# Efficiency of dynamos from autonomous generation of chiral asymmetry

Jennifer Schober<EMAIL_ADDRESS>Institute of Physics, Laboratory of
Astrophysics, École Polytechnique Fédérale de Lausanne (EPFL), 1290 Sauverny,
Switzerland  Igor Rogachevskii  Department of Mechanical Engineering, Ben-
Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel
Nordita, KTH Royal Institute of Technology and Stockholm University, 10691
Stockholm, Sweden  Axel Brandenburg  Nordita, KTH Royal Institute of
Technology and Stockholm University, 10691 Stockholm, Sweden  The Oskar Klein
Centre, Department of Astronomy, Stockholm University, AlbaNova, SE-10691
Stockholm, Sweden  School of Natural Sciences and Medicine, Ilia State
University, 0194 Tbilisi, Georgia  McWilliams Center for Cosmology and
Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania
15213, USA

###### Abstract

At high energies, the dynamics of a plasma with charged fermions can be
described in terms of chiral magnetohydrodynamics. Using direct numerical
simulations, we demonstrate that chiral magnetic waves (CMWs) can produce a
chiral asymmetry $\mu_{5}=\mu_{\mathrm{L}}-\mu_{\mathrm{R}}$ from a spatially
fluctuating (inhomogeneous) chemical potential
$\mu=\mu_{\mathrm{L}}+\mu_{\mathrm{R}}$, where $\mu_{\mathrm{L}}$ and
$\mu_{\mathrm{R}}$ are the chemical potentials of left- and right-handed
electrically charged fermions, respectively. If the frequency of the CMW is
less than or comparable to the characteristic growth rate of the chiral dynamo
instability, the magnetic field can be amplified on small spatial scales. The
growth rate of this small-scale chiral dynamo instability is determined by the
spatial maximum value of $\mu_{5}$ fluctuations. Therefore, the magnetic field
amplification occurs during periods when $\mu_{5}$ reaches temporal maxima
during the CMW. If the small-scale chiral dynamo instability leads to a
magnetic field strength that exceeds a critical value, which depends on the
resistivity and the initial value of $\mu$, magnetically-dominated turbulence
is produced. Turbulence gives rise to a large-scale dynamo instability, which
we find to be caused by the magnetic alpha effect. Our results have
consequences for the dynamics of certain high-energy plasmas, such as the
early Universe or proto-neutron stars.

††preprint: NORDITA-2024-008

##  I Introduction

In the Standard Model of particle physics, the chirality of high-energy
fermions can lead to macroscopic quantum effects, which are a result of the
chiral anomaly. A prominent example is the chiral magnetic effect (CME) [1],
which is relevant at high energies and can lead to a magnetic field-aligned
electric current, if there is an asymmetry between the number density of left-
and right-handed electrically charged fermions. The emergence of the CME and
other novel quantum phenomena in non-equilibrium relativistic quantum matter
can be derived from first principles [2, 3, 4, 5, 6, 7, 8, 9]. However, to
improve the usability of the models, lots of effort has been put into the
development of a quantum kinetic theory for massless fermions often referred
to as “chiral kinetic theory” [10, 11, 12, 13]. The additional electric
current caused by the CME can also be incorporated into an effective
description of a relativistic plasma. Such models have become known as “chiral
(or anomalous) magnetohydrodynamics” (chiral MHD) [14, 15, 16, 17, 18]. This
paper is based on chiral MHD as its theoretical framework.

Chiral phenomena occur in plasmas with fermions that are effectively massless.
In the context of astrophysics and cosmology (see [19] for a recent review),
this limits the applications to high-energy plasma in which the temperature is
above $10$ MeV [20]. A prime example is the hot and dense plasma that fills
the early Universe. It was first suggested in Ref. [21] that the CME can lead
to an instability in the primordial magnetic field, which is now known as the
chiral plasma instability [19] or the small-scale chiral dynamo instability
111This instability is referred to differently in the literature of different
fields. In high-energy physics, it is often called the “chiral plasma
instability”, but also the names “$v_{\mu}$-” or “$v_{5}$-”dynamo exist. In
this paper, we only use the notion “small-scale chiral dynamo instability”..
If the dynamo is excited, strong helical magnetic fields can be generated
[23], which can drive magnetically-dominated turbulence that gives rise to
mean-field dynamos [17, 24]. These primordial magnetic fields can potentially
explain the baryon asymmetry of the Universe [25, 26], produce relic
gravitational waves [27, 28], and affect the properties of the global 21 cm
signal [29] and dwarf galaxies [30].

A second domain within astrophysics and cosmology where the chiral anomaly
becomes relevant is core-collapse supernovae. Here, a chiral imbalance is
generated through the emission of neutrinos which are, in the Standard Model
of particle physics, only left-handed. Chiral effects have been included in
modelling the magnetic field evolution in core-collapse supernovae [31, 32],
and were suggested to play a role in the generation of magnetars [33, 34, 35,
36] and the occurrence of pulsar kicks [37, 38]. These ideas have recently
been extended by possible implications of the chiral anomaly in magnetospheres
of pulsars [39], where the produced chiral asymmetry can be substantial. It
can trigger the small-scale chiral dynamo which, in turn, can produce
circularly polarized electromagnetic radiation in a wide range of frequencies,
spanning from radio to near-infrared. This can affect some features of fast
radio bursts.

Beyond the extreme environments in the Universe, chiral effects can be studied
more directly in heavy-ion colliders [40]. However, the existence of the CME
has not yet been confirmed in experiments conducted at the Large Hadron
Collider [41] or the Relativistic Heavy Ion Collider [42]. At low energies,
chiral effects can emerge in new materials that include massless quasi-
particles [43, 44, 45]. The detection of the CME in Dirac semimetals [46, 47]
opens up the possibilities of novel technological developments [e.g., in the
field of quantum computing 48].

Chiral MHD differs from classical MHD by an additional term in the induction
equation, which describes the evolution of the magnetic field. This term stems
directly from the additional contribution to the electric current from the CME
and is proportional to the chiral chemical potential
$\mu_{5}\equiv\mu_{\mathrm{L}}-\mu_{\mathrm{R}}$, where $\mu_{\mathrm{L}}$ and
$\mu_{\mathrm{R}}$ are the chemical potentials of left- and right-handed
fermions, respectively. This additional term leads to an instability in the
magnetic field on small spatial scales [21], the (small-scale) chiral dynamo
instability, if $\mu_{5}$ is non-zero. The amplification of magnetic energy in
the nonlinear stage of the chiral dynamo instability can cause the production
of magnetically-dominated turbulence, making exact analytical treatment
unfeasible. Nevertheless, mean-field theory allows for exploring the effects
of turbulence in chiral MHD. In particular, the occurrence of a new mean-field
dynamo, i.e., the $\alpha_{\mu}$ dynamo, was predicted in Ref. [17]. With
direct numerical simulations (DNS), it has been shown that, in the nonlinear
evolutionary stage, a mean-field dynamo instability can occur [24]. In recent
studies [49, 50], it has been demonstrated that the chiral dynamo instability
even occurs in a plasma with an initial spatial fluctuating chiral chemical
potential with zero mean. A necessary condition for a chiral dynamo
instability is that the effective correlation length of chiral chemical
potential fluctuations is larger than the corresponding instability length
scale, which is given by the inverse of the spatial maximum value of
$\mu_{5}$.

The aforementioned studies have explored the role of the CME in the evolution
of magnetic fielxds. However, the CME is not the only macroscopic quantum
effect that results from the chiral anomaly. Another prominent example is the
chiral separation effect (CSE) [51]. The CSE is a complementary transport
phenomenon to the CME in which a nonzero chemical potential
$\mu=\mu_{\mathrm{L}}+\mu_{\mathrm{R}}$ generates an axial current along an
external magnetic field. A consequence of the CSE is the possibility of
exciting new collective modes, most notably the chiral magnetic wave (CMW)
[52]. These waves imply periodic conversion between $\mu_{5}$ and $\mu$, in
the presence of a small background magnetic field and non-vanishing gradients
of $\mu_{5}$ and $\mu$. Chiral magnetic waves in chiral plasma have been
studied in a number of publications [53, 54, 55]. Simulating the CMW in a
Cartesian domain, it has been recently shown [56] that the chiral dynamo
instability and even mean-field dynamos can occur for vanishing initial chiral
asymmetry if initial spatial fluctuations of the chemical potential are
inhomogeneous $({\bm{\nabla}}\mu\not=0)$. In this study, we explore the
parameter space of CMWs (for which the chemical potential is nonuniform) and
identify the conditions under which the chiral dynamo instability and mean-
field dynamos can be exited for plasmas with vanishing initial chiral
asymmetry.

The outline of this paper is as follows. In Sec. II we present the system of
equations that describe plasma with relativistic fermions including the CMS
and CSE, and discuss the initial conditions that we consider. In Sec. III the
evolution of the system is modelled phenomenologically and we make some
predictions for different scenarios. The system of equations is solved
numerically in Sec. IV, where we compare our predictions with the numerical
results. Finally, discussion and conclusions are drawn in Sec. V.

