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\begin{document}
\jvol{00} \jnum{00} \jyear{2012} %\jmonth{February}
\markboth{Schober, Brandenburg, Rogachevskii \& Kleeorin}{Generation of turbulence by chiral MHD dynamos in the early Universe}
%JS: replaced
%\articletype{Proceedings}
%AB: ok, I don't have strong views on this
\articletype{GAFD Special issue on ``Recent Developments in Natural Dynamos''}
% %\title{{\textit{Properties of chiral-magnetically driven turbulence and chiral dynamos}}}
% %AB: I still think "Magnetic Prandtl number dependence of turbulence from chiral MHD dynamos" is good
% %IR: agree with Axel
% %JS: I think this version is clearly enough different from our previous papers:
% %\title{{\textit{Magnetic Prandtl number dependence of turbulence from chiral MHD dynamos}}}
% \title{{\textit{Properties of turbulence driven by chiral MHD dynamos}}}
% %JS: It's short, different from previous titles, and includes the word "dynamo".
%AB: but it doesn't say what is new here.
%JS: I'd still prefer to replace "from" by "generated by", especially since the title is two lines anyways.
%\title{{\textit{Magnetic Prandtl number dependence of turbulence from chiral MHD dynamos}}}
%AB: ok, like so then?
\title{{\textit{Magnetic Prandtl number dependence of turbulence generated by chiral MHD dynamos}}}
\author{J. SCHOBER, A. BRANDENBURG, I. ROGACHEVSKII, \& N. KLEEORIN ${\dag}$$^{\ast}$\thanks{$^\ast$Email: jennifer.schober@epfl.ch}}
\vspace{6pt}\received{\today,~ $ $Revision: 1.75 $ $}
\maketitle
\begin{abstract}
%An asymmetry between the number density of left- and right-handed fermions
%AB: I think one can say "in"
%JS: not sure, but probably ok
An asymmetry in the number density of left- and right-handed fermions
%gives rise to a new term in the induction equation that can result in a
%AB: gives -> is known to give
%JS: ok
is known to give rise to a new term in the induction equation that can result in a
small-scale instability.
This is a microphysical effect and is mathematically similar to the
$\alpha$ effect, which is a turbulent or macrophysical effect.
%At high temperatures, the only regime where a chiral asymmetry can survive, these
%AB: the only regime where -> when, added "for long enough"
%JS: ok
At high temperatures, when chiral asymmetry can survive for long enough, these
chiral MHD dynamos can amplify magnetic fields efficiently, which in turn drive
turbulence via the Lorentz force.
While it has been demonstrated in numerical simulations that chiral magnetically
driven turbulence exists and modifies the evolution of the plasma, the details
of this process remain unclear.
The goal of this paper is to shed new light on the properties of chiral magnetically
driven turbulence using numerical simulations with the \textsc{Pencil Code}.
We explore the generation of turbulence for different initial conditions,
%IR: what is the meaning of "the initial chiral instability"?
%IR: Do you mean variations of $\mu_0$?
%JS: Yes, but I wanted to avoid defining mu_0 in the abstract.
including a variation of the initial chiral instability and the
magnetic Prandtl number, $\Pm$.
In particular, we determine
the ratio of kinetic to magnetic energy, $\Upsilon^2$,
that can be reached in chiral magnetically driven turbulence.
Within the parameter space explored in this study, $\Upsilon$ reaches a value of
approximately $0.24$--$0.27$---independently of the initial chiral asymmetry and
for $\Pm=1$.
Our simulations suggest, that $\Upsilon$ decreases as a power law when
increasing $\Pm$.
While the exact scaling depends on the details of the fitting criteria and the
Reynolds number regime, an approximate result of $\Upsilon(\Pm)=0.3\ \Pm^{-0.2}$
is reported.
Using the findings from our numerical simulations, we estimate the properties of
chiral magnetically driven turbulence in the early Universe.
\begin{keywords}
Relativistic magnetohydrodynamics (MHD); Chiral MHD dynamos; Turbulence; Early Universe
\end{keywords}
\end{abstract}
%%%%%%%%%%%%
% SECTION %
\section{Introduction}
%%%%%%%%%%%%
Turbulence and magnetic fields are closely connected in many
geophysical and astrophysical flows:
Magnetohydrodynamic (MHD) dynamos are often caused by
turbulence, so for example in the cases of the small-scale \citep{Kazantsev1968,KulsrudAnderson1992} and large-scale
dynamos, especially those driven by a turbulent $\alpha$ effect \citep{P55,SKR66}.
On the other hand, the Lorentz force resulting from magnetic fields can
drive turbulent motions.
How much magnetic energy can be converted into kinetic energy depends on
various properties of the plasma,
characterised by the fluid and magnetic Reynolds numbers,
and the structure of the magnetic field.
Hence, turbulence is a key ingredient for understanding the origin and
evolution of cosmic magnetic fields.
In recent years, the nature of primordial cosmic magnetic fields
has been more and more constrained.
%The puzzling observational indications for large-scale intergalactic magnetic
%AB: why puzzling? Need to talk about lower limits
%JS: "puzzling" refers more to the interpretation of these observations.
The lower limits on the strength of intergalactic magnetic
%JS: here we don't need a comma now
%fields \citep{NV10,DCRFCL11},
fields \citep{NV10,DCRFCL11}
%the possible remains of primordial fields, can be explained by the following
%AB: the -> might be due to, added "which"
%JS: ok
might be due to possible remains of primordial fields, which can be explained by the following
scenario:
Seed fields are generated on small spatial scales, below the
co-moving Hubble radius of the early Universe, and subsequently
cascaded to larger scales in decaying MHD turbulence either with magnetic helicity
%JS: added Field & Carroll 2000:
%\citep{BEO96,BM99PhRvL, KTBN13,BKMPTV17} or without \citep{BKT15,Zra14}.
\citep{BEO96,BM99PhRvL,FC2000,KTBN13,BKMPTV17} or without \citep{BKT15,Zra14}.
%AB: do BM99 talk about cosmology? This ref is relevant for B ~ t^1/3
%JS: No, I don't see a discussion on cosmology in BM99. Didn't you add this
%JS: reference for general decaying MHD turbulence?
%AB: should maybe add Field & Carroll 1999 or so
%JS: done. this one fits in better than BM99PhRvL.
Cosmological seed fields, however, are
a highly debated topic in modern cosmology, see e.g.\
\citet{GrassoRubinstein2001,KZ08,Subramanian16}, and have been
%connected to a microphysical effect related to the different handedness
%AB: I think we need plural, so "handednesses"
%JS: not sure. I think both might be ok. E.g.: "These cars have different color."
%JS: vs. "These cars have different colors."
connected to a microphysical effect related to the different handednesses
of fermions.
In the presence of an external magnetic field, the momenta of fermions
align along the field lines according to their spin:
right-handed fermions move along the field lines, while left- handed ones move
in the opposite direction.
Consequently, an asymmetry in the number density of left- and right-handed
particles leads to a net current along the magnetic field. This effect is
called the \textit{chiral magnetic anomaly}
\citep{Vilenkin:80a,RW85,Tsokos:85,AlekseevEtAl1998,Frohlich:2000en,
Frohlich:2002fg,Kharzeev:07,Fukushima:08,Son:2009tf}
%and can lead to a magnetic instability \citep{JS97}.
%AB: inserted "the resulting current"
%JS: ok
and the resulting current can lead to a magnetic instability \citep{JS97}.
Especially the studies of the chiral inverse magnetic cascade and
the evolution of a non-uniform chiral chemical potential
by \citet{BFR12,BFR15} who found that a chiral asymmetry can, in principle,
%survive down to energies of the order of 10 MeV, made this effect an interesting
%AB: wouldn't it be good to remind some readers of the temperature here?
%JS: yes, ok
survive down to energies of the order of $10\MeV$ ($10^{11}\K$), made this effect an interesting
candidate for cosmological applications.
Recently, a systematic analytical analysis of the system of chiral MHD
equations, including the back-reaction of the magnetic field on the
chiral chemical potential, and the coupling to the plasma velocity field has
been performed by \citet{REtAl17}.
High-resolution numerical simulations, presented in \citet{Schober2017},
confirm results from mean-field theory,
in particular the existence of a new chiral $\alpha$ effect
that is not related to the kinetic helicity, the so-called $\alpha_\mu$ effect.
Spectral properties of chiral MHD turbulence have been analyzed in
\citet{BSRKBFRK17}.
A key result from these direct numerical simulations (DNS) is
that turbulence can be magnetically driven by the Lorentz force
due to a small-scale chiral dynamo instability.
In particular, a new three-stage-scenario of the magnetic field
evolution has been found in \citet{Schober2017}.
The small-scale chiral dynamo instability is followed by a phase in which
magnetically-produced turbulence triggers a large-scale dynamo
instability, which eventually saturates due to the decrease of the chiral
chemical potential.
%In this paper we explore the properties of this chiral magnetically produced
%turbulence using high-resolution numerical simulations.
%We measure how much kinetic
%energy is generated for different initial conditions and plasma parameters.
%In particular, we determine the ratio between the rms velocity, $u_\mathrm{rms}$,
%and the rms magnetic field strength, $B_\mathrm{rms}$, for various initial
%chemical potentials and for different magnetic Prandtl numbers
%AB: I don't think it is fair to pretend that we are here the first to
%AB: "explore the properties of this chiral magnetically produced turbulence"
%AB: We review it to some extent and should say this, but we should first
%AB: say what is new, so how about this:
In this paper we extend the work of \citet{Schober2017} and explore
%JS: It might be better to talk about energies here (the ratio of urms over Brms is only dimensionless if we normalize these quantities. it might be confusing at this point)
%for the first time the dependence of the ratio of the rms velocity,
%$u_\mathrm{rms}$, to rms magnetic field strength, $B_\mathrm{rms}$,
for the first time the dependence of the ratio of the kinetic energy over the
magnetic energy
%JS: why not mentioning that we also explore the dependence on mu_0?
%JS: we don't find a dependence on mu_0, but I think this was not clear before
%JS: doing the simulations (e.g. our referee for paper II asks this question)
%on the magnetic Prandtl number,
on the initial chiral asymmetry and the magnetic Prandtl number,
%JS.
%AB.
$\Pm\equiv \nu/\eta$, where $\nu$ is the kinematic viscosity and $\eta$ is the magnetic
diffusivity.
