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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}
\articletype{Proceedings}
%\title{{\textit{Generation of turbulence by mean-field chiral MHD dynamos in the early Universe}}}
%AB: make title more focussed and different from earlier work
\title{{\textit{Magnetic Prandtl number dependence of turbulence %generation in chiral MHD}}}
%JS: added the word "dynamos" to the title
%AB: I was hoping to have the title fill one line and thought that dynamo is
%AB: implied by the word "generated"
%JS: I slightly prefer my version of the title and don't care very much about the 2-line title.
%generated by chiral MHD dynamos}}}
%AB: It seems that the final layout of GAFD will be different from what we see here,
%AB: but it a compact title might still be worth considering. How about this:
from 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: 1.32 $ $}
\maketitle
\begin{abstract}
%A chiral asymmetry in the early Universe can give rise to a new type of dynamos.
%Magnetic field is amplified exponentially.
%Lorentz force in the Navier-Stokes equation drives turbulence.
%We anaylse high-resolution numerical simulations.
%For a chiral chemical potential of the order of the photon density,
%we estimate Reynolds numbers.
\begin{keywords}
Relativistic magnetohydrodynamics; Turbulence; Early Universe; Chiral dynamos
\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 powered 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, like the viscous and magnetic diffusion rates
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 magnetic fields
has been more and more constrained.
The puzzling observational indications for large-scale intergalactic magnetic fields \citep{NV10,DCRFCL11},
the possible remains of primordial fields, 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 helicity
\citep{BEO96,BM99PhRvL, KTBN13,BKMPTV17} or without \citep{BKT15,Zra14}.
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
of fermions \citep{JS97}.
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}.
Especially the studies of 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
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 a new turbulent $\alpha$ effect that is not related to a 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 indeed be driven effectively by the Lorentz force of the magnetic field
amplified by chiral MHD dynamos.
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
$\Pm\equiv \nu/\eta$, where $\nu$ is the kinematic viscosity and $\eta$ is the magnetic
diffusivity.
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 one reference run in detail.
In Section~\ref{subsec_urmsBrms} we analyze the ratio
$u_\mathrm{rms}/B_\mathrm{rms}$ 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/(\kB T)^2,
\end{eqnarray}
which is related to the difference of the number densities of left- and
right-chiral fermions, $n_\mathrm{L}$ and $n_\mathrm{R}$, respectively.
Here $\alphaem \approx 1/137$ is the fine structure constant,
$T$ 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 the current
\begin{eqnarray}
\label{eq_CME}
\JJ_{\rm CME} = \frac{\alphaem}{\pi \hbar} \mu_5 \BB.
\end{eqnarray}
This standard model physics effect results in
an additional term in the Maxwell equations.
Based on these modified Maxwell equations,
\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 $\BB^2/2$ (without the $4\pi$ factor), and
%AB: maybe this way is more useful. Could also say both, but it's obvious that there is no 4pi.
%JS: ok
%AB: The superduper potential is also just S_P, without 4pi.
%JS: what does "S_P" stand for?
%AB: S_P = *s*uperduper *p*otential
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
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}
Further, $\eta$ is the microscopic magnetic diffusivity,
$D_5$ 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
speed.
The last term of equation~(\ref{mu-DNS}) describing the chiral flipping reactions
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.
\end{eqnarray}
${\rm Ma}_\mu$, measuring the relevance of the chiral term in the induction
equation~(\ref{ind-DNS}), determines the dynamo growth rate.
The nonlinear back reaction from the magnetic field on $\mu$ is determined by
$\lambda_\mu$ which affects the strength of the saturation magnetic field and
thereby the strength of turbulence.
In this paper we consider only cases with $\lambda_\mu \ll 1$, i.e., when turbulence
is excited efficiently due to strong magnetic fields produced by chiral dynamos.
\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 $\mu$ in chiral MHD and the
$\alpha$ in MFE.