##  II System of equations

###  II.1 Chiral MHD equations with CSE

In this paper, we study effects of relativistic fermions applying an effective
fluid description for plasma motions. As in our previous study [56], we
consider the following set of equations which includes both the chiral
magnetic effect and the CSE:

$\displaystyle\frac{\partial\bm{B}}{\partial t}$ $\displaystyle=$
$\displaystyle\bm{\nabla}\times\left[{\bm{U}}\times{\bm{B}}+\eta\,\left(\mu_{5}{\bm{B}}-\bm{\nabla}\times{\bm{B}}\right)\right],$
(1) $\displaystyle\rho{D\bm{U}\over Dt}$ $\displaystyle=$
$\displaystyle(\bm{\nabla}\times{\bm{B}})\times\bm{B}-\bm{\nabla}p+\bm{\nabla}{\bm{\cdot}}(2\nu\rho\mbox{\boldmath${\sf
S}$}{}),$ (2) $\displaystyle\frac{D\rho}{Dt}$ $\displaystyle=$
$\displaystyle-\rho\,\bm{\nabla}\cdot\bm{U},$ (3)
$\displaystyle\frac{D\mu}{Dt}$ $\displaystyle=$
$\displaystyle-\mu\,\bm{\nabla}\cdot\bm{U}-\mathcal{D}_{\mu}\,\nabla^{4}\mu-
C_{\mu}(\bm{B}{\bm{\cdot}}\bm{\nabla})\mu_{5},$ (4)
$\displaystyle\frac{D\mu_{5}}{Dt}$ $\displaystyle=$
$\displaystyle-\mu_{5}\,\bm{\nabla}\cdot\bm{U}-\mathcal{D}_{5}\,\nabla^{4}\mu_{5}-C_{5}({\bm{B}}{\bm{\cdot}}\bm{\nabla})\mu$
(5)
$\displaystyle+\lambda\,\eta\,\left[{\bm{B}}{\bm{\cdot}}(\bm{\nabla}\times{\bm{B}})-\mu_{5}{\bm{B}}^{2}\right].$

Here, $\bm{B}$ and $\bm{U}$ are the magnetic field and the velocity field,
respectively, $\eta$ is the microscopic magnetic diffusivity, $p$ is the
pressure, $\nu$ is the viscosity, $\rho$ is the mass density, ${\sf S}$ is the
trace-free strain tensor with components ${\sf
S}_{ij}=(\partial_{j}U_{i}+\partial_{i}U_{j})/2-\delta_{ij}({\bm{\nabla}}{\bm{\cdot}}\bm{U})/3$.
In Eq. (5), $\lambda=3\hbar c(8\alpha_{\rm em}/k_{\rm B}T)^{2}$ is the chiral
feedback parameter, where $\hbar$ is the reduced Planck constant, $c$ is the
speed of light, $\alpha_{\rm em}\approx 1/137$ is the fine structure constant,
$k_{\rm B}$ is the Boltzmann constant, and $T$ is the temperature. To close
the system of equations, we use an isothermal equation of state $p=\rho c_{\rm
s}^{2}$, where $c_{\rm s}$ is the sound speed. For numerical stability, the
evolution equations for $\mu_{5}$ and $\mu$ also include (hyper-)diffusion
terms with the diffusion coefficients $\mathcal{D}_{5}$ and
$\mathcal{D}_{\mu}$ [50]. The coupling between $\mu_{5}$ and $\mu$, the
strength of which is determined by the coupling constants $C_{5}$ and
$C_{\mu}$, leads to chiral magnetic waves (CMWs) [52]. When considering the
coupled linearized equations (4) and (5), the frequency of CMWs is found to be

$\displaystyle\omega_{\rm
CMW}=\pm\left[C_{5}\,C_{\mu}({\bm{k}}\cdot{\bm{B}}_{\mathrm{ex}})^{2}-{1\over
4}\left(\lambda\eta\,B_{\mathrm{ex}}^{2}\right)^{2}\right]^{1/2},$ (6)

where ${\bm{B}}_{\mathrm{ex}}$ is the external magnetic field and ${\bm{k}}$
is the wave vector. As long as the magnetic fluctuations are smaller than
${\bm{B}}_{\mathrm{ex}}$, the characteristic timescale of these waves is half
of the period

$\displaystyle P_{\mathrm{CMW}}=\frac{2\pi}{\omega_{\mathrm{CMW}}},$ (7)

since this is the timescale on which the sign of $\mu_{5}$ changes. The
damping rate of the CMW is

$\displaystyle\gamma_{\rm CMW}=-{1\over 2}\lambda\eta\,B_{\mathrm{ex}}^{2}.$
(8)

###  II.2 Initial conditions

We consider initial conditions where $\mu_{5}(t_{0})=0$ and $\mu(t_{0})$ are
spatially random fields consisting of Gaussian noise with a power law
spectrum, $E_{\mu}(k,t_{0})\propto\left(k/k_{1}\right)^{s}$, where $k_{1}$ is
the minimum wavenumber in the system. The initial magnetic field is weak and
in the form of Gaussian noise. Additionally, we consider an external very weak
uniform magnetic field with $\bm{B}_{\mathrm{ex}}=(B_{\mathrm{ex}},0,0)$ to
support CMWs, which effectively produce the chiral asymmetry, i.e., a
difference in the left- and right-handed chemical potentials. The initial
velocity field vanishes.

##  III Phenomenology

###  III.1 Production of $\mu_{5}$

In the initial phase, a chiral asymmetry is generated via the term involving
$C_{5}$ in Eq. (5). For times less than the period of a CMW, i.e., $t\ll
2\pi\omega_{\mathrm{CMW}}^{-1}$, the evolution of $\mu_{5}$ can be
approximated as

$\displaystyle\mu_{5}(t)\approx
C_{5}({\bm{B}_{\mathrm{ex}}}{\bm{\cdot}}\bm{\nabla})\mu\,t.$ (9)

Assuming an initial condition where $\mu(t_{0})$ has a characteristic wave
number $k_{\mu,\mathrm{eff}}$, we find

$\displaystyle\mu_{5}(t)\approx
C_{5}B_{\mathrm{ex}}k_{\mu,\mathrm{eff}}\mu(t_{0})\,t.$ (10)

Note that, even for $t\ll 2\pi\omega_{\mathrm{CMW}}^{-1}$, $\mu$ can be a
function of time due to the dissipation term in Eq. (4).

Given that the chemical potential has a spectrum $E_{\mu}\propto k^{s}$, we
can write for its $k$-dependent value

$\displaystyle\mu^{2}(k)\approx E_{\mu}(k)k\propto k^{1+s}.$ (11)

Inserting $\mu(k)\approx k^{(1+s)/2}$ in Eq. (10), yields

$\displaystyle\mu_{5}(k,t)$ $\displaystyle\propto$ $\displaystyle
C_{5}B_{\mathrm{ex}}k^{(3+s)/2}~{}t$ (12)

The spectrum of $\mu_{5}$ is then

$\displaystyle E_{5}(k)=\frac{\mu_{5}(k)^{2}}{k}\propto
C_{5}^{2}B_{\mathrm{ex}}^{2}k^{2+s}~{}t^{2}.$ (13)

###  III.2 Chiral tangling

Nonuniform fluctuations of the chemical potential $\mu$ produce nonuniform
fluctuations of the chiral chemical potential $\mu_{5}$ due to the term
$-C_{5}({\bm{B}}{\bm{\cdot}}\bm{\nabla})\mu$ in Eq. 5. This can lead to a
linear in time growth of magnetic fluctuations, which is analogous to tangling
of an external magnetic field by velocity fluctuations. The relevant term in
the induction equation is

$\displaystyle\frac{\partial\bm{B}}{\partial
t}=\eta{\bm{\nabla}}\mu_{5}\times\boldsymbol{B}_{\mathrm{ex}}.$ (14)

We call this effect “chiral tangling” and expect it to be only relevant in
early phases, or in cases where the generation of $\mu_{5}$ is not efficient
enough to lead to a chiral dynamo instability.

###  III.3 Small-scale chiral dynamo instability

If $\mu_{5}$ exceeds a critical value, a chiral dynamo instability amplifies
the magnetic field exponentially with the maximum growth rate

$\displaystyle\gamma_{5}=\frac{\eta\mu_{5,\mathrm{max}}^{2}}{4},$ (15)

where $\mu_{5,\mathrm{max}}$ is the spatial maximum of $\mu_{5}$ [49, 50]. The
expression given in Eq. (15) is the maximum possible growth rate and is only
reached if the instability wave number

$\displaystyle k_{5}=\frac{\mu_{5,\mathrm{max}}}{2},$ (16)

is much larger than the effective correlation wave number
$k_{\mu_{5},\mathrm{eff}}$ of $\mu_{5}$ [49], where

$\displaystyle k_{\mu_{5},\mathrm{eff}}^{-1}(t)=\frac{\int
E_{5}(k)k^{-1}~{}\mathrm{d}k}{\int E_{5}(k)~{}\mathrm{d}k}.$ (17)

Note that during this phase, $\mu_{5}$ continues to grow similar to Eq. (10),
but the external constant field $B_{\mathrm{ex}}$ is being replaced by
$B_{\mathrm{ex}}+B_{\mathrm{rms(t)}}$, once $B_{\mathrm{rms}}(t)\gtrsim
B_{\mathrm{ex}}$. Therefore, the produced chiral chemical potential $\mu_{5}$
depends on the magnetic fluctuations $B_{\mathrm{rms}}$ and the chiral dynamo
instability becomes nonlinear.

Whether a large enough chiral asymmetry can be produced to trigger a chiral
dynamo instability depends on the initial $\mu$ as well as on the
characteristic parameters of the system. The first necessary condition for a
dynamo is that the maximum value of $\mu$, $\mu_{\mathrm{max}}$, needs to be
much larger than its effective correlation length, $k_{\mu,\mathrm{eff}}$.
Only then, a large enough $\mu_{5,\mathrm{max}}$ can be produced such that the
dynamo instability scale, $\mu_{\mathrm{max}}/2$, exceeds
$k_{\mu_{5},\mathrm{eff}}$. The second necessary condition for the dynamo
instability in CMWs is that chiral dynamo needs to operate on a timescale that
is less than half of the period of the CMW,
$P_{\mathrm{CMW}}/2=\pi/\omega_{\mathrm{CMW}}$. In other words, the (minimum
possible) dynamo timescale

$\displaystyle t_{\mathrm{D}}=\frac{4}{\eta\mu^{2}_{\mathrm{max}}(t_{0})}.$
(18)

needs to be shorter than $P_{\mathrm{CMW}}/2$. In Eq. 18 we assume that, at
times when $\mu_{\mathrm{5,max}}$ reaches its maxima, its value corresponds to
$\mu_{\mathrm{max}}(t_{0})$ [which implies that the numerical dissipation of
$\mu_{5}$ and $\mu$ is insignificant] and therefore the maximum possible
growth rate $\gamma_{5}$ is determined by $\mu_{\mathrm{max}}(t_{0})$.

