23}\,{\rm cm}^{2}\,{\rm
s}^{-1}$, which is three orders of magnitude below the canonical estimates
(Brandenburg et al., 1993, SS22). We see about six windings at $1\,{\rm Gyr}$
with a thirty-fold increase of $|\bm{B}|$. This amount of winding is not
observed in any galaxy.

Mean-field dynamo theory was originally developed in the solar context
(Steenbeck & Krause, 1969) and can predict both axisymmetric and
nonaxisymmetric magnetic fields (Baryshnikova et al., 1987). Parker (1971) was
the first to show that the most easily excited axisymmetric large-scale
magnetic fields in oblate bodies such as galaxies have an azimuthal component
that is symmetric about the midplane, i.e., the fields have quadrupolar
symmetry – in contrast to the dipolar symmetry that is often found in
spherical bodies such as the Earth.

Figure 2:  Snapshots of field lines together with representations of
$|\bm{B}|$ color-coded (in units of its original value) at $0.1$ and $1\,{\rm
Gyr}$ for the wind-up problem described in the text. The inset in the upper
left corner of the left panel shows the field lines at the time $0.01\,{\rm
Gyr}$.

While dynamo theory can also produce BSS-type fields (Krasheninnikova et al.,
1989), they are not the most easily excited modes (Elstner et al., 1990,
Brandenburg et al., 1990), unless the turbulence is strongly anisotropic or
the dynamo is controlled by strongly anisotropic flow structures (Moss et al.,
1993). Today, the significance of primordial magnetic fields is still not
resolved, and global 3-D numerical simulations suggest that both primordial
and magnetic fields of astrophysical origin may be present in typical galaxies
(e.g., Martin-Alvarez et al., 2021, see Figure 3).

Figure 3: Section of Figure 1 from Martin-Alvarez et al. (2021), showing a
cosmological galaxy model evolved with two different initial magnetic
fields:primordial or “injected” on small scales by stellar feedback. The left
panel shows a color-composite mock observation in the optical, the middle
panel shows dust absorption along the line of sight, and the one on the right
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.

###  3.2 The need for magnetic diffusivity: the example of steady flows

The evolution of the magnetic field $\bm{B}$ is governed by the usual
induction equation,

$\frac{\partial\bm{B}}{\partial
t}={\bm{\nabla}}\times\left(\bm{U}\times\bm{B}-\bm{J}/\sigma\right),$ (2)

where $\bm{U}$ is the velocity and $\bm{J}={\bm{\nabla}}\times\bm{B}/\mu_{0}$
is the current density with $\mu_{0}$ being the vacuum permeability. Equation
2 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 resistivity is $1/\sigma$, and the microphysical magnetic diffusivity
is then given by $\eta=1/\sigma\mu_{0}$.

[t]

##  4 Characteristic nondimensional numbers

Fluid and magnetic Reynolds numbers and their ratio, the magnetic Prandtl
number, are defined as

$\mbox{\rm Re}=u_{\rm rms}/\nu k_{\rm f},\quad\mbox{\rm Re}_{\rm M}=u_{\rm
rms}/\eta k_{\rm f},\quad\mbox{\rm Pr}_{\rm M}=\nu/\eta,$ (SB1)

where $u_{\rm rms}$ is the root-mean square velocity, $\nu$ is the
(microphysical) kinematic viscosity, $\eta$ is the (microphysical) magnetic
diffusivity, and $k_{\rm f}$ is the characteristic flow wavenumber. Note that
the Reynolds numbers are sometimes based on the length scale, $2\pi/k_{\rm
f}$, 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 enters as another nondimensional number.

Dynamos convert kinetic energy into magnetic energy through what is termed the
dynamo instability. It occurs when the magnetic Reynolds number, $\mbox{\rm
Re}_{\rm M}$, exceeds a certain critical value (for the definition of
$\mbox{\rm Re}_{\rm M}$, see the text box titled “Characteristic
nondimensional numbers”). Here, $k_{\rm f}$ 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:
$\bm{B}(\bm{x},t)=\bm{B}_{\lambda}(\bm{x})\,e^{\lambda t}$, where $\lambda$ is
the eigenvalue and $\bm{B}_{\lambda}(\bm{x})$ the eigenfunction. For steady,
mass conserving, compressible flows, Moffatt & Proctor (1985) 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 $\eta\to 0$, which are called fast dynamos (Soward, 1987), but it is
important to stress that the limit $\eta\to 0$ is quite different from the
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 of academic interest only, although it can be described using Euler
potentials, $\Phi(\bm{x},t)$ and $\Psi(\bm{x},t)$, such that
$\bm{B}(\bm{x},t)={\bm{\nabla}}\Phi\times{\bm{\nabla}}\Psi$, where
$\Phi(\bm{x},t)$ and $\Psi(\bm{x},t)$ obey (e.g., Rosswog & Price, 2007),

$\frac{{\rm D}{}\Phi}{{\rm D}{}t}=\frac{{\rm D}{}\Psi}{{\rm
D}{}t}=0\quad\Longleftrightarrow\quad\frac{\partial\bm{B}}{\partial
t}={\bm{\nabla}}\times\left(\bm{U}\times\bm{B}\right).$ (3)

We see that in the special case of 2-D, this equation agrees with the
advection–diffusion equation Equation 1, where $\Phi=A_{z}$ and $\Psi=z$ have
been assumed. In this special case, it is possible to recover the induction
equation in the presence of microphysical magnetic diffusion.

Dynamos have not been found in this formulation – even for 3-D turbulent or
other flows that allow for dynamo action in the limit $\eta\to 0$
(Brandenburg, 2010). The method forbids even a very weakly diffusive advection
of $\Phi$ and $\Psi$, which would be needed in any numerical simulation to
prevent the formation of infinitely sharp gradients.

To understand the problem with the case $\eta=0$, let us now discuss instead
the limit $\eta\to 0$. The tangling of a pre-existing magnetic field 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 2-D field,
$\bm{B}=\bm{B}(x,y)$, as we have seen in Figure 2. The magnetic field is
amplified by perpetual stretching, so it continuously develops smaller scale
structures. This continuous change in the field structure makes it impossible
to describe the evolving field by an eigenfunction of the form
$\bm{B}_{\lambda}(\bm{x})$, which is independent of time. The actual solution
$\bm{B}(\bm{x},t)$ in the case of $\eta=0$ 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.

###  4.1 Dynamos in turbulent and time-dependent flows

All astrophysically relevant flows are time-dependent. Turbulent flows can be
statistically steady, so one can still determine an eigenvalue problem by
averaging over the fluctuations; see Subramanian & Brandenburg (2014) for
detailed studies of kinematic dynamos in helical and fractionally helical
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—both for
decaying turbulence (Brandenburg et al., 2019) and for turbulent gravitational
collapse (Sur et al., 2012, Xu & Lazarian, 2020) has been reported, as is
discussed in Section 13.1 of this review.

[t]

##  5 The $\alpha$ effect: example of a mean-field dynamo

The $\alpha$ effect quantifies how a systematic twist (or swirl) in a
turbulent flow produces secondary magnetic fields around a primary field in a
specific direction. An example is the production of a poloidal field from a
toroidal field, as is believed to occur through cyclonic convection in the Sun
(Parker, 1955). Mathematically, this is described by a contribution to the
averaged electromotive force, $\overline{\bm{u}\times\bm{b}}$, in the
direction of the main magnetic field ${\overline{\bm{B}}}$, i.e.,
$\overline{\bm{u}\times\bm{b}}=\alpha{\overline{\bm{B}}}+$ higher-order
derivatives. It is called $\alpha$ effect, because of the historically chosen
coefficient $\alpha$. Here, $\bm{u}$ and $\bm{b}$ are fluctuations of $\bm{U}$
and $\bm{B}$, respectively. The type of averaging depends on the problem at
hand and will be discussed later in Section 6.1. Using
${\overline{\bm{B}}}={\bm{\nabla}}\times{\overline{\bm{A}}}$, and ignoring for
now mean flows such as the galactic differential rotation, the averaged
uncurled induction equation 2 takes the form

$\partial{\overline{\bm{A}}}/\partial
t=\alpha{\bm{\nabla}}\times{\overline{\bm{A}}}+\eta\nabla^{2}{\overline{\bm{A}}},$
(SB2)

where $\eta={\rm const}{}$ has been assumed. For $\alpha={\rm const}{}$,
solutions are proportional to the eigenfunctions of the curl operator, for
example ${\overline{\bm{A}}}=(\sin kz,\,\cos kz,\,0)$, which satisfies
${\bm{\nabla}}\times{\overline{\bm{A}}}=k{\overline{\bm{A}}}$. Seeking
solutions of the form ${\overline{\bm{A}}}\propto\bm{A}_{0}e^{{\rm
i}kz+\lambda t}$, with $\bm{A}_{0}$ being the eigenfunction, yields the
dispersion relation $\lambda=\alpha k-\eta k^{2}$, and therefore self-excited
solutions for $\alpha>\eta k$.

The $\alpha$ effect thus explains the exponential growth of a weak mean
magnetic field. We recall that the full magnetic field has fluctuations, but
they are usually growing at the same rate as the mean field. Since the
magnetic field is a pseudovector, but the electromotive force is an ordinary
vector, $\alpha$ must be a pseudoscalar, i.e., its sign changes when viewed in
a mirror. An $\alpha$ effect can occur when the system is governed by a
specific pseudoscalar (Krause & Rädler, 1980). As an example, systems governed
by gravity $\bm{g}$ and angular velocity $\bm{\Omega}$ have a finite
pseudoscalar given by $\bm{g}\cdot\bm{\Omega}$. The existence of this
pseudoscalar is is what caused the systematic twist or swirl in the flow
which, in turn, produces the $\alpha$ effect in galaxies. Twist or swirl can
also occur through corresponding driving and through initial conditions. It is
then characterized by the kinetic helicity.

One of the higher-order derivative contributions to
$\overline{\bm{u}\times\bm{b}}$ is from turbulent diffusion, so one has
$\overline{\bm{u}\times\bm{b}}=\alpha{\overline{\bm{B}}}-\eta_{\rm
t}\mu_{0}{\overline{\bm{J}}}$, where $\eta_{\rm t}$ is the turbulent magnetic
diffusivity. The second term nearly balances the former and is therefore
important. We also note that two further generalizations to this formulation
will be discussed in Section 9.1: (i) $\alpha$ and $\eta_{\rm t}$ become
tensors and (ii) the multiplications become convolutions.

###  5.1 Early examples of dynamos

Cowling (1933) formulated an anti-dynamo theorem, concluding that “The theory
proposed by Sir Joseph Larmor, that the magnetic field of a sunspot is
maintained by the currents it induces in moving matter, is examined and shown
to be faulty.” At that time, there was no hint that the solution to the
problem could lie in the third dimension. It was only later that the use of a
2-D analysis in the work of Cowling (1933) was understood as not just a
simplification, but as a crucial restriction precluding dynamo action. Even
after Parker’s discovery (Parker, 1955) of what is now called the $\alpha$
effect (see the text box on “The $\alpha$ effect”), it was not generally
accepted that dynamos could work even in principle. For example, Chandrasekhar
(1956) found that particular flow geometries could prolong the resistive decay
time to half a billion years when using the magnetic diffusivity of the
Earth’s outer core. He speculated that the Earth’s magnetic field could be
explained in that way, rather than by a dynamo. His speculation suggests that
the existence of dynamos was far from being widely accepted at that time.

[] Small-scale dynamosgenerate magnetic fields at the resistive scale in the
kinematic regime, but at a fraction of the forcing scale otherwise

[] Large-scale dynamoscreate coherent structures on large spatial and temporal
scales

The first rigorous examples of dynamos were presented by Herzenberg (1958) and
Backus (1958). The former, consisting of two rotors (eddies) with an angle
between their axes, was also realized experimentally (Lowes & Wilkinson,
1963). However, the length scale of those magnetic fields was only comparable
to that of the rotors. This property could classify the Herzenberg result as a
small-scale dynamo. During the kinematic phase, small-scale dynamos produce a
field at the resistive scale and can later grow to the scale of turbulent
eddies as the dynamo saturates. They do not possess a mean field.

