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\newcommand{\AAA}{\bm{A}}
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\newcommand{\mnras}{Mon.\ Not.\ Roy.\ Astron.\ Soc.}
\newcommand{\prd}{Phys.\ Rev. D}
\newcommand{\pre}{Phys.\ Rev. E}
\newcommand{\apj}{Astrophys.\ J.}
\newcommand{\apjl}{Astrophys.\ J.\ Lett.}
\newcommand{\physrep}{Phys.\ Rep.}
\newcommand{\aap}{Astro.\ and Astrophys.}




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% \newcommand{\initasymRnin}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/chiral_dynamo_modes/128_2D_eta1e-3_diffmu5hyper2245e-7_lambda51e4_mu50gaussian_amplmu570/postproc/}
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\newcommand{\initconRnin}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/128_2D_eta5e-4_lambda51e4_diffmu5hyper122e-7_B0Belkx1ampl1e-9_muSsin300kxmuS6_Bex_fidis/postproc/}
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\title{Chiral dynamos sourced by chiral magnetic waves}


\begin{document}
\noindent



\maketitle

\tableofcontents

%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%


% Outline:
% \begin{itemize}
% \item{Section~\ref{sec_background}: Theoretical background and setup
% of the direct numerical simulations.}
% \item{Section~\ref{sec_initialmu5}: Analysis of influence of $\mu_5$ power
% spectrum on chiral dynamo amplification of
% the magnetic field.
% Tested $\mu_5$ initial conditions:
% Constant (has been considered in all previous studies), sine waves and sum of
% multiple sine waves, and Gaussian noise.}
% \item{Section~\ref{sec_CSE}: Generation of $\mu_5$ from gradients in $\mu$ via the
% chiral separation effect (CSE).
% In most of our tested scenarios, $\mu_5$ is generated with
% an extended power spectrum.
% This leads to a growth of the magnetic field that is very difficult to
% control in DNS. }
% \item{Section~\ref{sec_CSE}: If a weak external magnetic field is applied and the initial energy
% spectra of the magnetic field and $\mu$ are located in single modes, the
% $\mu_5$ power spectrum also is localized on a single wavenumber.
% In this case, we observe an exponential increase of the magnetic
% field strength over many orders of magnitude.}
% \item{Section~\ref{sec_CSE}: Larger Reynolds numbers:...}
% \item{Section~\ref{sec_conclusion}: Conclusions.}
% \end{itemize}



Central question:\\
\textit{Can chiral magnetic waves generate enough $\mu_5$
to self-consistently drive chiral dynamos?}
\\


Plan for this paper:

\begin{itemize}
\item{
Analysis of the influence of the initial $\mu_5$ power spectrum on the chiral
dynamo instability (Section~\ref{sec_initialmu5}).
How is the chiral dynamo affected if $\mu_5$ is not constant, but has an extended
power spectrum?
To answer this question, we perform DNS without the chiral separation effect (CSE).
}
\item{
In Section~\ref{sec_CSE_spectra} we explicitly include the CSE and
consider scenarios without an initial chiral asymmetry.
However, a generation of $\mu_5$ results from gradients
in $\mu$.
In most of our tested scenarios, a $\mu_5$ is generated with
an extended power spectrum.
We will identify initial conditions that generate $\mu_5$
on individual modes only.
}
\item{
In Section~\ref{sec_CSE_singlemodes} we focus on scenarios where
the $\mu_5$ power spectrum is generated on a single wavenumber
(This can be achieved if a weak external magnetic field is applied and the initial energy
spectra of the magnetic field and $\mu$ are located in single modes.).
We will analyze the exponential increase of the magnetic
field strength over many orders of magnitude and the subsequent
steady state phase.
}
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Physical background and methods}
\label{sec_background}
%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Chiral magnetic effect}
%%%%%%%%%%%%%%%%%%%%%%%%%%




%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Numerical setup}
%%%%%%%%%%%%%%%%%%%%%%%%%%


%general
We use the \textsc{Pencil Code}\footnote{\textit{http://pencil-code.nordita.org/}} to
solve equations~(\ref{ind-DNS})--(\ref{mu-DNS})
in a three-dimensional periodic domain of size $L^3 = (2\pi)^3$ with a resolution of
up to $224^3$.
This code employs a third-order accurate time-stepping method of \cite{Wil80}
and sixth-order explicit finite differences in space \citep{BD02,Bra03}.