###  III.4 Maximum possible magnetic field strength generated by the chiral
dynamo

Figure 1: Dependence of the maximum magnetic field strength that can result from a chiral magnetic wave, $B_{*}$ as given by Eq. (20), on the coupling constants $C_{5}$ and $C_{\mu}$. The different panels indicate different combinations of the initial value of the maximum chemical potential, $\mu_{\mathrm{max}}(t_{0})$, and the magnetic resistivity, $\eta$. Colors indicate the value of the feedback parameter $\lambda$. The horizontal black lines indicate the approximate threshold for the production of turbulence, as estimated in Eq. (25), and the dashed black line shows the approximate minimum value above which the velocity of the CMW becomes supersonic, see Appendix A. Overplotted are the values obtained according to Eq. (20) for all the runs presented in this paper; see Tab. 1.  Table 1: Summary of the simulations. The three reference runs are in bold. Runs $A3$ and $A10b$ have been presented also in Schober  _et al._ [56], where they were named $R1$ and $R2$, respectively. Run $R$–$2$ is a comparison run without the CSE, which has been presented in Schober  _et al._ [50].  | Parameters | Initial conditions | Phenomenology | DNS results (maximum values)
---|---|---|---|---
Run | Res. | $B_{\mathrm{ex}}$ | $\eta$ | $\lambda$ | $C_{5}=C_{\mu}$ | $\mu_{\mathrm{max}}$ | $E_{\mu}$ | $E_{5}$ | $\frac{P_{\mathrm{CMW}}/2}{t_{\mathrm{D}}}$ | $B_{*}$ | $B_{\mathrm{crit}}$ | $\mu_{\mathrm{5,max}}$ | $B_{\mathrm{rms}}$ | ${\rm Re}_{{}_{\mathrm{M}}}$ | $\mathrm{Lu}$
$A1$ | $672^{3}$ | $1\times 10^{-4}$ | $1\times 10^{-4}$ | $4\times 10^{2}$ | $1$ | $45$ | $\propto k^{-4}$ | 0 | $1400.99$ | $0.14$ | $0.0022$ | $59$ | $0.041$ | $58$ | $200$
$A3$ | $1024^{3}$ | $1\times 10^{-4}$ | $1\times 10^{-4}$ | $4\times 10^{2}$ | $3$ | $45$ | $\propto k^{-4}$ | 0 | $467.63$ | $0.047$ | $0.0023$ | $68$ | $0.12$ | $77$ | $440$
$\boldsymbol{A5}$ | $\boldsymbol{1024^{3}}$ | $\boldsymbol{1\times 10^{-4}}$ | $\boldsymbol{1\times 10^{-4}}$ | $\boldsymbol{4\times 10^{2}}$ | $\boldsymbol{5}$ | $\boldsymbol{45}$ | $\boldsymbol{\propto k^{-4}}$ | $\boldsymbol{0}$ | $\boldsymbol{280.60}$ | $\boldsymbol{0.028}$ | $\boldsymbol{0.0023}$ | $\boldsymbol{63}$ | $\boldsymbol{0.15}$ | $\boldsymbol{110}$ | $\boldsymbol{610}$
$A5b$ | $720^{3}$ | $1\times 10^{-4}$ | $1\times 10^{-4}$ | $4\times 10^{4}$ | $5$ | $48$ | $\propto k^{-4}$ | 0 | $338.65$ | $0.034$ | $0.0024$ | $50$ | $0.011$ | $8.4$ | $40$
$A10$ | $720^{3}$ | $1\times 10^{-4}$ | $1\times 10^{-4}$ | $4\times 10^{2}$ | $10$ | $48$ | $\propto k^{-4}$ | 0 | $169.32$ | $0.017$ | $0.0024$ | $42$ | $0.0077$ | $0.89$ | $9.3$
$A10b$ | $672^{3}$ | $1\times 10^{-4}$ | $1\times 10^{-4}$ | $4\times 10^{4}$ | $10$ | $44$ | $\propto k^{-4}$ | 0 | $133.03$ | $0.013$ | $0.0022$ | $38$ | $0.024$ | $36$ | $98$
$A80a$ | $448^{3}$ | $1\times 10^{-4}$ | $1\times 10^{-4}$ | $4$ | $80$ | $44$ | $\propto k^{-4}$ | 0 | $17.17$ | $0.0017$ | $0.0022$ | $66$ | $0.00043$ | $0.054$ | $0.45$
$\boldsymbol{A80}$ | $\boldsymbol{448^{3}}$ | $\boldsymbol{1\times 10^{-4}}$ | $\boldsymbol{1\times 10^{-4}}$ | $\boldsymbol{4\times 10^{2}}$ | $\boldsymbol{80}$ | $\boldsymbol{44}$ | $\boldsymbol{\propto k^{-4}}$ | $\boldsymbol{0}$ | $\boldsymbol{17.17}$ | $\boldsymbol{0.0017}$ | $\boldsymbol{0.0022}$ | $\boldsymbol{66}$ | $\boldsymbol{0.00043}$ | $\boldsymbol{0.054}$ | $\boldsymbol{0.45}$
$A80b$ | $448^{3}$ | $1\times 10^{-4}$ | $1\times 10^{-4}$ | $4\times 10^{4}$ | $80$ | $44$ | $\propto k^{-4}$ | 0 | $17.17$ | $0.0017$ | $0.0022$ | $66$ | $0.00043$ | $0.055$ | $0.45$
$A80c$ | $432^{3}$ | $1\times 10^{-4}$ | $1\times 10^{-4}$ | $4\times 10^{6}$ | $80$ | $41$ | $\propto k^{-4}$ | 0 | $15.00$ | $0.0015$ | $0.0021$ | $38$ | $0.00014$ | $0.0077$ | $0.47$
$A80d$ | $432^{3}$ | $1\times 10^{-4}$ | $1\times 10^{-4}$ | $4\times 10^{10}$ | $80$ | $41$ | $\propto k^{-4}$ | 0 | $-$ | $0.00018$ | $0.0021$ | $6.3$ | $0.0001$ | $0.0039$ | $0.6$
$A200$ | $448^{3}$ | $4\times 10^{-4}$ | $1\times 10^{-4}$ | $4\times 10^{2}$ | $2\times 10^{2}$ | $44$ | $\propto k^{-4}$ | 0 | $1.70$ | $0.00068$ | $0.0022$ | $65$ | $0.00044$ | $0.023$ | $2.2$
$A400$ | $448^{3}$ | $4\times 10^{-4}$ | $1\times 10^{-4}$ | $4\times 10^{2}$ | $4\times 10^{2}$ | $44$ | $\propto k^{-4}$ | 0 | $0.85$ | $0.00034$ | $0.0022$ | $74$ | $0.00042$ | $0.015$ | $1.5$
$\boldsymbol{A1000}$ | $\boldsymbol{448^{3}}$ | $\boldsymbol{4\times 10^{-4}}$ | $\boldsymbol{1\times 10^{-4}}$ | $\boldsymbol{4\times 10^{2}}$ | $\boldsymbol{1\times 10^{3}}$ | $\boldsymbol{44}$ | $\boldsymbol{\propto k^{-4}}$ | $\boldsymbol{0}$ | $\boldsymbol{0.35}$ | $\boldsymbol{0.00014}$ | $\boldsymbol{0.0022}$ | $\boldsymbol{94}$ | $\boldsymbol{0.00041}$ | $\boldsymbol{0.0039}$ | $\boldsymbol{2.2}$
$B40a$ | $720^{3}$ | $1\times 10^{-4}$ | $2\times 10^{-4}$ | $4$ | $40$ | $48$ | $\propto k^{-4}$ | 0 | $84.77$ | $0.0085$ | $0.0048$ | $45$ | $0.0045$ | $0.69$ | $5$
$B40$ | $720^{3}$ | $1\times 10^{-4}$ | $2\times 10^{-4}$ | $4\times 10^{2}$ | $40$ | $48$ | $\propto k^{-4}$ | 0 | $84.70$ | $0.0085$ | $0.0048$ | $45$ | $0.0044$ | $0.68$ | $4.8$
$B40b$ | $720^{3}$ | $1\times 10^{-4}$ | $2\times 10^{-4}$ | $4\times 10^{4}$ | $40$ | $48$ | $\propto k^{-4}$ | 0 | $84.70$ | $0.0085$ | $0.0048$ | $45$ | $0.0037$ | $0.61$ | $4$
$B40c$ | $720^{3}$ | $1\times 10^{-4}$ | $2\times 10^{-4}$ | $4\times 10^{6}$ | $40$ | $48$ | $\propto k^{-4}$ | 0 | $84.77$ | $0.0085$ | $0.0048$ | $41$ | $0.001$ | $0.17$ | $0.92$
$B40d$ | $720^{3}$ | $1\times 10^{-4}$ | $2\times 10^{-4}$ | $4\times 10^{8}$ | $40$ | $48$ | $\propto k^{-4}$ | 0 | $85.15$ | $0.0022$ | $0.0048$ | $30$ | $0.00027$ | $0.016$ | $0.3$
$C80c$ | $720^{3}$ | $1\times 10^{-4}$ | $1\times 10^{-4}$ | $4\times 10^{6}$ | $80$ | $96$ | $\propto k^{-4}$ | 0 | $84.88$ | $0.0085$ | $0.0048$ | $90$ | $0.0015$ | $0.24$ | $1.2$
$R$–$2$ | $672^{3}$ | $0$ | $2\times 10^{-4}$ | $4\times 10^{2}$ | $0$ | $-$ | 0 | $\propto k^{-2}$ | $-$ | $-$ | $-$ | $50$ | $0.18$ | $140$ | $460$

Within one period of the wave, $2\pi/\omega_{\mathrm{CMW}}$, the sign of the
produced $\mu_{5}$ oscillates between positive and negative. Therefore a
chiral dynamo instability can amplify the magnetic field significantly as long
as the timescale on which $\mu_{5}$ changes sign,