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 mean-field electrodynamics, 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 and, assuming it to be constant in time, we can write
\citep[see equation 18 of][]{BB02}
\begin{equation}
\frac{\partial \alpha}{\partial t} =
\lambda_{\rm MFE} \, \eta_{\rm t} \left[\eta_{\rm t} {\BB} {\bm \cdot} (\nab \times {\BB})
- \alpha {\BB}^2\right]
-\Gamma_{\rm MFE}\, (\alpha-\alpha_{\rm K}),
\label{eq_alpha}
\end{equation}
where $\eta_{\rm t}$ is the turbulent magnetic diffusivity.
In MFE, the coupling coefficient is given by
$\lambda_{\rm MFE}=2\eta_{\rm t}k_{\rm f}^2/B_{\rm eq}^2$ 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 $\alpha$
effect, that in turn results in a slow-down of the decay.
\subsection{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.
Based on the evolution of the magnetic field, we can distinguish
three different phases: \\
\textbf{Phase 1}: laminar small-scale chiral dynamo instability with the
maximum growth rate given in equation~(\ref{gamma-max}); \\
\textbf{Phase 2}: turbulent large-scale dynamo instability, caused by
chiral magnetically produced turbulence, with the growth rate given in
equation~(\ref{gammamax_turb}); \\
\textbf{Phase 3}: saturation of the magnetic field and reduction of $\mu$
according to the conservation law~(\ref{CL}). \\
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.
The time evolution of $\mu$ is presented as the black line and the one of $\BB$
as the blue line.
Further, the initial value of $\mu$, $\mu_0$, and the saturation value of
$\BB$, $(\mu_0/(\lambda \xi_\mathrm{M}))^{1/2}$, are indicated as
horizontal dotted lines.
The individual phases of the evolution are marked by black dashed-dotted
lines and discussed in Section~\ref{sec_overview}.
\textit{Right panel:} Evolution of magnetic energy spectra with an initial instability on the
scale $k_\mu$ (black lines).
The peak of the spectrum moves to smaller wavenumbers for
$E_\mathrm{M}>C_\mu \rho \mu_0\eta^2$ and reaches it's maximum value at
$E_\mathrm{M}=C_\lambda \mu_0/\lambda$ at $k_\lambda$, leaving behind a
$k^{-2}$ spectrum.
The blue lines show a case with a larger $\mu_0$ than for the black lines and
the red lines a case with a smaller $\lambda$ 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}
In \textbf{Phase 1} the velocity field can be considered vanishing and a laminar
dynamo operates.
The growth rate found from the linearized equation~(\ref{ind-DNS}) is
laminar $v_\mu^2$ dynamo:
\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
\begin{eqnarray}
\gamma^{\rm max}_\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}
In \textbf{Phase 2}, turbulence produced by the Lorentz force can no longer be
neglected and affects the evolution of the magnetic field.
It has been shown in \citet{BSRKBFRK17} that the magnetic spectrum reaches
a value of
\begin{equation}
E_\mathrm{M}^{1,2} = C_\mu\meanrho \mu_0 \eta^2.
\end{equation}
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}
Now equation~(\ref{ind-DNS}) can be 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}
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,
the turbulent magnetic diffusivity $\eta_\mathrm{T}$ reduces $\gamma$.
In the mean-field equation, the chiral $v_\mu$ effect is replaced by the
mean chiral $v_\mu$ 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 $v_\mu$ effect.
It can be shown that the growth rate of the large-scale dynamo instability is
\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}).
\subsubsection{Saturation magnetic field}
Saturation of the chiral dynamo, \textbf{Phase 3}, is determined by the
conservation law of 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
in $\BB = {\bm \nabla} {\bm \times} {\bm A}$, ${\bm \EE}=
- c^{-1} \, \{ {\bm \UU} {\bm \times} {\BB} + \eta \, \big(\mu {\BB} -
{\bm \nabla} {\bm \times} {\BB} \big) \}$ 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 \langle \AAA \cdot \BB\rangle + \bar\mu = \mu_0 = \mathrm{const},
\end{equation}
where $\bar\mu$ is spatially averaged value of the chemical potential and
$ \langle \AAA \cdot \BB\rangle$ is the magnetic helicity.