%AB: can say something here about review part
We also determine the dependence of the kinetic to magnetic energy
dissipation rate on the magnetic Prandtl number.
%JS: not sure if the following sentence is needed. we say the same in the next paragraph
We begin by briefly reviewing some essential findings regarding
chiral magnetically produced turbulence.
%AB.
The paper is structured as follows: In section~\ref{sec_ChiralMHD} we review
the chiral MHD equations, the growth rates of their instabilities, and the
saturation magnetic fields expected from the conversation law in chiral MHD.
The different stages of chiral magnetically driven turbulence are discussed.
The setup of our numerical simulations is described in section~\ref{sec_DNS}
and compared with those presented in \citet{Schober2017}.
We present the evolution of the velocity field and the magnetic field
for a reference run in detail.
In section~\ref{subsec_urmsBrms} we analyze the ratio
of kinetic over magnetic energy for different dynamo growth rates and
different magnetic Prandtl numbers.
In section~\ref{sec_EU}, we estimate the magnetic Prandtl and Reynolds numbers
in the relativistic plasma of the early Universe and apply our results on the
magnetic Prandtl number dependence.
%%%%%%%%%%%%
% SECTION %
\section{Chiral magnetohydrodynamics}
\label{sec_ChiralMHD}
%%%%%%%%%%%%
\subsection{System of equations}
In the following, we review the basic equations of chiral MHD, as derived
by \citet{REtAl17}.
We keep only terms that are linear in the microscopic
magnetic diffusivity, $\eta$, which is the relevant regime for astrophysical
applications.
The chiral asymmetry is described by the chiral chemical potential,
\begin{eqnarray}
\mu_5=6\,(n_{\rm L}-n_{\rm R})\,{(\hbar c)^3\over(\kB T)^2},
\end{eqnarray}
which is proportional to the difference in the number densities of left- and
right-chiral fermions, $n_\mathrm{L}$ and $n_\mathrm{R}$, respectively.
Here $T$ is the temperature, $\kB$ is the Boltzmann constant,
$c$ is the speed of light, and $\hbar$ is the reduced Planck constant.
In an external magnetic field, $\mu_5$ gives rise to a current
due to the chiral magnetic effect (CME)
\begin{eqnarray}
\label{eq_CME}
\JJ_{\rm CME} = \frac{\alphaem}{\pi \hbar} \mu_5 \BB ,
\end{eqnarray}
where $\alphaem \approx 1/137$ is the fine structure constant.
This standard model physics effect results in
an additional term in the Maxwell equations.
Based on these modified Maxwell equations,
%\citep{BFR15} and \citet{REtAl17} derived the following set of chiral MHD equations:
%AB: citep -> cite
%JS: cite -> citet (even though it looks the same in my pdf)
%\cite{BFR15} and \citet{REtAl17} derived the following set of chiral MHD equations:
\citet{BFR15} and \citet{REtAl17} derived the following set of chiral MHD equations:
\begin{eqnarray}
\frac{\partial \BB}{\partial t} &=& \nab \times \left[{\UU} \times {\BB}
- \eta \, \left(\nab \times {\BB}
- \mu {\BB} \right) \right] ,
\label{ind-DNS}\\
\rho{D \UU \over D t}&=& (\nab \times {\BB}) \times \BB
-\nab p + \nab {\bm \cdot} (2\nu \rho \SSSS)
+\rho \ff ,
\label{UU-DNS}\\
\frac{D \rho}{D t} &=& - \rho \, \nab \cdot \UU ,
\label{rho-DNS}\\
\frac{D \mu}{D t} &=& D_5 \, \Delta \mu
+ \lambda \, \eta \, \left[{\BB} {\bm \cdot} (\nab \times {\BB})
- \mu {\BB}^2\right]
-\Gamma_{\rm\!f}\mu,
\label{mu-DNS}
\end{eqnarray}
where the magnetic field $\BB$ is normalized such that the magnetic energy
density is $\BB^2/2$ (so the magnetic field in Gauss is $\sqrt{4\pi}\,\BB$), and
$D/D t = \partial/\partial t + \UU \cdot \nab$ is the
advective derivative.
Further, a normalization of $\mu_5$ is used such that
$\mu = (4 \alphaem /\hbar c) \mu_5$ and
%the chiral feedback parameter $\lambda$ characterizes the strength of the
%AB: inserted "has been introduced, which"
%JS: ok
the chiral feedback parameter $\lambda$ has been introduced, which characterizes the strength of the
back-reaction from the electromagnetic
field on the evolution of $\mu$.
For hot plasmas, when $\kB T \gg \max(|\mu_L|,|\mu_R|$),
it is given by \citep{BFR15}
\begin{eqnarray}
\lambda=3 \hbar c \left({8 \alphaem \over \kB T} \right)^2.
\label{eq_lambda}
\end{eqnarray}
In equations~(\ref{ind-DNS})--(\ref{mu-DNS}),
%$\eta$ is the microscopic magnetic diffusivity,
%AB: omitted this line, because eta has already been defined.
%JS: ok
$D_5$ is a chiral diffusion coefficient, $p$ is the fluid pressure,
${\sf S}_{ij}=\half(U_{i,j}+U_{j,i})-\onethird\delta_{ij} {\bm \nabla}
{\bm \cdot} \UU$
are the components of the trace-free strain tensor, where commas denote partial
spatial differentiation, $\nu$ is the kinematic viscosity,
and $\ff$ is the turbulent forcing function.
For an isothermal equation of state, the pressure $p$ is related
%to the density $\rho$ via $p=c_{\rm s}^2\rho$, where $c_{\rm s}$ is the sound
%AB: inserted "isothermal" (as opposed to adiabatic sound speed)
%JS: ok
to the density $\rho$ via $p=c_{\rm s}^2\rho$, where $c_{\rm s}$ is the isothermal sound
speed.
The last term in equation~(\ref{mu-DNS}) describing
%the chiral flipping reactions,
%AB: need to define "Gamma_f"
%JS: ok
the chiral flipping reactions at a rate $\Gamma_{\rm\!f}$
is neglected in this work, because we consider situations where it is
subdominant in comparison with the remaining terms.
The system of equations is determined by several non-dimensional parameters.
In terms of chiral MHD dynamos, the most relevant ones are the chiral
Mach number
\begin{eqnarray}
{\rm Ma}_\mu = \frac{\eta\mu_0}{c_\mathrm{s}} \equiv \frac{v_{\mu}}
{c_\mathrm{s}},
\label{Ma_mu_def}
\end{eqnarray}
and the dimensionless chiral nonlinearity parameter:
\begin{eqnarray}
\lambda_\mu = \lambda \eta^2 \meanrho.
\label{eq_lambdamu}
\end{eqnarray}
The parameter ${\rm Ma}_\mu$ measures the relevance
of the chiral term in the induction
equation~(\ref{ind-DNS}) and determines
the growth rate of the small-scale chiral dynamo instability.
The nonlinear back reaction
of the magnetic field on the chiral chemical potential $\mu$
is characterized by $\lambda_\mu$,
which affects the strength of the saturation magnetic field and
the strength of the magnetically driven turbulence.
In this paper we consider only cases with $\lambda_\mu \ll 1$, i.e., when turbulence
is produced efficiently due to strong magnetic fields
generated by the small-scale chiral dynamo instability.
%AB: This gives the impression that large values were never looked at,
%AB: so therefore I added this:
The turbulent cascade properties have previously been studied by
\citet{BSRKBFRK17} in the range $2\times10^{-6}\leq\lambda_\mu \leq200$,
\citet{Schober2017} in the range $10^{-9}\leq\lambda_\mu \leq10^{-5}$.
%AB: or did you also do larger values?
%JS: no, you have performed the runs with the largest lambda_mu
%AB.
\subsection{Analogy with the $\alpha$ effect in mean-field electrodynamics}
Readers familiar with mean-field electrodynamics \citep[MFE, see][]{M78,KR80}
will have readily noticed the analogy between
$v_\mu=\mu \, \eta$ in chiral MHD and the $\alpha$ in MFE.
For $\eta\to0$, the analogy goes further in that even the evolution
equation \eq{mu-DNS} for $\mu$ corresponds to an analogous one
for $\alpha$ in what is known as the dynamical quenching formalism \citep{KR82}.
In chiral MHD, this implies that the total chirality is conserved.
In MFE, it implies that the total magnetic helicity is conserved,
i.e., the sum of the magnetic helicity of the mean-field and that
of the fluctuating or small-scale field, the latter of which constitutes an
additional time-dependent contribution to the $\alpha$ effect.
The other contribution to the $\alpha$ effect in MFE is proportional
to the kinetic helicity, which was here assumed to be constant in time, so we can write
\citep[see equation 18 of][]{BB02}
%JS: I find a slightly different equation, when starting from (18) of BB02
%JS: (eta_T -> eta and different lambda_MFE). The similarity between our eq (6)
%JS: equation (10) would be even better, if we divide (10) by eta_T, meaning
%JS: write the evolution equation of alpha/eta_T. Then the eta_T in front of B*J
%JS: disappears.
%JS: Also, shouldn't we use mean fields here?
\begin{equation}
\frac{\partial \alpha}{\partial t} =
%\lambda_{\rm MFE} \, \eta \left[\eta_{_{T}} {\meanBB} {\bm \cdot} (\overline{\nab \times {\BB}})
%AB: the overbar should not include nabla times
%JS: is there a reason for that? this term comes from \overbar{B}\cdot\overbar{J}, where J has been replaced by
%JS: curl B.
\lambda_{\rm MFE} \, \eta \left[\eta_{_{T}} {\meanBB} {\bm \cdot} (\nab \times {\meanBB})
- \alpha {\meanBB}^2\right]
-\Gamma_{\rm MFE}\, (\alpha-\alpha_{\rm K}),
\end{equation}
%\begin{equation}
% \frac{\partial \alpha}{\partial t} =
% \lambda_{\rm MFE} \, \eta_{_{T}} \left[\eta_{_{T}} {\BB} {\bm \cdot} (\nab \times {\BB})
% - \alpha {\BB}^2\right]
% -\Gamma_{\rm MFE}\, (\alpha-\alpha_{\rm K}),
%\label{eq_alpha}
%\end{equation}
where $\eta_{_{T}}$ is the turbulent magnetic diffusivity.