In the early 1970s, Roberts (1972) showed that several non-planar 2-D,
spatially periodic steady flows can exhibit dynamo action. These flows are now
called Roberts flows I–IV. They are large-scale dynamos and their properties
have been investigated with modern tools (Rheinhardt et al., 2014). The
expressions for the four Roberts flows are included in a dedicated text box.
Flow I has maximum kinetic helicity with
$\langle\bm{\omega}\cdot\bm{u}\rangle=k_{\rm f}\langle\bm{u}^{2}\rangle$,
where angle brackets denote volume averaging. Flow II has
$\bm{\omega}\cdot\bm{u}=0$ pointwise, while flows III and IV have vanishing
helicity only on average ($\langle\bm{\omega}\cdot\bm{u}\rangle=0$), but not
pointwise. [] Kinetic helicity$\langle\bm{\omega}\cdot\bm{u}\rangle$, where
$\bm{\omega}={\bm{\nabla}}\times\bm{u}$ is the vorticity

[t]

##  6 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:

$u_{x}=v_{0}\,\sin k_{0}x\,\cos k_{0}y,\quad u_{y}=-v_{0}\,\cos k_{0}x\,\sin
k_{0}y,$ (SB3)

but the $z$ components are different for each flow:

$u_{z}=w_{0}\left\\{\begin{array}[]{ll}\sin k_{0}x\,\sin k_{0}y\quad\mbox{(for
flow I)},\\\ \cos k_{0}x\,\cos k_{0}y\quad\mbox{(for flow II)},\\\ (\cos
2k_{0}x+\cos 2k_{0}y)/2\quad\mbox{(for flow III)},\\\ \sin
k_{0}x\quad\mbox{(for flow IV)},\end{array}\right.$ (SB4)

where $v_{0}$, $w_{0}$, and $k_{0}$ are constants. Particular solutions are
obtained by specifying the magnetic Reynolds number $\mbox{\rm Re}_{\rm
M}=v_{0}/\eta k_{0}$ and the ratio $w_{0}/v_{0}$. The magnetic field must
always be 3-D and varies in the $z$ direction like $e^{{\rm i}k_{z}z}$, where
$k_{z}$ is sometimes chosen such that it maximizes the growth rate.

We summarize the essential features of flows I–IV in Table 1. The resulting
mean fields for flow I can be interpreted in terms of an $\alpha$ effect; see
the text box on “The $\alpha$ effect”. The mean field for flow IV was
identified to be due to a negative turbulent magnetic diffusivity (Devlen et
al., 2013). The origin of the mean field for flows II and III involves the
combination of 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 ${\overline{\bm{B}}}$, which can here be an $xy$ planar average, so
${\overline{\bm{B}}}={\overline{\bm{B}}}(z,t)$ depends just on time and on one
spatial coordinate. This defines what we call the fluctuating field
$\bm{b}\equiv\bm{B}-{\overline{\bm{B}}}$. For the Roberts flows, there is no
mean flow, i.e., $\overline{\bm{U}}=0$, so the evolution of
${\overline{\bm{B}}}$ is only governed by the mean electromotive force
$\overline{\mathbfcal{E}}\equiv\overline{\bm{u}\times\bm{b}}$, consisting of
fluctuations only.

In all cases, the mean magnetic field along the $z$ axis,
${\overline{\bm{B}}}_{\|}\equiv(0,0,\overline{B}_{z})$, vanishes. The
perpendicular components,
${\overline{\bm{B}}}_{\perp}\equiv(\overline{B}_{x},\overline{B}_{y},0)$ are
finite and we only need to focus on the components
$\overline{\mathbfcal{E}}_{\perp}$, ${\overline{\bm{B}}}_{\perp}$, and
${\overline{\bm{J}}}_{\perp}$. For flow I, which is maximally helical, there
is a systematic swirl. As we have explained in the text box on “The $\alpha$
effect,” as a result of this systematic swirl, flow I produces an $\alpha$
effect, and thus, we have

$\overline{\mathbfcal{E}}=\alpha{\overline{\bm{B}}}-\eta_{\rm
t}\mu_{0}{\overline{\bm{J}}}.$ (4)

In Equation 4, $\eta_{\rm t}$ is the turbulent magnetic diffusivity, because
it adds to the microphysical magnetic diffusivity $\eta$ to give the total
magnetic diffusivity $\eta_{\rm T}=\eta_{\rm t}+\eta$. For flows II and III,
the situation is more complicated in that $\alpha$ is now a tensor with
vanishing diagonal components. For flow IV $\alpha$ is zero and $\eta_{\rm t}$
is negative, which can thus lead to exponential growth. For all those flows,
it is important to realize that $\alpha$ and $\eta_{\rm t}$ are in general
scale-dependent, and $\eta_{\rm t}$ becomes positive when
${\overline{\bm{B}}}(z,t)$ has high spatial Fourier components, i.e., for mean
fields of smaller scale in the $z$ direction. The dependence of
$\overline{\mathbfcal{E}}$ on the mean magnetic field ${\overline{\bm{B}}}$
and its associated mean current density,
${\overline{\bm{J}}}={\bm{\nabla}}\times{\overline{\bm{B}}}/\mu_{0}$, is
discussed below.

[] Scale dependence$\alpha$ and $\eta_{\rm t}$ decrease toward smaller scales

To determine all components of the tensors $\alpha_{ij}$ and $\eta_{ijk}$ in
the representation $\overline{\cal
E}{}_{i}=\alpha_{ij}\overline{B}_{j}+\eta_{ijk}\overline{B}_{j,k}$ with a rank
three 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 Table 1, we indicate the form of
$\overline{\mathbfcal{E}}_{\perp}$ for each of the four flows. [] Magnetic
Reynolds numberDefined here as $\mbox{\rm Re}_{\rm M}=u_{\rm rms}/\eta k_{\rm
f}$, where $k_{\rm f}$ is the wavenumber of velocity fluctuations We also
indicate the critical values of the magnetic Reynolds number $\mbox{\rm
Re}_{\rm M}^{\rm crit}$, above which dynamo action occurs. Here, $\mbox{\rm
Re}_{\rm M}^{\rm crit}$ is defined with $k_{\rm f}=k_{0}$, 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
$k_{z}=k_{0}$.

Table 1:  Robert flows I–IV as simple benchmarks, and their dynamo properties.

Flow | helicity | interpretation | $\overline{\mathbfcal{E}}_{\perp}$ | $\mbox{\rm Re}_{\rm M}^{\rm crit}$
---|---|---|---|---
I | yes, and constant |  $\alpha$ effect | $\alpha{\overline{\bm{B}}}_{\perp}-\eta_{\rm t}{\overline{\bm{J}}}_{\perp}$ | 1.99
II | pointwise zero $\alpha$ |  off-diagonal $\alpha$ tensor with memory effect | $\pmatrix{0&a\cr a&0}{\overline{\bm{B}}}_{\perp}-\eta_{\rm t}{\overline{\bm{J}}}_{\perp}$ | 6.86
III | zero only on average |  pumping effect with memory effect | $\pmatrix{0&\gamma\cr-\gamma&0}{\overline{\bm{B}}}_{\perp}-\eta_{\rm t}{\overline{\bm{J}}}_{\perp}$ | 3.92
IV | zero only on average |  negative “turbulent” diffusion |  $-\eta_{\rm t}{\overline{\bm{J}}}_{\perp}$ with $\eta_{\rm t}<0$ on large length scales | 4.55

In all cases, $k_{z}=k_{0}/2$ was used. The values of $u_{\rm rms}$ are 0.866
for flows I–III and unity for flow IV.

###  6.1 Large-scale dynamos and averaging

As 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. The
most suitable type of averaging depends on the type of the mean magnetic field
that can emerge in certain geometries and in certain parameter regimes; see
Gent et al. (2013) and Hollins et al. (2022) for a discussion. 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, as well as on time. This dependence may
still involve rapid variability, which can easily lead to a confusing
terminology when we want to split the magnetic field into mean fields and
fluctuations, $\bm{B}={\overline{\bm{B}}}+\bm{b}$. To avoid the temptation to
refer to the non-smoothness of ${\overline{\bm{B}}}$ as fluctuations, one
sometimes refers to ordered and random fields instead (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 ${\overline{\bm{B}}}$ also depends on $\phi$. This
mean field could be understood as a low Fourier mode filtering. However, then
the average of the product of a mean and a fluctuation vanishes only
approximately; see Zhou et al. (2018) for the related discussion on what is
known as Reynolds rules for averaging.

Regarding the periodic flow patterns in Cartesian coordinates discussed in
Section 5.1, it is important to stress that there can be examples where planar
$xy$ averaging is not suitable. An example is the Taylor–Green flow, where a
one-dimensional average (here a $z$ average) must be taken to demonstrate the
existence of a large-scale dynamo due to a negative turbulent magnetic
diffusivity (Andrievsky et al., 2015). In that case, the mean field depends on
$x$, $y$, and $t$.

###  6.2 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 (Steenbeck et al., 1966), but in its essence,
it was already obtained by Parker (1955) using a more phenomenological
approach. It is intrinsically connected with the presence of kinetic helicity
and is proportional to the pseudo-scalar $\bm{g}\cdot\bm{\Omega}$, as
discussed in the text box on “The $\alpha$ effect.” Dynamos can work with an
$\alpha$ effect alone, in which case one talks about an $\alpha^{2}$ dynamo.
Astrophysical dynamos often have strong shear, so there is an extra
$\overline{\bm{U}}\times{\overline{\bm{B}}}$ term on the right-hand side of
Equation SB2, 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.

We do not know whether galactic dynamos are of $\alpha\Omega$ type.
Alternatives include the incoherent $\alpha$–shear effect, but also the
(magnetic) shear–current effect has been discussed Section 6.2.3; see Table 2
for a summary of the different types of 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
$\mbox{\rm Re}_{\rm M}$. This is usually not the case for laminar flows,
unless the flow has chaotic streamlines.111The Galloway–Proctor flow (Galloway
& Proctor, 1992) is an example of a laminar flow that is fast. It is a Roberts
flow with time-dependent phases in the trigonometric functions, which causes
its streamlines to be chaotic.

Table 2:  Summary of different types of large-scale dynamos.

Flow | main dynamo effect | small-scale | fast
---|---|---|---
helical turbulence |  $\alpha^{2}$ dynamo | ✓ | ✓
Roberts flow I (laminar, helical) |  $\alpha^{2}$ dynamo | — | —
Roberts flows II and III (laminar, nonhelical) | time delay | — | —
Roberts flows IV | neg turb diff | — | —
Rädler effect with shear |  $\bm{\Omega}\times{\overline{\bm{J}}}$ effect | — | ✓
(magnetic) shear–current effect |  $\mbox{\boldmath${\sf S}$}{}{\overline{\bm{J}}}$ effect | — | ✓
incoherent $\alpha$–shear effect | fluctuating $\alpha$ effect | ✓ | ✓

‘Small-scale’ refers to possibility that a small-scale dynamo would operate
together with a large-scale dynamo. ‘Fast’ refers to the possibility that the
dynamo works in the limit $\eta\to 0$, as discussed in Section 3.2.

####  6.2.1 Helical dynamos

Roberts flow I is maximally helical. It is a prototype of an $\alpha^{2}$
dynamo, whereby the two nonvanishing horizontally averaged mean-field
components, $\overline{B}_{x}$ and $\overline{B}_{y}$, are being amplified by
the $\alpha$ effect. 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 (Parker, 1979, Stix, 1975). [] Roberts flow Iis a
prototype of an $\alpha^{2}$ dynamo; the flow is fully helical

####  6.2.2 Nonhelical large-scale dynamos

There are various examples of large-scale dynamos that do not involve magnetic
helicity. Three of the four Roberts flows have clearly demonstrated that
large-scale dynamos do not have to be helical and they can even have pointwise
zero helicity. Common to all three examples of Roberts flows II–IV is the fact
that the two components, $\overline{B}_{x}$ and $\overline{B}_{y}$, are
uncoupled from each other. In these examples, the two components have the same
growth rate, but there are other flows, such as the Willis flow (Willis,
2012), where the growth rates of $\overline{B}_{x}$ and $\overline{B}_{y}$ 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 ${\bm{\nabla}}\times(\alpha{\overline{\bm{B}}})$ on the
right-hand side of the evolution equation for ${\overline{\bm{B}}}$. In the
presence of shear, for example by a mean flow with constant shear
$S=\partial\overline{U}_{y}/\partial x$, one has
$\partial\overline{B}_{y}/\partial t=S\overline{B}_{x}+...$, where the
ellipsis denotes further terms not relevant to the present discussion. []
Pumping velocity $\bm{\gamma}$is a contribution to the mean electromotive
force given by the off-diagonal terms of the $\alpha$ tensor through
$\gamma_{i}=-\frac{1}{2}\epsilon_{ijk}\alpha_{jk}$

The reason for the decoupling of the two magnetic field components in some
examples is that the dynamo-active terms operate on each field component
separately (i.e., $\partial\overline{B}_{x}/\partial
t=-\mbox{\boldmath$\gamma$}{}\cdot{\bm{\nabla}}\overline{B}_{x}$ and
$\partial\overline{B}_{y}/\partial
t=-\mbox{\boldmath$\gamma$}{}\cdot{\bm{\nabla}}\overline{B}_{y}$ for dynamos
where the pumping velocity $\gamma$ has a memory effect). In its simplest
form, a memory effect has an exponential kernel proportional to
$e^{-(t-t^{\prime})/\tau}$ for $t>t^{\prime}$, and zero otherwise. Here, $t$
is the current time and $t^{\prime}$ the integration variable, covering all
earlier times. In Fourier space, it leads to a factor $1/(1-{\rm
i}\omega\tau)$, where $\omega$ is the frequency and $\tau$ is the turnover
time. When $\gamma\omega\tau>(\eta+\eta_{\rm t})k$, dynamo action becomes
possible.

