Turbulence or random motions, which is prevalent in all systems from stars to galaxy clusters, is thought to be crucial for amplification of seed magnetic fields to the observed levels, a process called “turbulent dynamo”. Turbulence is driven mostly by supernovae in the galactic interstellar medium [

82], although during the formation of a galaxy by collapse from the IGM, accretion shocks and flows along cold streams could also be important [

83,

84,

85,

86,

87]. In disk galaxies, shear due to the differential rotation also plays an important role in the dynamo amplification process. Turbulent dynamos are conveniently divided into two classes, the fluctuation or small-scale and mean-field or large-scale dynamos. This split depends respectively on whether the generated field is ordered on scales smaller or larger than the scale of the turbulent motions. Here we briefly outline their role in galactic magnetism focusing on the challenges that they present. Much of our current understanding of these dynamos come from their analysis using statistical methods or direct numerical simulations (DNS). We shall focus more on some conceptual issues here.

#### 4.1. Fluctuation or Small-Scale Dynamos

The fluctuation dynamo is generic to sufficiently highly conducting plasma which hosts random motions, perhaps due to turbulence. First, in such plasma, magnetic flux through any area moving with the fluid is conserved. Moreover, in any turbulent flow, fluid parcels random walk away from each other and so magnetic field lines get extended. Consider a flux tube with plasma of density $\rho $, magnetic field B, area of cross section A and linking fluid elements separated by length l. Flux conservation implies $BA=\mathrm{constant}$. Mass conservation in the flux tube gives $\rho Al=\mathrm{constant}$, which implies $B/\rho \propto l$. Thus if l increases due to random stretching and $\rho $ is roughly constant, then B increases. This of course comes at the cost of $A\propto 1/\rho l\propto 1/B$ decreasing, the field being concentrated on smaller and smaller scales till resistivity becomes important at a scale ${l}_{B}$. An estimate for this resistive scale gives ${l}_{B}\sim l{R}_{\mathrm{m}}^{1/2}$, got by balancing the decay rate due to resistive diffusion, $\eta /{l}_{B}^{2}$, with growth rate due to random stretching $v/l$. Here v and l are the velocity and its coherence scale respectively of turbulent eddies. As ${R}_{\mathrm{m}}$ is typically very large in astrophysical systems, the resistive scale ${l}_{B}\ll l$.

What happens when resistive dissipation balances random stretching can only be addressed by a quantitative calculation. The first such calculation was due to Kazantsev [

88], who considered an idealized random flow which is

$\delta $-function correlated in time. For such a flow one can write an exact evolution equation for the two-point magnetic correlator, which has exponentially growing solutions, or is a dynamo, when

${R}_{\mathrm{m}}$ exceeds a modest critical value

${R}_{\mathrm{c}}\sim 100$. The growth rate is a fraction of the eddy turn over rate

$v/l$, and at this kinematic stage, the field is shown to be concentrated on the scale

${l}_{B}$. From the idealized Kazantsev model, it also turns out that

${R}_{\mathrm{c}}$ is larger and the growth slower for compressible flows compared to the incompressible case [

89,

90,

91]. Moreover, for Kolmogorov turbulence where the flow is multi-scale, ranging from the outer scale to the small scales where viscosity dominates, the fastest amplification is by the smallest supercritical eddy motions. In the galactic ISM, the kinematic viscosity

$\nu $ is typically much larger than the resistivity

$\eta $, and then growth would be expected to occur first due to dynamo action by the smaller viscous scale eddies [

92,

93]. For the interstellar turbulence with an outer scale of turbulence

$l\sim 100$ pc and velocities

$v\sim 10\phantom{\rule{0.166667em}{0ex}}\mathrm{km}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$, we expect

${R}_{\mathrm{m}}\gg {R}_{\mathrm{c}}$ and a growth time scale

$l/v\sim {10}^{7}$ years even by the largest eddies. This time scale is so much smaller than ages of even young high redshift galaxies, say a few times

${10}^{9}$ years old, that the fluctuation dynamo is expected to rapidly grow even weak seed magnetic fields to micro Gauss levels. Moreover, as smaller eddies can grow the field faster, significant amplification occurs even earlier. However as

${l}_{B}\ll l$, the field in the growing phase is extremely intermittent and concentrated on the small resistive scales. The big challenge is then whether these fields can become coherent enough to explain for example observations of the Faraday rotation inferred in young galaxies.