In MFE, the coupling coefficient is given by
% %JS: also lambda is different in my calculation (there was a eta_t,0 missing):
% %$\lambda_{\rm MFE}=2\eta_{_{T}}k_{\rm f}^2/B_{\rm eq}^2$ and
% $\lambda_{\rm MFE}=2 \Rm k_{\rm f}^2/B_{\rm eq}^2$ and
%AB: how about this? I didn't quite follow what you said and may have missed something.
%AB: Avoid now Rm
%JS: then I would prefer putting eta on the other side:
%$\lambda_{\rm MFE}\eta=2 \eta_{_{T}} k_{\rm f}^2/B_{\rm eq}^2$ and
$\lambda_{\rm MFE}=2 \eta_{_{T}} k_{\rm f}^2/(\eta B_{\rm eq}^2)$ and
$\Gamma_{\rm MFE}=2\eta k_{\rm f}^2$, where
$k_{\rm f}$ is the wavenumber of the energy-carrying eddies and
$B_{\rm eq}$ is the equipartition field strength.
The applications of chiral MHD carry over to decaying MHD turbulence with
finite initial large-scale or small-scale magnetic helicity \citep{KBJ11}.
During the decay, some of the magnetic helicity is transferred between
%the large- and small-scale fields, which leads to a change of the $\alpha$
%AB: here also in?
%JS: might be ok
the large- and small-scale fields, which leads to a change in the $\alpha$
effect that in turn results in a slow-down of the decay.
%\subsection{The three stages of chiral magnetically driven turbulence}
%AB: added "Review of"
%JS: Ok
\subsection{Review of the three stages of chiral magnetically driven turbulence}
\label{sec_overview}
Recent simulations by \citet{Schober2017} have demonstrated the existence
of three distinct stages characterizing the growth and saturation of
chiral magnetically driven dynamos.
A schematic overview of a chiral MHD dynamo is presented in
figure~\ref{fig_sketch}.
It shows the evolution of an initially weak magnetic seed field in the presence
of a chemical potential with an initial value $\mu_0$.
We consider here the case of a small nonlinearity parameter $\lambda_\mu$, which
allows for an extended dynamo phase with enough time for the excitation of
turbulent motions.
%JS: give reference here
%Based on the evolution of the magnetic field, we can distinguish
Based on the evolution of the magnetic field, \citet{Schober2017} distinguish
three different phases: \\
%JS: added "a" and deleted reference to equation (it's not good to refer to an equation below)
%\textbf{Phase 1}: laminar small-scale chiral dynamo instability with the
%maximum growth rate given by equation~(\ref{gamma-max}); \\
\textbf{Phase 1}: a laminar small-scale chiral dynamo instability; \\
\textbf{Phase 2}:
%IR:
%turbulent large-scale dynamo instability,
%JS: why did you delete "turbulent" here?
%IR: because you write large-scale dynamo instability, caused by turbulence
%IR: why do you need to write twice "turbulence"?
%IR: also "turbulent large-scale dynamo" is not unique notion for "mean-field dynamo"
%JS: ok, we can skip the first "turbulent"
%JS: added "a"
%large-scale dynamo instability,
a large-scale dynamo instability,
%IR.
%AB: to me, "turbulent" sounds ok
%JS: also deleted reference to equation
%caused by chiral magnetically produced turbulence, with the growth rate
%given by equation~(\ref{gammamax_turb}); \\
caused by chiral magnetically produced turbulence; \\
\textbf{Phase 3}:
saturation of the growth of the large-scale magnetic field
%JS: also deleted reference to equation
%and reduction of $\mu$ according to the conservation law~(\ref{CL}). \\
and reduction of $\mu$ according to the conservation law in chiral MHD. \\
%JS.
With no additional energy input, dynamo saturation
is followed by decaying helical MHD turbulence, where the magnetic field
decreases as a power law \citep{BM99PhRvL, KTBN13}.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/sketch_ts__inf}
\includegraphics[width=0.49\textwidth]{figures/sketch_spec__inf}
\caption{
\textit{Left panel:} Schematic overview of the evolution an initially weak magnetic field
in the presence of a chiral chemical potential.
%JS: black -> red, and added u
%The time evolution of $\mu$ is presented as the black line and the one of $\BB$
The time evolution of $\mu$ is presented as the red line, the one of $\uu$ as
the black line, and the one of $\BB$ as the blue line.
%JS.
Further, the initial value of the chiral chemical potential,
$\mu_0$, and the saturation value of the magnetic field strength,
$(\mu_0/(\lambda \xi_\mathrm{M}))^{1/2}$, are indicated as
horizontal dotted lines.
%JS:
%The individual phases of the evolution are marked by black dashed-dotted
The transitions between individual phases of the evolution are marked by black dashed-dotted
%JS.
lines and discussed in section~\ref{sec_overview}.
\textit{Right panel:} Evolution of magnetic energy spectra,
where the maximum growth rate of the chiral dynamo is located at the wavenumber $k_\mu$
(black lines).
The peak of the spectrum moves to smaller wavenumbers for
$E_\mathrm{M}>C_\mu \rho \mu_0\eta^2$ and reaches its maximum value
$E_\mathrm{M}=C_\lambda \mu_0/\lambda$ at $k_\lambda$, leaving behind a
$k^{-2}$ spectrum.
Characteristic values of the wavenumber and the spectral energy are only
indicated at the axes for the black spectra.
The blue lines show a case with a larger $\mu_0$ than for the black lines
(but same $\lambda$) and
the red lines a case with a smaller $\lambda$ (but same $\mu_0$)
than for the black lines.
Thin lines show intermediate spectra, while thick lines show the spectra at
dynamo saturation.}%
\label{fig_sketch}
\end{center}
\end{figure}
\subsubsection{Laminar and turbulent dynamo amplification}
%JS: Phase -> phase
%In \textbf{Phase 1}, the velocity field is negligible
In \textbf{phase 1}, the velocity field is negligible
and a small-scale laminar dynamo operates.
%JS: made shorter
%The growth rate found from the linearized equation~(\ref{ind-DNS})
%for the laminar $v_\mu^2$ dynamo is:
%\begin{eqnarray}
% \gamma = |v_\mu \, k| - \eta k^2,
%\label{gamma}
%\end{eqnarray}
%where $k$ is the wavenumber.
%The dynamo instability is excited, i.e.\ $\gamma > 0$, for $k < |\mu_{0}|$ with
%the maximum growth rate
The growth rate found from the linearized equation~(\ref{ind-DNS}) has a
maximum value of \citep{REtAl17}
\begin{eqnarray}
%JS: later we refer to this rate as \gamma_\mu, so
% \gamma^{\rm max}_\mu = \frac{v_\mu^2}{4 \eta}
\gamma_\mu = \frac{v_\mu^2}{4 \eta}
\label{gamma-max}
\end{eqnarray}
being attained at
\begin{equation}
k_\mu =\frac{|\mu_0|}{2}.
\label{eq_kmax}
\end{equation}
%JS: added
During this phase, turbulence is driven by the Lorentz force with the rms
velocity increasing at a rate of approximately $2 \gamma_\mu$.
%JS.
%JS: Phase -> phase
%In \textbf{Phase 2}, turbulence produced by the Lorentz force can no longer be neglected and
%JS: modified to avoid repetition:
%In \textbf{phase 2}, turbulence produced by the Lorentz force can no longer be neglected and
In \textbf{phase 2}, the turbulent velocity field has become so strong that it
affects the evolution of the magnetic field.
It has been shown by \citet{BSRKBFRK17} that the peak of the magnetic energy
spectrum reaches a value of
\begin{equation}
%JS: full stop removed
% E_\mathrm{M}^{1,2} = C_\mu\meanrho \mu_0 \eta^2.
E_\mathrm{M}^{1,2} = C_\mu\meanrho \mu_0 \eta^2
\end{equation}
%JS: Phase -> phase
%with $C_\mu\approx 16$ at the transition from Phase 1 to Phase 2.
with $C_\mu\approx 16$ at the transition from phase 1 to phase 2.
This moment coincides with the beginning of the inverse transfer, when the
$k^{-2}$ spectrum starts to build up, i.e.\ when the peak of the magnetic energy
spectrum moves from $k_\mu$ to smaller wavenumbers.
The corresponding transition field strength can be estimated as
\begin{equation}
B_\mathrm{rms}^{1,2} \approx \left(E_\mathrm{M}^{1,2} k_\mu\right)^{1/2}
\approx \left(\frac{C_\mu\meanrho}{2}\right)^{1/2} \mu_0 \eta.
\end{equation}
%JS: rewritten this part:
%At this stage, the large-scale magnetic field is generated by
%magnetically driven turbulence.
%To describe the large-scale dynamo instability,
%equation~(\ref{ind-DNS}) is replaced
%by a mean field equation, as derived by \citet{REtAl17}:
%\begin{eqnarray}
% \frac{\partial \meanBB}{\partial t} &=&
% \nab \times \left[\meanUU \times \meanBB
% + (\meanv_\mu + \alpha_\mu) \meanBB
% - (\eta+ \, \eta_{_{T}})\nab \times \meanBB\right].
%\label{ind4-eq}
%\end{eqnarray}
%%JS: comma
%%Here $\meanv_\mu = \eta \meanmu_{0}$ and an equilibrium
%Here, $\meanv_\mu = \eta \meanmu_{0}$ and an equilibrium
%state is considered with $\meanmu_{\rm eq}=\meanmu_{0}=\const$ and
%${\bm \meanUU}_{\rm eq}=0$.
%In comparison to equation~(\ref{ind-DNS}), the mean-field equation~(\ref{ind4-eq})
%contains additional terms caused by turbulence.
%While the chiral $\alpha_\mu$ effect, which is given
%for large Reynolds numbers and a weak mean magnetic field as
%\begin{eqnarray}
% \alpha_\mu = - {2 \over 3} \meanv_\mu \ln \Rm,
%\label{alphamu}
%\end{eqnarray}
%increases the dynamo growth rate, $\gamma$,
%the turbulent magnetic diffusivity
%$\eta_{_{T}}$ reduces $\gamma$.
%In the mean-field equation, the chiral $v_\mu$ effect is replaced by the
%mean chiral $\meanv_\mu$ effect.