####  6.2.3 Rädler and shear–current effects

Early in the history of mean-field dynamo theory, Rädler (1969) found a novel
large-scale dynamo effect for rotating, but unstratified bodies, whereby
$\overline{\mathbfcal{E}}$ has a term proportional to
$\bm{\Omega}\times{\overline{\bm{J}}}$. Here, $\bm{\Omega}$ is a pseudovector
pointing along the rotation axis. The azimuthal velocity is then
$\bm{u}_{\phi}=\mbox{\boldmath$\varpi$}{}\times\bm{\Omega}$. However, it is
easy to see that the $\bm{\Omega}\times{\overline{\bm{J}}}$ term in
$\overline{\mathbfcal{E}}$ does not contribute to the generation of mean-field
energy proportional to ${\overline{\bm{B}}}^{2}$, because the dot product with
${\overline{\bm{J}}}$ vanishes. Therefore, additional effects are needed to
achieve dynamo action. Shear is one such effect, which can also generate
another large-scale dynamo, similar to the Rädler effect: the shear–current
effect. Most of the numerical evidence today shows that this effect does not
have a favorable sign for dynamo action (Brandenburg et al., 2008a). There is
the possibility that this finding would change when the shear-current effect
is strongly controlled by the magnetic field from a small-scale dynamo (Squire
& Bhattacharjee, 2015). While it is true that large-scale magnetic fields can
be generated, it is possible that the real reason behind this is actually the
incoherent $\alpha$ effect, as will be discussed below; see also Zhou &
Blackman (2021) for a detailed assessment of the different possibilities.

####  6.2.4 Incoherent and shear dynamo effects

Another important class of large-scale dynamos may explain the phenomenon of
large-scale magnetic field generation in shear flows without helicity. Such
nonhelical dynamo action was first found in a more complex shear flow
geometry, relevant to the solar tachocline at mid to low latitudes
(Brandenburg, 2005). In this environment, large-scale fields can be generated
both with and without helicity in the driving of the turbulence. Subsequent
studies by Yousef et al. (2008) and Brandenburg et al. (2008a) produced such
dynamos in simpler shearing box simulations, but gave different
interpretations, which we discuss below.

One interpretation involves helicity fluctuations, which lead to an incoherent
$\alpha$ effect and, in conjunction with shear, to large-scale dynamo action
(Vishniac & Brandenburg, 1997). An incoherent $\alpha$ effect can lead to a
negative turbulent magnetic diffusivity (Kraichnan, 1976). In that sense, the
incoherent $\alpha$ effect is actually similar to the dynamo effect in Roberts
flow IV. [] Incoherent $\alpha$ effectan $\alpha$ effect with frequent sign
changes

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 shear 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.

###  6.3 Small-scale dynamos

Under fully isotropic conditions and without helicity, dynamo action is still
possible—both for large and small values of the magnetic Prandtl number
$\mbox{\rm Pr}_{\rm M}$ (Kazantsev, 1968); see the detailed discussion by
Schekochihin et al. (2004). The existence of small-scale dynamos under
isotropic conditions implies that the concept of nonmagnetic Kolmogorov
turbulence hardly exists in astrophysics, where the medium is usually always
highly conducting.

####  6.3.1 Early work on the subject

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
$\mbox{\rm Pr}_{\rm M}\gg 1$. Kulsrud & Anderson (1992) found that the peak
occurs at a wavenumber $k_{\eta}$ that depends on the growth rate $\lambda$
through $k_{\eta}=\sqrt{4\lambda/15\eta}$. In the saturated stage, the peak of
the magnetic spectrum shifts closer to the forcing scale; see Figure 4. More
recent work on small-scale dynamos is numerical, and is covered in detail in
the following sections.

Figure 4:  Magnetic (red lines) and kinetic (blue lines) energy spectra during
the kinematic (left) and nonlinear saturated (right) phases. Here,
$k_{\nu}=(\epsilon_{\rm K}/\nu^{3})^{1/4}$ with $\epsilon_{\rm K}$ being the
dissipation rate. Note how the peak of $E_{\rm M}(k)$ shifts to larger scales
in the saturated case. Figure adapted from Run E of Brandenburg et al. (2022).

####  6.3.2 Effect of ambipolar diffusion

Xu & Lazarian (2016) found a strong similarity between the regime of large
magnetic Prandtl numbers and the regime of partial ionization. Their results
have been confirmed in numerical simulations (Xu et al., 2019). In those
simulations, the microphysical magnetic Prandtl number remained undetermined,
because no explicit viscosity or magnetic diffusivity were used. Two-fluid
direct numerical simulations (Brandenburg, 2019) showed that at large magnetic
Prandtl number, the kinetic energy spectra for neutrals and ions show
different slopes. The energy spectra of ions and neutrals depart from each
other only a small scales when $k/k_{\nu}>1$. For larger ambipolar diffusion
coupling, the kinetic energy spectra of neutrals decrease further while those
of the ions increase slightly.

##  7 CATASTROPHIC QUENCHING AND MAGNETIC HELICITY FLUXES

As long as the magnetic field is weak, the Lorenz force plays no significant
role. Many dynamo effects, including those discussed in Section 5.1, can then
be fully described by a given velocity field. 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.

###  7.1 Catastrophic quenching for uniform magnetic fields

The term catastrophic quenching was coined by Blackman & Field (2000) to
denote any type of detrimental $\mbox{\rm Re}_{\rm M}$ dependence of the
nonlinear feedback. Ignoring the effect of turbulent diffusion, i.e., the term
$-\eta_{\rm t}\mu_{0}{\overline{\bm{J}}}$ in Equation 4, Cattaneo & Hughes
(1996) found numerically that $\alpha\propto(1+\mbox{\rm Re}_{\rm
M}{\overline{\bm{B}}}^{2}/B_{\rm eq}^{2})^{-1}$, where $B_{\rm
eq}=\sqrt{\mu_{0}\rho_{0}}u_{\rm rms}$ is the equipartition field strength,
whose energy density is equal to the kinetic energy density. Evidently, owing
to the $\mbox{\rm Re}_{\rm M}$ factor in the expression for $\alpha$, this
dependence is “catastrophic”. This dependence was originally anticipated by
Vainshtein & Cattaneo (1992) based on earlier analogous results by Cattaneo &
Vainshtein (1991) for the suppression of just $\eta_{\rm t}$ in 2-D. Gruzinov
& Diamond (1996) explained these results as a consequence of the conservation
of magnetic helicity $\langle\bm{A}\cdot\bm{B}\rangle$ in 3-D, which is
routinely seen during laboratory plasma relaxation (Ji et al., 1995). However,
the dependence of $\alpha$ on $\mbox{\rm Re}_{\rm M}$ is a peculiar property
of the magnetic helicity equation in the presence of an imposed magnetic field
$\bm{B}_{0}$. In that case, the magnetic helicity corresponding to the
departure from the imposed field, $\bm{b}$, yields, in the steady state,
$0=\langle(\bm{u}\times\bm{B}_{0})\cdot\bm{b}\rangle-\eta\mu_{0}\langle\bm{j}\cdot\bm{b}\rangle$.
Since we define here mean fields as volume averages, and since
$\langle\bm{J}\rangle=0$ in Equation 4, we have
$\langle(\bm{u}\times\bm{B}_{0})\cdot\bm{b}\rangle=-(\langle\bm{u}\times\bm{b}\rangle)\cdot\bm{B}_{0}=-\alpha\bm{B}_{0}^{2}$,
and therefore
$\alpha=-\eta\mu_{0}\langle\bm{j}\cdot\bm{b}\rangle/\bm{B}_{0}^{2}$, so
$\alpha\to 0$ as $\eta\to 0$ or $\mbox{\rm Re}_{\rm M}\to\infty$. This agrees
with the heuristic quenching formula $\alpha\propto(1+\mbox{\rm Re}_{\rm
M}{\overline{\bm{B}}}^{2}/B_{\rm eq}^{2})^{-1}$, which also predicts
$\alpha\to 0$ as $\mbox{\rm Re}_{\rm M}\to\infty$. The analysis also shows
that the quenching is related to magnetic helicity conservation. A detailed
explanation of this derivation is reviewed in Brandenburg & Subramanian
(2005a).

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 Keinigs
(1983). However, it is not really relevant in practice, because it assumes
that the magnetic field can meaningfully be described by volume averages. This
is not the case, because a volume-averaged magnetic field is always constant
in a periodic domain.

A relevant mean field for this kind of problem can be defined as planar
averages, as discussed in Section 5.1. We denote that by overbars. The
diffusion term $\eta_{\rm t}\mu_{0}{\overline{\bm{J}}}$ can then not be
neglected and the relation of Keinigs (1983) can then be written in the form
$\alpha-\eta_{\rm t}k_{\rm
m}=-\eta\mu_{0}\langle\bm{j}\cdot\bm{b}\rangle/\bm{B}_{0}^{2}$, where $k_{\rm
m}=\mu_{0}{\overline{\bm{J}}}\cdot{\overline{\bm{B}}}/{\overline{\bm{B}}}^{2}$.
This would mean that only the difference $\alpha-\eta_{\rm t}k_{\rm m}$, not
$\alpha$ itself, is quenched catastrophically.

###  7.2 Catastrophically slow saturation in closed domains

In reality, even if the restriction to closed or periodic domains is retained,
neither $\alpha$ nor $\eta_{\rm t}$ are quenched in a catastrophic fashion
(Brandenburg et al., 2008b). Instead, the timescale for reaching ultimate
saturation is “catastrophically” prolonged, i.e., the final saturation obeys
(Brandenburg, 2001)

${\overline{\bm{B}}}^{2}\approx\langle\bm{b}^{2}\rangle\,(k_{\rm
f}/k_{1})\,\left[1-e^{-2\eta k_{1}^{2}(t-t_{\rm sat})}\right]\quad\mbox{(for
$t>t_{\rm sat}$)},$ (5)

where $k_{\rm f}$ 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 Equation 5, the
value of $\eta$ is the microphysical value, which is extremely small in
galaxies. This motivates the characterization as “catastrophically slow”.