This growth of random magnetic fields due to the fluctuation dynamo has been verified by direct numerical simulations of driven turbulence, albeit in the idealized setting of isothermal plasma, for both subsonic and supersonic flows [

94,

95,

96,

97,

98,

99,

100,

101]). Such simulations however have a modest values of

${R}_{\mathrm{m}}/{R}_{\mathrm{c}}\sim $ 10–20. The basic expectations of the idealized Kazantsev model during the kinematic phase are qualitatively verified. The field grows exponentially and is concentrated initially on the resistive scales. It is also found that the small-scale dynamo is less efficient for compressible compared to solenoidal forcing, as it generates less vorticity [

100,

102,

103]. Importantly, the DNS can now also follow the field evolution in to the nonlinear regime when Lorentz forces act to saturate the dynamo. By the time the dynamo saturates, the coherence length of the field increases to be a fraction of order 1/3–1/4 the scale of the driving, at least when the magnetic Prandtl number

${\mathrm{Pr}}_{\mathrm{m}}=\nu /\eta $ is of order unity [

94,

96,

97,

98,

104]. These DNS have resolutions from

${512}^{3}$ upto

${2048}^{3}$. More modest resolution (

${256}^{3}$) DNS with large

${\mathrm{Pr}}_{\mathrm{m}}$ but small fluid Reynolds number Re found the magnetic energy spectrum to be still peaked at the resistive scale

${l}_{B}$ even at saturation [

95]. It is difficult to directly simulate the case expected in the interstellar medium, of both a large

${R}_{\mathrm{m}}/{R}_{\mathrm{c}}$ and large Re, as one then has to resolve both the widely separated resistive and viscous dissipation scales. Clearly the saturated state of the fluctuation dynamo deserves further study, especially in this highly turbulent and

${\mathrm{Pr}}_{\mathrm{m}}\gg 1$ regime.

We have also directly determined Faraday rotation measures (RMs), in simulations of the fluctuation dynamo with various values of

${R}_{\mathrm{m}}$, fluid Reynolds number Re and up to rms Mach number of

$\mathcal{M}=2.4$ [

97,

101,

105]. At dynamo saturation, for a range of parameters, we find an rms RM contribution which is about half the value expected if the field is coherent on the turbulent forcing scale. Interestingly, in subsonic and transonic cases, the general sea of volume filling fields, dominates in determining the strength of RM. The rarer, strong field structures, contribute only about 10%–20% to the RM signal, indicating that perhaps the coherence of the generated fields is associated with more typical volume filling magnetic field regions. However, when the turbulence is supersonic significant contributions to the RM also comes from strong field regions as well as moderately over dense regions. How exactly the field orders itself during saturation is at present an open problem.

One may wonder if magnetic reconnection is important for dynamo action. We note that the reconnection speed, depends inversely on the magnetic field strength even when it is efficient. Thus it would be too long compared to the dynamo growth rate to be relevant during the kinematic stage of the dynamo. It could however play a role once the field becomes dynamically important. Some interesting aspects of a reconnecting flux rope dynamo have been explored in [

106]. In nearly collisionless plasmas like galaxy clusters, plasma effects could set transport properties even for weak fields and small scale dynamo action in such a context is just beginning to be explored [

107,

108].

Simulations of galaxy formation from cosmological initial conditions have also showed evidence for amplification by the fluctuation dynamo, over and above the result of amplification by flux freezing during the compressive collapse to form the galaxy [

87,

109,

110,

111,

112]. One of the main limitations of such cosmological simulations is the resolution; that it will be very difficult to capture both the galactic scale and the dissipative scales, to predict correctly the rate of growth of magnetic energy and the coherence scale of the saturated field. Intriguingly, some of the direct simulations of SNe driven turbulence, which have possibility of a multiphase medium, do not yet show a strong fluctuation dynamo [

113,

114,

115], although they do show large-scale dynamo action (except for [

116]).

All in all, one expects energy of random, intermittent magnetic fields to generically grow rapidly in the turbulent ISM of galaxies. This turbulence could be driven by supernovae in disk galaxies. Galactic disks would then host significant fields, and a line of sight going through the disk could have a significant RM [

97,

101]. This can partly explain the statistical detection of excess RM in MgII absorption systems [

4,

5], which are thought to be associated with young galaxy disks at redshifts

$z\sim 1$. However, the abundance of these systems gives evidence that the MgII absorption arises not only in line of sights through the disk, but also in extended gaseous halos [

117]. Thus one would need the halo to be also magnetized and produce a significant RM. This could occur through outflows from the disk which also carry cold magnetized "clouds". More work is required to firm up such a speculation.