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}
where $C_\mu\approx 16$ and $C_\lambda\approx 1$, and $k_\mu$ as given
in 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}
Here, $E_{\rm M}(k,t)$ is normalized such that
%$\mathcal{E}_{\rm M} = \int E_{\rm M}(k)~\mathrm{d}k=\bra{\BB^2}/2$
%AB: rewritten, because we haven't defined \mathcal{E}_{\rm M} yet
%AB: added $k_\lambda of, and added ~
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.
Form 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.
\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
$u_\mathrm{rms}/B_\mathrm{rms}$ stays approximately constant and
%AB: this ratio appears all the time and the notation is a bit awkward.
%AB: We could define the kinetic to magnetic energy ratio, or even
%AB: the inverse of it, others call it beta or betaˆ2.
%JS: Yes, it would be worth introducing a shortcut for this ratio. I don't like
%JS: beta, because it reminds me of the plasma beta. Actually, I started using
%JS: an epsilon, which will become also confusion once I add epsilon_M and
%JS: epsilon_K plots. What about \beta_{MK} for Magnetic and Kinetic?
%JS: I reformulate everything in terms of energies, 1/2 rho u_rms^2 and
%JS: B_rms^2/2, so
%JS: \beta_MK = (1/2 rho u_rms^2) / (B_rms^2/2)
%AB: We could also work with {\cal E}_{\rm K}/{\cal E}_{\rm M}, which would be very
%AB: clear, especially when writing \langle{\cal E}_{\rm K}/{\cal E}_{\rm M}\rangle_t
%AB: for the time average (which is right this way, right), but this would require
%AB: you working now with a squared quantity, but this may be better, or
%AB: \langle{\cal E}^{\rm rms}_{\rm K}/{\cal E}_{\rm M}^{\rm rms}\rangle_t.
%AB: Maybe introduce it all as a few macros, maybe first just with beta.
%AB: The MK is a bit funny too. Or how about \Upsilon or something similar?
%JS: I like \Upsilon, because I don't associate anything with it. By the way,
%JS: \beta is not good because it is used for two other things relevant to
%JS: our studies: 1) plasma beta, 2) beta = 1/(k_B T) in statistical physics.
%JS:
%JS: When using the ratio {\cal E}_{\rm K}/{\cal E}_{\rm M}, I would need to
%JS: use the var files for calculating __. This would require some
%JS: time, and I'm not sure if we have enough time left. Also I need to check
%JS: if I have enough var files for all runs to do a good analysis.
%JS: By {\cal E}^{\rm rms}_{\rm K} and {\cal E}_{\rm M}^{\rm rms}, do you mean
%JS: {\cal E}^{\rm rms}_{\rm K} = 1/2 rho u_rms^2
%JS: and
%JS: {\cal E}^{\rm rms}_{\rm M} = 1/2 rho B_rms^2 ?
%JS: This would be a straight forward change in my plotting and fitting scripts.
%AB: I didn't mean to cause any extra work, just a new name, so \Upsilon is ok
decreases as soon as the peak of the magnetic energy spectrum reaches the
box wavenumber $k_1$.
Afterwards, turbulence is not driven by the Lorentz force anymore and
the kinetic energy dissipates.
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 $u_\mathrm{rms}/B_\mathrm{rms}$ 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, expect
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 slightly 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 $u_\mathrm{rms}/B_\mathrm{rms}$ 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
$u_\mathrm{rms}/B_\mathrm{rms} \approx 0.2-0.3$.
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.
This line style of the curve 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 is completely inside the box, and
hence kinetic energy is not dissipated 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.
