%Especially the observational indications for intergalactic magnetic fields
%on scales of the order of a $\Mpc$ are puzzling.
%Here, constrains on the upper limit of primordial magnetic fields exist,
%for instance from observations of Faraday rotation \citep{BBO1999,PTU15}
%and the cosmic microwave background \citep{JKO2000,TSS14}.
%Recently, a lower limit for intergalactic magnetic fields has
%been proposed using Fermi observations of TeV blazars.
%\citet{NV10} report a minimum field strength of $3\times10^{16} \G$ at a
%scale of $1 \Mpc$, while while \citet{DCRFCL11} claim a lower limit of
%$5 \times10^{19} \G$ of the intergalactic magnetic field.
%While these lower bounds are frequently interpreted as remains of primordial
%magnetic fields we note, that also alternative interpretation of the Fermi data
%have been proposed \citep{SKS13}.
%One scenario for the primordial origin of these large-scale fields is the
%generation of
%strong, but small-scale, seed fields and the subsequent transport to larger
%scales in decaying magnetohydrodynamics (MHD) turbulence
%\citep{BM99PhRvL, KTBN13,BKMPTV17}.
%Cosmological seed fields, formed shorty after the Big Bang, are
%a highly debated topic in modern cosmology, see e.g.\
%\citet{GrassoRubinstein2001,KZ08,Subramanian16}.
%In the last years, a new effect related to the different handedness
%of fermions has been revealed that influences the dynamics of the
%relativistic primordial plasma.
%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,Son:2009tf,Fukushima:08}
%and can lead to a magnetic instability \citep{JS97}.
%The evolution of non-uniform chiral chemical potential has been studied in
%\citet{BFR12,BFR15} who found that a chiral asymmetry can, in principle,
%survive down to energies of the order of 10 MeV
%due to coupling with an effective axion field.
%These studies initiated various investigations related to chiral MHD turbulence
%\citep{Pavlovic:2016gac}
%and its role in the early Universe \citep{TVV12,DvornikovSemikoz2017}
%as well as in neutron stars \citep{DvornikovSemikoz2015,SiglLeite2016}.
%%turbulence and magnetic fields in cosmology & bounds on magnetic fields
%Turbulence and magnetic fields are closely coupled and central features of many
%geophysical and astrophysical flows.
%However, the origin of cosmic magnetic fields remains unsolved.
%Especially the observational indications for intergalactic magnetic
%fields on scales of the order of a $\Mpc$ are puzzling.
%Here, constrains on the upper limit of primordial magnetic fields exist,
%for instance from observations of Faraday rotation \citep{BBO1999,PTU15}
%and the cosmic microwave background \citep{JKO2000,TSS14}.
%Recently, a lower limit for intergalactic magnetic fields has
%been proposed using Fermi observations of TeV blazars.
%\citet{NV10} report a minimum field strength of $3\times10^{16} \G$ at a
%scale of $1 \Mpc$, while while \citet{DCRFCL11} claim a lower limit of
%$5 \times10^{19} \G$ of the intergalactic magnetic field.
%While these lower bounds are frequently interpreted as remains of primordial
%magnetic fields we note, that also alternative interpretation of the Fermi data
%have been proposed \citep{SKS13}.
% understanding the EU: a new ingredient at high energies
%One scenario for the primordial origin of these large-scale fields is the
%generation of
%strong, but small-scale, seed fields and the subsequent transport to larger
%scales in decaying magnetohydrodynamics (MHD) turbulence
%\citep{BM99PhRvL, KTBN13,BKMPTV17}.
%Cosmological seed fields, formed shorty after the Big Bang, are
%a highly debated topic in modern cosmology, see e.g.\
%\citet{GrassoRubinstein2001,KZ08,Subramanian16}.
%In the last years, a new effect related to the different handedness
%of fermions has been revealed that influences the dynamics of the
%relativistic primordial plasma.
%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,Son:2009tf,Fukushima:08}
%and can lead to a magnetic instability \citep{JS97}.
%The evolution of non-uniform chiral chemical potential has been studied in
%\citet{BFR12,BFR15} who found that a chiral asymmetry can, in principle,
%survive down to energies of the order of 10 MeV
%due to coupling with an effective axion field.
%These studies initiated various investigations related to chiral MHD turbulence
%\citep{Pavlovic:2016gac}
%and its role in the early Universe \citep{TVV12,DvornikovSemikoz2017}
%as well as in neutron stars \citep{DvornikovSemikoz2015,SiglLeite2016}.
% state of the art
%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}.
% this paper
%In this manuscript we explore the turbulent velocity field which is generated
%indirectly by chiral MHD dynamos.
%Namely, the Lorentz force drives turbulence efficiently on the correlation scale
%of the magnetic field.
%We study the velocity field in
%direct numerical simulations (DNS), where the full set of the chiral MHD
%equations is solved.
%A key parameter measured in DNS is the ratio of kinetic to magnetic energy at
%dynamo saturation.
%Using the findings from DNS we estimate the Reynolds number in the relativistic
%plasma of the early Universe.
%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}).
%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.