#### 4.2. Mean-Field or Large-Scale Dynamos and Galactic Magnetism

Remarkably, when turbulence is helical, magnetic fields on scales larger than the coherence scale of the turbulence can be amplified. In any rotating, stratified system like the ISM of a disk galaxy random motions driven by supernovae do become helical due to the Coriolis force, with one sign of helicity in the northern hemisphere and the opposite sign in the southern hemisphere. Such helical turbulent motions of the plasma draw out toroidal fields in the galaxy into a twisted loop generating poloidal components (called the $\alpha $-effect). Differential rotation of the disk shears the radial component of the poloidal field to generate back a toroidal component (the $\omega $-effect). These two can combine to exponentially amplify the large-scale field provided that the generation terms can overcome an extra resistivity due to the turbulence. This is quantified by a dimensionless dynamo number being supercritical. Turbulent resistivity also allows the mean-field flux to be changed.

Quantitatively, in mean-field dynamo theory, the total magnetic field is split as

$B}=\overline{{\scriptstyle B}}+{\textstyle b$, the sum of a mean (or the large-scale) field

$\overline{{\scriptstyle B}}$ and fluctuating (or the small-scale) field

$b$. A similar split of the velocity field gives

$V}=\overline{{\scriptstyle V}}+{\textstyle v$. The mean is defined by some form of averaging on scales larger than the turbulence coherence scale, ideally but not necessarily satisfying Reynolds rules for such averaging. These rules are [

118]:

$\overline{{{\scriptstyle B}}_{1}+{{\scriptstyle B}}_{2}}={\overline{{\scriptstyle B}}}_{1}+{\overline{{\scriptstyle B}}}_{2}$,

$\overline{\overline{{\scriptscriptstyle B}}}=\overline{{\scriptstyle B}}$, so

$\overline{{\scriptstyle b}}=0$,

$\overline{\overline{{\scriptscriptstyle B}}{\scriptstyle b}}=0$,

$\overline{{\overline{{\scriptscriptstyle B}}}_{1}\phantom{\rule{0.277778em}{0ex}}{\overline{{\scriptscriptstyle B}}}_{2}}={\overline{{\scriptstyle B}}}_{1}\phantom{\rule{0.277778em}{0ex}}{\overline{{\scriptstyle B}}}_{2}$ and averaging commutes with both time and space derivatives. The induction equation Equation (

1) then averages to give

Here a new term quadratic in the fluctuating fields arises, the mean electromotive force (EMF),

$\mathcal{E}}=\overline{{\scriptstyle v}\times {\scriptstyle b}$. To express this in terms of the mean fields themselves presents a closure problem, even when Lorentz forces are not yet important. The simplest such closure, which is valid when the correlation time

$\tau $ is small compared to

$l/v$ gives

$\mathcal{E}}={\alpha}_{K}\overline{{\scriptstyle B}}-{\eta}_{t}\nabla \times \overline{{\scriptstyle B}$, where the turbulent motions are also assumed to be isotropic. Here

${\alpha}_{K}=-{\textstyle \frac{1}{3}}\tau \langle {\textstyle v}\xb7{\textstyle \mathit{\omega}}\rangle $ with

$\mathit{\omega}}=\nabla \times {\textstyle v$, depending on the kinetic helicity of the turbulence and is the

$\alpha $-effect mentioned above while

${\eta}_{t}={\textstyle \frac{1}{3}}\tau \langle {{\textstyle v}}^{2}\rangle $ is a turbulent diffusivity and depends on the kinetic energy of the turbulence. In disk galaxies we also have a

$\overline{{\scriptstyle V}}=r\Omega \left(r\right){\textstyle \mathit{\varphi}}$ corresponding to its differential rotation with frequency

$\Omega $ along the toroidal direction

$\mathit{\varphi}$. The mean-field dynamo equation (

9) with this form for

$\mathcal{E}$ and

$\overline{{\scriptstyle V}}$, has exponentially growing solutions provided a dimensionless dynamo number has magnitude

$D=|{\alpha}_{0}S{h}^{3}/{\eta}_{t}^{2}|>{D}_{crit}\sim 6$ [

75,

119,

120]. Here

h is the disk scale height and

$S=rd\Omega /dr$ the galactic shear,

${\alpha}_{0}$ a typical value of

$\alpha $, and we have defined

D to be positive. This condition can be satisfied in disk galaxies and the mean field typically grows on the rotation time scale,

$\sim {10}^{8}-{10}^{9}$ years. A detailed account of mean-field theory predictions for galactic dynamo theory and its comparisons to observations is done by other authors in this volume. We focus on the challenges for this general paradigm, in our view.