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 $u_\mathrm{rms}/B_\mathrm{rms}$ remains
approximately constant for times larger than $\approx 12~\gamma_\mu^{-1}$.
\begin{figure}
\begin{center}
\subfigure{\includegraphics[width=0.49\textwidth]{figures/urmsBrms_t__mu}}
\subfigure{\includegraphics[width=0.49\textwidth]{figures/urmsBrms_mu}}
\caption{
\textit{Left panel:} Rms velocity over magnetic field strength 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
nonlinearity parameter; see Table~\ref{table}.
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} abscissa
$u_\mathrm{rms}/B_\mathrm{rms}$ as a function of $\mu_0/k_1$ (lower abscissa,
%black data points) and a function of $\mathrm{Ma}_\mu$ (upper x axes, blue
%AB: x axes => abscissa
black data points) and a function of $\mathrm{Ma}_\mu$ (upper abscissa, blue
data points) for all runs with $\Pm=1$.
For the results shown as filled dots, the average has been performed for the
%time interval for which $u_\mathrm{rms}/B_\mathrm{rms}$ is at least 50 \% of
%AB: no space
time interval for which $u_\mathrm{rms}/B_\mathrm{rms}$ is at least 50\% of
%it's maximum value.
%AB: isn't it "its"? "It's" always means "it is", which is not right.
its maximum value.
For open dots, the time average is taken for all
$u_\mathrm{rms}/B_\mathrm{rms}> 0.9~\mathrm{max}(u_\mathrm{rms}/B_\mathrm{rms})$.
%JS: this text belongs to the figure below:
%AB: OK to delete commented text then.
%\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_urmsBrms_t}
\end{center}
\end{figure}
In the right panel of Figure~\ref{fig_urmsBrms_t}, we show the mean ratio
$\overline{u_\mathrm{rms}/B_\mathrm{rms}}$ 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
$u_\mathrm{rms}/B_\mathrm{rms}$ larger than 50\% of its maximum value.
The open dots are obtained by using all values for which
$u_\mathrm{rms}/B_\mathrm{rms}> 0.9~\mathrm{max}(u_\mathrm{rms}/B_\mathrm{rms})$,
which obviously results in a larger average value.
There is no significant dependence of $\overline{u_\mathrm{rms}/B_\mathrm{rms}}$
on the values of $\mu_0$ and $\mathrm{Ma}_\mu$ for the parameter space explored
here.
\begin{figure}
\begin{center}
\subfigure{\includegraphics[width=0.49\textwidth]{figures/Rm_t__mu}}
\subfigure{\includegraphics[width=0.49\textwidth]{figures/Rmmax_mu}}
\caption{
%JS: this text needs to go up:
%AB: OK to delete commented text then.
%\textit{Left panel:} The time averaged ratio
%$u_\mathrm{rms}/B_\mathrm{rms}$ as a function of $\mu_0/k_1$ (lower x axes,
%black data points) and a function of $\mathrm{Ma}_\mu$ (upper x axes, blue
%data points) for all runs with $\Pm=1$.
%For the results shown as filled dots, the average has been performed for the
%time interval for which $u_\mathrm{rms}/B_\mathrm{rms}$ is at least 50 \% of
%it's maximum value.
%For open dots, the time average is taken for all
%$u_\mathrm{rms}/B_\mathrm{rms}> 0.9~\mathrm{max}(u_\mathrm{rms}/B_\mathrm{rms})$.
%JS: Right -> left
%\textit{Right panel:} The magnetic Reynolds number $\Rm$ as a function of time
\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.
%JS: copied from above:
%AB: OK
\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 $\overline{u_\mathrm{rms}/B_\mathrm{rms}}$ 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 in 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
$u_\mathrm{rms}/(\eta k_1)$.
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.
% main conclusions:
% in simulations $u$ dissipates once, kM=k1
% $u_\mathrm{rms}/B_\mathrm{rms} \approx 0.2-0.3$, independent of $\mathrm{Ma}_\mu$ and $\lambda_\mu$
% in simulations $urms$ and $kM$ are limited by the size of the box
% in simulations changing $mu_0$ changes urms and and hence Rm; eta does not affect Rm.