%Note, however, that at large magnetic Reynolds numbers
%$\Rm=u_0 \ell_0/\eta = 3 \eta_{_{T}}/\eta$,
%the $\alpha_\mu$ effect becomes dominant in comparison
%with the $\meanv_\mu$ effect.
%The growth rate of the large-scale dynamo instability is \citep{REtAl17}
%\begin{eqnarray}
% \gamma = |(\meanv_\mu + \alpha_\mu)\, k| - (\eta+ \, \eta_{_{T}}) \, k^2
%\label{gamma_turb}
%\end{eqnarray}
%with $k^2=k_x^2 + k_z^2$.
%The maximum growth rate of the large-scale dynamo instability, attained at
%the wavenumber
%\begin{equation}
% k_\alpha = {|\meanv_\mu + \alpha_\mu| \over 2(\eta+ \, \eta_{_{T}})},
%\label{kmax_turb}
%\end{equation}
%is given by
%\begin{eqnarray}
%\gamma^{\rm max}_\alpha
%= {(\meanv_\mu + \alpha_\mu)^2\over 4 (\eta+ \, \eta_{_{T}})}
%= {(\meanv_\mu + \alpha_\mu)^2\over 4 \eta \, (1 + \, \Rm/3)}.
%\label{gammamax_turb}
%\end{eqnarray}
%For vanishing $\Rm$, this equation
%yields the correct result for the laminar $v_\mu^2$ dynamo;
%see equation~(\ref{gamma-max}).
\citet{REtAl17} have studied the evolution of the large-scale magnetic using
a mean-field approach.
They find that the maximum growth rate of the field strength reaches a value of
\begin{equation}
%JS: also omited "max" here (we only show the maximnum growth rate)
%\gamma^{\rm max}_\alpha
\gamma_\alpha
= {(\meanv_\mu + \alpha_\mu)^2\over 4 (\eta+ \, \eta_{_{T}})}
= {(\meanv_\mu + \alpha_\mu)^2\over 4 \eta \, (1 + \, \Rm/3)}.
\label{gammamax_turb}
\end{equation}
Despite the contribution from the chiral alpha effect, given by the term
\begin{eqnarray}
\alpha_\mu = - {2 \over 3} \meanv_\mu \ln \Rm,
\label{alphamu}
\end{eqnarray}
the overall growth rate is reduced as compared to the laminar dynamo.
Here, $\Rm$ is the magnetic Reynolds number defined by
$\Rm=u_0 \ell_0/\eta = 3 \eta_{_{T}}/\eta$ and $\eta_{_{T}}$ is the turbulent
magnetic diffusivity.
The maximum growth rate of the large-scale chiral dynamo is attained at
the wavenumber
\begin{equation}
k_\alpha = {|\meanv_\mu + \alpha_\mu| \over 2(\eta+ \, \eta_{_{T}})}.
\label{kmax_turb}
\end{equation}
%JS.
\subsubsection{Saturation magnetic field}
%JS: shortened this:
%Saturation of the chiral dynamo, \textbf{phase 3}, is determined by the
%conservation law
%following from equations~(\ref{ind-DNS})--(\ref{mu-DNS}):
%\begin{equation}
%\frac{\partial }{\partial t} \left({\lambda \over 2} {\bm A} {\bm \cdot} \BB
% + \mu \right) + \nab {\bm \cdot} \FF_{\rm tot} = 0,
%\label{CL}
%\end{equation}
%where $\FF_{\rm tot} = \lambda/2 \left({\bm \EE} \times
%{\bm A} + \BB \, \Phi\right) - D_5 \nab \mu$
%is the flux of total chirality with the vector potential ${\bm A}$
%being defined as
%$\BB = {\bm \nabla} {\bm \times} {\bm A}$, ${\bm \EE}=
%- c^{-1} \, \{ {\bm \UU} {\bm \times} {\BB} + \eta \, \big(\mu {\BB} -
%{\bm \nabla} {\bm \times} {\BB} \big) \}$ being the electric field, $\Phi$ being the
%electrostatic potential, and $\lambda$ is assumed to be constant.
%This implies that the total chirality, is a conserved quantity:
%\begin{equation}
% \label{cons_law}
% \frac\lambda 2 \meanAB + \bar\mu = \mu_0 = \mathrm{const},
%\end{equation}
%where $\bar\mu$ is spatially averaged value of the chemical potential and
%$\meanAB$ is the magnetic helicity.
Saturation of the chiral dynamo, \textbf{phase 3}, is determined by the
conservation law following from equations~(\ref{ind-DNS})--(\ref{mu-DNS}), which
implies that the total chirality
\begin{equation}
\frac\lambda 2 \meanAB + \bar\mu = \mu_0 = \mathrm{const},
\label{CL}
\end{equation}
is a conserved quantity; see \citet{REtAl17} for more details.
Here, $\bar\mu$ is spatially averaged value of the chemical potential and
$\meanAB$ is the magnetic helicity.
%JS.
According to the conservation law (\ref{CL}), the magnetic field reaches the
following value at
dynamo saturation:
\begin{eqnarray}
B_\mathrm{sat} = \left(\frac{\mu_0}{\lambda\xi_\mathrm{M}}\right)^{1/2},
\label{eq_Bsat}
\end{eqnarray}
where $\xi_\mathrm{M}$ is the correlation length of the magnetic field.
The magnetic energy spectrum $E_{\rm M}(k,t)$ in chiral MHD turbulence has been
studied in \citet{BSRKBFRK17}.
In particular, it was found that $E_{\rm M}$ is proportional to $k^{-2}$
between the wavenumber
\begin{equation}
k_\lambda=\sqrt{\meanrho\lambda \frac{C_\mu}{C_\lambda}}\,\eta\mu_0,
\label{klambda}
\end{equation}
with $C_\mu\approx 16$, $C_\lambda\approx 1$, and $k_\mu$ given
by equation~(\ref{eq_kmax}).
We note that the only case considered here is $\lambda_\mu \ll 1$, which implies
$k_\lambda \ll k_\mu$.
Using dimensional arguments and numerical simulations,
\citet{BSRKBFRK17} found that for chiral magnetically driven turbulence,
the saturation magnetic energy spectrum $E_{\rm M}(k,t)$ obeys
\begin{equation}
E_{\rm M}(k,t)=C_\mu\,\meanrho\mu_0^3\eta^2k^{-2}
\label{Cmu}
\end{equation}
in $k_\lambda\mu_\mathrm{black}$ and
$\lambda_\mathrm{blue}=\lambda_\mathrm{black}$.
%%%%%%%%%%%%%%%%%
%SECTION
\section{Chiral magnetically driven turbulence in direct numerical simulations}
\label{sec_DNS}
%%%%%%%%%%%%%%%%%
\subsection{Numerical setup}
We solve equations~(\ref{ind-DNS})--(\ref{mu-DNS}) in a three dimensional
periodic box of size $L^3 = (2\pi)^3$ with the
\textsc{Pencil Code}\footnote{\textit{http://pencil-code.nordita.org/}}.
This code is well suited for MHD studies; it employs a third-order
accurate time-stepping method and sixth-order explicit finite differences
in space \citep{BD02,Bra03}.
The smallest wavenumber covered in the numerical domain is $k_1 = 2\pi/L = 1$ and
the resolution is varied between $480^3$ and $1216^3$.
Some of these simulations have already been presented previously
\citep{Schober2017}, but we now include additional runs for $\Pm\neq1$.
The sound speed in the simulations is set to $\cs = 1$ and the
mean fluid density to $\meanrho = 1$.
If not indicated otherwise, the magnetic Prandtl number is $1$,
i.e.\ the magnetic diffusivity equals the viscosity.
However, we do consider cases between $\Pm=0.5$ and $\Pm=10$, where the value of $\eta$
is fixed and $\nu$ changes.
No external forcing is applied to drive turbulence in the simulations, i.e.\
the velocity field is then driven entirely by magnetic fields.
All runs are initialized with a weak magnetic seed field in form of
Gaussian noise, with constant $\mu$, and vanishing velocity.
The main parameters of all simulations presented in this paper are summarized in
table~\ref{table}.
%%%%%%%%%%%%%%%%%%
% run directories:
%%%%%%%%%%%%%%%%%%
%
% Run A: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/576_3d_eta1e-4_nu1e-4_l2e1_mu020_Gaussian
%
% Run B: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/1216_3d_eta75e-5_nu75e-5_l16e3_mu088_Gaussian
%
% Run C: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/480_3d_eta5e-5_nu5e-5_l2e3_mu030_Gaussian
%
% Run D05: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/480_3d_eta5e-5_nu2.5e-5_l2e3_mu040_Gaussian
% Run D1: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/480_3d_eta5e-5_nu5e-5_l2e3_mu040_Gaussian_2
% Run D2: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/480_3d_eta5e-5_nu1e-4_l2e3_mu040_Gaussian_2
% Run D5: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/480_3d_eta5e-5_nu2.5e-4_l2e3_mu040_Gaussian
% Run D10: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/480_3d_eta5e-5_nu5e-4_l2e3_mu040_Gaussian
%
% Run E: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/480_3d_eta5e-5_nu5e-5_l2e3_mu050_Gaussian
%
% Run F1: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/576_3d_eta5e-5_nu5e-5_l2e3_mu060_Gaussian
% Run F2: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/576_3d_eta5e-5_nu1e-4_l2e3_mu060_Gaussian
% Run F5: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/576_3d_eta5e-5_nu25e-4_l2e3_mu060_Gaussian
% Run F10: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/576_3d_eta5e-5_nu5e-4_l2e3_mu060_Gaussian
%
% Run G05: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/480_3d_eta1e-4_nu5e-5_l2e3_mu040_Gaussian
% Run G1: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/480_3d_eta1e-4_nu1e-4_l2e3_mu040_Gaussian
% Run G2: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/480_3d_eta1e-4_nu2e-4_l2e3_mu040_Gaussian
% Run G51: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/480_3d_eta1e-4_nu5e-4_l2e3_mu040_Gaussian
% Run G10: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/480_3d_eta1e-4_nu1e-3_l2e3_mu040_Gaussian
%
% Run H02: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/480_3d_eta2e-4_nu4e-5_l5e2_mu040_Gaussian_beskow
% Run H05: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/480_3d_eta2e-4_nu1e-4_l5e2_mu040_Gaussian_beskow
% Run H1: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/480_3d_eta2e-4_nu2e-4_l5e2_mu040_Gaussian_beskow
% Run H2: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/480_3d_eta2e-4_nu4e-4_l5e2_mu040_Gaussian_beskow
% Run H5: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/480_3d_eta2e-4_nu1e-3_l5e2_mu040_Gaussian_beskow
% Run H10: pencil-code/jenny/chiral_fluids/laminar_dynamos/alpha2/480_3d_eta2e-4_nu2e-3_l5e2_mu040_Gaussian_beskow
%
%%%%%%%%%%%%%%%%%%
\begin{table}
\centering
\caption{Overview of the simulations discussed in this paper. Runs C, D, and E have
been repeated with different magnetic Prandtl numbers, i.e.\ Run C05 has $\Pm=0.5$,
Run C1 has $\Pm=1$, up to Run C10 with $\Pm=10$.