#### 4.2.1. Magnetic Helicity Conservation

The first potential difficulty, which has already received considerable attention, arises due to the conservation of magnetic helicity in the highly conducting galactic plasma. Magnetic helicity is usually defined as

$H={\int}_{V}{\textstyle A}\xb7{\textstyle B}\phantom{\rule{4pt}{0ex}}dV$ over a ’closed’ volume

V, with

$A$ the vector potential satisfying

$\nabla \times {\textstyle A}={\textstyle B}$. It is invariant under a gauge transformation

${{\textstyle A}}^{\prime}={\textstyle A}+\nabla \Lambda $ only if the normal component of the field on the boundary to volume

V goes to zero. Magnetic helicity measures the linkages between field lines [

121,

122], is an ideal invariant and is better conserved than total energy in many contexts, even when resistivity is included. The mean-field dynamo works by generating poloidal from toroidal field and vice-versa and thus automatically generates links between these components, and thus a large-scale magnetic helicity. To conserve the total magnetic helicity, corresponding oppositely signed helicity must then be transferred to the small-scale field, which as we shall see is done by the turbulent emf

$\mathcal{E}$.

In fact, when helical motions writhe the toroidal field to generate a poloidal field, an oppositely signed twist must develop on smaller scales, to conserve magnetic helicity. For the same magnitude of magnetic helicity on small and large scales, the Lorentz force

$({\textstyle J}\times {\textstyle B})/c$ is generally stronger on small-scales (since

$J$ the current density has two more derivatives compared to the vector potential which determines magnetic helicity). Thus Lorentz forces associated with this twist helicity can unwind the field while turbulent motions writhe it. According to closure models like the Eddy damped quasi linear Markovian (EDQNM) approximation [

123] or the

$\tau $ approximation [

81,

124,

125], Lorentz forces then lead to an additional effective magnetic

$\alpha $-effect,

${\alpha}_{M}=\frac{1}{3}\tau \overline{{\scriptstyle j}\xb7{\scriptstyle b}}/4\pi \rho $, with the total

$\alpha ={\alpha}_{K}+{\alpha}_{M}$. The generated magnetic

${\alpha}_{M}$ opposes the kinetic

${\alpha}_{K}$ produced by the helical turbulence and quenches the

$\alpha $-effect and the dynamo, making it subcritical, much before the large-scale field grows strong enough to itself affect the turbulence. For avoiding such quenching, small-scale helicity must be shed from the galactic interstellar medium. In principle resistivity can dissipate small-scale magnetic helicity but this takes a time longer than the age of the universe! For large-scale dynamos to work small-scale helicity must be lost more rapidly, through magnetic helicity fluxes [

81,

122,

126].

Magnetic helicity being a topological quantity, one may wonder how to define its density and its flux! A Gauge invariant definition of helicity density was given by Subramanian and Brandenburg [

127] using the Gauss linking formula for the magnetic field [

121,

128]. They proposed that the magnetic helicity density

h of a random magnetic field

$b$ is the density of correlated links of the magnetic field [

127]. This definition by construction involves only the random field

$b$, works if this field has a small correlation scale compared to the system scale, and is closest to the helicity density defined using the vector potential in Coulomb gauge. An evolution equation can then be derived for this density of helicity which now also involves a helicity flux density

$\overline{{\scriptstyle F}}$ [

127],

This equation involves transfer of magnetic helicity from large to small scales by the turbulent emf along the mean field (

$-2{\textstyle \mathcal{E}}\xb7\overline{{\scriptstyle B}}$ term), the dissipation by resistivity (

$-2\eta \overline{\nabla \times {\scriptstyle b}\xb7{\scriptstyle b}}$) and the spatial transport by the helicity flux (

$\nabla \xb7\overline{{\scriptstyle F}}$). In the absence of such a flux, and in the steady state we see that