[t]

##  8 Derivation of Equation 5

In periodic domains, the slow saturation behavior after $t=t_{\rm sat}$ is
governed by magnetic helicity conservation. The uncurled induction equation
reads, $\partial\bm{A}/\partial t=-\bm{E}-{\bm{\nabla}}\varphi$, where
$\bm{E}=-\bm{U}\times\bm{B}+\eta\mu_{0}\bm{J}$ is the electric field and
$\varphi$ is the electrostatic potential. The evolution of the magnetic
helicity density $\bm{A}\cdot\bm{B}$ is then given by

$\frac{\partial}{\partial
t}(\bm{A}\cdot\bm{B})=\cancel{2(\bm{U}\times\bm{B})\cdot\bm{B}}-2\eta\mu_{0}\bm{J}\cdot\bm{B}-{\bm{\nabla}}\cdot\mathbfcal{F},$
(SB5)

where $\mathbfcal{F}=\bm{E}\times\bm{A}+\varphi\bm{B}$ is the magnetic
helicity flux density. (Note the analogy with the Poynting flux
$\bm{E}\times\bm{B}/\mu_{0}$ of magnetic energy density.) The equations
involving $\bm{A}$ and $\mathbfcal{F}$ 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
${\overline{\bm{B}}}={\bm{\nabla}}\times{\overline{\bm{A}}}$ and
${\overline{\bm{J}}}={\bm{\nabla}}\times{\overline{\bm{B}}}/\mu_{0}$, along
with the resulting fluctuations, $\bm{a}=\bm{A}-{\overline{\bm{A}}}$,
$\bm{b}=\bm{B}-{\overline{\bm{B}}}$, and $\bm{j}=\bm{J}-{\overline{\bm{J}}}$,
so, after averaging, Equation SB5 becomes

$\frac{\partial}{\partial
t}({\overline{\bm{A}}}\cdot{\overline{\bm{B}}}+\overline{\bm{a}\cdot\bm{b}})=-2\eta\mu_{0}\left({\overline{\bm{J}}}\cdot{\overline{\bm{B}}}+\overline{\bm{j}\cdot\bm{b}}\right)-{\bm{\nabla}}\cdot\overline{\mathbfcal{F}}_{\rm
m},$ (SB6)

where $\overline{\mathbfcal{F}}_{\rm m}$ is the magnetic helicity flux for the
mean field. Our analysis concerns only the phase when the small-scale dynamo
has already saturated (for $t>t_{\rm sat}$), so $\overline{\bm{a}\cdot\bm{b}}$
is approximately constant in time. Assuming the field to be helical with
negative helicity at small scales and positive at large scales, we have
$\mu_{0}\overline{\bm{j}\cdot\bm{b}}\approx-k_{\rm f}\overline{\bm{b}^{2}}$
and
${\overline{\bm{A}}}\cdot{\overline{\bm{B}}}\approx\overline{\bm{B}^{2}}/k_{1}\approx\mu_{0}{\overline{\bm{J}}}\cdot{\overline{\bm{B}}}/k_{1}^{2}$.
Inserting this into Equation SB6 and performing volume averaging over the
whole domain, indicated by angle brackets, so that the flux divergence term
vanishes, one obtains

$\frac{{\rm d}{}}{{\rm d}{}t}\langle{\overline{\bm{B}}}^{2}\rangle=-2\eta
k_{1}^{2}\langle{\overline{\bm{B}}}^{2}\rangle+2\eta k_{1}k_{\rm
f}\langle\bm{b}^{2}\rangle,$ (SB7)

the solution of which for $\langle\bm{b}^{2}\rangle={\rm const}{}$ is given by
Equation 5.

The derivation of Equation 5 is based on just the magnetic helicity equation,
i.e., no mean field theory was invoked; see the text box on the “Derivation of
Equation 5”. 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 (Blackman &
Brandenburg, 2002), so the magnetic field (in a periodic or closed domain) can
still be strong, but only after a resistively long time; see, again,
Brandenburg & Subramanian (2005a) for a review.

###  8.1 Alleviating catastrophic quenching by magnetic helicity fluxes

It has long been hypothesized that the action of magnetic helicity fluxes can
overcome what would otherwise be an extremely slow approach to saturation (as
described in Section 7.2) or a saturation at a very low amplitude (Gruzinov &
Diamond, 1996). The latter has been demonstrated in Brandenburg et al. (2002),
who discussed the preferential removal of small scale magnetic fields. In this
experiment, they periodically removed magnetic field at high Fourier modes
from the simulation. After each removal, the small-scale field was then no
longer saturated and was thus allowed to grow and regain old strength, while
the large-scale field grew to larger strength than before. This continued with
each removal step. While the idea is simple and convincing, there is as yet no
conclusive demonstration from simulations that this also works with actual
magnetic helicity fluxes.

For assessing the role of magnetic helicity fluxes, the decisive equation is
that for the magnetic helicity of the fluctuating field,
$\overline{\bm{a}\cdot\bm{b}}$. The fluctuating field can be determined from
the equation for the mean field, which, under the Weyl gauge, can be written
as

$\frac{\partial{\overline{\bm{A}}}}{\partial
t}=\overline{\bm{U}}\times{\overline{\bm{B}}}+\overline{\mathbfcal{E}}-\eta\mu_{0}{\overline{\bm{J}}},$
(6)

where we recall that
$\overline{\mathbfcal{E}}\equiv\overline{\bm{u}\times\bm{b}}$ is the mean
electromotive force. This expression results in the following equation for the
magnetic helicity of the mean magnetic field:

$\frac{\partial}{\partial
t}({\overline{\bm{A}}}\cdot{\overline{\bm{B}}})=2\overline{\mathbfcal{E}}\cdot{\overline{\bm{B}}}-2\eta\mu_{0}{\overline{\bm{J}}}\cdot{\overline{\bm{B}}}-{\bm{\nabla}}\cdot\overline{\mathbfcal{F}}_{\rm
m}.$ (7)

The equation for $\overline{\bm{a}\cdot\bm{b}}$ must also have a corresponding
$\overline{\mathbfcal{E}}$ term,
$-2\overline{\mathbfcal{E}}\cdot{\overline{\bm{B}}}$,

$\frac{\partial}{\partial
t}\overline{\bm{a}\cdot\bm{b}}=-2\overline{\mathbfcal{E}}\cdot{\overline{\bm{B}}}-2\eta\mu_{0}\overline{\bm{j}\cdot\bm{b}}-{\bm{\nabla}}\cdot\overline{\mathbfcal{F}}_{\rm
f},$ (8)

so that the sum of both equations yields Equation SB5. Here,
$\overline{\mathbfcal{F}}_{\rm f}$ is the magnetic helicity flux for the
fluctuating field.

Figure 5:  $\mbox{\rm Re}_{\rm M}$ dependence of terms on the right-hand side
of the small-scale magnetic helicity equation 8. Note that
${\bm{\nabla}}\cdot\overline{\mathbfcal{F}}_{\rm f}$ becomes comparable to
$2\overline{\mathbfcal{E}}\cdot{\overline{\bm{B}}}$ and
$2\eta\overline{\bm{j}\cdot\bm{b}}$ only for $\mbox{\rm Re}_{\rm M}>1000$.
Adapted from Del Sordo et al. (2013).

In the steady state, there are three terms on the right hand side,
$2\overline{\mathbfcal{E}}\cdot{\overline{\bm{B}}}$,
$2\eta\mu_{0}\overline{\bm{j}\cdot\bm{b}}$, and
${\bm{\nabla}}\cdot\overline{\mathbfcal{F}}_{\rm f}$. Simulations by Del Sordo
et al. (2013) and Rincon (2021) showed that the helicity flux divergence
begins to become more important than the resistive terms only at $\mbox{\rm
Re}_{\rm M}$ of the order of 1000 (see Figure 5). Both works showed the
presence of turbulent diffusive magnetic helicity fluxes in the simulations.
Those fluxes were proportional to the negative gradient of the local magnetic
helicity density. In the work of Del Sordo et al. (2013), there was also a
galactic wind contributing to an advective magnetic helicity flux proportional
to the wind speed. One could expect the saturation behavior to become
independent of $\mbox{\rm Re}_{\rm M}$. However, simulations still show that
${\overline{\bm{B}}}^{2}$ declines with increasing $\mbox{\rm Re}_{\rm M}$.
This could mean that $\mbox{\rm Re}_{\rm M}$ 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 $2\eta\mu_{0}\overline{\bm{j}\cdot\bm{b}}$ does become clearly
subdominant.

##  9 MEAN-FIELD COEFFICIENTS AND NONLOCALITY

###  9.1 Parameterization of the mean electromotive force

The mean electromotive force, $\overline{\mathbfcal{E}}$, in Equation 6 can be
expressed nonlocally in terms of the mean magnetic field as

$\overline{\cal
E}{}_{i}=\alpha_{ij}\ast\overline{B}_{j}+\eta_{ijk}\ast\partial\overline{B}_{j}/\partial
x_{k},$ (9)

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., $\partial\overline{B}_{j}/\partial
x_{k}=\overline{B}_{j,k}$. For planar averages that depend on just one
direction, we can write $\overline{\cal
E}{}_{i}=\alpha_{ij}\ast\overline{B}_{j}-\eta_{ij}\ast\overline{J}_{j}$, 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 we refer
the reader to Warnecke et al. (2018) for a study in the context of 3-D
convection in a sphere.

Most of the published literature ignores the fact that $\alpha_{ij}$ and
$\eta_{ij}$ are integral kernels, and approximates the convolution by a
multiplication. This approximation then assumes a local connection between
$\overline{\mathbfcal{E}}$ and the mean fields. It ignores the effect of
strong variations of the mean field in space and time. In Fourier space, the
convolution in Equation 9 becomes a multiplication, so it describes the
combined response of all Fourier modes. This becomes relevant when measuring
the mean-field coefficients for sinusoidal mean fields; see Section 9.4. []
Convolutionan operation that becomes a multiplication in Fourier space

###  9.2 Mean-field coefficients

One of the major advances in mean-field dynamo theory is the development of
numerical methods to avoid the limitations imposed by using analytic
approaches. This concerns mainly the linearization of the evolution equations
for the magnetic and velocity fluctuations in a turbulent flow.

To obtain expressions for $\alpha_{ij}$ and $\eta_{ijk}$, one has to solve the
equations for the fluctuations $\bm{u}$ and $\bm{b}$. The most important one
is that for $\bm{b}$ and is obtained by subtracting the equation for
${\overline{\bm{B}}}$ from that for $\bm{B}$. The equations are nonlinear in
the fluctuations. In analytic approaches, those nonlinear terms are often
ignored (SS22), which is termed the second order correlation approximation,
but this restriction is no longer required in the numerical evaluations of
$\overline{\mathbfcal{E}}\equiv\overline{\bm{u}\times\bm{b}}$. This
approximation is only valid when $\mbox{\rm Re}_{\rm M}\ll 1$, or when the
correlation time is short (which is even for supernova-driven turbulence
hardly the case). Neither of the two is relevant to astrophysics, so we focus
here on a numerical, 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 qualitatively new effects
emerge: turbulent pumping in the Galloway–Proctor flow, or the effect of
kinetic helicity on the turbulent magnetic diffusivity, although those effects
remain mainly of academic interest (for a review, see Brandenburg, 2018).

###  9.3 Methods for measuring $\alpha$ and other effects

One approach is to use a nonlinear simulation to obtain $\bm{u}$ and $\bm{b}$
in the presence of an additional imposed magnetic field. The resulting
$\overline{\bm{u}\times\bm{b}}$ can be related to ${\overline{\bm{B}}}$ by
ignoring $\eta_{ij}$ and ${\overline{\bm{J}}}$. This is termed the imposed-
field method, but it can only be used when ${\overline{\bm{J}}}$ vanishes, for
example when the averages are zero-dimensional, i.e., volume averages.

Another approach is to relate $\overline{\bm{u}\times\bm{b}}$ to the actual
${\overline{\bm{B}}}$ and ${\overline{\bm{J}}}$ 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 (Brandenburg & Sokoloff, 2002, Bendre & Subramanian, 2022) and the
coefficients in the local version (Simard et al., 2016). The reliability of
this approach is unclear and it has not yet been verified for the simple
examples of the Roberts flow mean-field dynamos discussed in Section 5.1. This
method is sometimes called the correlation method. The occurrence of
unphysical results with this method (e.g., $\eta_{\rm t}<0$) can sometimes be
alleviated by using the singular value decomposition (Simard et al., 2016).

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 Brandenburg et al. (2010).

###  9.4 Using test fields

The TFM was originally applied by Schrinner et al. (2005, 2007) to determine
the dependence of all transport coefficients in a sphere using longitudinal
averages. In that case, one has 9 coefficients for $\alpha_{ij}$ and 18
nonvanishing coefficients for rank three tensor $\eta_{ijk}$ in the
representation $\overline{\cal
E}{}_{i}=\alpha_{ij}\overline{B}_{j}+\eta_{ijk}\overline{B}_{j,k}$. (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 Brandenburg (2005) and
Brandenburg et al. (2008a) 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
$\overline{\cal
E}{}_{i}=\alpha_{ij}\overline{B}_{j}-\eta_{ij}\mu_{0}\overline{J}_{j}$,
because there are only two nonvanishing components of $\overline{B}_{j,k}$
that can be expressed as the two components of the mean current density with
$\mu_{0}\overline{J}_{x}=-\overline{B}_{y,z}$ and
$\mu_{0}\overline{J}_{y}=\overline{B}_{x,z}$. In that case, one can use four
sinusoidal test fields $(\sin kz,0,0)$, $(\cos kz,0,0)$, as well as $(0,\sin
kz,0)$, $(0,\cos kz,0)$.