%normaliztion and code units
The smallest wavenumber covered in the numerical domain is $k_1 = 2\pi/L = 1$ which we use for
normalization of length scales.
All velocities are normalized to the sound speed $\cs = 1$ and further the
mean fluid density to $\meanrho = 1$.
Further, the magnetic Prandtl number is $1$,
i.e.\ the magnetic diffusivity equals the viscosity.
Time is normalized by the diffusion time $t_\eta = (\eta k_1^2)^{-1}$.

An overview of all runs presented in this paper is given
in Tab.~\ref{tab_DNSoverview_noCSE} and Tab.~\ref{tab_DNSoverview_CSE} 
in Appendix~\ref{sec_appendix}.







%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Generation of $\mu_5$ via the chiral separation effect and subsequent chiral dynamo}
\label{sec_CSE}
%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%


In this section we are extending the system of
equations~(\ref{ind-DNS})--(\ref{rho-DNS}) plus (\ref{mu-DNS})
to include also the chiral separation effect (CSE).

We then consider scenarios in which there is no initial
chiral asymmetry but $\mu_\mathrm{5}$ can be generated
by gradients in $\mu$
via the CSE term in
the evolution equation of $\mu_\mathrm{5}$.

The goal is to explore whether a $v_5$ or even an
$\alpha_5$ dynamo can be driven in such a scenario.



%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Theoretical background}
\label{sec_CSE_theory}
%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsubsection{System of equations and analytical solutions}


The CSE couples the chiral chemical potential ($\mu_5$) with the chemical potential ($\mu$).
The respective coupling constants are $C_5$ and $C_\mu$.
In this section we consider the following set of equations:
%For most runs we use
\begin{eqnarray}
  \frac{\partial \BB}{\partial t} &=& \nab   \times   \left[{\UU}  \times   {\BB}
  - \eta \, \left(\nab   \times   {\BB}
  - \mu_5 {\BB}  \right) \right]
\label{ind-DNS_CSE}\\
  \rho{D \UU \over D t}&=& (\nab   \times   {\BB})  \times   \BB
  -\nab  p + \nab  {\bm \cdot} (2\nu \rho \SSSS)
\label{UU-DNS_CSE}\\
  \frac{D \rho}{D t} &=& - \rho \, \nab  \cdot \UU
\label{rho-DNS_CSE}\\
  \frac{D \mu_5}{D t} &=& - \mathcal{D}_5 \, \nabla^4 \mu_5
  + \lambda \, \eta \, \left[{\BB} {\bm \cdot} (\nab   \times   {\BB})
  - \mu_5 {\BB}^2 \right]
  - C_5 ({\BB} {\bm \cdot} \nab) \mu,
\label{mu5-DNS_CSE} \\
  \frac{D \mu}{D t}  &=& - \mathcal{D}_\mu \, \nabla^4 \mu - C_\mu (\BB {\bm \cdot} \nab)  \mu_5
\label{mu-DNS_CSE}
\end{eqnarray}
The coupling between $\mu_5$ and $\mu$ leads to the interesting
phenomenon of chiral magnetic waves \citep[CMWs; see][]{KY11}.
When considering the coupled linearized equations~(\ref{mu5-DNS_CSE})
and (\ref{mu-DNS_CSE}), the frequency of CMWs is found to be
\begin{eqnarray}
%IR:
%   \omega_{\rm CMW} = (C_5 \, C_\mu)^{1/2} \left|{\bm k} {\bm \cdot} {\bm B}_0 \right|.
\omega_{\rm CMW} =  \left[C_5 \, C_\mu (k_x B_0)^2 - {1 \over 4} \left(\lambda \eta \,B_0^2
- (D_5 + D_\mu) k^2\right)^2 \right]^{1/2} ,
\label{eq_CMW}
\end{eqnarray}
where the external magnetic field ${\bm B}_0$ is along $x$ axis.
The damping rate of these waves is
\begin{eqnarray}
\gamma_{\rm CMW} = - {1 \over 2} \left[\lambda \eta \,B_0^2 + (D_5 + D_\mu) k^2\right] ,
\label{damp_CMW}
\end{eqnarray}
In the case of the hyper-diffusion $(D_5 + D_\mu) k^2$ should be replaced by $(D_5 + D_\mu) k^4$.
%IR.