$\displaystyle t_{\mathrm{CMW,nl}}$ $\displaystyle\approx$
$\displaystyle\frac{\pi}{\left[C_{5}\,C_{\mu}(k_{\mu,\mathrm{eff}}B_{\mathrm{rms}})^{2}-{1\over
4}(\lambda\eta B_{\mathrm{rms}}^{2})^{2}\right]^{1/2}},\;\;$ (19)

is longer than the dynamo timescale in Eq. (18). In Eq. (19), we assume that
the system is at a stage where magnetic fluctuations are larger than the
imposed field. With increasing $B_{\mathrm{rms}}$, $t_{\mathrm{CMW,nl}}$
decreases and eventually becomes comparable with $t_{\mathrm{D}}$. This allows
estimating the maximum strength of a magnetic field produced by CMWs.
Comparing Eqs. (19) and (18) yields a maximum magnetic field strength of

$\displaystyle|B_{*}|$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{\eta\lambda}\sqrt{C_{5}\,C_{\mu}}k_{\mu,\mathrm{eff}}\biggl{[}\pm\left(1-\xi^{2}\right)^{1/2}+1\biggr{]}^{1/2}$
(20)

with

$\displaystyle\xi\equiv\frac{\pi\lambda\eta^{2}\mu^{2}_{\mathrm{max}}(t_{0})}{4C_{5}\,C_{\mu}k_{\mu,\mathrm{eff}}^{2}}$
(21)

and where we used the frequency of the CMW as given in Eq. (6). This
expression for $B_{*}$ is based on the assumptions that (i) a complete
conversion of $\mu$ to $\mu_{5}$ is possible and (ii) that there is no
turbulence in the system. It is worth noting that Eq. (20) approaches

$\displaystyle|B^{-}_{*}|$ $\displaystyle=$
$\displaystyle\frac{\pi\eta\mu^{2}_{\mathrm{max}}(t_{0})}{4\sqrt{C_{5}\,C_{\mu}}k_{\mu,\mathrm{eff}}},$
(22)

or

$\displaystyle|B^{+}_{*}|$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{C_{5}\,C_{\mu}}k_{\mu,\mathrm{eff}}}{\eta\lambda},$
(23)

if $\xi\ll 1$. The physically relevant value is
$|B_{*}|=\mathrm{min}(|B^{-}_{*}|,|B^{+}_{*}|)$, because as soon as the
magnetic field strength reaches the lower branch of the solutions, the sign of
$\mu_{5}$ changes on a timescale that is shorter than $t_{\mathrm{D}}$. In the
limit of $\xi\gg 1$ Eq. (20) becomes

$\displaystyle|B_{*}|=\frac{\sqrt{\pi}}{2\sqrt{\lambda}}~{}\mu_{\mathrm{max}}(t_{0}).$
(24)

However, in this limit the damping of the CMW can be significant, see Eq. (8),
and $\mu_{5,\mathrm{max}}$ never reaches the maximum possible value of
$\mu_{5,\mathrm{max}}=\mu_{\mathrm{max}}(t_{0})$. Therefore, the expression in
Eq. (24) can be considered as an upper limit. The expression given by Eq. (20)
is plotted for different parameters in Fig. 1.

###  III.5 Production of turbulence and mean-field dynamo instability

Magnetic fluctuations generated by the chiral dynamo instability, produce
velocity fluctuations $U_{\mathrm{rms}}$ through the Lorentz force. This leads
to an increase of the Reynolds number
$\mathrm{Re}_{\mathrm{M}}=U_{\mathrm{rms}}/(k_{\mathrm{f}}\eta)$, where
$k_{\mathrm{f}}$ is the forcing wave number. In such magnetically-driven
turbulence, $k_{\mathrm{f}}$ is roughly equal to the wavenumber on which the
magnetic energy peaks. For a chiral dynamo instability, this corresponds to
$k_{\mathrm{f}}\approx k_{\mathrm{5}}=\mu_{5,\mathrm{max}}/2$. Using the rough
assumption that $\sqrt{\langle{\rho}\rangle}\,U_{\mathrm{rms}}\approx
B_{\mathrm{rms}}$, we can estimate the critical magnetic field strength
$B_{\mathrm{crit}}$ which is necessary for the production of turbulence, i.e.
for reaching a value of $\mathrm{Re}_{\mathrm{M}}$ above unity. We find that
the critical magnetic field strength is estimated as

$\displaystyle
B_{\mathrm{crit}}\approx\sqrt{\langle{\rho}\rangle}\,\eta\frac{\mu_{\mathrm{max}}(t_{0})}{2}.$
(25)

The value of $B_{\mathrm{crit}}$ is presented as horizontal black lines in
Fig. 1. It can be used to illustrate the regions of the parameter regime in
which turbulence can be produced.

If the small-scale chiral dynamo leads to a magnetic field that exceeds
$B_{\mathrm{crit}}$, a mean-field dynamo instability can be excited. The
maximum growth rate of the mean-field dynamo is

$\displaystyle\gamma_{\alpha}=\frac{(\eta\langle{\mu_{5}}\rangle+\alpha_{\mathrm{M}})^{2}}{4(\eta+\eta_{\mathrm{T}})},$
(26)

where $\langle{\mu_{5}}\rangle$ is the mean chiral chemical potential and
$\alpha_{\mathrm{M}}$ is the magnetic $\alpha$ effect [49, 50]. Here
$\alpha_{\mathrm{M}}=2(q-1)/(q+1)\,\tau_{\rm c}\,\chi_{\rm
c}/\langle{\rho}\rangle$ is the magnetic $\alpha$ effect, which is determined
by the current helicity $\chi_{\rm
c}=\langle{\bm{b}}{\bm{\cdot}}({\bm{\nabla}}\times{\bm{b}})\rangle\approx\langle{\bm{a}}\cdot{\bm{b}}\rangle\,k_{\mathrm{f}}^{2}$,
where $q$ is the exponent of the magnetic energy spectrum $E_{\rm M}\propto
k^{-q}$, and ${\bm{a}}$ and ${\bm{b}}$ are the fluctuations of the vector
potential and the magnetic field, respectively. The correlation time of the
magnetically-driven turbulence is $\tau_{\rm
c}\approx(U_{\mathrm{A}}k_{\mathrm{f}})^{-1}$, where the Alfvén speed is
$U_{\rm
A}=\sqrt{\langle{\bm{b}}^{2}\rangle}/\sqrt{\langle{\rho}\rangle}\approx
B_{\mathrm{rms}}/\sqrt{\langle{\rho}\rangle}$. The turbulent diffusion
coefficient $\eta_{\mathrm{T}}$ is estimated as
$\eta_{\mathrm{T}}=U_{\mathrm{rms}}/(3k_{\mathrm{f}})$. The characteristic
wavenumber on which the mean-field dynamo occurs is

$\displaystyle
k_{\alpha}=\frac{|\eta\langle{\mu_{5}}\rangle+\alpha_{\mathrm{M}}|}{2(\eta+\eta_{\mathrm{T}})}.$
(27)

Figure 2: Exemplary runs from the three different regimes: A high-frequency
CMW with just chiral tangling (Run $A1000$, left panels), inefficient small-
scale chiral dynamo due to a CMW with moderate frequency (Run $A80$, middle
panels), and a low-frequency CMW with a small-scale chiral and mean-field
dynamo (Run $A5$, right panels). The top panels show the time evolution of the
rms and maximum values of the chemical potential $\mu$ and the chiral chemical
potential $\mu_{5}$, respectively, as well as the volume average of $\mu$. The
bottom panels show the evolution of the rms values of the magnetic and the
velocity fields, $B_{\mathrm{rms}}$ and $U_{\mathrm{rms}}$ as well as the
external field strength $B_{\mathrm{ex}}$ and the maximum possible magnetic
field strength $B_{*}$ if no turbulence is produced. For Run $A5$ the time
evolution of the mean magnetic field strength $\langle
B\rangle_{\mathrm{int}}$ is presented for comparison.  Figure 3: Snapshots of
Run $A1000$. The surface of the cubic domain is shown for $\mu$ (upper row),
$\mu_{5}$ (middle row), and the $x$ component of the magnetic field $B_{z}$
(lower row). The snapshots cover the different evolutionary phases, from the
initial $\mu_{5}$ production phase ($t=0.05~{}t_{\eta}$, first column), to the
onset of the chiral dynamo instability ($t=0.10~{}t_{\eta}$, second column),
to the early mean-field dynamo stage ($t=0.15~{}t_{\eta}$, third column), and
the end of the mean-field dynamo stage ($t=0.20~{}t_{\eta}$, forth column).
Figure 4: Similar to Fig. 4 but for Run $A5$.  Figure 5: Exemplary runs from
the three different regimes: A high-frequency CMW with just chiral tangling
(Run $A1000$, left panels), inefficient small-scale chiral dynamo due to a CMW
with moderate frequency (Run $A80$, middle panels), and a low-frequency CMW
with small-scale chiral and mean-field dynamos (Run $A5$, right panels). From
top to bottom, the spectrum of fluctuations of chemical potential,
$E_{\mu}(k)$, chiral chemical potential, $E_{5}(k)$, and the magnetic energy
spectrum $E_{\mathrm{M}}(k)$ are shown. The vertical dotted line indicates the
highest possible value of the small-scale dynamo instability scale,
$\mu_{\mathrm{max}}(t_{0})/2$, on which the magnetic field is amplified if all
of the initial $\mu$ has been converted to $\mu_{5}$.

##  IV Numerical simulations

In this section, we use simulations to verify the phenomenology discussed
above. Using direct numerical simulations (DNS), the conditions for chiral
dynamo instabilities, efficient magnetic field amplification and, in
particular, the mean-field dynamo phase can be analyzed qualitatively.

###  IV.1 Setup and analysis tools

We use the Pencil Code [57] to solve equations (1)–(5) in a three-dimensional
periodic domain of size $L^{3}=(2\pi)^{3}$ with a resolution of up to
$1024^{3}$. This code employs a third-order accurate time-stepping method [58]
and sixth-order explicit finite differences in space [59, 60]. The smallest
wave number covered in the numerical domain is $k_{1}=2\pi/L=1$ which we use
for normalization of length scales. All velocities are normalized to the sound
speed $c_{\rm s}=1$ and the mean fluid density is set to
$\langle{\rho}\rangle=1$. Further, the magnetic Prandtl number is $1$, i.e.
the magnetic diffusivity equals the viscosity. Time is normalized either by
the diffusion time $t_{\eta}=(\eta k_{1}^{2})^{-1}$ or by the period of the
chiral magnetic wave $P_{\mathrm{CMW}}$.