Reference runs are highlighted by bold font.}
\begin{tabular}{l|lllllll}
\hline
\hline
simulation & resolution & ${\rm Ma}_\mu$ & $\lambda_\mu$ & $(\mu_0/\lambda)^{1/2}$ & $k_\mu/k_1$ & $k_\lambda/k_1$ & $\Pm$\\
\hline
\textbf{Run A} & $\mathbf{576^3}$ & $\mathbf{2\times10^{-3}}$& $\mathbf{2\times10^{-7}}$ & $\mathbf{1.00}$ & $\mathbf{10}$ & $\mathbf{0.036}$ & $\mathbf{1.0}$\\
Run B & $1216^3$ & $6.6\times10^{-3}$ & $9\times10^{-6}$ & $0.12$ & $44$ & $1.1$ & $1.0$ \\
Run C & $480^3$ & $1.5\times10^{-3}$ & $5\times10^{-6}$ & $0.12$ & $15$ & $0.27$ & $1.0$ \\
\textbf{Runs D05...10} & $\mathbf{480^3}$ & $\mathbf{2\times10^{-3}}$ & $\mathbf{5\times10^{-6}}$ & $\mathbf{0.14}$ & $\mathbf{20}$ & $\mathbf{0.36}$ & $\mathbf{0.5...10}$ \\
Run E & $480^3$ & $2.5\times10^{-3}$ & $5\times10^{-6}$ & $0.16$ & $25$ & $0.45$ & $1.0$ \\
Runs F1...10 & $576^3$ & $3\times10^{-3}$ & $5\times10^{-6}$ & $0.17$ & $30$ & $0.54$ & $0.5...10$ \\
Runs G05...10 & $480^3$ & $4\times10^{-3}$ & $2\times10^{-5}$ & $0.14$ & $20$ & $0.72$ & $0.5...10$ \\
Runs H05...10 & $480^3$ & $8\times10^{-3}$ & $2\times10^{-5}$ & $0.28$ & $20$ & $0.72$ & $0.5...10$ \\
\hline
\hline
\end{tabular}
\label{table}
\end{table}
%JS: modified
\subsection{Effects from the finite numerical domain}
%\subsection{Finite size effects}
Due to the finiteness of the simulation domain, the evolution of the magnetic field
and turbulence is slightly modified in comparison to
the scenario discussed in section~\ref{sec_overview}.
The evolution of the chemical potential, the magnetic field strength, and
the velocity, as well as the time evolution of the magnetic energy spectrum
in finite box simulations is presented in figure~\ref{fig_sketch_box}.
First of all, the chemical potential does not vanish at dynamo saturation,
but reaches a finite value which is equal to the minimum
wavenumber possible in the box, $k_1$.
The magnetic field reaches the saturation value given in
equation~(\ref{eq_Bsat}).
However, the evolution of the magnetic energy spectrum differs as compared to
an infinite system.
In the laminar dynamo phase, we expect an instability on the scale $k_\mu$,
as predicted by theory; see the black curves in the right panel of
figure~\ref{fig_sketch_box}.
With the onset of turbulence, the peak of the energy spectrum moves to larger
%spatial scales, leaving behind a $k^{-2}$ spectrum.
%AB: how about this? Could you try to determine C_WT for your case?
%JS: yes, but I'm not sure if I have enough time today (I will be busy with teaching..)
spatial scales through inverse transfer.
%JS: Actually, this has been said above, where the constant was introduced as C_\mu.
%JS: So it's better to change this again:
%The turbulence is magnetically dominated and, as argued by \cite{BKT15},
%this implies a weak turbulence scaling with a spectrum
%$E_{\rm M}(k,t)=C_{\rm WT}(v_{\rm A}\epsilon k_{\rm M})^{1/2}k^{-2}$,
%where $C_{\rm WT}$ is a weak turbulence coefficient of the order of unity.
As discussed above, we expect a scaling of the magnetic energy spectrum
proportional to $k^{-2}$; see equation~(\ref{Cmu}).
%AB.
Once, the peak reaches the size of the box, however, we observe a steepening
of the spectrum, as indicated in the schematic figure~\ref{fig_sketch_box}.
This steepening is caused by the growth of the magnetic field on the smallest
possible wavenumber, until the spectrum reaches its saturation value
$C_\lambda \mu_0/\lambda$.
For large values of $\mu_0$ the initial instability occurs on smaller spatial
scales, i.e.\ larger $k$, and thus the $k^{-2}$ spectrum can extend over a
larger range; see the blue curves in the right panel of
figure~\ref{fig_sketch_box}.
%We note that we do expect the evolution of the magnetic energy spectrum to be,
%in principle, exactly as presented in the right panel of figure~\ref{fig_sketch}
%AB: modified
%JS: ok
When $k_\lambda saturation
%JS: no, I mean that gamma becomes smaller, due to the generation of turbulence.
%JS: After $4$ orders of magnitude we see the transition from the laminar to the
%JS: turbulent dynamo. Saturation occurs once the magnetic field has increased
%JS: by two more orders of magnitude.
%to saturation. Saturation occurs at $t \approx 0.4~t_\eta$.
to the generated turbulence. Saturation occurs at $t \approx 0.4~t_\eta$.
%AB.
Both, the velocity $u_\mathrm{rms}$ (gray dashed line) and the
magnetic helicity $\meanAB$ (blue dotted line) increase at a rate twice the one
of $B_\mathrm{rms}$.
The value of $\mu_\mathrm{rms}$ (orange dashed-dotted line), here shown with a
constant factor $2/\lambda$, decreases above $t \approx 0.3~t_\eta$,
switching off the dynamo instability.
In accordance with the conservation law~(\ref{CL}), the sum
$\meanAB + 2\mu_\mathrm{rms}/\lambda$ (purple dashed-dotted line) is constant
throughout the simulation time.
In the right panel of figure~\ref{fig_ts}, it can be seen that the magnetic
field is initially dominated by fluctuations (blue line).
During the mean-field dynamo phase a large-scale field (gray line) is produced.
\begin{figure}
\begin{center}
\subfigure{\includegraphics[width=0.49\textwidth]{figures/spec}}
\subfigure{\includegraphics[width=0.49\textwidth]{figures/spech}}
\caption{Normalized energy and helicity spectra for the reference run, Run A.
The timesteps between two spectra are equi-distant and the last spectra are
presented by solid lines.
The left panel above is equivalent to figure~10 of \citet{Schober2017}.}
\label{fig_spec}
\end{center}
\end{figure}
The kinetic and magnetic energy (left panel) and helicity spectra (right panel)
are presented in figure~\ref{fig_spec}.
All spectra grow from initially low values and the final spectra are presented
as solid lines.
From here we confirm that the laminar chiral dynamo injects energy at the
wavenumber $k_\mu$, as given in equation~($\ref{eq_kmax}$).
Once turbulence has been generated, the magnetic correlation length moves
to smaller wavenumbers.
%
%IR: no new paragraph with one short sentence
%
%JS: However, this sentence refers to the full section, not only to the last
%JS: paragraph. Moved it above now, in a modified version:
%For a more detailed discussion of this simulation we refer to section~4 of
%\citet{Schober2017}.
\subsection{Turbulence in different scenarios}
\label{subsec_urmsBrms}
In the analysis of the reference Run A, we have seen that the kinetic
energy reaches a certain percentage of the magnetic energy.
With the onset of the turbulent dynamo phase, phase 2, the ratio
\begin{eqnarray}
\Upsilon \equiv \left(\frac{\rho u_\mathrm{rms}^2/2}{B_\mathrm{rms}^2/2}\right)^{1/2},
\label{eq_Upsilon}
\end{eqnarray}
%IR: here kinetic energy is normalised by the total magnetic energy.
%IR: Additional important information about turbulence can be obtained
%IR: for another quantity: when kinetic energy is normalised by the
%IR: energy of magnetic fluctuations.
%JS: This could be done, I have the time series of the mean-field components
%JS: of the magnetic field.
stays approximately constant and
decreases as soon as the peak of the magnetic energy spectrum reaches the
box wavenumber $k_1$.
%AB: I think you mean we enter the regime of decaying turbulence.
%AB: Turbulence always dissipates, even if it is not decaying.
%JS: I agree. So what about:
%Afterwards, turbulence is not driven by the Lorentz force anymore and
%the kinetic energy dissipates.
Afterwards, turbulence is not driven by the Lorentz force anymore and
the plasma enters the regime of decaying MHD turbulence.
%JS.
In this section we explore how the details of this scenario are effected by the
properties of the plasma.
Therefore, we study a broad parameter space, varying the chiral
parameters as well as the magnetic Prandtl number.
\subsubsection{Dependence on the chiral parameters $\mathrm{Ma}_\mu$ and $\lambda_\mu$}
The time evolution of $\Upsilon$ is presented in
the left panel of
figure~\ref{fig_urmsBrms_t} for runs with different values of $\mathrm{Ma}_\mu$
and $\lambda_\mu$.
Time is normalized here by the inverse of the laminar dynamo growth
rate~(\ref{gamma-max}), allowing a better comparison between runs with different
$v_\mu$.
The evolution up to $t\approx 12\,\gamma_\mu^{-1}$ of all runs is similar, except
for a minor time delay of Run A.
This can be explained by the magnetic diffusivity which is larger than the
one in Run C by a factor of two.
Phase 2, when turbulence affects the evolution of the magnetic field, begins
approximately at $t\approx 12 - 14 ~\gamma_\mu^{-1}$ for the runs considered
here.