In the following, we are considering the following initial conditions where $\mu_5(t=0) = 0$ and
$\mu(t=0)$ is either in form of Gaussian noise or sin waves in $x$ direction.
The initial magnetic field is weak and form of Gaussian noise or Beltrami fields.
In some of the DNS in this section we additionally apply an external magnetic field with
$\BB_\mathrm{ex}=(B_{\mathrm{ex},x}, 0, 0)$ and $B_{\mathrm{ex},x}=10^{-6}$.
The there is no initial velocity field.


Let us analyze the system of equations~(\ref{ind-DNS_CSE})--(\ref{mu-DNS_CSE})
with such initial conditions.
In
\textbf{Phase 1},
a chiral asymmetry is generated via the last term in Eq.~(\ref{mu5-DNS_CSE}).
For times less than the periode of a CMW, we can approximate the evolution of $\mu_5$ as
\begin{eqnarray}
  \mu_5(t) \approx C_5 ({\BB} {\bm \cdot} \nab) \mu  ~ t.
\end{eqnarray}
In all our DNS the external field will be in $x$ direction.
When we then consider $\mu(t=0) = \mu_0 \mathrm{sin}(k_x x)$ we find
\begin{eqnarray}
  \mu_5(t) \approx C_5 B_{\mathrm{ex},x} k_x \mu(t) ~ t.
\label{eq_mu5_init_gen}
\end{eqnarray}
Note that $\mu$ is a function of time due to the dissipation term in
Eq.~(\ref{mu-DNS_CSE}).
We define the start of
\textbf{Phase 2},
when $\mu_5$ exceeds $1$ and a $v_5$ amplifies the magnetic field
exponentially with the growth rate~(\ref{eq_gammalam_max}).
Note that during this phase, $\mu_5$ continues to grow
according to Eq.~(\ref{eq_mu5_init_gen}).
In the beginning of
\textbf{Phase 3}
the generation of $\mu_5$ stops.
This is the case, once the terms
$\lambda \, \eta \, \left[{\BB} {\bm \cdot} (\nab   \times   {\BB}) - \mu_5 {\BB}^2 \right]$
and $C_5 ({\BB} {\bm \cdot} \nab) \mu$ become comparable.
It is important to note that the generation term of $\mu_5$
is determined by the external large-scale magnetic field,
while the destruction term is determined mostly by the
rms magnetic fields that is much larger than $B_{\mathrm{ex},x}$
during phase 3.
An equilibrium state is then reached when
\begin{eqnarray}
  \lambda \, \eta \, \mu_5 {\BB}^2 \approx C_5 B_{\mathrm{ex},x} k_x \mu.
\label{eq_mu5generation}
\end{eqnarray}
Hence for a given set of parameters, a finial value of the production efficiency
\begin{eqnarray}
   \frac{\mu_5 {\BB}^2}{\mu} \approx \frac{C_5 B_{\mathrm{ex},x} k_x}{\lambda \, \eta}
\label{eq_mu5Bumusat}
\end{eqnarray}
can be reached.
We can also introduce the magnetic field strength $B_\mathrm{eq}$ that
corresponds to the terms $\lambda \, \eta \, \left[{\BB} {\bm \cdot} (\nab   \times   {\BB}) - \mu_5 {\BB}^2 \right]$
and $C_5 ({\BB} {\bm \cdot} \nab) \mu$ being comparable:
\begin{eqnarray}
   {\BB}_\mathrm{eq} \equiv \left(\frac{C_5 B_{\mathrm{ex},x} k_x}{\lambda \, \eta}\right)^{1/2}.
\label{eq_Beq}
\end{eqnarray}