The simulation parameters have been selected to cover the three different
regimes: the “chiral tangling regime” ($t_{\mathrm{D}}\gtrsim
P_{\mathrm{CMW}}/2$), the “small-scale chiral dynamo regime”
($t_{\mathrm{D}}\lesssim P_{\mathrm{CMW}}/2$ and $B_{*}\lesssim
B_{\mathrm{crit}}$) and the “mean-field dynamo regime”
($t_{\mathrm{D}}\lesssim P_{\mathrm{CMW}}/2$ and $B_{*}\gtrsim
B_{\mathrm{crit}}$). We also perform a comparison with the results obtained in
our previous study Ref. [49] (see Run $R$–$2$ there), where chiral dynamo
instabilities were found for an initial $\mu_{5}\not=0$ with zero mean but
spatial fluctuations. For this comparison, the spatial maximum value of the
chemical potential at the initial time $t_{0}$, $\mu_{\mathrm{max}}$, and its
spectrum $E_{\mu}(k,t_{0})$ have been chosen to eventually (before the onset
of the small-scale chiral instability) result in a state of the system that is
comparable to the initial conditions in the Run $R$–$2$. In particular, in the
Run $R$–$2$ the initial $\mu_{5,\mathrm{max}}$ was $\approx 50$ and the
initial spectrum was $E_{5}(k,t_{0})\propto k^{-2}$. We therefore choose, for
most runs of this study, $\mu_{\mathrm{max}}(t_{0})\approx 50$ and
$E_{\mu}(k,t_{0})\propto k^{-4}$ which results in the spectrum
$E_{5}(k,t_{0})\propto k^{-2}$ according to Eq. (13).

The range of parameters chosen for this study is also based on numerical
aspects. The parameter space that we explore includes the regime where the
magnetic field strength becomes larger than the critical value
$B_{\mathrm{crit}}$ for the production of turbulence and the subsequent
excitation of mean-field dynamos. According to the estimate in Eq. (20), which
is illustrated in Fig. 1, the maximum magnetic field strength $B_{*}$ is
higher for lower frequencies $\omega_{\mathrm{CMW}}$ of the CMW. However, for
low $\omega_{\mathrm{CMW}}$ and therefore low values of $C_{5}$, the initial
linear production of $\mu_{5}$ becomes very slow, as can be seen in Eq. (10).
Increasing the initial value of $\mu$ increases the initial production rate of
$\mu_{5}$, but this also leads to a larger value $B_{*}$, which can cause the
characteristic velocity of the CMW to become comparable or larger than the
sound speed. Additionally, larger values of the initial $\mu$ lead to larger
values of $\mu_{5}$ and therefore a higher characteristic wave number of the
chiral dynamo instability. Hence sufficient spatial resolution is required.
More details on the numerical criteria are given in Appendix A.

Due to the numerical constraints discussed above, and also to allow for an
appropriate comparison with the DNSs presented in [49, 50], we initiate most
of the simulations with $\mu_{\mathrm{max}}(t_{0})\approx 50$ and use
$\eta=10^{-4}$. We name this main series of simulations Series A. Series B has
$\mu_{\mathrm{max}}(t_{0})\approx 50$ and $\eta=2\times 10^{-4}$ and Series C
has $\mu_{\mathrm{max}}(t_{0})\approx 100$ and $\eta=10^{-4}$. A summary of
all runs of this study is given in Table 1 and the values for the
corresponding estimates of $B_{*}$ is shown in Fig. 1. Comparing the estimates
$B_{*}$ and $B_{\mathrm{crit}}$, we can expect the occurrence of turbulence in
Runs $A1$, $A3$, $A5$, $A5b$, and potentially in Runs $A10$ and $A10b$. All
other runs are expected to result in values of $B_{*}$ that are comparable or
below $B_{\mathrm{crit}}$.

For runs in which turbulence develops, we perform a mean-field analysis. To
this end, an averaging of the instantaneous fields needs to be performed in
the DNS. Since turbulence is driven magnetically, the forcing scale $k_{f}$
corresponds to the integral scale of the magnetic field which we determine via
the magnetic energy spectrum $E_{\mathrm{M}}(k)$ as

$\displaystyle k_{\mathrm{int}}\equiv\left[\frac{1}{{\cal E}_{\rm M}}\int
E_{\mathrm{M}}(k)\,k^{-1}~{}\mathrm{d}k\right]^{-1}.$ (28)

Magnetic energy density ${\cal E}_{\rm M}$ and magnetic spectrum
$E_{\mathrm{M}}(k)$ are connected as

$\displaystyle{\cal E}_{\rm M}\equiv\frac{B_{\mathrm{rms}}^{2}}{2}=\int
E_{\mathrm{M}}(k)\,\mathrm{d}k.$ (29)

To take into account that the magnetically-driven turbulence exists in the
range of the wave numbers $k\geq k_{\mathrm{int}}$, we define the mean
quantity $X$ in simulations as

$\displaystyle\langle X\rangle_{\mathrm{int}}\equiv\left[\int
E_{\mathrm{X}}(k)f(k)\,\mathrm{d}k\right]^{1/2},$ (30)

where we use the function

$\displaystyle f(k)\equiv\left[1-\mathrm{tanh}(k-k_{\mathrm{int}})\right]/2$
(31)

to filter out the scale $k\gtrsim k_{\mathrm{int}}$. The result of taking the
average $\langle X\rangle_{\mathrm{int}}$ is typically different from the
volume average, which is denoted by $\langle X\rangle_{\mathrm{V}}$.

###  IV.2 Results for the reference runs

In this section, we present three reference runs that have the same initial
chemical potential, but different frequencies of the CMW. Run $A1000$ is the
run in our sample with the highest frequency of the CMW. With a ratio of
$P_{\mathrm{CMW}}/(2t_{\mathrm{D}})\approx 0.35$, no dynamo activity is
expected in Run $A1000$. The second reference run is $A80$, which has
$P_{\mathrm{CMW}}/(2t_{\mathrm{D}})\approx 17$. Therefore, a small-scale
chiral dynamo can occur. However since the expected maximum magnetic field
strength $B_{*}\approx 0.0017$ is lower than the critical value
$B_{\mathrm{crit}}\approx 0.0017$ that is necessary for the production of
turbulence, no mean-field dynamo is expected in Run $A80$. A mean-field dynamo
can occur in the third reference run, Run $A5$, which has
$B_{*}/B_{\mathrm{crit}}\approx 12$. Run $A5$ is the run with the third to the
highest value of $P_{\mathrm{CMW}}/(2t_{\mathrm{D}})$ in our sample. We
discuss the results of the reference runs in the following and confront them
with the estimates based on the phenomenological estimates presented in Sec.
III.

The left panels of Fig. 2 show the time evolution of various parameters of Run
$A1000$. In the top left panel, the oscillatory behavior of
$\mu_{\mathrm{rms}}$ and $\mu_{5,\mathrm{rms}}$ is clearly seen and the time
evolution governs almost $20$ periods of the CMW. In systems like this, where
the CMW has a very high frequency and the initial chiral chemical potential is
small, the timescale of the chiral dynamo $t_{\mathrm{D}}$ is much longer than
$P_{\mathrm{CMW}}$. In this case, magnetic fluctuations can only be amplified
by chiral tangling. This phenomenon alone leads to the production of magnetic
fluctuations that are of the order of the imposed magnetic field
$B_{\mathrm{ex}}$. The magnetic field evolution in Run $A1000$ can be seen in
the lower left panel of Fig. 2. The maximum value of $B_{\mathrm{rms}}$
produced by chiral tangling alone is approximately less than half of
$B_{\mathrm{ex}}$. At $t\gtrsim 0.02~{}t_{\eta}$, both $\mu_{\mathrm{rms}}$
and $\mu_{5,\mathrm{rms}}$ decay, and therefore $B_{\mathrm{rms}}$ decreases.

Snapshots of Run $A1000$ are presented in Fig. 3. While the magnetic field is,
as in all simulations of this paper, set up as weak and random fluctuations,
the magnetic fluctuations quickly develop into patches that are stretched
along the $x$ axis. At the forth snapshot shown here (at $t\approx
7.7~{}P_{\mathrm{CMW}}$), the $B_{x}$ patches stretch out through the entire
numerical domain. The magnetic field structure produced by chiral tangling is
therefore very different from what is expected when a small-scale chiral
dynamo instability is excited. In linear theory, the magnetic field
instability is expected to occur on a characteristic wavenumber that is half
of the value of $\mu_{5,\mathrm{max}}$. This leads to the formation of
isotropic patches of high absolute values of $B_{x}$ on the surface of the
numerical domain, at the locations where $\mu_{5}$ reaches the maximum value.

The middle panels of Fig. 2 show the time evolution of various parameters of
Run $A80$, where a small-scale chiral dynamo instability occurs. Two
oscillations between $\mu_{\mathrm{rms}}$ and $\mu_{5,\mathrm{rms}}$ are seen
in the upper middle panel. The initial production of the spatial maximum value
of $\mu_{5}(t)$, $\mu_{5,\mathrm{max}}(t)$, proceeds linear in time and
follows the prediction given by Eq. (10) until the instant $t\approx
0.015\,t_{\eta}$. Temporal maxima of $\mu_{5,\mathrm{rms}}$ are reached at
$t\approx 0.025\,t_{\eta}$ and $t\approx 0.055\,t_{\eta}$. These times
coincide, as expected, with an increased growth rate of the magnetic field;
see the time evolution of magnetic fluctuations in the lower middle panel.
However, the magnetic field fluctuations, $B_{\mathrm{rms}}$, never reach a
field strength that is much larger than the one of the imposed field
$B_{\mathrm{ex}}$. At its maximum, the rms magnetic field strength is
approximately $0.2~{}B_{*}$ in Run $A80$. At later times, $t\gtrsim
1.0\,P_{\mathrm{CMW}}$, the quantities $\mu_{\mathrm{rms}}$,
$\mu_{5,\mathrm{rms}}$, and $B_{\mathrm{rms}}$ decay.