Figure 6:  Dynamo coefficients from supernova-driven turbulence. Shown here
are the off-diagonal components, $\alpha_{\phi R}$ (black line) and
$-\alpha_{R\phi}$ (gray line), contributing to the pumping velocity
$\gamma_{z}=(\alpha_{\phi R}-\alpha_{R\phi})/2$. The mean flow velocity is
also shown. Courtesy of Gressel et al. (2013)

In Figure 6 we reproduce results from the work of Gressel et al. (2008b), who
performed simulations of dynamos from supernova-driven turbulence in a portion
of a stratified galactic disk. Using the TFM with the Nirvana code, they found
that $\eta_{\rm t}$ increases away from the midplane and that this leads to
turbulent pumping toward the midplane, which is given by
$\mbox{\boldmath$\gamma$}{}\approx-(\tau/2){\bm{\nabla}}\eta_{\rm t}$. The
pumping velocity $\gamma$ 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 the subscript $R$ denotes cylindrical radius. The pumping, also termed
turbulent diamagnetism (SS22), pushes the magnetic field toward the midplane
and thereby strengthens the dynamo (Brandenburg et al., 1993). Surprisingly,
this pumping increases toward smaller scales (Gressel & Elstner, 2020).

###  9.5 Nonlocality in space and time

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 “Evolution equation
for nonlocality in space and time”. For many turbulent flows, the components
of both $\alpha_{ij}$ and $\eta_{ij}$ decline with increasing values of $k$ in
a Laplacian fashion approximately proportional to $[1+(ak/k_{\rm
f})^{2}]^{-1}$, where $a$ depends on details of the flow. In this relation,
the value of the empirical coefficient $a$ varied between 0.1 and 0.5,
depending on the nature of the turbulent flow (Rheinhardt & Brandenburg,
2012).

The significance of nonlocality is that the transport coefficients become
effectively quenched when the mean field is of small scale, i.e., smaller than
the integral scale of the turbulence. Especially near boundaries, where sharp
boundary layers may occur in calculations that ignore nonlocality, the actual
field would be smoother. In fact, sharp contrasting structures have been found
in earlier galactic dynamo simulations (Moss, 1996). Such results would need
to be revisited in view of the importance of nonlocality effects.

[t]

##  10 Evolution equation for nonlocality in space and time

In Fourier space, the simplest empirical approximations to the spatial and
temporal nonlocalities, as obtained with the test-field method, can be
combined to a single expression, which reads

$\tilde{\cal
E}_{i}(k,\omega)=\frac{\tilde{\alpha}_{ij}(k,\omega)\tilde{B}_{j}(k,\omega)-\tilde{\eta}_{ij}(k,\omega)\mu_{0}\tilde{J}_{j}(k,\omega)}{1-{\rm
i}\omega\tau+\ell^{2}k^{2}},$ (SB8)

where $\ell=O(1/k_{\rm f})$ and $\tau=O(1/u_{\rm rms}k_{\rm f})$. Moving the
denominator to the left-hand side, the equation becomes

$\left(1-{\rm i}\omega\tau+\ell^{2}k^{2}\right)\tilde{\cal
E}_{i}(k,\omega)=\tilde{\alpha}_{ij}(k,\omega)\tilde{B}_{j}(k,\omega)-\tilde{\eta}_{ij}(k,\omega)\mu_{0}\tilde{J}_{j}(k,\omega),$
(SB9)

which, back in real space, becomes a simple evolution equation with a
diffusion term on the right-hand side:

$\tau\frac{\partial\overline{\cal E}{}_{i}}{\partial
t}=\alpha_{ij}\overline{B}_{j}-\eta_{ij}\mu_{0}\overline{J}_{j}+\ell^{2}\nabla^{2}\overline{\cal
E}{}_{i}-\overline{\cal E}{}_{i}.$ (SB10)

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.

Even more important than spatial nonlocality is temporal nonlocality. It is
also termed a 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 $(1-{\rm i}\omega\tau)^{-1}$, 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; see the box box on “Dynamos from the memory effect”. Whether those
effects play a role in turbulent dynamos is unclear.

Although the memory effect may not be strong enough to produce new dynamo
effects in turbulent flows, it is strong enough to produce significant phase
shifts between the generation of magnetic fields in galactic arm and interarm
regions. This has been studied in detail by Shukurov (1998) and Chamandy et
al. (2013). Including a memory effect in numerical simulations is, in general,
very cumbersome, because it requires storing the full spatial form of the mean
field for many earlier times in order to evaluate the convolution integral in
Equation 9. In the present case, however, and to leading order, the
convolution integral can be converted into an evolution equation for the
electromotive force, which is computationally much easier to solve; see
Equation SB10. This approach was first proposed by Rheinhardt & Brandenburg
(2012) and was applied to dynamos in spheres (Brandenburg & Chatterjee, 2018).
This formalism also reproduces the dynamo effect from a time delay for Roberts
flows II and III; see Section 5.1, as was demonstrated by Rheinhardt et al.
(2014). This is explained in the box “Dynamos from the memory effect”.

[t]

##  11 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 the memory 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 now easy
to explain this effect.

The dispersion relation for a problem with turbulent pumping $\gamma$ and
turbulent magnetic diffusion $\eta_{\rm t}$ is given by $\lambda=-{\rm
i}k\gamma-\eta_{\rm t}k^{2}$. Since $\mbox{\rm Re}\lambda<0$, the solution can
only decay, but it is oscillating with the frequency $\omega=-\mbox{\rm
Im}\lambda=k\gamma$. In the presence of a memory effect, $\gamma$ is replaced
by $\gamma/(1-{\rm i}\omega\tau)$, where $\tau$ is the memory time. Then,
$\lambda\approx-{\rm i}k\gamma\,(1-{\rm i}\omega\tau)-\eta_{\rm t}k^{2}$, and
$\mbox{\rm Re}\lambda$ can be positive. This is the case for the Roberts flow
discussed in Section 5.1.

##  12 SETTING THE SCENE FOR DYNAMO ACTION IN REAL GALAXIES

###  12.1 Possibilities for seed magnetic fields

The conditions in the early Universe provide several possibilities for seeding
galactic dynamos. The seeds could be primordial, which generally means that
they were generated during inflation or phase transitions, or they could
originate from a cosmic battery. Other theories also involve later seeding
from astrophysical processes. We examine these possibilities in the following
sections. [] Cosmic batteryA mechanism that can generate magnetic field even
when there is none initially

####  12.1.1 The need for sufficiently strong seed magnetic fields

In Section 3.1 we calculated an average of 70 revolutions in the Galaxy’s
lifetime. This is not very much, so we have to be concerned about possible
effects on the strength and shape of the initial magnetic field. Typical
estimates for the growth rate of the Galactic dynamo are of the order of
$\Gamma\approx 2\,{\rm Gyr}^{-1}$ (Beck et al., 1996). This means that the
mean magnetic field could be amplified by up to 12 orders of magnitude in
about $14\,{\rm Gyr}$. To reach the current level of the mean magnetic field
of about $3\,\mu{\rm G}$, we would need a seed magnetic field of about
$10^{-18}\,{\rm G}$. This is just about the level that can be expected from
the Biermann battery mechanism (Rees, 1987). The text box “Battery mechanisms”
provides some more information about the Biermann battery and other mechanisms
that can generate magnetic fields in an unmagnetized plasma.

Even though the growth rate of a galactic large-scale dynamo may just be large
enough for explaining the current level of the mean field of $\approx
3\,\mu{\rm G}$ at the present time, it would be insufficient for explaining
large-scale magnetic fields in very young (redshift $z=1$) galaxies.
Observationally, however, such fields are believed to exist. Kronberg et al.
(1992) found evidence for strong magnetic fields in a $z=0.395$ galaxy. In a
more systematic search, Bernet et al. (2008) found strong RMs in quasar
sightlines passing from $z\simeq 1$ galaxy halos. More recently, Mao et al.
(2017) estimated a $\mu$G, kpc-coherent magnetic field in a lensing galaxy at
$z\simeq 0.46$. The fact that the RMs of the lensed images are similar
provides evidence for large-scale magnetic coherence.

To explain smaller-scale magnetic fields at equipartition levels of
$3\,\mu{\rm G}$ 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.

[t]

##  13 Battery mechanisms

##### The Biermann battery.

When the density and the temperature gradient in a plasma are misaligned, the
electrons move down the pressure gradient, generating an electromotive force
that gives rise to a magnetic field. The resulting time derivative of the
magnetic vector potential is then

$\frac{\partial\bm{A}}{\partial t}=\frac{c}{qn_{\rm e}}{\bm{\nabla}}p_{\rm e}$
(SB11)

where $n_{\rm e}$ and $p_{\rm e}$ are the electron number density and
pressure, $c$ the speed of light, and $q$ the electron charge.

##### The Durrive battery.

Massive stars are surrounded by a region of ionized gas. Durrive & Langer
(2015) proposed that an electromotive force should be created by the surplus
momentum transferred to the electron after the ionization of an atom. Then,
the uncurled induction equation for zero initial magnetic field becomes:

$\frac{\partial\bm{A}}{\partial t}=\frac{c}{qn_{\rm e}}{\bm{\nabla}}p_{\rm
e}-\frac{c}{qn_{\rm e}}\dot{\bf{p}}_{\rm e}$ (SB12)

where $\dot{\bf{p_{e}}}$ is the rate of momentum transfer to the electrons and
Equation SB12 also includes the Biermann battery.

In the early Universe, the Biermann battery can appear from local fluctuations
in the sound speed right after recombination (Naoz & Narayan, 2013) and later,
around rippled shocks, while both battery mechanisms should operate around
ionization fronts (Subramanian et al., 1994, Kulsrud et al., 1997, Gnedin et
al., 2000). Garaldi et al. (2021) performed cosmological simulations testing,
among other scenarios, the efficiency of the Biermann and Durrive battery
terms through cosmic time. They found that, although the two batteries behave
similarly, the Durrive term produces systematically weaker magnetic fields by
approximately three orders of magnitude.

####  13.0.1 Small-scale dynamos as a seed for the large-scale dynamo

All dynamos require seed magnetic fields—even small-scale ones. 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 seeds. The idea has been discussed by Beck et al. (1994).

Another related idea is to produce galactic seed magnetic fields in the first
stars. The simplest form of this idea is that stars could pick up a Biermann
seed, which would then be amplified though a stellar dynamo and get ejected
with a supernova explosion at the end of the star’s life. This scenario has
been explored in many cosmological simulations (Beck et al., 2013, Katz et
al., 2019, Vazza et al., 2017, Martin-Alvarez et al., 2021), and shows that it
can, in fact, magnetize galaxies very efficiently. However, these simulations
use unrealistically high values for the supernova-injected magnetic field
(Ntormousi et al., 2022). Those fields could then well be large-scale ones,
i.e., on the scale of stars, but those would grow on an even shorter
timescale, because stars are much smaller than the envisaged turbulent eddies
in the interstellar medium. Young stars and also their surrounding accretion
disks can host powerful dynamos that also drive magnetized winds; see the
estimates in Brandenburg (2000) and corresponding mean-field simulations by
von Rekowski et al. (2003). Those winds could magnetize the surrounding
interstellar medium and could well produce much more efficient seeds for the
galactic dynamo than any battery mechanism. The wind-based injection model
would be a viable alternative to the uncertain supernova seeding often used in
cosmological simulations.

####  13.0.2 Battery versus plasma instabilities

When the electron distribution is anisotropic and the magnetic field is not
too weak, the Weibel instability (Weibel, 1959) 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. This is also in an
agreement with recent laser plasma experiments that have accessed a regime
relevant to astrophysical dynamos (Schoeffler et al., 2016).

[] Weibel instabilityOccurs in a nearly homogeneous plasma when there is an
anisotropy in velocity space

####  13.0.3 Primordial seed magnetic fields

In the early Universe, inflation and phase transitions, such as the decoupling
of the weak force and the electromagnetic force or the formation of hadrons
from quarks, may have produced hydromagnetic turbulence (Widrow, 2002). Owing
to the lack of further energy input, any magnetic field generated at that time
would be slowly decaying. The dilution of the magnetic field due to the
expansion of the Universe is always scaled out by talking about the comoving
magnetic field, which is $\tilde{\bm{B}}=a^{2}\bm{B}$, where $a$ is the scale
factor of the Universe. When time is being replaced by conformal time,
$\tilde{t}=\int{\rm d}{}t/a(t)$, the MHD equations, during the radiative era,
have their usual form without expansion factors (Brandenburg et al., 1996).
Hereafter, the tildes are therefore dropped.

Not much is known about the strength of the comoving magnetic field today.
There are only constraints. Upper limits can be derived from Big Bang
nucleosynthesis (BBN) constraints; see Grasso & Rubinstein (1995) and
Kahniashvili et al. (2022) for recent work taking into account the decay of
the magnetic field between the moment of generation and the time of BBN. A
lower limit on the present-day magnetic field strength has been proposed on
the grounds that magnetic fields would prevent the reconnection of pair-
created electrons and positrons when the TeV photons from powerful blazars
interact with the extragalactic background light (e.g., Neronov & Vovk, 2010).
Note, however, that the validity of this technique may have a systematic
uncertainty in that plasma instabilities could potentially also provide an
explanation for the non-observation of GeV halos (Broderick et al., 2012).