\begin{figure}[h!]
\begin{minipage}[t]{0.19\textwidth}
\textbf{[Run B1]} \\
  %\includegraphics[width=0.95\textwidth]{./figures/initconRone/ts.ps}\\
  \includegraphics[width=0.95\textwidth]{./figures/initconRone/ts}\\
  \includegraphics[width=0.95\textwidth]{./figures/initconRone/gamma_t.ps}\\
%  \includegraphics[width=0.95\textwidth]{./figures/initconRone/kmax_t.ps}\\
  \includegraphics[width=0.95\textwidth]{./figures/initconRone/spec_muS2.ps}\\
  \includegraphics[width=0.95\textwidth]{./figures/initconRone/spec_mag.ps}\\
  \includegraphics[width=0.95\textwidth]{./figures/initconRone/spec_mu52.ps}
\end{minipage}
%
\begin{minipage}[t]{0.19\textwidth}
\textbf{[Run B2]} \\
  \includegraphics[width=0.95\textwidth]{./figures/initconRfiv/ts.ps}\\
  \includegraphics[width=0.95\textwidth]{./figures/initconRfiv/gamma_t.ps}\\
%  \includegraphics[width=0.95\textwidth]{./figures/initconRfiv/kmax_t.ps}\\
  \includegraphics[width=0.95\textwidth]{./figures/initconRfiv/spec_muS2.ps}\\
  \includegraphics[width=0.95\textwidth]{./figures/initconRfiv/spec_mag.ps}\\
  \includegraphics[width=0.95\textwidth]{./figures/initconRfiv/spec_mu52.ps}
\end{minipage}
%
\begin{minipage}[t]{0.19\textwidth}
\textbf{[Run B3]} \\
  \includegraphics[width=0.95\textwidth]{./figures/initconRsix/ts.ps}\\
  \includegraphics[width=0.95\textwidth]{./figures/initconRsix/gamma_t.ps}\\
%  \includegraphics[width=0.95\textwidth]{./figures/initconRsix/kmax_t.ps}\\
  \includegraphics[width=0.95\textwidth]{./figures/initconRsix/spec_muS2.ps}\\
  \includegraphics[width=0.95\textwidth]{./figures/initconRsix/spec_mag.ps}\\
  \includegraphics[width=0.95\textwidth]{./figures/initconRsix/spec_mu52.ps}
\end{minipage}
%
\begin{minipage}[t]{0.19\textwidth}
\textbf{[Run B4]} \\
  \includegraphics[width=0.95\textwidth]{./figures/initconRthr/ts.ps}\\
  \includegraphics[width=0.95\textwidth]{./figures/initconRthr/gamma_t.ps}\\
%  \includegraphics[width=0.95\textwidth]{./figures/initconRthr/kmax_t.ps}\\
  \includegraphics[width=0.95\textwidth]{./figures/initconRthr/spec_muS2.ps}\\
  \includegraphics[width=0.95\textwidth]{./figures/initconRthr/spec_mag.ps}\\
  \includegraphics[width=0.95\textwidth]{./figures/initconRthr/spec_mu52.ps}
\end{minipage}
%
\begin{minipage}[t]{0.19\textwidth}
\textbf{[Run B5]} \\
  \includegraphics[width=0.95\textwidth]{./figures/initconRfou/ts.ps}\\
  \includegraphics[width=0.95\textwidth]{./figures/initconRfou/gamma_t.ps}\\
%  \includegraphics[width=0.95\textwidth]{./figures/initconRfou/kmax_t.ps}\\
  \includegraphics[width=0.95\textwidth]{./figures/initconRfou/spec_muS2.ps}\\
  \includegraphics[width=0.95\textwidth]{./figures/initconRfou/spec_mag.ps}\\
  \includegraphics[width=0.95\textwidth]{./figures/initconRfou/spec_mu52.ps}
\end{minipage}
%
\caption{Generation of $\mu_5$ by the CSE using different initial conditions in 2D DNS.
In the top row, solid lines show positive quantities and dashed lines shows the absolute value of
a quantity with negative sign.
\textit{1st column:} The initial $\mu$ and $\BB$ are Gaussion noise.
\textit{2nd column:} The initial $\mu$ is Gaussion noise, the initial $\BB$ is a Beltrami field on $k=1$.
\textit{3rd column:} The initial $\mu$ is a sin wave field on $k=6$, the initial $\BB$ is Gaussion noise.
\textit{4th column:} The initial $\mu$ is a sin wave field on $k=6$, $\BB$ is a Beltrami field on $k=1$.
\textit{5th column:} Same as 4th column, but with weak external magnetic field.
}
\label{fig_initcond}
\end{figure}