The estimated maximum value of $B_{\mathrm{rms}}$, $B_{*}$, is $16$ times
higher in Run $A5$ than in $A80$. Contrary to the other reference runs, in Run
$A5$, the maximum value $B_{*}$ of magnetic field exceeds $B_{\mathrm{crit}}$,
which implies that turbulence can be produced. The time evolution of Run $A5$
is shown in the right panels of Fig. 2. Due to the smaller value of
$\omega_{\mathrm{CMW}}$, the production of $\mu_{5}$ is much slower than that
in Run $A80$. Here, the threshold for the small-scale chiral dynamo
instability only being exceeded at $t\approx 0.08\,t_{\eta}$. After the
magnetic field has been amplified by more than two orders of magnitude through
the small-scale chiral dynamo, a mean-field dynamo instability is excited at
$t\approx 0.11~{}t_{\eta}$ with a growth rate of the magnetic field that is
less than that for the small-scale chiral dynamo instability. In Run $A5$, the
magnetic field strength exceeds $B_{*}$ by a factor of $\approx 5.4$. The
analysis of the mean-field dynamo phase of Run $A5$ and the other runs in
which the value of the Reynolds number eventually exceeds unity will be
discussed in more detail in Sec. IV.4.

The time evolution of the simulation snapshots for Run $A5$ is presented in
Fig. 4. Here, the values for the quantities $\mu$, $\mu_{5}$, and the $x$
component of the magnetic field, $B_{x}$, on the surface of the cubic domain
are shown for $t=0.05~{}t_{\eta}-0.172~{}t_{\eta}$. The snapshots show that,
as expected, $\mu_{5}$ grows fastest where the gradient of $\mu$ is largest.
At $t=0.05~{}t_{\eta}$, the fastest production of $\mu_{5}$ occurs
approximately in the middle of the front $x$-$z$ plane (where $\mu_{5}$ is
produced with a positive sign) and in the middle of the front of the $x$-$y$
plane (where $\mu_{5}$ is produced with a negative sign). These are the two
locations on the shown surface of the domain, where also the magnetic field
instability kicks in the fastest.

In the snapshot at time $t=0.10~{}t_{\eta}$, the magnetic field grows
approximately on the length scale $k_{5}^{-1}\approx(\mu_{5}/2)^{-1}\approx
1/20$. At $t=0.20~{}t_{\eta}$, the simulation is at the end of the mean-field
dynamo stage and the characteristic length scale of the magnetic field has
increased. At late times, we also observe that both $\mu$ and $\mu_{5}$
develop small-scale fluctuations, especially in locations where the magnetic
field is the strongest. These small-scale structures are symmetric in $\mu$
and $\mu_{5}$, but with opposite sign.

For a quantitative analysis of the evolution of the characteristic scales, the
evolution of the energy spectra is presented in Fig. 5 for Runs $A1000$
(middle panels), $A80$ (middle panels), and $A5$ (right panels). In all cases,
the initial spectrum of $\mu_{5}$, $E_{5}(k)$, scales with the wave number as
$k^{-2}$, as expected for an initial $E_{\mu}(k)$ spectrum that is
proportional to $k^{-4}$; see Eq. (13). For Runs $A1000$ $A80$, the initial
$k^{-2}$ scaling of $E_{5}(k)$ is less visible due to the fast production of
$\mu_{5}$. At later times, the spectra $E_{5}(k)$ and $E_{\mu}(k)$, approach a
scaling of $k^{-1}$, as has been reported in [49]. The evolution of the
magnetic energy spectra $E_{\mathrm{M}}(k)$ is shown in the lowest panels of
Fig. 5. In the case of Run $A80$, a short phase of amplification on
$k=\mu_{\mathrm{max}}(t_{0})/2\approx 22$ is seen, but at $t\gtrsim
0.07\,t_{\eta}$ the magnetic energy decays and a $E_{\mathrm{M}}\propto
k^{-3}$ develops. The magnetic field amplification is much more efficient in
Run $A5$. Here the initial instability occurs also on $k=\mu(t_{0})\approx
22$. Due to the production of turbulence, however, the peak of the magnetic
energy spectrum moves to smaller wave numbers. Eventually, a
$E_{\mathrm{M}}\propto k^{-3}$ develops in Run $A5$ as well.

Figure 6: Time series for runs with different values of the ratio of the CMW
period over the resistive time, $P_{\mathrm{CMW}}/t_{\eta}$, as indicated by
the colorbar. (a) Maximum values of chemical and chiral chemical potential,
$\mu_{5,\mathrm{max}}$ and $\mu_{\mathrm{max}}$. (b) Rms value of the magnetic
field strength $B_{\mathrm{rms}}$. (c) Measured growth rate
$\gamma_{\mathrm{rms}}$ of $B_{\mathrm{rms}}$. (d) The measured scale
separation $k_{5}/k_{\mu_{5},\mathrm{eff}}$.  Figure 7: Time series for runs
with different values of $\lambda$, as indicated by the colorbar. (a) Maximum
values of chemical and chiral chemical potential, $\mu_{5,\mathrm{max}}$ and
$\mu_{\mathrm{max}}$. (b) Rms value of the magnetic field strength
$B_{\mathrm{rms}}$. (c) Measured growth rate $\gamma_{\mathrm{rms}}$ of
$B_{\mathrm{rms}}$, and the theoretical value $\eta\mu_{5,\mathrm{max}}^{2}/4$
of the maximum growth rate, which would be reached for an infinite scale
separation ($k_{5}\gg k_{\mu_{5},\mathrm{eff}}$). (d) The measured scale
separation $k_{5}/k_{\mu_{5},\mathrm{eff}}$.  Figure 8: The maximum value of
$B_{\mathrm{rms}}$ measured in all runs as a function of the
phenomenologically derived maximum, $B_{*}$. Colors indicate (a) the value of
$\lambda$ and (b) the ratio of the resistive time $t_{\eta}$ over the period
$P_{\mathrm{CMW}}$ of the CMW [Note, that Run $A80d$ is not shown in panel b
since there is no CMW]. The size of the symbols increases with increasing
maximum Reynolds number obtained in the individual runs.

###  IV.3 Exploration of the parameter space

A direct comparison between runs with different CMW frequencies
$\omega_{\mathrm{CMW}}$, including Runs $A5$, $A80$ and $A1000$, is presented
in Fig. 6. All of the runs in Fig. 6 have the same initial values of $\mu$,
and the same $\lambda$ and $\eta$. Even though the temporal maximum values of
$\mu_{5,\mathrm{max}}$ is higher for runs with higher $\omega_{\mathrm{CMW}}$,
the maximum value of the produced magnetic field decreases with increasing
$\omega_{\mathrm{CMW}}$. Therefore, CMWs with higher frequencies are less
efficient in amplifying the magnetic field. This stems from the small-scale
chiral dynamo being less efficient when the period of the CMW is small.

Figure 7 shows a comparison between runs with different chiral feedback
parameter, $\lambda$. As expected from Eq. (20), larger values of $\lambda$
lead to lower magnetic field strengths. In Fig 7a, it can be seen that
$\mu_{5,\mathrm{max}}$ in all runs with low $\lambda$ reach a value that is
comparable to (or even slightly exceeds) the initial value of
$\mu_{\mathrm{max}}$. This leads to three instances of magnetic field
amplification, see Fig 7b. In Run $A80c$, which has $\lambda=4\times 10^{6}$,
lower values of $\mu_{\mathrm{max}}$ are reached, which is caused by the
damping of the CMW according to Eq. (8). However, phases of magnetic field
amplification can still be seen for Run $A80c$. This is different for Run
$A80d$, which has $\lambda=4\times 10^{8}$. Here, no CMW occurs since the
frequency of the wave is imaginary.

The maximum magnetic field strength found in DNS agrees well with the
prediction given by Eq. (20), as is illustrated in Fig. 8. Here, the predicted
value $B_{*}$ is plotted against the temporal maximum of the magnetic field
strength in all DNS of this study. The agreement between phenomenology and DNS
is better for runs with lower values of $\lambda$. This follows from the fact
that Eq. (20) is based on two assumptions: (i) the effective correlation wave
number of $\mu$, $k_{\mu,\mathrm{eff}}$, stays constant until the maximum
magnetic field strength is reached, and (ii) $\mu_{5,\mathrm{max}}$ can reach
the same value as $\mu_{\mathrm{max}}(t_{0})$. As can be seen in Fig 7a, the
conversion between $\mu$ and $\mu_{5}$ becomes less efficient when $\lambda$
increases. Even though all runs in Fig. 7 have similar values of
$\mu_{\mathrm{max}}(t_{0})\approx 45$, in the run $A80d$ the temporal maximum
of $\mu_{\mathrm{5,max}}$ never exceeds $6.3$. Therefore, in the limit of
large $\lambda$, the expression in Eq. (20) has to be considered as an upper
limit for the maximum possible magnetic field strength.