The onset of phase 2 is weakly dependent on $\eta$ and, in principle, also on
the initial value of the magnetic field strength, which is the same for all
runs presented in this paper.
During phase 2, the ratio $\Upsilon$ is comparable for all
3 runs considered here, even
though $\mathrm{Ma}_\mu$ and $\lambda_\mu$ are different.
Once the mean-field dynamo phase begins, we observe a ratio
$\Upsilon \approx 0.2$--$0.3$.
Run A, the reference run discussed in the previous section,
has the lowest value of $\lambda$ in our sample, leading to a small value of
$k_\lambda$ in comparison to the maximum wavenumber in the box:
$k_\lambda \approx 0.036 k_1$.
This implies that $k_1$ is reached early, much before
dynamo saturation, and the kinetic energy dissipates.
The line style of the curves in the left panel of figure~\ref{fig_urmsBrms_t} is
changed from solid to dashed at the time when
$k_\mathrm{M} = k_1$ is reached.
For a scenario in which the inverse cascade takes place completely inside the box, and
%hence kinetic energy is not dissipated within phase 2, we have performed Run B.
%AB: it must be superfluid if it doesn't dissipate; I think you mean decay
%JS: yes, i agree
hence kinetic energy does not decay within phase 2, we have performed Run B.
The choice of the parameters for this run, listed in table~\ref{table}, can be
justified as follows:
As for all of the runs presented in this paper, the ratio between $k_\mu$ and
%$k_\lambda$ needs to be large, in order to observe a turbulent dynamo phase,
%phase 2.
%AB: I would just put it in parentheses
%JS: ok
$k_\lambda$ needs to be large, in order to observe a turbulent dynamo phase
(phase 2).
%AB.
Using equations~(\ref{eq_kmax}) and (\ref{klambda}), we see that
\begin{equation}
\frac{k_\mu}{k_\lambda} = \left(\frac{C_\lambda}{4 C_\mu} \frac{1}{\meanrho \eta^2 \lambda}\right)^{1/2}
= \left(\frac{C_\lambda}{4 C_\mu} \frac{1}{\lambda_\mu}\right)^{1/2},
\label{eq_kmuklambda}
\end{equation}
which is independent of $\mu_0$.
However, $\mu_0$ needs to be chosen high enough to ensure that $k_\lambda>k_1$.
Here we use $\mu_0/k_1 = 88$, which implies that the laminar dynamo
instability occurs on small spatial scales and a high numerical resolution
is required.
Run B is presented in the left panel of figure~\ref{fig_urmsBrms_t} as a gray
solid line, for which the ratio $\Upsilon$ remains
approximately constant for times larger than $\approx 12~\gamma_\mu^{-1}$.
\begin{figure}
\begin{center}
%AB: I think one could clip the left panel at t*gamma=30.
%JS: Ok, I will change the plot ranges.
%AB: One could also write into the plot kM > k1 on the top line,
%AB: and kM = k1 on the lower dashed line.
%JS: This might be confusing, because the transition from kM = k1 to kM > k1
%JS: takes place at different time for different runs.
%AB: The black dashed line looks somewhat grayish too on my printout.
%JS: not sure what to do about this. I don't want to change the line styles,
%JS: because I use different ones already. I could use a lighter gray for Run B, maybe.
\subfigure{\includegraphics[width=0.49\textwidth]{figures/urmsBrms_t__mu}}
\subfigure{\includegraphics[width=0.49\textwidth]{figures/urmsBrms_mu}}
\caption{
\textit{Left panel:} The square root of the ratio of kinetic over magnetic
energy, $\Upsilon$, as a
function of time, normalized by the inverse laminar dynamo growth rate $\gamma_\mu$
%as given in equation~(\ref{gamma-max}), for different runs with $\Pm=1$.
%The Runs A -- C differ in their values of the chiral Mach number and the
%AB: combine
%JS: ok
as given in equation~(\ref{gamma-max}), for Runs~A--C with $\Pm=1$
and different values of the chiral Mach number and the
%AB.
nonlinearity parameter; see table~\ref{table}.
The time during which $k_\mathrm{M}$ is inside the numerical
box, i.e., $k_\mathrm{M}>k_1$, is marked by solid line style.
For $k_\mathrm{M} 0.9~\mathrm{max}(\Upsilon)$.
The solid gray line shows the mean value of $\UpsilonTimeAve$ resulting from the
first time averaging condition and the dashed gray for the latter condition.
}%
\label{fig_urmsBrms_t}
\end{center}
\end{figure}
In the right panel of figure~\ref{fig_urmsBrms_t}, we show the time-averaged ratio
$\UpsilonTimeAve$ for all our runs with $\Pm=1$ as a
function of $\mu_0/k_1$ as black data points.
The blue data points refer to the upper $x$ axes and indicate the
corresponding value of $\mathrm{Ma}_\mu$.
For the time averaging procedure, we consider two different criteria: For solid
data points the time average is performed for all values of
$\Upsilon$ larger than 50\% of its maximum value.
The open dots are obtained by using all values for which
$\Upsilon > 0.9~\mathrm{max}(\Upsilon)$,
which obviously results in a larger average value.
Error bars represent the standard deviation of $\UpsilonTimeAve$.
There is no significant dependence of $\UpsilonTimeAve$
on the values of $\mu_0$ and $\mathrm{Ma}_\mu$ for the parameter space explored
here:
When averaging over all $\Upsilon > 0.5~\mathrm{max}(\Upsilon)$ we find a mean
$\UpsilonTimeAve\approx0.24\pm0.01$ and when employing the criterion
$\Upsilon > 0.9~\mathrm{max}(\Upsilon)$ we find $\UpsilonTimeAve\approx0.27\pm0.01$.
\begin{figure}
\begin{center}
\subfigure{\includegraphics[width=0.49\textwidth]{figures/Rm_t__mu}}
\subfigure{\includegraphics[width=0.49\textwidth]{figures/Rmmax_mu}}
\caption{
\textit{Left panel:} The magnetic Reynolds number $\Rm$ as a function of time
for Runs A -- C.
The solid lines indicate the result using $k_\mathrm{M}$ as the integral scale
of turbulence and dashed lines show the result using $k_1$.
At late times $k_\mathrm{M}=k_1$ in our DNS, which are confined within a finite
box.
\textit{Right panel:} The maximum magnetic Reynolds number found in our
simulations verses $\mu_0/k_1$, for both cases $\Rm = u_\mathrm{rms}/(\eta k_1)$
(black dots) and $\Rm = u_\mathrm{rms}/(\eta k_\mathrm{M})$ (blue diamonds).
The solid line indicates the scaling $\mu_0^{3/2}$ which is expected for finite
box simulations with $k_\mathrm{box}>k_\lambda$.
}
\label{fig_Rmmax_mu}
\end{center}
\end{figure}
Knowing the amount of turbulent kinetic energy that can be produced by
Lorentz force driving, is essential for estimating the Reynolds number in
the plasma.
The latter is determined by the rms velocity, the magnetic diffusivity, and
the correlation length of the magnetic field.
In numerical simulations, both $u_\mathrm{rms}$ and $k_\mathrm{M}$ can be
limited by the size of the box, while $\eta$ is an input parameter.
We have seen that $\UpsilonTimeAve$ has an
approximately fixed value in the mean-field dynamo phase for $\Pm=1$ and as
long as $k_\mathrm{M} > k_1$.
This implies that the value of $u_\mathrm{rms}$ is proportional to
$B_\mathrm{rms}$ and reaches a maximum at the time $t_\mathrm{box}$,
which is defined as the time
when the peak of the energy spectrum reaches the size of the box,
i.e., when $k_\mathrm{M}= k_1$.
The energy spectrum at this time is given by equation~(\ref{Cmu}) and has a
maximum value of
$E_\mathrm{M}(k_1, t_\mathrm{box}) = C_\mu \meanrho \eta^2 \mu_0^3/2$.
The magnetic field strength corresponding to this energy spectrum can be
estimated as $(E_\mathrm{M}(k_1, t_\mathrm{box})k_1)^{1/2}
= (C_\mu \meanrho/2)^{1/2} \eta \mu_0^{3/2}$.
Hence we expect a scaling of the maximum velocity in our simulation
$\propto \eta \mu_0^{3/2}$.
The Reynolds number, $\Rm = u_\mathrm{rms}/(\eta k_\mathrm{M})$ with
$k_\mathrm{M}=k_1$ at late times, is thus independent of $\eta$ and scales
$\propto \mu_0^{3/2}$.
This scaling is observed in our simulations; see the right panel of
figure~\ref{fig_Rmmax_mu}.
The Reynolds number as a function of time for runs with different $\mu_0/k_1$
is presented in the left panel of this figure.
Here we show two different ratios, $u_\mathrm{rms}/(\eta k_\mathrm{M})$ and
$u_\mathrm{rms}/(\eta k_1)$.
%JS: comma
%For the first case $k_\mathrm{M}$ is measured as a function of time using the
For the first case, $k_\mathrm{M}$ is measured as a function of time using the
magnetic energy spectra.
At late times, once the peak of the magnetic energy spectrum has reached the
box wavenumber, $k_\mathrm{M}=k_1$, and the dashed and solid curves for individual
runs coincide.
\subsubsection{Dependence on the magnetic Prandtl number}
\begin{figure}
\begin{center}
\subfigure{\includegraphics[width=0.49\textwidth]{figures/urmsBrms_t__Pm}}
\subfigure{\includegraphics[width=0.49\textwidth]{figures/Rm_t__Pm}}
%AB: I think one should clip the lower parts which are all the same.
%AB: We should maybe also focus on the time interval when there are plateaus.
%JS: Ok, I will change the plot ranges.
%AB: We'd need their values and plot epsK/epsM vs PrM, which I thought you did.
%JS: I did not add this figure, because the interpretation is not clear yet.
%AB: Also, it would be good to know to what extent epsM and ujxbm agree.
\subfigure{\includegraphics[width=0.49\textwidth]{figures/ujxb_t__Pm}}
\subfigure{\includegraphics[width=0.49\textwidth]{figures/eKeM_t__Pm}}
%JS: added (maybe ujxb_t__Pm.pdf could be replaced by ujxbOeM_t__Pm.pdf?)