%IR added:
\subsubsection{Conservation of total chirality}

We use the induction equation and the equation for the vector potential:
\begin{eqnarray}
&&\frac{\partial \BB}{\partial t} = -
{\bm \nabla} \times {\bm \EE} ,
\label{ind-2}\\
&&\frac{\partial {\bm A}}{\partial t} = - {\bm \EE} + {\bm \nabla} \Phi ,
\label{AA-2}
\end{eqnarray}
where $\BB = {\bm \nabla}  \times   {\bm A}$ and $\Phi$ is the electrostatic potential.
Multiplying Equation (\ref{ind-2}) by ${\bm A}$ and Equation (\ref{AA-2}) by $\BB$, and adding them, we
obtain an evolution equation for the magnetic helicity density,
${\bm A} {\bm \cdot} \BB$:
\begin{eqnarray}
&&\frac{\partial {\bm A} {\bm \cdot} \BB}{\partial t} + {\bm \nabla}  {\bm \cdot}
\left({\bm \EE} \times   {\bm A} + \BB \, \Phi\right) = - 2 {\bm \EE} {\bm \cdot} \BB .
\label{AB}
\end{eqnarray}
Since ${\bm \EE} {\bm \cdot} \BB \propto \eta$,
the density of magnetic helicity, ${\bm A} {\bm \cdot} \BB$ , is conserved
in the limit $\eta \to 0$.
Equation~(\ref{mu5-DNS_CSE}) for $\mu_5$ can be rewritten in the form
\begin{eqnarray}
\frac{\partial (2 \mu_5/\lambda)}{\partial t} + {\bm \nabla}  {\bm \cdot}
\big[C_5 \BB \mu - (2D_5/\lambda) \, {\bm \nabla}  \mu_5\big] = 2 {\EE} {\bm \cdot} {\BB} ,
\label{mu-2}
\end{eqnarray}
where we have assumed that $D_5$ and $\lambda$ are constant.

Adding Equations~(\ref{AB}) and~(\ref{mu-2}), we find the conservation law for total chirality as
\begin{eqnarray}
\frac{\partial }{\partial t} \left({\bm A} {\bm \cdot} \BB
+ \frac{2\mu_5}{\lambda}\right) + {\bm \nabla} {\bm \cdot}
\left[{\bm \EE} \times   {\bm A} + \BB \, \Phi - \frac{2 D_5}{\lambda}
{\bm \nabla}  \mu \right] = 0 . \quad
\label{CL}
\end{eqnarray}
Thus, ${\bm A} {\bm \cdot} \BB +2 \mu_5/\lambda$
is conserved for arbitrary $\eta$.
This implies that when ${\bm A} {\bm \cdot} \BB$ increases, during the dynamo action,
the chiral chemical potential $\mu_5$ must decrease.