$\mathcal{E}}\xb7\overline{{\scriptstyle B}}=-2\eta \overline{\nabla \times {\scriptstyle b}\xb7{\scriptstyle b}$ and so the emf along the field, which is important for the dynamo, is resistively suppressed for

${R}_{\mathrm{m}}\gg 1$. Even in the time dependent case, as the

$\overline{{\scriptstyle B}}$ builds up,

h also grows and produces an

${\alpha}_{M}$ which cancels

${\alpha}_{K}$ to suppress the net

$\alpha $ effect. In the presence of helicity fluxes however,

h can be transported out of the system allowing mean-field dynamos to work efficiently [

126,

129,

130].

One such flux is simply advection of the gas and its magnetic field out of the disk, i.e.,

$\overline{{\scriptstyle F}}=h\overline{{\scriptstyle V}}$ [

127,

129]. Several other types of helicity fluxes have been calculated like the Vishniac-Cho flux depending on shear and the mean field [

81,

131] and a flux involving inhomogeneous

$\alpha $ [

132]. A diffusive flux

$\overline{{\scriptstyle F}}=-\kappa \nabla h$ was postulated by [

133] and subsequently measured in DNS [

134]. A new type of helicity flux which depends on purely an inhomogeneous random magnetic field and rotation or shear has been worked out by Vishniac [

135], and could be potentially important to drive a large-scale dynamo purely from random fields in the galaxy, but has not yet been studied in detail. Both the diffusive flux and the later Vishniac flux have been derived from the irreducible triple correlator contribution to

$\overline{{\scriptstyle F}}$ by [

136] using a simple

$\tau $-closure theory, but they also find several other terms which cannot be reduced to either of these forms. A detailed study of magnetic helicity fluxes still remains one of the important challenges of the future.

As an interesting application of these ideas, Chamandy et al. [

137] solved the mean-field dynamo equation incorporating both an advective flux and a diffusive flux in Equation (

10). Advection can be larger from the optical spiral, where star formation and galactic outflows are expected to be enhanced. The helicity fluxes allow the mean-field dynamo to survive, but stronger outflow along spiral arms led to a relative suppression of mean field generation there and an interlaced pattern of magnetic and gaseous arms as seen in the galaxy NGC6946 [

138]. Interestingly a wide spread magnetic spiral only results if the optical spiral is allowed to wind up and thus here we are constraining spiral structure theory using magnetic field observations [

137,

139]! In another direction, the cosmic evolution of large-scale magnetic fields during hierarchical clustering in the universe to form galaxies, has also been extensively explored [

140,

141].

#### 4.2.2. Mean-Field Dynamo in Presence of the Fluctuation Dynamo

We have discussed possibilities of both fluctuation and mean-field dynamos in the turbulent interstellar medium. However random magnetic fields due to the fluctuation dynamo grow much faster on time scale of

${10}^{7}$ years, at least a factor 10 faster than the mean-field. Lorentz forces can then become important to saturate the field growth much before the mean field has grown significantly. Will, then, these strong fluctuations make mean field theory invalid? Can the large-scale field then grow at all? Earlier work [

93] suggested that perhaps the intermittency of the small-scale dynamo generated field on saturation still allows the Lorentz force to be sub dominant in the bulk, and thus allow large-scale field growth. Bhat et al. [

142] examined this issue using direct simulations of magnetic field amplification due to fully helical turbulence in a periodic box, following up earlier work on the kinematic stage by [

143]. Turbulence was forced at about

$1/4$ th the scale of the box, so that in principle both scales smaller and larger than forcing can grow. Initially all scales grow together as a shape invariant eigen function dominated by power on small-scales. This behaviour is akin to what happens in fluctuation dynamos. But crucially on saturation of small scales due to the Lorentz force, larger and larger scales continue to grow, and come to dominate due to the mean-field dynamo action. Finally system scale fields (here the scale of the box) develop provided small-scale magnetic helicity can be efficiently removed, which in this simulation is due to resistive dissipation. Recent work by Bhat et al. [

144] in fact now finds evidence for two stages of exponential growth, the sequential opertaion of both the small-scale dynamo, and as it saturates, a quasi-kinematic large-scale dynamo, which is indeed exciting! This issue of how the small- and large- scale dynamos come to terms with each other deserves much more attention including a better analytic understanding.