Simulations by Sironi & Giannios (2014), confirm that plasma instabilities do
indeed operate, but they only account for about 90% of the loss of GeV photons
and the suppression of the remaining 10% would still need to be explained by
the presence of magnetic fields. Similar conclusions were reached by Alves
Batista et al. (2019), who performed detailed simulations for individual
blazars.

The lower limits derived by Neronov & Vovk (2010), which become less stringent
for larger length scales, provide an exciting motivation for primordial
magnetogenesis scenarios. At the present time, those primordial magnetic
fields may have strengths in the range of $10^{-16}\,{\rm G}$ to
$10^{-9}\,{\rm G}$ (see Subramanian, 2016, for a review), and could act as
seed magnetic fields for any subsequent dynamo processes – once sufficient
kinetic energy becomes available. These seeds, if confirmed, would not only be
stronger than those from batteries, but they would also be present in the
voids. Figure 7 shows the expected magnetic field ranges as a function of the
typical scale $\lambda_{B}$ of the magnetic field. For a nonhelical magnetic
field, there are still magnetic helicity fluctuations. They constrain the
decay such that the correlation integral of the local magnetic helicity is
conserved (Hosking & Schekochihin, 2021). This leads to a decay with
$\bm{B}^{2}\sim t^{-10/9}$ and $\lambda_{B}\sim t^{4/9}$, so that
$\bm{B}^{4}\lambda_{B}^{5}={\rm const}{}$ (Hosking & Schekochihin, 2022).

The first stars are expected to form about $10^{8}\,{\rm yr}$ after the Big
Bang, marking the beginning of the reionization epoch. After that, galaxies
start growing through continuous gas accretion and mergers (Dayal & Ferrara,
2018). Since dynamo action is fastest at small length scales, the magnetic
field generation during the formation of the first collapsing structures is
potentially important and may have produced a stronger seed magnetic field for
the subsequent global galactic dynamo. Strong magnetic fields may also affect
galaxy and large-scale structure formation of the Universe (Kahniashvili et
al., 2013).

Figure 7:  Summary of lower and upper magnetic field limits as a function of
correlation length. The white solid lines describe the decay of a helical
magnetic field ($\bm{B}^{2}\sim t^{-2/3}$) along with the increase of its
typical length scale ($\lambda_{B}\sim t^{2/3}$), so that
$\bm{B}^{2}\lambda_{B}={\rm const}{}$. Only the narrowly hashed region
indicates a few permissible strengths. Courtesy of Korochkin et al. (2021), in
which we have added the prediction of Hosking & Schekochihin (2022)
$B\propto\lambda_{B}^{-5/4}$. The asterisk shows the scale where 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
Alfvén time.

####  13.0.4 Primordial fields during structure formation

Modern numerical simulations of cosmological large-structure formation are
taking into account the evolution of the magnetic field, seeded by the various
mechanisms outlined above. A central question in these studies is whether the
topology and strength of these primordial fields leave measurable signatures
on the cluster or galaxy structures.

Vazza et al. (2017) performed a comprehensive suite of cosmological
simulations using different magnetogenesis mechanisms: a uniform seed, meant
to simulate the magnetic field created by inflation, a seed that follows the
distribution of density perturbations to approximate the magnetic field
generation by a Biermann battery, a seed that approximates the turbulent
dynamo amplification, and an astrophysical seed that simulates the injection
of magnetic fields by stellar sources. They find that, at $z=0$, all
mechanisms agree on the cluster magnetization (which they were designed to
reproduce). However, there are large differences in the magnetic field
structure both on galaxy scales and in the voids. Recently, Mtchedlidze et al.
(2022) explored a more diverse set of primordial magnetic fields, 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 Figure 8, where we show maps of Faraday rotation at the
present time. The simulations started at a redshift of $z=50$ with the four
initial conditions discussed above.

Figure 8:  Faraday rotation maps from four cosmological simulations of
Mtchedlidze et al. (2022) using inflationary uniform and scale-invariant
fields (upper and lower left), as well as phase-transitional helical and
nonhelical initial fields (upper and lower right). Courtesy of Mtchedlidze et
al. (2022).

The above works use a uniform spatial resolution, which offers the advantage
of an unbiased view of cosmological magnetic field evolution. However, models
with adaptive resolution can give a more detailed view of the magnetic field
on galaxy scales, while following their cosmological history. One recent
example is the work of Garaldi et al. (2021), who explored the evolution of
cosmological volumes and zoom-ins using four different mechanisms for magnetic
field generation: primordial, Biermann battery, Durrive battery, and stellar
seeds. They report, contrary to the findings of the uniform-resolution, large-
volume works mentioned above, that the initial conditions are forgotten by
redshift $z\sim 2$. However, none of their initial conditions contained
magnetic helicity, which should not have decayed.

Marinacci & Vogelsberger (2016) and Martin-Alvarez et al. (2020) focused on
the effects of cosmological magnetic fields on galaxy formation. They found
that fewer, smaller galaxies form for stronger primordial fields. As mentioned
in Section 3.1, Martin-Alvarez et al. (2021), traced the evolution of the
primordial field and the field injected by stellar sources separately (see
Figure 3). 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 supernova
(SN)-enriched gas containing mostly fields of stellar origin with small-scales
coherence. The cold, star-forming gas contains a mixture of the two. However,
in agreement with the Garaldi et al. (2021) result, the origin of the galactic
magnetic field becomes practically indistinguishable very early on without the
tracers. All these simulations result in microgauss magnetic fields, but their
length scales are typically too small to explain the fields seen in actual
galaxies.

The results of these comprehensive simulations point to a complex picture in
which various seeding mechanisms combine to give the initial and boundary
conditions for dynamos on different scales and different epochs. They also
point to cluster scales – rather than galaxy scales for an answer regarding
the origin of cosmic magnetic fields.

####  13.0.5 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
Roettiger et al. (1999).

The study of the relevance of cluster mergers to dynamos has not been followed
up in recent years. In the meantime, there have been many relevant advances in
dynamo theory in connection with time dependence of the flow and in the
context of measuring field transport coefficients. In view of these advances,
this approach might deserve more detailed follow-up studies in the future.
However, there are similarities with recent studies of gravitational collapse
dynamos that will be discussed next.

###  13.1 Dynamos from gravitational collapse and other instabilities

By the time the first gravitationally bound structures (stars or galaxies)
formed, any primordial turbulent velocities from the processes we mention in
Section 13.0.3 had already decayed. However, the assembly into these first
structures certainly generated large amounts of turbulent kinetic energy,
which could have triggered dynamo action.

Several numerical works show that the formation of the first stars is ideal
for amplifying nG (Sur et al., 2010, 2012, Federrath et al., 2011), or even
just $10^{-20}\,{\rm G}$ (Schober et al., 2012) fields to equipartition values
through a small-scale dynamo. Gravitational compression can amplify the field
further, although in the presence of turbulence, the resulting dependence of
the magnetic field on the density is weaker than the prediction from ideal
flux freezing (Sur et al., 2012). While flux freezing predicts
$|\bm{B}|\propto\rho^{2/3}$, the theory of Xu & Lazarian (2020), which
includes the turbulent dynamo, predicts that
$|\bm{B}|/\rho^{2/3}\propto\rho^{2/57-1/6}\approx\rho^{-0.13}$, where the
scaling of the small-scale dynamo enters as an assumption. This remarkable
agreement with the simulation results of Sur et al. (2012) is taken to be
suggestive of the importance of reconnection diffusion and the breakdown of
flux-freezing (Xu & Lazarian, 2020), which makes compressional amplification
less efficient.

However, the results obtained so far 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 Xu & Lazarian (2020)
does not really address when the dynamo is excited, but focusses on the
discussion of the nonlinear regime of a supercritical dynamo.

####  13.1.1 Nature of collapse dynamos

An important tool for characterizing dynamo action in time-dependent flows
such as decaying turbulence or gravitational collapse, is to compare the work
done against the Lorentz force with the Joule dissipation rate, and to look at
different contributions to the Lorentz work term. These work and dissipation
terms emerge when deriving the evolution equation for the magnetic energy
density. Taking the dot product of Equation 2 with $\bm{B}$, averaging, and
ignoring surface terms, we obtain

$\frac{{\rm d}{}}{{\rm
d}{}t}\langle\bm{B}^{2}/2\mu_{0}\rangle=\langle\bm{J}\cdot(\bm{U}\times\bm{B})\rangle-\eta\mu_{0}\langle\bm{J}^{2}\rangle.$
(10)

Using $\bm{J}\cdot(\bm{U}\times\bm{B})=-\bm{U}\cdot(\bm{J}\times\bm{B})$, one
can write the first term on the right-hand side of Equation 10 as the work
against the Lorentz force. Two further refinements can then be employed
(Brandenburg & Ntormousi, 2022). First, one can decompose
$(\bm{U}\times\bm{B})_{i}=-U_{j}\partial_{j}A_{i}+U_{j}\partial_{i}A_{j}$ to
write $W_{\rm L}=W_{\rm L}^{\rm 2D}+W_{\rm L}^{\rm 3D}$, where $W_{\rm L}^{\rm
2D}=\langle J_{i}U_{j}\partial_{j}A_{i}\rangle$ and $W_{\rm L}^{\rm
3D}=-\langle J_{i}U_{j}\partial_{i}A_{j}\rangle$. The second term, $W_{\rm
L}^{\rm 3D}$, vanishes for 2-D magnetic fields oriented in the plane and
describes therefore the work term associated with 2-D compression, stretching,
and bending, such as in Equation 1. Second, one can decompose
$\bm{J}\times\bm{B}$ 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, Brandenburg & Ntormousi (2022) computed
the aforementioned terms that enter the magnetic energy balance. The basic
conclusion is that there is indeed dynamo action during the early phase of the
collapse while the initial turbulence is slowly decaying, but that dynamo
action diminishes when the flow becomes dominated by 3-D compression toward
the various collapsing potential minima, where only the irrotational flow
component gains in strength, which, however, does not (or not much) contribute
to dynamo action in their simulations. In Figure 9 we visualize the collapsing
magnetic field from Brandenburg & Ntormousi (2022), the diminishing of the
vorticity, expressed here as a wavenumber $k_{\bm{\omega}}=\omega_{\rm
rms}/u_{\rm rms}$, the gain of compressive motions, here expressed through
$k_{\,{\bm{\nabla}}\cdot\bm{u}}=({\bm{\nabla}}\cdot\bm{u})_{\rm rms}/u_{\rm
rms}$, along with several other quantities,
$k_{p{\bm{\nabla}}\cdot\bm{u}}=-\langle
p{\bm{\nabla}}\cdot\bm{u}\rangle/p_{0}u_{\rm rms}$ and
$k_{\bm{\omega}\cdot\bm{u}}=|\langle\bm{\omega}\cdot\bm{u}\rangle|/u_{\rm
rms}^{2}$, characterizing the work done by compression and the amount of
kinetic helicity, respectively. A potential problem with the simulations of
Brandenburg & Ntormousi (2022) is the relatively short collapse time compared
with the turnover time of the turbulence.

Figure 9:  Left: Visualization of the magnetic field. Right: Characteristic
wavenumbers $k_{\omega}$ (blue), $k_{\bm{\omega}\cdot\bm{u}}$ (red),
$k_{\,{\bm{\nabla}}\cdot\bm{u}}$ (solid black), and
$k_{p{\bm{\nabla}}\cdot\bm{u}}$ (dotted black).

####  13.1.2 Magneto-buoyancy and magneto-rotational instabilities

These instabilities can drive turbulence and may play important roles in parts
of the galaxy. Magneto-rotational instabilityarises in a magnetized, rotating
disk when the angular velocity decreases with distance. Buoyancy may be driven
by cosmic rays inflating flux tubes and are thought to speed up the dynamo
(Parker, 1992, Hanasz et al., 2013). The magneto-rotational instability
(Balbus & Hawley, 1991) can drive turbulence from the kinetic energy in the
shear. It can also play a role in the outer parts of the galaxy were supernova
driving is less efficient (Piontek & Ostriker, 2007).

##  14 GALACTIC MEAN-FIELD DYNAMOS

###  14.1 Global magnetic field structure

One of the strongest existing tests for dynamo theories is the predicted
structure of the large-scale magnetic field contrasted to observations. In the
next sections we outline the predictions from different models.

####  14.1.1 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 (Vainshtein &
Ruzmaikin, 1971, Parker, 1971), just after the first successful mean-field
models were proposed for the Sun and 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 (Krause & Rädler, 1980).]