\subsubsection{Budget equation of total energy}

Multiplying momentum equation~(\ref{UU-DNS_CSE}) by $\UU$, continuity equation~(\ref{rho-DNS_CSE}) by $\UU^2/2$ and induction equation~(\ref{ind-DNS_CSE}) by $\BB$, and adding them, we obtain budget equation for the kinetic plus magnetic energy densities,
\begin{eqnarray}
&&\frac{\partial }{\partial t} \biggl({\rho \, \UU^2 \over 2} + {\BB^2 \over 2}\biggr) + {\bm \nabla} \cdot
\biggl[\UU \biggl({\rho \, \UU^2 \over 2} + p + \BB^2 - 2\nu \rho \SSSS \biggr)
\nonumber\\
&& \quad \quad - \BB (\UU \cdot \BB) -
\eta \, \BB \times ({\bm \nabla}  \times \BB) \biggr] = - Q + \eta \, \mu_5 \, \BB \cdot ({\bm \nabla}  \times \BB),
\label{KME}
\end{eqnarray}
where $Q=\eta ({\bm \nabla}  \times \BB)^2 + 2\nu \rho \SSSS (\nabla_j U_i) - p ({\bm \nabla} \cdot \UU)$.
To derive equation~(\ref{KME}), we use identity:
${\rm div} \, (\tilde{\EE} \times \BB) = \BB \cdot {\bf curl} \, \tilde{\EE} - \tilde{\EE} \cdot {\bf curl} \, \BB$.

The budget equation for the internal energy density $E_{\rm in} = c_{\rm v} T$ is
\begin{eqnarray}
{\partial (\rho \, E_{\rm in})\over \partial t} +
\nab \cdot \left(\rho \,{\bm U} W - {\bm U} P - K \nab T \right) = Q,
\label{INE}
\end{eqnarray}
where $\rho$, $T$ and $P$ are the density, temperature and pressure, respectively,
which satisfy the equation of state for a perfect gas,
$K$ is the coefficient of the molecular heat conductivity,
$W=c_{\rm p} T = c_{\rm v} T + p/\rho= E_{\rm in} + p/\rho$ is the enthalpy,
where $c_{\rm v}$ and $c_{\rm p}$ are the specific heats
at the constant volume and pressure.
Adding equations~(\ref{KME}) and~(\ref{INE}) we obtain budget equation for the total energy,
\begin{eqnarray}
&&\frac{\partial }{\partial t} \biggl({\rho \, \UU^2 \over 2} + {\BB^2 \over 2} + \rho \, E_{\rm in} \biggr) + {\bm \nabla} \cdot \biggl[\UU \biggl({\rho \, \UU^2 \over 2} + p + \BB^2 - 2\nu \rho \SSSS \biggr)
\nonumber\\
&& \quad \quad - \BB (\UU \cdot \BB) - \eta \, \BB \times ({\bm \nabla}  \times \BB) \biggr] = \eta \, \mu_5 \, \BB \cdot ({\bm \nabla}  \times \BB) .
\label{CLTE}
\end{eqnarray}
This implies that the source term caused by a nonzero chiral chemical potential in equations~(\ref{CLTE})
can produce helical magnetic field, and, therefore, it can produce magnetic and kinetic energies.
%IR.




\begin{figure}[h!]
\centering
  \includegraphics[width=0.3\textwidth]{./figures/gammatheomax_omegaCMW.ps} 
  \includegraphics[width=0.3\textwidth]{./figures/gammamax_omegaCMW.ps} 
  \includegraphics[width=0.3\textwidth]{./figures/gammatheomax_dampCMW.ps} \\
  \includegraphics[width=0.3\textwidth]{./figures/mu5maxOmu0_omegaCMW.ps}  
  \includegraphics[width=0.3\textwidth]{./figures/Bmax_omegaCMW.ps} 
  \caption{
  \textit{Upper left:} Coverage of the parameter space. 
  \textit{Upper right:} Maximum of the measured growth rate of $B$ vs.~frequency of CMW.
  \textit{Lower left:} Maximum measured $\mu_\mathrm{5,rms}$ over the initial amplitude of $\mu$ vs.~frequency of CMW.
  \textit{Lower right:} Maximum measured $B_\mathrm{rms}$ vs.~frequency of CMW.
  Filled dots: Magnetic field is amplified by a chiral dynamo.
  Open dots: No dynamo amplification is observed.}
\end{figure}




%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Exploring the generation of $\mu_5$ spectra}
\label{sec_CSE_spectra}
%%%%%%%%%%%%%%%%%%%%%%%%%%


In this section we present a series of 2D runs where
different initial conditions of $\mu$ are tested.
In particular, we are interested in the resulting $\mu_5$
spectra.
Based on the results reported in Sec.~\ref{sec_initialmu5},
we then can understand the evolution of $B_\mathrm{rms}$.