\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}}
\subfigure{\includegraphics[width=0.49\textwidth]{figures/ujxb_t__Pm}}
\subfigure{\includegraphics[width=0.49\textwidth]{figures/eKeM_t__Pm}}
%% \subfigure{\includegraphics[width=0.49\textwidth]{figures/urmsBrms_t__Pm_eta1e-4}}
%% \subfigure{\includegraphics[width=0.49\textwidth]{figures/Rm_t__Pm_eta1e-4}}
%% \subfigure{\includegraphics[width=0.49\textwidth]{figures/urmsBrms_t__Pm_eta2e-4}}
%% \subfigure{\includegraphics[width=0.49\textwidth]{figures/Rm_t__Pm_eta2e-4}}
\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 $k_\mathrm{M} 0.5~\mathrm{max}(u_\mathrm{rms}/B_\mathrm{rms})$
and open symbols averaging is performed for all data with
$u_\mathrm{rms}/B_\mathrm{rms} > 0.9~\mathrm{max}(u_\mathrm{rms}/B_\mathrm{rms})$
\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}
The maximum magnetic Reynolds numbers obtained in Runs D are presented in the
right panel of Figure~\ref{fig_urmsBrms_Pm__mean}.
It can be seen that a dependence of $\mathrm{Re}_\mathrm{M,max}$ 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 $\mu_0$.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/slope_Rm__05}
\includegraphics[width=0.49\textwidth]{figures/slope_Rm}
\caption{
The slope resulting from a fit of the function
$\overline{u_\mathrm{rms}/B_\mathrm{rms}} = a \Pm^b$ 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}.
The vertical bars indicate the range of $\mathrm{Re}_\mathrm{M,max}$ 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.
\textit{Left panel:} $\overline{u_\mathrm{rms}/B_\mathrm{rms}}$
has been obtained using all data with
$u_\mathrm{rms}/B_\mathrm{rms} > 0.5~\mathrm{max}(u_\mathrm{rms}/B_\mathrm{rms})$.
\textit{Right panel:} $\overline{u_\mathrm{rms}/B_\mathrm{rms}}$
has been obtained using all data with
$u_\mathrm{rms}/B_\mathrm{rms} > 0.8~\mathrm{max}(u_\mathrm{rms}/B_\mathrm{rms})$.
}
\label{fig_slope_Rm}
\end{center}
\end{figure}
We now perform power-law fits to the data with a function
$\overline{u_\mathrm{rms}/B_\mathrm{rms}} = a \Pm^b$ with the fit parameters
$a$ and $b$.
Both averaging conditions, using all data points for which
$u_\mathrm{rms}/B_\mathrm{rms} > 0.5~\mathrm{max}(u_\mathrm{rms}/B_\mathrm{rms})$
and
$u_\mathrm{rms}/B_\mathrm{rms} > 0.9~\mathrm{max}(u_\mathrm{rms}/B_\mathrm{rms})$
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 $u_\mathrm{rms}/B_\mathrm{rms} > 0.5~\mathrm{max}(u_\mathrm{rms}/B_\mathrm{rms})$
condition are shown in the left and the case of
$u_\mathrm{rms}/B_\mathrm{rms} > 0.9~\mathrm{max}(u_\mathrm{rms}/B_\mathrm{rms})$
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.
For all cases considered here, $b$ has a value between $\approx -0.27$ and
$\approx -0.16$.
The value of $b$ becomes more negative, 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 $\overline{u_\mathrm{rms}/B_\mathrm{rms}}$
might change in the transition from small to large $\Pm$.
\begin{table}
\centering
%AB: here it would be very handy to have defined a parameter beta
%JS: I will update everything once we agreed on the definition and the name of
%JS: this parameter. beta might be confusing because it makes people think of
%JS: thermal pressure.