\subfigure{\includegraphics[width=0.49\textwidth]{figures/ujxbOeM_t__Pm}}
\caption{
\textit{Top left panel:} Rms velocity over magnetic field strength as a function of
time for Run series D (see table~\ref{table}), in which
the magnetic Prandtl number is varied between $\Pm=0.5$ (Run D05) and
$\Pm=10$ (Run D10), respectively, through variation of the viscosity.
Time is normalized by the inverse laminar dynamo growth rate $\gamma_\mu$
given in equation~(\ref{gamma-max}).
The simulation time during which $k_\mathrm{M}$ is inside the numerical
box, i.e., $k_\mathrm{M}>k_1$, is marked by solid line style.
For $k_\mathrm{M} phase
%There is sufficient data for Phase 2 available for this run to include it the
There is sufficient data for phase 2 available for this run to include it the
$\Pm=0.5$ runs in our analysis.
However, a quantitative study of the low Prandtl number regime is inaccessible
to our current simulations and beyond the scope of this paper.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/urmsBrms_Pm__mean}
\includegraphics[width=0.49\textwidth]{figures/Rmmax_Pm}
%JS: added this here for now:
\includegraphics[width=0.49\textwidth]{figures/eKeM_Pm__mean}
\caption{
\textit{Left panel:} The time averaged value of $\Upsilon$
for different run series.
Filled symbols represent the result from an time averaging of all data points
with $\Upsilon > 0.5~\mathrm{max}(\Upsilon)$
and open symbols averaging is performed for all data with
%JS: full stop and comment on errors
%$\Upsilon > 0.9~\mathrm{max}(\Upsilon)$
$\Upsilon > 0.9~\mathrm{max}(\Upsilon)$.
Errors are of the the order of 10\% for the filled symbols and 5\% for the open
symbols, but not presented in the figure for better visualization.
%JS.
\textit{Right panel:} The maximum magnetic Reynolds number in the different
DNS as a function of $\Pm$.
}
\label{fig_urmsBrms_Pm__mean}
\end{center}
\end{figure}
%JS: added $\Rm^\mathrm{max}$
%The maximum magnetic Reynolds numbers obtained in Runs D are presented in the
The maximum magnetic Reynolds numbers, $\Rm^\mathrm{max}$, obtained in Runs D
are presented in the
%JS: top
%right panel of figure~\ref{fig_urmsBrms_Pm__mean}.
top right panel of figure~\ref{fig_urmsBrms_Pm__mean}.
%JS: $\mathrm{Re}_\mathrm{M,max}$ -> $\Rm^\mathrm{max}$ and comma
%It can be seen that a dependence of $\mathrm{Re}_\mathrm{M,max}$ on $\Pm$ exists
It can be seen that a dependence of $\Rm^\mathrm{max}$ on $\Pm$ exists,
which is caused by the decrease of $u_\mathrm{rms}$ for increasing $\Pm$.
As discussed in the previous section, a change of $\Rm$ in DNS with
$k_\lambda< k_1$ can only be achieved by changing $\mu_0$.
In the bottom panels of figure~\ref{fig_urmsBrms_t__Pm}, we present the time
evolution of two more quantities describing the energy flow from kinetic to
magnetic energy.
We find that both, the work done by the Lorentz force,
$\langle\mathbf{U}\cdot(\mathbf{J}\times\mathbf{B})\rangle$, and the
ratio of viscous over Joule dissipation,
$\epsilon_\mathrm{K}/\epsilon_\mathrm{M}$ depend on $\Pm$.
The latter dissipation ratio is expected to increase with $\Pm$ for large-scale
and small-scale dynamos in classical MHD; see \citet{B14}.
This trend of $\epsilon_\mathrm{K}/\epsilon_\mathrm{M}$ with $\Pm$ is also
observed in our DNS of chiral MHD.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/slope_Rm__05.pdf}
\includegraphics[width=0.49\textwidth]{figures/slope_Rm.pdf}
\caption{
The slope resulting from a fit of the function
$\UpsilonTimeAve = a \Pm^b$ to the data points
presented in the left panel of figure~\ref{fig_urmsBrms_Pm__mean} as a function
of the range of maximum magnetic Reynolds numbers found in simulations
shown in the right panel of figure~\ref{fig_urmsBrms_Pm__mean}.
%JS: comma and $\mathrm{Re}_\mathrm{M,max}$ ->$\Rm^\mathrm{max}$
%The vertical bars indicate the range of $\mathrm{Re}_\mathrm{M,max}$ which
The vertical bars indicate the range of $\Rm^\mathrm{max}$, which
decreases from small to large $\Pm$.
Results for the fitting to all data are shown as black data points and results
for using $\Pm>1$ as blue diamonds.
Horizontal lines indicate the average values of the slopes.
\textit{Left panel:} $\UpsilonTimeAve$
has been obtained using all data with
$\Upsilon > 0.5~\mathrm{max}(\Upsilon)$.
\textit{Right panel:} $\UpsilonTimeAve$
has been obtained using all data with
$\Upsilon > 0.9~\mathrm{max}(\Upsilon)$.
}
\label{fig_slope_Rm}
\end{center}
\end{figure}
We now perform power-law fits to the data with a function
$\UpsilonTimeAve = a \Pm^b$ with the fit parameters
$a$ and $b$.
Both averaging conditions, using all data points for which
$\Upsilon > 0.5~\mathrm{max}(\Upsilon)$ and
%JS: comma
%$\Upsilon > 0.9~\mathrm{max}(\Upsilon)$
$\Upsilon > 0.9~\mathrm{max}(\Upsilon)$,
are considered.
The results for the full range of $\Pm$ as well as for $\Pm>1$ can be
found in table~\ref{table_fits}.
Additionally, we present the slopes $b$ as a function of the corresponding
range of $\mathrm{Re}_\mathrm{M,max}$ in figure~\ref{fig_slope_Rm}.
Fit results to the $\Upsilon > 0.5~\mathrm{max}(\Upsilon)$
condition are shown in the left and the case of
$\Upsilon > 0.9~\mathrm{max}(\Upsilon)$
in the right panel.
The obtained value of $b$ is presented for fitting to all available data as
black data points and for $\Pm>1$ as blue ones.
We do not find a clear dependence of $b$ on the Reynolds number range.
When using data for the full $\Pm$ regime explored in this paper, we find mean
slopes between $b=-0.16$ and $-0.20$.
The slope of the function $\UpsilonTimeAve(\Pm)$ becomes steeper with values
%JS: updated
%between $b=-0.20$ and $b=-0.24$, when fitting only to data with $\Pm>1$.
between $b=-0.19$ and $b=-0.27$, when fitting only to data with $\Pm>1$.
The latter should be a better description for the large Prandtl number regime,
since the scaling of $\UpsilonTimeAve$
%might change in the transition from small to large $\Pm$.
%AB: this is more precises
%JS: ok
might change in the transition from $\Pm<1$ to $\Pm>1$.
\begin{table}
\centering
\caption{Fit results for $\UpsilonTimeAve = a \Pm^b$}
\begin{tabular}{l|ll|ll}
\hline
\hline
Run series & $a$ (all $\Pm$) & $b$ (all $\Pm$) & $a$ ($\Pm>1$)& $b$ ($\Pm>1$)\\
\hline
%JS: updated (simulations have continued running..)
%D (using $\Upsilon >0.5~\mathrm{max}(\Upsilon)$) &0.24 &$-0.17 $&0.25&$-0.19$\\
%D (using $\Upsilon >0.9~\mathrm{max}(\Upsilon)$) &0.27 &$-0.16 $&0.29 &$-0.22$\\
D (using $\Upsilon >0.5~\mathrm{max}(\Upsilon)$) &0.23 &$-0.20$&0.24&$-0.22$\\
D (using $\Upsilon >0.9~\mathrm{max}(\Upsilon)$) &0.27 &$-0.16$&0.29&$-0.22$\\
%JS.
\hline
F (using $\Upsilon >0.5~\mathrm{max}(\Upsilon)$) &0.23 &$-0.20 $&0.26&$-0.27$\\
F (using $\Upsilon >0.9~\mathrm{max}(\Upsilon)$) &0.27 &$-0.17 $&0.29&$-0.20$\\
\hline
G (using $\Upsilon >0.5~\mathrm{max}(\Upsilon)$) &0.25 &$-0.19 $&0.27&$-0.25$\\
G (using $\Upsilon >0.9~\mathrm{max}(\Upsilon)$) &0.27 &$-0.16 $&0.28&$-0.19$\\
\hline
H (using $\Upsilon >0.5~\mathrm{max}(\Upsilon)$) &0.25 &$-0.18$&0.28&$-0.25$\\
H (using $\Upsilon >0.9~\mathrm{max}(\Upsilon)$) &0.27 &$-0.17$&0.29&$-0.20$\\
\hline
\hline
%JS: also updated:
%mean (using $\Upsilon >0.5~\mathrm{max}(\Upsilon)$) &0.24 &$-0.18$ &0.27 &$-0.24$ \\
%mean (using $\Upsilon >0.9~\mathrm{max}(\Upsilon)$) &0.27 &$-0.16$ &0.29 &$-0.20$ \\
mean (using $\Upsilon >0.5~\mathrm{max}(\Upsilon)$) &0.24 &$-0.19$ &0.26 &$-0.25$ \\
mean (using $\Upsilon >0.9~\mathrm{max}(\Upsilon)$) &0.27 &$-0.16$ &0.29 &$-0.20$ \\
%JS.
\hline
\hline
\end{tabular}
\label{table_fits}
\end{table}
% conclusions:
% 1) we find a power law scaling of the ratio u/B with Pm
% 2) we do not find a dependence of the slope on the Rm regime explored in this paper.
%%%%%%%%%%%%%
% SECTION
\section{Chiral magnetically driven turbulence in the early Universe}
\label{sec_EU}
%%%%%%%%%%%%%
The findings from DNS presented above can be leveraged to estimate the
turbulent velocities and the
Reynolds number in the early Universe.
As we have seen in the previous section, the ratio of kinetic to
magnetic energy depends on the magnetic Prandtl number.
Hence, as a first step we estimate the value of $\Pm$ in the early Universe.
Afterwards we estimate $u_\mathrm{rms}$ and the magnetic Reynolds number
for chiral magnetically driven turbulence.
\subsection{Magnetic Prandtl number}
The magnetic Prandtl number has been defined before as the ratio of viscosity
over magnetic diffusivity.
Hence it measures the relative strength of these two transport coefficients.