Figure 9:  Time evolution of all runs in which magnetically-dominated
turbulence is produced. Different colored lines indicate the different runs
(A1, A3, A5, A5b, and A10b) as indicated in panel (c). (a) Time evolution of
$\mu_{\mathrm{max}}$ (dashed lines) and $\mu_{5,\mathrm{max}}$ (solid lines).
(b) Time evolution of the magnetic Reynolds number $\mathrm{Re}_{\mathrm{M}}$.
(c) Time evolution of the magnetic field strength $\langle
B\rangle_{\mathrm{int}}$ averaged on the integral scale. The color bars in the
panels highlight the time range in which the Reynolds number is larger than
unity. This time range is shown in Fig. 10.  Figure 10: Mean-field dynamo
analysis for all runs in which magnetically-dominated turbulence is produced.
Different rows of panels indicate the different runs (A1, A3, A5, A5b, and
A10b) as indicated on the top. The time axis is reduced to the phase of the
DNS, where the Reynolds number is larger than unity up to the final time of
the individual simulations. This time range is also indicated by the color
blocks in panels (a)-(c) of Fig. 9. (Upper row) Time evolution of dynamo
coefficients, $\alpha_{\mathrm{M,int}}$ (dotted lines),
$\alpha^{\mathrm{sat}}_{\mathrm{M,int}}$ (dashed lines), and
$\eta+\eta_{\mathrm{T}}$ multiplied by the integral wave number
$k_{\mathrm{int}}$ (solid lines). The evolution of
$\eta\langle\mu_{5}\rangle_{\mathrm{M,int}}$ (dashed-dotted lines) is also
shown, which can be relevant for the mean-field dynamo. (Middle row) Time
evolution of the integral scale of the magnetic field $k_{\mathrm{int}}$ as
measured in the DNS (solid lines). This evolution is compared to theoretical
estimates of the mean-field dynamo theory that is based on the turbulent
transport coefficients. Different expressions for the magnetic $\alpha$ effect
are used: $\alpha_{\mathrm{M,int}}$ (dotted lines) and
$\alpha^{\mathrm{sat}}_{\mathrm{M,int}}$ (dashed lines and dashed-dotted
lines). (Bottom row) Same as middle row, but for the measured growth rate of
$\langle B\rangle_{\mathrm{int}}$ in DNS, $\gamma_{\mathrm{int}}$, and
different theoretical estimates.

###  IV.4 DNS with mean-field dynamo activity

As can be seen in Fig. 8, the maximum magnetic field tends to exceed the
estimate from Eq. (20) for runs that develop turbulence. This is
understandable, since during the mean-field dynamo phase, the sign of
$\mu_{5}$ does not affect the magnetic field amplification. Therefore, the
comparison between the timescale of the CMW and the chiral dynamo instability
that leads to the estimate given by Eq. (20), is not applicable in the
presence of turbulence. The magnetic field can grow to higher strengths, until
saturation occurs due to non-linear effects.

In runs with low-frequency CMWs, the magnetic field strength reaches the
highest values, leading to efficient driving of magnetically dominated
turbulence. In this case, a large-scale magnetic field is generated via a
mean-field dynamo instability, as can be seen in the snapshots of Run $A5$ in
Fig. 4. Out of all the runs presented here, the ones in which
$\mathrm{Re}_{\mathrm{M}}$ exceeds unity, i.e. in which turbulence develops,
are Runs $A1$, $A3$, $A5$, $A5b$, and $A10b$. The time series of various
quantities of these runs are directly compared in Fig. 9. After the turbulence
production phase, the value of $\mu_{5,\mathrm{max}}$ is comparable in Runs
$A1$, $A3$, $A5$, and $A5b$. In Run $A10b$, $\mu_{5,\mathrm{max}}$ never
exceeds $40$, which is due to the higher frequency of the CMW. In all runs,
the magnetic Reynolds number exceeds unity after less than a resistive time;
see Fig. 9b. With $\mathrm{Re}_{\mathrm{M}}$ becoming larger than one, the
type of dynamo instability changes from a small-scale chiral dynamo to a mean-
field dynamo. This transition, which is accompanied by a change in the growth
rate, can be seen in Fig. 9c, where the time evolution of the mean magnetic
field strength is presented.

The theoretically expected growth rate during the mean-field dynamo phase is
given by Eq. (26). In the simulations, we estimate the magnetic $\alpha$
effect as $\alpha_{\mathrm{M,int}}=2(q-1)/(q+1)\,\tau_{\rm c}\,\chi_{\rm
c}\approx\tau_{\rm
c}\langle{\bm{a}}\cdot{\bm{b}}\rangle_{\mathrm{int}}\,k_{\mathrm{int}}^{2}$,
assuming that the forcing scale is $k_{\mathrm{f}}\approx k_{\mathrm{int}}$
and that the exponent of the magnetic energy spectrum $q\approx 3$. The
correlation time of the magnetically-driven turbulence is $\tau_{\rm
c}\approx(U_{\mathrm{A}}k_{\mathrm{int}})^{-1}$, where the Alfvén speed is
$U_{\rm A}=\sqrt{\langle{\bm{b}}^{2}\rangle}\approx B_{\mathrm{rms}}$. The
mean fluid density $\langle{\rho}\rangle$ entering in $U_{\rm A}$ and
$\alpha_{\mathrm{M}}$ is set to unity in the DNS. The turbulent diffusion
coefficient $\eta_{\mathrm{T}}$ is estimated as
$\eta_{\mathrm{T}}=U_{\mathrm{rms}}/(3k_{\mathrm{int}})$. The time evolution
of $\alpha_{\mathrm{M,int}}$ and $\eta_{\mathrm{T}}$ for all turbulent runs is
presented in the upper panels of Fig. 10. The time range shown in Fig. 10 is
the moment when $\mathrm{Re}_{\mathrm{M}}$ exceeds unity up to the final time
of the simulation, i.e. it governs the turbulent phase of the simulation.
Right after the onset of turbulence, $\alpha_{\mathrm{M,int}}$ is the dominant
transport coefficient for all runs presented in Fig. 10. However,
$\eta_{\mathrm{T}}$ grows constantly with time.

The magnetic $\alpha$ effect can also be estimated from the evolutionary
equation for the magnetic helicity $\langle{\bm{a}}\cdot{\bm{b}}\rangle$ of
the small-scale field ${\bm{b}}={\bm{\nabla}}\times{\bm{a}}$ in chiral MHD
[17]:

$\displaystyle{\partial\over\partial
t}\langle{\bm{a}}{\bm{\cdot}}{\bm{b}}\rangle+\nabla\cdot{\bm{F}}$
$\displaystyle=$ $\displaystyle
2\eta\,\langle\mu_{5}\rangle\langle{\bm{b}}^{2}\rangle-2\langle\mbox{\boldmath{${\cal{E}}$}}\rangle{}\cdot\langle\mbox{\boldmath$B$}\rangle{}$
(32)
$\displaystyle-2\eta\,\langle{\bm{b}}\,({\bm{\nabla}}\times{\bm{b}})\rangle,$

where ${\bm{F}}$ is the flux of $\langle{\bm{a}}{\bm{\cdot}}{\bm{b}}\rangle$
that is given by

$\displaystyle{\bm{F}}$ $\displaystyle=$
$\displaystyle\left\langle{\bm{u}}\,a_{j}\right\rangle\left\langle
B_{j}\right\rangle-\langle\mbox{\boldmath$B$}\rangle{}\left\langle{\bm{a}}\cdot{\bm{u}}\right\rangle-\eta\left\langle{\bm{a}}\times({\bm{\nabla}}\times{\bm{b}})\right\rangle$
(33)
$\displaystyle+\left\langle{\bm{a}}\times({\bm{u}}\times{\bm{b}})\right\rangle,$

and
$\langle\mbox{\boldmath{${\cal{E}}$}}\rangle{}\equiv\langle{\bm{u}}{\bm{\times}}{\bm{b}}\rangle=\alpha_{\mathrm{M}}\langle\mbox{\boldmath$B$}\rangle{}-\eta_{T}\,({\bm{\nabla}}\times\langle\mbox{\boldmath$B$}\rangle{})$
is the turbulent electromotive force. In the steady-state, two leading
source/sink terms in Eq. (32),
$2\eta\,\langle\mu_{5}\rangle\langle{\bm{b}}^{2}\rangle-2\alpha_{\mathrm{M}}\langle\mbox{\boldmath$B$}\rangle{}^{2}$,
compensate each other, so that the magnetic $\alpha$ effect reaches [49, 50]

$\displaystyle\alpha_{\mathrm{M}}^{\rm
sat}=\eta\,\langle\mu_{5}\rangle\,\frac{\langle{\bm{b}}^{2}\rangle}{\langle\mbox{\boldmath$B$}\rangle{}^{2}}.$
(34)

The time evolution of $\alpha_{\mathrm{M,int}}^{\rm
sat}=\eta\langle\mu_{5}\rangle_{\mathrm{int}}B_{\mathrm{rms}}^{2}/\langle
B\rangle_{\mathrm{int}}^{2}$ is compared to the one of
$\alpha_{\mathrm{M,int}}$ in the upper panels of Fig. 10. We note that the
values of $\alpha_{\mathrm{M,int}}^{\rm sat}$ are consistently lower than
$\alpha_{\mathrm{M,int}}$, which could result from the fact that the
divergence of the magnetic helicity fluxes is ignored in the estimate of
$\alpha_{\mathrm{M,int}}^{\rm sat}$.

In the middle and lower panels of Fig. 10, the estimates of the turbulent
transport coefficients are used to calculate the theoretically expected
characteristic wavenumber $k_{\alpha}$ and growth rate $\gamma_{\alpha}$ of
the mean-field dynamo, respectively. The theoretical estimates, given by Eqns.
(26) and (27), are compared with the measured characteristic wavenumber of the
magnetic field, $k_{\mathrm{int}}$ and the measured growth rate
$\gamma_{\mathrm{int}}$ of the mean magnetic field strength $\langle
B\rangle_{\mathrm{int}}$. In the middle row of Fig. 10 we compare the measured
$k_{\mathrm{int}}$ to
$|\alpha_{\mathrm{M,int}}|/\left[2(\eta+\eta_{\mathrm{T}}))\right]$ and
$|\alpha_{\mathrm{M,int}}^{\mathrm{sat}}|/\left[2(\eta+\eta_{\mathrm{T}})\right]$,
respectively, and in the bottom row, we compare the measured
$\gamma_{\mathrm{int}}$ to
$|\alpha_{\mathrm{M,int}}|^{2}/\left[4(\eta+\eta_{\mathrm{T}})\right]$ and
$|\alpha_{\mathrm{M,int}}^{\mathrm{sat}}|^{2}/\left[4(\eta+\eta_{\mathrm{T}})\right]$,
respectively. Using the $\alpha_{\mathrm{M,int}}$ to estimate $k_{\alpha}$ and
$\gamma_{\alpha}$, tends to lead to slightly higher values than the measured
$k_{\mathrm{int}}$ and $\gamma_{\mathrm{int}}$, while using
$\alpha_{\mathrm{M,int}}^{\rm sat}$, leads to slightly lower values.