In view of many early claimed discoveries of BSS fields (for a review, see
Sofue et al., 1986), an important question in those early days concerned the
possibility of preferred nonaxisymmetric magnetic fields. Such modes 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 [] ASS and BSS fieldsdefined as $m=0$ and $m=1$ magnetic fields and vary
with azimuth $\phi$ like $e^{{\rm i}m\phi}$. neglected in the generation of
the toroidal magnetic field, nonaxisymmetric modes where found to be excited,
although the growth rates of the corresponding ASS fields were always larger;
see Section 3.1. This approximation turned out to be not permissible, when the
magnetic field is nonaxisymmetric.

####  14.1.2 Boundary conditions

Standard dynamo problems are usually formulated with vacuum boundary
conditions, i.e., the magnetic field is current-free and extends to infinity
outside the domain (Krause & Rädler, 1980). However, such boundary conditions
can only be formulated for spheres or ellipsoids, but not for cylinders, for
example. Stix (1975) employed ellipsoidal coordinates and obtained an
axisymmetric solution. Contrary to Parker (1971), he found that oscillatory
solutions occur only at substantially larger dynamo numbers. Dynamo
numberDefined 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. Unfortunately, the
implementation of ellipsoidal coordinates in a numerical code is rather
cumbersome. This led to the approach of embedding the galaxy in a sufficiently
large poorly conducting halo, which itself is then contained either in a
cylinder with perfectly conducting boundaries (Elstner et al., 1990), or in a
sphere with vacuum boundaries (Brandenburg et al., 1990). These two
alternatives are rather different from each other, but the hope is that these
boundaries are far enough away from the physical boundaries that these
differences are without consequence.

###  14.2 Dynamo models for specific galaxies

Various attempts have been made to produce dynamo models of individual
galaxies. One such example is M31, i.e., the Andromeda galaxy. Its magnetic
field is often described as a ring field. It is also often regarded as an
analog of the Milky Way. Corresponding models have been presented by Poezd et
al. (1993) using nonlinear $\alpha$ quenching. An important challenge here is
to reproduce the right pitch angle of the magnetic field and its radial
dependence; see Shukurov (2000) and Fletcher et al. (2004) for detailed
discussions.

Another interesting case is M81, whose magnetic field is possibly
predominantly nonaxisymmetric. This was difficult to explain. Moss et al.
(1993) showed however that such a field could result from an initial magnetic
field that might have survived for long enough times, at least in the outer
parts of the galaxy. Yet another very different case is NGC 6946, whose field
may consist of structures usually termed magnetic arms. Magnetic arms are
often interlaced with the stellar arms, but can also be phase-shifted relative
to them; see Shukurov (1998) and Chamandy et al. (2015) for a more detailed
discussion, and Beck et al. (2019) for recent updates.

Finally, we mention the magnetic fields in the halos of the edge-on galaxies
NGC 891 and NGC 4631. Brandenburg et al. (1993) and Elstner et al. (1995)
found that the observed polarization vectors could only be reproduced when
there is a strong enough outflow. We return to this in Section 14.5.

###  14.3 Galactic models with magnetic helicity flux

In Section 8.1 we have discussed the potential importance of magnetic helicity
fluxes. It is often believed that they would be required to explain strong
magnetic fields in galaxies. Here, we demonstrate the effect of magnetic
helicity fluxes in specific models. Shukurov et al. (2006) have presented
nonlinear models in a one-dimensional geometry using the magnetic helicity
flux associated with a galactic fountain flow. The main nonlinearity was here
given through the dynamical quenching formalism with advective magnetic
helicity fluxes included, similarly to what was discussed in Section 8.1. 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 $300\,{\rm m}\,{\rm s}^{-1}$. More
detailed studies have been performed by Prasad & Mangalam (2016), who also
included advective and diffusive magnetic helicity fluxes.

Figure 10:  Left: Evolution of the energy of the mean magnetic field in a
model with a shear-induced magnetic helicity flux for different values of
$\mbox{\rm Re}_{\rm M}$ (here denoted by $R_{\rm m}$). Adapted from
Brandenburg & Subramanian (2005b). Right: Magnetic field evolution in models
with advective magnetic helicity fluxes for $\mbox{\rm Re}_{\rm M}=10^{5}$ and
different values strength of advection. The advection velocity is
characterized by the parameters $C_{U}$. Courtesy of Shukurov et al. (2006).

There is some uncertainty regarding the main contributors to the magnetic
helicity flux; see the more recent study by Vishniac & Shapovalov (2014). In
addition to advection, shear could modify the turbulent correlations in such a
way as to transport magnetic helicity efficiently outward. First proposed by
Vishniac & Cho (2001), this can lead to episodic magnetic field amplification,
especially as $\mbox{\rm Re}_{\rm M}$ is increased; see Brandenburg &
Subramanian (2005b) and the left-hand side of Figure 10. On the right-hand
side of Figure 10, we reproduce the simulation result of Shukurov et al.
(2006) with an advective magnetic helicity flux. In the models with
insufficient advective flux, the magnetic energy decreases to very small
values. The earlier simulations by Brandenburg & Subramanian (2005b), their
Figure 7, with somewhat smaller values of the magnetic Reynolds number showed
that the magnetic field can recover after some time, but then, again, it
begins to fall off, just like what is seen in Figure 10. However, one may want
to remain sceptical about whether these fluxes really do alleviate the
catastrophic quenching, because so far this has been seen only in mean-field
models and not yet in actual turbulence simulations.

###  14.4 Galactic rotation measure signature

In Section 3.1 we mentioned the historical importance of RM studies for
distinguishing between an ASS field, characteristic of dynamo models, and a
BSS field, characteristic of a wound-up primordial fields. The subsequent
findings of RM studies indicate that galactic magnetic field evolution might
be more complex than this simple dichotomy.

At the time of the review of Sofue et al. (1986), most 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 for only M81 (Krause et al., 1989). In a later review, Beck et
al. (1996) listed the field structures for 33 galaxies. The picture became
more complicated, with four examples primarily of ASS fields (albeit two where
marked as uncertain). The dominance of ASS over BSS continues to persist even
today. For M33, Tabatabaei et al. (2008) found an axisymmetric field in the
inner regions and a superposition of axisymmetric and bisymmetric fields in
the outer regions. Beck (2015b) observed a weak ($0.5\,\mu{\rm G}$)
axisymmetric field in IC 342, and Beck et al. (2020) found a dominating ASS
field in M31 combined with a six-times weaker BSS component.

Figure 11: Full-sky view of the Galactic magnetic field in (A) RM of
extragalactic sources (Hutschenreuter et al., 2022, where the range of RMs is
saturated at $\rm\lvert RM\rvert=250~{}rad/m^{2}$) and (B) synchrotron
emission (Bennett et al., 2013).

Classifying the Milky Way’s magnetic field structure is much harder. The
existing parametric models of the Galactic magnetic field are largely based on
full-sky RM maps of extragalactic sources (e.g., Oppermann et al., 2015,
Hutschenreuter et al., 2022) and synchrotron emission maps of the Milky Way
(mainly from the Wilkinson Microwave Anisotropy Probe; see Page et al., 2007,
Bennett et al., 2013, see also Figure 11). The model of Sun et al. (2008)
assumes an axisymmetric spiral with a reversal in the inner 5 kpc. Jansson &
Farrar (2012) include magnetic spiral arms and an X-shaped field in the halo
(see also Section 14.5 for the observational motivation). Jaffe et al. (2010)
also fit magnetic spiral arms to the disk data, also including the random
magnetic field component. Terral & Ferrière (2017), using analytic forms for
the 3-D field, conclude that a bisymmetric ($m=1$) halo field best fits the RM
data. However, West et al. (2020) found evidence for an axisymmetric ($m=0$)
quadrupolar magnetic field with a small net vertical component in RM. A newer
analysis (Dickey et al., 2022) shows that a combination of an axisymmetric and
a bisymmetric mode, based on analytical galactic dynamo models from Henriksen
et al. (2018), best explains the large-scale morphology of the Galactic RM
data. As the quality of the observational data improves, a complex picture of
galactic magnetic field morphology emerges, including that of the Milky Way.

###  14.5 Synchrotron emission from mean-field models

Synchrotron emission provides an important means of measuring the magnetic
field in galaxies and comparing models with simulations (see Beck et al.,
2019, for a review on bridging dynamo models and observations).

Early attempts of computing the polarized synchrotron emission from models
were presented by Donner & Brandenburg (1990), who computed the linearly
polarized emission from galactic mean-field models, which contained both
axisymmetric and nonaxisymmetric magnetic fields. They confirmed the idea of
distinguishing these modes by measuring the RM along a ring around the galaxy.
Another example was the computation of linear polarization from the magnetic
field on both sides of the midplane in edge-on galaxies. In galaxies seen
edge-on, synchrotron emission reveals X-shaped halo magnetic fields (e.g.,
Golla & Hummel, 1994, Tüllmann et al., 2000, Krause et al., 2006, Stein et
al., 2019, Krause et al., 2020). Although the 3-D morphology of these fields
is unknown, the X-shaped signature can be reproduced by dynamo models that
include an outflow (Brandenburg et al., 1993, Elstner et al., 1995). The
resulting magnetic fields were thought to have quadrupolar symmetry also in
the halo, but this now seems to be ruled out by new observations (Mora-
Partiarroyo et al., 2019). An alternative would be a dynamo in the halo
itself, which could produce dominant dipole modes (Brandenburg et al., 1992,
Moss & Sokoloff, 2008).

At long radio wavelengths, the synchrotron emission from even just a uniform
magnetic field suffers depolarization from the superposition of Faraday-
rotated contributions. However, if the magnetic field is helical, the
polarized intensity can either enhance the depolarization if helicity and RM
have opposite signs, or it can cancel it if they have the same sign
(Brandenburg & Stepanov, 2014, Horellou & Fletcher, 2014). This leads to a
correlation between polarized intensity and RM (Volegova & Stepanov, 2010),
which has now been used by West et al. (2020) to characterize the Galactic
magnetic helicity. Future observations with the Square Kilometre Array are
expected to reveal much more detailed information on magnetic helicity using a
continuous band of wavelengths (Beck et al., 2015).

The synchrotron intensity gives an indication about the magnetic field
strength. It is proportional to the product of the density of relativistic
cosmic ray electrons and a power close to 2 of the local magnetic field
component perpendicular to the line of sight. However, the relativistic cosmic
ray electron density may itself depend on the local magnetic energy density,
because cosmic rays and magnetic fields have supernova explosions as a common
source of energy. These arguments have been reviewed by Seta & Beck (2019),
who also make comparisons with numerical simulations of cosmic ray confinement
by a local dynamo-generated magnetic field, similar to what was done earlier
by Snodin et al. (2006). Seta & Beck (2019) conclude that the commonly made
assumption of an equipartition between cosmic ray and magnetic energy
densities is not valid on scales smaller than at least $100\,{\rm pc}$. They
argue that ignoring the nonlinear dependence of the synchrotron emission on
the plane-of-the sky magnetic field component can lead to an overestimation of
the actual magnetic field by up to a factor of 1.5.

An interesting comparison between radio synchrotron and dust polarization in
emission can be found in Borlaff et al. (2021) for M51. 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.

###  14.6 $E$ and $B$ polarizations

The linear polarization described by Stokes $Q$ and $U$ can also be expressed
in terms of the rotationally invariant parity-even $E$ polarization and the
parity-odd $B$ polarization, as is commonly done in cosmology (Seljak &
Zaldarriaga, 1997, Kamionkowski et al., 1997). Here, the symbols $E$ and $B$
have nothing to do with electric and magnetic fields, except that both can
qualitatively be described as gradient-like and curl-like fields. It is
important to stress, however, that $E$ and $B$ are only defined on a 2-D
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

$R\equiv E+{\rm
i}B=\sum_{\ell=2}^{N_{\ell}}\sum_{m=-\ell}^{\ell}\tilde{R}_{\ell m}Y_{\ell
m}(\theta,\phi).$ (11)

and $\tilde{R}_{\ell m}$ are coefficients that have been computed as

$\tilde{R}_{\ell m}=\int_{4\pi}(Q+{\rm i}U)\,_{2}Y_{\ell
m}^{\ast}(\theta,\phi)\,\sin\theta\,{\rm d}{}\theta\,{\rm d}{}\phi,$ (12)

with ${}_{2}Y_{\ell m}(\theta,\phi)$ being the spin-2 spherical harmonics and
the asterisk denoting the complex conjugate.