%AB: ok
\caption{Fit results for $\overline{u_\mathrm{rms}/B_\mathrm{rms}} = a \Pm^b$}
\begin{tabular}{l|ll|ll}
\hline
\hline
Run series & $a$ (all $\Pm$) & $b$ (all $\Pm$) & $a$ ($\Pm>1$) & $b$ ($\Pm>1$) \\
\hline
D (using $u_\mathrm{rms}/B_\mathrm{rms} >0.5~\mathrm{max}(u_\mathrm{rms}/B_\mathrm{rms})$) &0.24 &$-0.17 $&0.247&$-0.188$\\
D (using $u_\mathrm{rms}/B_\mathrm{rms} >0.9~\mathrm{max}(u_\mathrm{rms}/B_\mathrm{rms})$) &0.266&$-0.16 $&0.29 &$-0.215$\\
\hline
F (using $u_\mathrm{rms}/B_\mathrm{rms} >0.5~\mathrm{max}(u_\mathrm{rms}/B_\mathrm{rms})$) &0.234&$-0.199$&0.263&$-0.268$\\
F (using $u_\mathrm{rms}/B_\mathrm{rms} >0.9~\mathrm{max}(u_\mathrm{rms}/B_\mathrm{rms})$) &0.27 &$-0.167$&0.286&$-0.201$\\
\hline
G (using $u_\mathrm{rms}/B_\mathrm{rms} >0.5~\mathrm{max}(u_\mathrm{rms}/B_\mathrm{rms})$) &0.246&$-0.189$&0.273&$-0.254$\\
G (using $u_\mathrm{rms}/B_\mathrm{rms} >0.9~\mathrm{max}(u_\mathrm{rms}/B_\mathrm{rms})$) &0.266&$-0.158$&0.279&$-0.188$\\
\hline
H (using $u_\mathrm{rms}/B_\mathrm{rms} >0.5~\mathrm{max}(u_\mathrm{rms}/B_\mathrm{rms})$) &0.252&$-0.178$&0.278&$-0.247$\\
H (using $u_\mathrm{rms}/B_\mathrm{rms} >0.9~\mathrm{max}(u_\mathrm{rms}/B_\mathrm{rms})$) &0.269&$-0.165$&0.285&$-0.203$\\
\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 $\epsilon(\Pm)$ of the saturation
%magnetic field strength $B_\mathrm{sat}$, we estimate in physical units
%AB: we -> it is customary to. Maybe we should refer here to earlier such work, e.g
%AB: Dvornikov, I thought, but maybe others
magnetic field strength $B_\mathrm{sat}$, it is customary to estimate in physical units
\begin{equation}
u_\mathrm{rms} \approx \epsilon(\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 \epsilon(\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 ~\epsilon(\Pm).
%AB: is all this still consistent with our ApJL paper?
\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 $\epsilon(\Pm) = 0.3\, \Pm^{-0.2}$,
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}
%%%%%%%%%%%%%
%AB: could start with something like this
%JS: This will be extended soon.
Our main findings may be summarized as follows.
%AB.
\begin{itemize}
\item{Chiral MHD dynamos drive turbulence via the Lorentz force.} \\
\item{For $\Pm=1$, the ratio
$\epsilon= u_\mathrm{rms}/B_\mathrm{rms}\approx 0.2-0.3$,
where $B_\mathrm{rms}$ is measured in terms of
the Alfv\'en velocity.} \\
\item{For $\Pm>1$,
$\epsilon$ decreases. With our DNS we find a scaling of
$\epsilon(\Pm) = 0.3\, \Pm^{-0.2}$.} \\
\item{We do not find a change of the function $\epsilon(\Pm)$ 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 $\Rm$.}\\
\item{The value of $\Pm$ in the early Universe has been estimated to be of the
order of $10^7$.
Based on finding from DNS, a magnetic Reynolds number of
$\Rm \mathcal{O}(10^3)$ can be expected for chiral magnetically driven
turbulence. }
\end{itemize}
%AB: do you plan to say more here?
%JS: Yes, of course. I wanted to write the conclusions (and the abstract)
%JS: after the main part has been finalized.
%AB: ok
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