One issue that arises in the comparison with theory is that while computing
the mean value of $\langle\mu_{5}\rangle_{\mathrm{int}}$ the information about
the sign is lost, as the averaging process is based on the spectrum of
$\mu_{5}^{2}$. This is a problem because the expressions of $k_{\alpha}$ and
$\gamma_{\alpha}$, as given in Eqs. (27) and (26), include the sum of
$\eta\langle{\mu_{5}}\rangle$ and $\alpha_{\mathrm{M}}$. For strong
turbulence, we expect that
$\alpha_{\mathrm{M}}\gg\eta\langle{\mu_{5}}\rangle$, and therefore we neglect
the $\eta\langle{\mu_{5}}\rangle_{\mathrm{int}}$ term in the estimates. But
for systems with low Reynolds numbers, the sign of
$\langle\mu_{5}\rangle_{\mathrm{int}}$ can be relevant in the comparison
between DNS and mean-field theory. As can be seen in the upper row of Fig. 10,
indeed, in our simulations, the contribution of
$\eta\langle{\mu_{5}}\rangle_{\mathrm{int}}$ can be relevant as it is not much
smaller than the values of $\alpha_{\mathrm{M,int}}$ and
$\alpha_{\mathrm{M,int}}^{\mathrm{sat}}$ As $\alpha_{\mathrm{M,int}}^{\rm
sat}$ is proportional to $\langle\mu_{5}\rangle_{\mathrm{int}}$, for this case
the sign of $\langle\mu_{5}\rangle_{\mathrm{int}}$ is irrelevant in the
expression $|\alpha_{\mathrm{M}}+\eta\langle{\mu_{5}}\rangle|$, and we can use
the full expressions from Eqs. (27) and (26), which are shown as dashed-dotted
lines in the middle and bottom rows of Fig. 10. The contribution of
$\eta\langle{\mu_{5}}\rangle$ leads to slightly higher characteristic
wavenumbers and growth rates, which generally agree better with the directly
measured values of $k_{\mathrm{int}}$ and $\gamma_{\mathrm{int}}$.

All of the turbulent runs presented in Figs. 9 and 10 reach saturation
eventually, i.e. the mean magnetic field stops growing. This can be seen in
the time evolution of $\langle B\rangle_{\mathrm{int}}$ in Fig. 9 and in the
bottom row of Fig. 10, where $\gamma_{\mathrm{int}}$ vanishes towards the end
of the individual runs. In case of mean-field dynamos, the maximum magnetic
field strength cannot be estimated by $B_{*}$ as given by Eq. 20, because the
characteristic timescale are different from that of the small-scale chiral
dynamo. Instead, we expect non-linear effects to play a role in the mean-field
dynamo saturation. In particular, the growth rate of this dynamo vanishes when
$|(\eta+\eta_{\mathrm{T}})k_{\mathrm{int}}|\approx|\alpha_{\mathrm{M}}+\eta\langle{\mu_{5}}\rangle|\approx|\alpha_{\mathrm{M}}|$.
Indeed, we find that $(\eta+\eta_{\mathrm{T}})k_{\mathrm{int}}$ becomes
comparable to the different estimates of $\alpha_{\mathrm{M}}$ (see the top
row of Fig. 10) at the same time when $\gamma_{\mathrm{int}}$ vanishes (see
the bottom row of Fig. 10).

Figure 11:  Summary of the three different regimes in systems with vanishing
initial chiral asymmetry, in which $\mu_{5}$ is generated through the chiral
separation effect. The maximum magnetic field strength is the lowest in the
regime where just chiral tangling occurs (left side of the sketch). When the
small-scale chiral dynamo instability is excited, the maximum field strength
is either given by $B_{*}$ (see Eq. 20; middle part of the sketch) or larger
if magnetically-driven turbulence is produced (right side of the sketch).

##  V Discussion and Conclusions

In this paper, we have studied a high-energy plasma with joint action of the
chiral magnetic effect (CME) and the chiral separation effect (CSE). We
considered a very weak initial magnetic field in the form of Gaussian
fluctuations plus a weak external magnetic field $B_{\mathrm{ex}}$. The
initial chiral chemical potential $\mu_{5}$ is zero, but there is a strong
initial gradient of the chemical potential fluctuations $\mu$. Through the
CSE, chiral magnetic waves (CMWs) generate inhomogeneous fluctuations of
$\mu_{5}$. As there is no (initial) velocity field in the system, the only way
for the magnetic field to get amplified in this scenario is through the
produced chiral asymmetry (i.e., a non-zero $\mu_{5}$). The generation of the
magnetic field is caused by the second term on the right-hand side of the
induction equation (1). However, this term can only lead to a magnetic field
instability if the produced $\mu_{5}$ becomes large enough. In this paper, we
have identified the parameter space in which such an instability can occur.

Depending on the parameters and the initial conditions of the system, we have
found three different evolutionary branches which are summarized in Fig. 11.
If the timescale of the small-scale chiral dynamo instability $t_{\mathrm{D}}$
[see Eq. (18), and remember that it is based on the assumption that the total
initial $\mu$ can be converted to $\mu_{5}$ through the CMW] is smaller than
the characteristic period of the CMW
$t_{\mathrm{CMW}}=2\pi/\omega_{\mathrm{CMW}}$, the magnetic field fluctuations
can only be produced due to chiral tangling (see Sec. III.2). In this case,
the maximum magnetic field is limited by the value of the small imposed
magnetic field $B_{\mathrm{ex}}$.

If $t_{\mathrm{D}}<t_{\mathrm{CMW}}$, the small-scale chiral dynamo can occur
and amplify the magnetic fluctuations to values
$B_{\mathrm{rms}}>B_{\mathrm{ex}}$. We have estimated the maximum magnetic
field strength $B_{*}$ for a given set of initial conditions in Eq. 20, and
find that it depends on the values of the coupling parameters $C_{5}$ and
$C_{\mu}$, the initial strength and correlation length of the chiral chemical
potential, as well as on the microscopic resistivity $\eta$ and the chiral
feedback parameter $\lambda$. Generally, we expect more efficient magnetic
field amplification for CMWs with lower frequencies; see Fig. 1.

For systems in which $B_{*}>B_{\mathrm{crit}}$, the Reynolds number eventually
exceeds unity and the produced turbulence leads to mean-field effects. Using
direct numerical simulations we have shown in Sec. IV.4 that a mean-field
dynamo, caused by the magnetic alpha effect, can amplify the magnetic
fluctuations to $B_{\mathrm{rms}}>B_{*}$. We concluded that saturation of the
mean-field dynamo is caused by an increasing turbulent diffusivity
$\eta_{\mathrm{T}}$ in the system.

With our study, we have shown that chiral dynamo instabilities and even mean-
field dynamos are universal mechanisms for high-energy plasma, even in the
absence of an initial chiral asymmetry. Our results may have important
consequences for the plasma of the early Universe, proto-neutron stars, heavy
ion collision experiments, and the understanding of quantum materials.

###### Acknowledgements.

J.S. acknowledges the support of the Swiss National Science Foundation under
Grant No. 185863. A.B. was supported in part through a grant from the Swedish
Research Council (Vetenskapsrådet, 2019-04234).

##  Appendix A Numerical constraints for simulations with CMWs

In the simulations presented in this study, two crucial criteria need to be
satisfied. As in any simulation of chiral MHD, the resolution needs to be high
enough to resolve the small-scale chiral instability. The instability is
attained on the wave number $k_{5}$ given in Eq. (16). With the minimum wave
number in the numerical domain with resolution $N$ being $2\pi N/L$, the
criterion for chiral MHD simulations is

$\displaystyle\frac{2\pi N}{L}\gtrsim\frac{\mu_{5}}{2}.$ (35)

If $\mu_{5}$ is produced from CMWs, the approximate maximum value of $\mu_{5}$
is the initial value of the chemical potential, $\mu(t_{0})$, and therefore
the criterion in Eq. (35) becomes

$\displaystyle\frac{2\pi N}{L}\gtrsim\frac{\mu(t_{0})}{2}.$ (36)

Another constraint on the parameter space that is accessible with DNS is
related to the time step. As discussed in [61], the time step contribution
from the terms including $\mu_{5}$ and $\mu$ is

$\displaystyle\delta t_{\mathrm{chiral}}=c_{\mathrm{\delta
t,chiral}}~{}\mathrm{min}(\delta t_{\lambda_{5}},\delta t_{D_{5}},\delta
t_{\mathrm{CMW}},\delta t_{D_{\mu}},\delta t_{v_{\mu}})$

with

$\displaystyle\delta t_{v_{\mu}}=\,$ $\displaystyle\frac{\delta
x}{\eta\mu_{5}},\,$ $\displaystyle\delta t_{\mathrm{CMW}}=\,$
$\displaystyle\frac{\delta x}{|\bm{B}|\sqrt{C_{5}C_{\mu}}},\,$
$\displaystyle\delta t_{\lambda_{5}}=\,$
$\displaystyle\frac{1}{\lambda\eta\bm{B}^{2}},$ $\displaystyle\delta
t_{D_{5}}=\,$ $\displaystyle\frac{\delta x^{4}}{\mathcal{D}_{5}},\,$
$\displaystyle\delta t_{D_{\mu}}=\,$ $\displaystyle\frac{\delta
x^{4}}{\mathcal{D}_{\mu}},\,$ (38)

and with the scaling parameter $c_{\mathrm{\delta t,chiral}}$. For CMWs with
large frequencies, the contribution from $\delta t_{\mathrm{CMW}}$ becomes the
most relevant one. With $B$ increasing through the chiral dynamo instability,
the CMW frequency increases, in other words, the characteristic velocity of
the CMWs

$\displaystyle v_{\mathrm{CMW}}\approx|\bm{B}|\sqrt{C_{5}C_{\mu}}$ (39)

becomes larger. If $v_{\mathrm{CMW}}$ becomes larger than the sound speed
$c_{\mathrm{s}}=1$, shocks develop and the numerical solution becomes
unstable. Therefore, the parameters should be chosen such that the maximum
magnetic field strength $B_{*}$ generated self-consistently through CMWs [see
Eq. (39)], is less than $(C_{5}C_{\mu})^{-1/2}$.

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