Figure 12: Left: Galactic $B$ mode polarization. Right: longitudinally
averaged $B$ mode polarization. Here, $\theta$ and $\phi$ are Galactic
colatitude ($=90\hbox{${}^{\circ}$}-\mbox{latitude}$) and longitude.

Brandenburg & Brüggen (2020) found that the $B$ polarization averaged over
Galactic longitude is very small owing to longitudinal cancelation, but there
is a small net hemispheric antisymmetry. This is shown in Figure 12, where we
plot the Galactic $B$ mode polarization together with the longitudinally
averaged $B$ mode polarization. It may be tempting to associate this
hemispheric dependence with that anticipated for the $\alpha$ effect, which is
also a parity-odd quality with hemispheric sign change. However, 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 $B$.

##  15 TURBULENCE SIMULATIONS OF GALACTIC DYNAMOS

###  15.1 Physical parameters of the ISM

The interstellar gas can be found in various phases, characterized by
different temperatures, densities, and degrees of ionization. The neutral,
atomic gas is found in a cold ($50\,{\rm K}<T<100\,{\rm K}$) and a warm
($10^{3}\,{\rm K}<T<10^{4}\,{\rm K}$) phase, usually termed the cold and warm
neutral media (CNM and WNM, respectively). The ionized gas is also found in a
hot ($T\simeq 10^{6}\,{\rm K}$) and a warm ($T\simeq 10^{4}\,{\rm K}$) phase
(HIM and WIM, respectively). Finally, the densest and coldest ($10\,{\rm
K}<T<20\,{\rm K}$) gas is mostly molecular medium (MM), with a very low
ionization fraction.

Dynamos can easily be excited in the interstellar medium, since all phases
(apart from the MM) are characterized by large values of Re, $\mbox{\rm
Re}_{\rm M}$, and $\mbox{\rm Pr}_{\rm M}$, although these vary greatly between
phases; see Table 3, with order-of-magnitude values taken from Ferrière (2020)
and Draine (2011). This vast range of parameters also poses a challenge for
accurately modeling interstellar turbulence. At large $\mbox{\rm Pr}_{\rm M}$,
the small-scale magnetic energy tends to dominate and the dynamo returns much
of the magnetic energy back into kinetic energy (Brandenburg & Rempel, 2019).
However, no effect on the large-scale dynamo has been reported as yet.

Table 3:  Parameters of the interstellar medium.  Phase | HIM | WIM | WIM | CNM | WNM | MM
---|---|---|---|---|---|---
Re | $10^{2}$ | $10^{7}$ | $10^{7}$ | $10^{10}$ | $10^{7}$ | $10^{7}$
$\mbox{\rm Re}_{\rm M}$ | $10^{23}$ | $10^{19}$ | $10^{18}$ | $10^{11}$ | $10^{18}$ | $10^{4}$
$\mbox{\rm Pr}_{\rm M}$ | $10^{21}$ | $10^{11}$ | $10^{11}$ | $10^{4}$ | $10^{11}$ | $10^{5}$

Most of the numerical work on ISM turbulent dynamos so far has been isothermal
(e.g., Schekochihin et al., 2004, Seta et al., 2020). In an interesting
extension, Seta & Federrath (2022) modelled a small-scale dynamo in driven
turbulence simulations of a two-phase ISM and identified the processes
responsible for vorticity generation in each phase. They found that the
magnetic to turbulent kinetic energy ratio is lower in the cold phase. This
will be discussed in more detail in Section 15.3.

###  15.2 Numerical approaches

Including magnetic fields in simulations of astrophysical systems is a non-
trivial task because the chosen discretization must obey the zero divergence
constraint. In Eulerian codes, a commonly used approach is the Constrained
Transport (CT) scheme (Evans & Hawley, 1988), which ensures
$\nabla\cdot\bm{B}=0$ by defining the magnetic field components on cell faces.
However, no similar scheme is applicable to Lagrangian codes such as Smoothed
Particle Hydrodynamics (SPH), which rely on divergence-cleaning algorithms
(e.g., Brackbill & Barnes, 1980, Powell et al., 1999, Dedner et al., 2002).

Table 4:  Overview of the numerical codes mentioned in this review, indicated
by the references.

Code name | Approach | Other properties
---|---|---
AREPO1 |  ${\bm{\nabla}}\cdot\bm{B}$-clean |  | Finite volume, unstructured moving mesh
---
ENZO2 | CT or ${\bm{\nabla}}\cdot\bm{B}$-clean |  | AMR, Riemann, split and unsplit schemes
---
FLASH3 | CT or ${\bm{\nabla}}\cdot\bm{B}$-clean |  | AMR, Riemann, split and unsplit schemes
---
Gadget4 |  ${\bm{\nabla}}\cdot\bm{B}$-clean | SPH
Nirvana5 | CT | AMR, Godunov (Riemann)
Pencil Code6 | $\bm{B}={\bm{\nabla}}\times\bm{A}$ | centered finite diff., sixth order
RAMSES 7 | CT | AMR, Godunov (Riemann)

  * 1

Pakmor et al. (2017), van de Voort et al. (2021), Whittingham et al. (2021)

  * 2

Vazza et al. (2017), Mtchedlidze et al. (2022)

  * 3

Sur et al. (2010), Federrath et al. (2011), Sur et al. (2012)

  * 4

Steinwandel et al. (2019)

  * 5

Gressel et al. (2008a)

  * 6

Brandenburg (2001, 2005, 2010), Brandenburg et al. (2010), Brandenburg (2019),
Gent et al. (2021), Brandenburg & Ntormousi (2022)

  * 7

Rieder & Teyssier (2016, 2017a, 2017b), Ntormousi et al. (2020)

Figure 13: Figure 7 from Mocz et al. (2016), where the same disk is evolved
with CT and a Powell-like divergence cleaning scheme. The winding pattern of
the magnetic field is better captured with CT, while an artifact is formed at
the center with divergence cleaning.

Lagrangian codes are particularly well-suited for modeling galaxies and
cosmological volumes due to the natural adaptation of the resolution to areas
of interest. They are also naturally Galilean-invariant. However, their
dependence on divergence cleaning poses a significant drawback when modeling
astrophysical dynamos. This was clearly demonstrated by Mocz et al. (2016),
who compared CT and divergence cleaning approaches, both implemented on the
moving-mesh code AREPO. They found that divergence cleaning systematically
creates artifacts that mimic physical effects. Some of these artifacts are
illustrated in Figure 13.

Out of these effects, particularly notorious for dynamo studies is an
artificial increase of the magnetic energy when using scalar divergence-
cleaning schemes, as pointed out by Balsara & Kim (2004) for supernova-driven
turbulence. Another very relevant example of an artifact caused by divergence
cleaning is the spontaneous production of magnetic helicity, which has been
found by Brandenburg & Scannapieco (2020), who compared simulations that
employed a divergence cleaning algorithm with one that advances instead the
magnetic vector potential, $\bm{A}$, so that
$\bm{B}={\bm{\nabla}}\times\bm{A}$ is always divergence-free. They found that,
for a helically driven flow in a periodic domain, spurious net magnetic
helicity is generated on dynamical timescales. An interesting experiment by
Tricco et al. (2016) compared an SPH code to the FLASH grid code, both using
divergence cleaning, in simulations of turbulent dynamos. They found very good
agreement between the codes, both in the growth rates and the saturation level
of the dynamo. This suggests that potential problems with divergence cleaning
may not be severe.

Yet another method of dealing with the ${\bm{\nabla}}\cdot\bm{B}=0$ constraint
is to employ the Euler or Clebsch potentials. However, this method only works
in the strictly ideal case when the microphysical magnetic diffusivity
vanishes; see Section 3.2. As we have stressed in Section 3.2, the addition of
an almost negligibly small diffusivity to the evolution equations for the
Euler potentials does not correspond to any physical magnetic diffusivity and
leads to wrong results where no dynamo is possible—even for flows that are
fast dynamos (Brandenburg, 2010).

Table 4 outlines some characteristics of the codes mentioned in this review.
Next to each code we have mentioned the works cited in this review that use
it. The fact that many of them have to rely on divergence cleaning methods is
evidence of the difficulty in dealing with the divergence problem, but also a
sign to use caution when interpreting the results in the context of dynamo
action.

###  15.3 Local dynamo simulations of galaxy portions

Simulating the magnetic field evolution over an entire galactic disk is
another challenging task, due to the vast range of dynamical scales of the
problem and the large shearing velocities involved. One approach to this
challenge that can successfully capture many aspects of the problem is
simulating galaxy portions. The local approach has been rather successful in
the context of accretion disks, where simulations have been performed in what
is known as shearing boxes. This means that the radial boundary is “shearing-
periodic”, i.e., it is periodic with respect to an azimuthal position that
shifts in time following the background shear flow.

Using a shearing box, Gressel et al. (2008a) performed the first simulation of
a galactic dynamo, including supernova-induced turbulence. It was similar to
earlier multiphase simulations of supernova-driven turbulence of Korpi et al.
(1999), where the magnetic Reynolds number was still too low to permit dynamo
action. Gressel et al. (2008a) found that the rotation frequency of the
considered galaxy portion is the dominant factor in determining the dynamo
efficiency, while the supernova rate did not significantly affect the
efficiency of the dynamo. This finding suggests that the simulations were able
to capture large-scale dynamo action, but not small-scale dynamo action.
Interestingly, they also found no evidence of catastrophic quenching in the
range of $\mbox{\rm Re}_{\rm M}$ values explored by varying the rotation
frequency of the galaxy portion. They hypothesize that this could be due to
helicity fluxes. In a subsequent paper, Gressel et al. (2013) speculated about
various quenching scenarios based on the magnetic field dependence, but that
was just for one value of the microphysical magnetic diffusivity.

Using the Pencil Code and a similar setup, Gent et al. (2013) showed that the
mean and fluctuating fields have different growth rates, indicating a co-
existence of small- and large-scale dynamos. Theoretically, however, the
possibility of large-scale and small-scale dynamos having different growth
rates in one and the same system is not well understood (Subramanian &
Brandenburg, 2014).

Recently, Gent et al. (2021) sought to derive criteria for the appearance of a
small-scale dynamo in simulations of interstellar turbulence. By not employing
a shearing-box setup or stratification, they focused only on the effects of
the supernova-driven turbulence. They confirm that, below a critical physical
resistivity (i.e., a sufficiently high $\mbox{\rm Re}_{\rm M}$), a small-scale
dynamo is easily excited by ISM turbulence, a result that appears to converge
at resolutions below 1 pc.

Seta & Federrath (2022) have shown that the multiphase aspect of the ISM tends
to have a detrimental effect on the small-scale dynamo. This is mostly because
of the stronger Lorentz force in the cold regions. Their simulations show that
with solenoidal forcing, the magnetic field is mostly decoupled from the
density behavior; see Figure 14. Simple compression along magnetic field lines
($\rho^{0}$), perpendicular to to magnetic field lines ($b\propto\rho^{1/2}$
for cylindrical/filamentary geometry and $b\propto\rho^{1}$ for disc-like/slab
geometry), and spherical compression ($b\propto\rho^{2/3}$) hardly occur. One
might argue, however, that in the compressive case, the cold phase shows a
higher slope than the warm phase.

Figure 14:  2D PDFs of magnetic field and density for (a) solenoidal and (b)
compressive forcing. The dashed black lines show various $b$–$\rho$ relations
for simple gas compressions. Especially the solenoidal shows very little
similarity with any of the simple relations.

###  15.4 Global isolated galaxy simulations

The increasing efficiency and complexity of numerical codes and the
availability of resources have led to a number of works studying the magnetic
field evolution in global galaxy models. In contrast to the shearing-box
approach, such models more naturally allow for the study of the large-scale
dynamo. However, the limited resolution is still problematic for
simultaneously capturing the small-scale dynamo, as outlined above.

Wang & Abel (2009) performed disk galaxy simulations including $\,{\rm nG}$
ordered seeds with a code similar to Enzo, using Dedner et al. (2002)
divergence-cleaning. They found that the tiny seed was amplified to $\mu$G
levels over 500 Myr. They also noticed that the magnetic field in the cold gas
saturated first. Their setup did not include stellar feedback, so the
amplification process was driven by differential rotation only.

In a series of papers, Rieder & Teyssier (2016, 2017a, 2017b) performed multi-
component simulations of the magnetic field evolution in dwarf and Milky-Way-
like galaxies using the RAMSES code and including supernova feedback. Their
setup includes dark matter and stars as collisionless particles, coupled to an
Adaptive Mesh Refinement (AMR) grid on which the MHD equations are solved. In
the first paper of the series, the authors found signatures of (Stellar)
FeedbackThe energy and momentum deposited to the ISM by stars through