The derivation of transport coefficients in weakly
coupled high temperature gauge theories has been presented in
\citet{ArnoldEtAl2000} for various matter field content.
For the electric conductivity, \citet{ArnoldEtAl2000} found the leading log term
(converted from natural to cgs units)
\begin{equation}
\sigma_\mathrm{el} = \frac{\kappa_\sigma}{4\pi\, \alpha \log\left((4\pi \alpha)^{-1/2}\right)}
\frac{\kB T}{\hbar}
\end{equation}
with $\kappa_\sigma = 11.9719$ for the largest number of species considered.
The magnetic resistivity in the early Universe follows as
\begin{equation}
\eta(T) = \frac{c^2}{4\pi\, \sigma_\mathrm{el} }
= \frac{\alpha}{\kappa_\sigma} \log\left((4\pi \alpha)^{-1/2}\right) \frac{\hbar c^2}{\kB T}
\approx 7.3\times 10^{-4} \frac{\hbar c^2}{\kB T}
\approx {4.3\times10^{-9}}~T_{100}^{-1}\cm^2\s^{-1}.
\label{eq_etaEU}
\end{equation}
Here $T_{100}=1.2\times10^{15}\K$, so that $\kB T_{100} = 100 \GeV$.
For the shear viscosity, \citet{ArnoldEtAl2000} report
\begin{equation}
\nu_\mathrm{shear} = \frac{\kappa_\mathrm{shear}}{\alpha^2 \log\left(\alpha^{-1}\right)} \frac{(\kB T)^3}{\hbar^2 c^3}
\label{eq_nushear}
\end{equation}
with $\kappa_\mathrm{shear}\approx 147.627$ for the largest number of species
considered.
The kinematic viscosity is obtained by $\nu = \nu_\mathrm{shear}/\meanrho$ with
the mean density in the early Universe being
\begin{equation}
\meanrho = \frac{\pi^2}{30}\,g_*\frac{(\kB T)^4}{\hbar^3c^5}
\approx 7.6\times10^{26}g_{100}T_{100}^4\g\cm^{-3}.
\label{eq_meanrho}
\end{equation}
Dividing equation~(\ref{eq_nushear}) by (\ref{eq_meanrho}) we find
\begin{equation}
\nu = \frac{30 \kappa_\mathrm{shear}}{\pi^2 g_* \alpha^2 \log\left(\alpha^{-1}\right)} \frac{\hbar c^2}{\kB T}
\approx 1.6\times10^{4} \frac{\hbar c^2}{\kB T}
\approx {9.4\times10^{-2}}~T_{100}^{-1}\cm^2\s^{-1} .
\label{eq_nuEU}
\end{equation}
The ratio of equations~(\ref{eq_nuEU}) and (\ref{eq_etaEU}) yields the
magnetic Prandtl number
\begin{equation}
\Pm = \frac{\nu}{\eta}
\approx 2.2\times10^{7}.
\label{eq_PmEU}
\end{equation}
\subsection{Reynolds numbers}
Assuming $u_\mathrm{rms}$ reaches a fraction $\Upsilon(\Pm)$ of the saturation
magnetic field strength $B_\mathrm{sat}$, it is customary to estimate in physical units
\begin{equation}
u_\mathrm{rms} \approx \Upsilon(\Pm) \frac{B_\mathrm{sat}}{\sqrt{4\pi \meanrho}}.
\label{eq_uEU1}
\end{equation}
The mean density in the early Universe is given in equation~(\ref{eq_meanrho})
and the value of $B_\mathrm{sat}$ depends on the chiral nonlinearity parameter
\begin{equation}
\lambda=3 \hbar c\,\left({8\alphaem\over\kB T}\right)^2\approx1.3\times10^{-17}\,T_{100}^{-2}\,\cm\erg^{-1}
\label{eq_lambda_1}
\end{equation}
and the initial chiral chemical potential $\mu_0$.
Since the latter is unknown, we estimate it via the
thermal energy density:
\begin{equation}
\mu_0 = \vartheta~4\alphaem~\frac{\kB T}{\hbar c}
\approx 1.5\times10^{14}~ \vartheta~ T_{100} \cm^{-1}.
\label{eq_mu0_EU}
\end{equation}
Due to the uncertainties in $\mu_0$ we introduce the free parameter $\vartheta$,
allowing us to explore different initial conditions.
The magnetic field produced by chiral dynamos as discussed in this paper reaches
a maximum value of
\begin{equation}
B_\mathrm{sat} = \left(4\pi \frac{\mu_0 k_\lambda}{\lambda}\right)^{1/2}
\approx 6.2\times10^{21}~ \vartheta~ T_{100}^2 \G,
\label{}
\end{equation}
where we use as an estimate of the inverse magnetic correlation length
equation~(\ref{klambda}), that results in
\begin{equation}
k_\lambda \approx 2.6\times10^{11}~ \vartheta~ T_{100} \cm^{-1}.
\label{eq_klamEU}
\end{equation}
Following equation~(\ref{eq_uEU1}), the magnetic field drives turbulence with an
rms velocity of
\begin{equation}
u_\mathrm{rms} =6.1\times 10^7 \Upsilon(\Pm) \vartheta \cm \s^{-1}.
\label{eq_uEU}
\end{equation}
Finally, using equations~(\ref{eq_uEU}), (\ref{eq_klamEU}), and (\ref{eq_etaEU}),
we find the following value for the magnetic
Reynolds number in the early Universe:
\begin{equation}
\Rm \approx \frac{u_\mathrm{rms}}{k_\lambda\eta}
\approx 5.5\times 10^4 ~\Upsilon(\Pm).
\label{eq_uEU}
\end{equation}
We stress that the size of the inertial range is independent of $\vartheta$,
and hence of $\mu_0$, because both, the forcing scale $k_\lambda$ and the
initial energy input scale $k_\mu$, scale linear with $\mu_0$.
Using an extrapolation of $\Upsilon(\Pm) = 0.3\, \Pm^{-0.2}$,
motivated by DNS, see table~\ref{table_fits}, we find $\Rm = \mathcal{O}(10^3)$
when using $\Pm = \mathcal{O}(10^7)$.
%%%%%%%%%%%%%
% SECTION
\section{Conclusions}
%%%%%%%%%%%%%
In this paper we have explored the dependence of chiral magnetically driven
turbulence on initial conditions and magnetic Prandtl numbers.
%AB: this sentence above sounds so general
%JS: one could replace "initial conditions" by "initial chiral asymmetries"
This process is caused by chiral MHD dynamos---an instability that originates
from an asymmetry between left- and right-handed fermions and which shows
mathematical similarities to classical $\alpha$ dynamos.
However, while classical $\alpha$ dynamos require an energy input by turbulence,
chiral dynamos can operate also in laminar flows, where they can eventually
induce turbulent motions via the Lorentz force.
By solving the set of chiral MHD equations in numerical simulations, we have
gained new insights into the properties of chiral magnetically driven
turbulence.
Our main findings from DNS may be summarized as follows:
\begin{itemize}
\item{For a large range of parameters, it has been shown that the chiral
magnetic instability generates turbulence in simulations.
In this paper we have focused on the case of small chiral nonlinearity
parameters $\lambda_\mu$, defined in equation~(\ref{eq_lambdamu}), where turbulence
becomes strong enough to affect the evolution of the magnetic field.
}
\item{A central parameter explored in our simulations is $\Upsilon$, the square root
of which is the ratio of kinetic over magnetic energy; see definition in
equation~(\ref{eq_Upsilon}).
Due to the nonlinearity of the Lorentz force, the velocity field grows at a
rate that is twice the one of the magnetic field strength.
As a result, $\Upsilon$ increases initially as a power law.
Once there is a back reaction of the velocity field on the magnetic field,
$\Upsilon$ stays approximately constant, see e.g.\ the left panel of
figure~\ref{fig_urmsBrms_t}.
}
\item{For magnetic Prandtl numbers $\Pm=1$,
the time average of $\Upsilon$, taken after its exponential growth phase
and referred to here as $\UpsilonTimeAve$,
has been determined to be between $0.24$ and $0.27$.
This value seems to be independent of the initial chiral asymmetry.
%JS: removed \\
%} \\
}
\item{For $\Pm>1$,
$\Upsilon$ decreases.
With our DNS we find a scaling of approximately
%JS: removed \\
%$\Upsilon(\Pm) = 0.3\, \Pm^{-0.2}$.} \\
$\Upsilon(\Pm) = 0.3\, \Pm^{-0.2}$.}
\item{We do not find a change of the function $\Upsilon(\Pm)$ for different
regimes of $\Rm$, however, only a small variation of $\Rm$ has been considered
and this might change when increasing the statistics and the extending the
range of $\Rm$.}
\end{itemize}
A chiral dynamo instability and hence chiral magnetically driven turbulence
can only occur in extreme astrophysical environments, because a high
temperature is required for the existence of a chiral asymmetry.
At low energies chiral flipping reactions destroy any difference in number
density between left- and right handed fermions.
As an astrophysically relevant regime, we have discussed the plasma of the
early Universe.
We have presented an approximation of the magnetic Prandtl number and used it,
together with our findings from DNS, to estimate the magnetic Reynolds number.
We find that a value of $\Rm \mathcal{O}(10^3)$ can be expected for
chiral magnetically driven turbulence, if the chiral asymmetry is generated
by thermal processes.
\section*{Acknowledgements}
This project has received funding from the
European Union's Horizon 2020 research and
innovation program under the Marie Sk{\l}odowska-Curie grant
No.\ 665667.
%We thank for support by the \'Ecole polytechnique f\'ed\'erale de Lausanne, Nordita,
%AB: uppercase?
%JS: apparently it is lower case (also for polytechnique):
%We thank for support by the \'Ecole Polytechnique F\'ed\'erale de Lausanne, Nordita,
We thank for support by the \'Ecole polytechnique f\'ed\'erale de Lausanne, Nordita,
and the University of Colorado through
the George Ellery Hale visiting faculty appointment.
Support through the NSF Astrophysics and Astronomy Grant Program (grants 1615100 \&
1615940), the Research Council of Norway (FRINATEK grant 231444),
and the European Research Council (grant number 694896) are
gratefully acknowledged.
Simulations presented in this work have been performed
with computing resources
provided by the Swedish National Allocations Committee at the Center for
Parallel Computers at the Royal Institute of Technology in Stockholm.
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