In Fig.~\ref{fig_initcond}, we directly compare five different
DNS (B1--5) which have different initial $\mu$ and $E_\mathrm{mag}$.
To allow for significant dynamo amplification, we use low
initial magnetic fields in all cases, see Tab.~\ref{tab_DNSoverview_CSE}.
In B1, both the $\mu$ and $\BB$ spectra are in form of Gaussian noise,
which leads to the generation of a Gaussian noise in $\mu_5$.
The high-$k$ part of the $\mu_5$ spectrum is dissipated quickly and
hence no subsequent chiral dynamo is observed.
In B2, a initial Beltrami magnetic field is applied while $\mu$
is again Gaussian.
This leads to an extended $\mu_5$ spectrum with many modes
contributing to an efficient dynamo amplification.
The magnetic field produced by this dynamo increase $\mu_5$
even more, leading to a catastrophic magnetic field amplification
all across the resolved wavenumbers.
Eventually B2 crashes.
Similarly, B3, where $\mu$ is set up as a sine wave at $k_x=6$
and $E_\mathrm{mag}$ is Gaussian noise, leads to an
extended $\mu_5$ spectrum and
catastrophic magnetic field amplification.
When both, $E_\mathrm{mag}$ and the $\mu$ power spectrum are
initially localized on one scale only, $\mu_5$ is produced
on a single scale (B4), but even this run crashed eventually.

Only when at the same time a week external magnetic field is
applied (B5) that ensures a constant rate of $\mu_5$ generation,
the subsequent chiral dynamo is evolving in a moderate way.
During the entire run time of B5, $\mu_5$ remains localized on
a single scale $k=6$, however at later times, other modes
in the $\mu_5$ spectrum are growing as well.
An approximately constant dynamo growth rate is observed
in the time interval of $t\approx1.5-2~t_\eta$.
The dynamo saturates at $t\approx2~t_\eta$ after having
amplified $B_\mathrm{rms}$ by five orders of magnitude.
Both a mean $\mu_5$ and a mean  $\mathbf{A}\cdot\mathbf{B}$ are generated during
the dynamo amplification, with alternating opposite signs.





% %%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Single mode $\mu_5$ spectra}
% \label{sec_CSE_singlemodes}
% %%%%%%%%%%%%%%%%%%%%%%%%%%

% Having found a way to generate $\mu_5$ spectra that consist of
% one single mode only by applying a weak external magnetic field,
% we will focus on that case in the remaining part of the paper.
% The goal of this section is to explore self-consistently generated
% chiral dynamos in 3D simulations and test the influence of
% different run parameters, in particular the influence of the
% magnetic Reynolds number.




% % \begin{figure}[h!]
% % %
% % \begin{minipage}[t]{\textwidth}
% %   \includegraphics[width=0.245\textwidth]{./figures/RmRsix/ts.ps}
% %   \includegraphics[width=0.245\textwidth]{./figures/RmRsix/gamma_t.ps}
% %   \includegraphics[width=0.245\textwidth]{./figures/RmRsix/kmax_t.ps}
% %   \includegraphics[width=0.245\textwidth]{./figures/RmRsix/Rm_t.ps}\\
% %   \includegraphics[width=0.245\textwidth]{./figures/RmRsix/spec_muS2.ps}
% %   \includegraphics[width=0.245\textwidth]{./figures/RmRsix/spec_mu52.ps}
% %   \includegraphics[width=0.245\textwidth]{./figures/RmRsix/spec_mag.ps}
% %   \includegraphics[width=0.245\textwidth]{./figures/RmRsix/spec_kin.ps}
% % \end{minipage}
% % \caption{\textbf{[Run D1]} 3D DNS of a chiral dynamo driven by chiral magnetic waves.
% % The initial $\mu_5$ is zero, but is quickly generated by gradients in
% % $\mu$ and then leads to a chiral $v_5$ dynamo that amplifies the
% % magnetic field by approximately five orders of magnitude before
% % saturation.
% % Both $\langle \mathbf{A}\cdot \mathbf{B} \rangle$ and $\langle \mu_5 \rangle$
% % are generated with alternating signs.
% % During the entire run $\Rm\ll1$.
% % }
% % \label{fig_lowRm}
% % \end{figure}
% % \subsubsection{A $v_5$ dynamo initiated by chiral magnetic waves}


% Fig.~\ref{fig_lowRm} shows a 3D DNS (D1) in which a $\mu_5$ is
% generated from a gradient in $\mu$ and a weak external magnetic
% field.
% The parameters of D1 are exactly the same as in B5, but now
% we consider 3D and use slightly higher resolution
% ($140^3$ instead of $128^2$).

% The evolution of D1 is comparable to B5, except for the time
% after dynamo saturation.
% At $t\gtrsim 2 t_\eta$ in B5, the values of
% $\langle \mathbf{A}\cdot \mathbf{B} \rangle$ and
% $\langle \mu_5 \rangle$ stay constant, while in D1, the sign
% of these quantities continues flipping at a constant frequency.



% Additionally, in the 3D run, the quantity
% $\langle \mathbf{A}\cdot \mathbf{B} \rangle + \langle 2 \mu_5/\lambda_5 \rangle$
% is better conserved as in 2D.





\begin{figure}[h!]
\centering
   \includegraphics[width=0.19\textwidth]{./figures/TwoDa/ts.ps}
   \includegraphics[width=0.19\textwidth]{./figures/TwoDb/ts.ps}
   \includegraphics[width=0.19\textwidth]{./figures/TwoDc/ts.ps}  \\
%
   \includegraphics[width=0.19\textwidth]{./figures/TwoDa/gamma_t.ps}
   \includegraphics[width=0.19\textwidth]{./figures/TwoDb/gamma_t.ps}
   \includegraphics[width=0.19\textwidth]{./figures/TwoDc/gamma_t.ps} \\
%
   \includegraphics[width=0.19\textwidth]{./figures/TwoDa/kmax_t.ps}
   \includegraphics[width=0.19\textwidth]{./figures/TwoDb/kmax_t.ps} 
   \includegraphics[width=0.19\textwidth]{./figures/TwoDc/kmax_t.ps} \\
%
%    \includegraphics[width=0.3\textwidth]{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/448_2D_eta1e-5_lambda51e2_diffmu5hyper199e-10_B0Belkx1ampl1e-9_muSsin200kxmuS6_Bex1e-8/postproc/spec_muS2.ps}
%    \includegraphics[width=0.3\textwidth]{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/448_2D_eta1e-5_lambda51e4_diffmu5hyper199e-10_B0Belkx1ampl1e-9_muSsin200kxmuS6_Bex1e-6/postproc/spec_muS2.ps}
%    \includegraphics[width=0.3\textwidth]{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/448_2D_eta1e-5_lambda51e6_diffmu5hyper199e-10_B0Belkx1ampl1e-9_muSsin200kxmuS6_Bex1e-4/postproc/spec_muS2.ps} \\
% %
   \includegraphics[width=0.19\textwidth]{./figures/TwoDa/spec_mu52.ps}
   \includegraphics[width=0.19\textwidth]{./figures/TwoDb/spec_mu52.ps}
   \includegraphics[width=0.19\textwidth]{./figures/TwoDc/spec_mu52.ps}  \\
%
   \includegraphics[width=0.19\textwidth]{./figures/TwoDa/spec_mag.ps}
   \includegraphics[width=0.19\textwidth]{./figures/TwoDb/spec_mag.ps}
   \includegraphics[width=0.19\textwidth]{./figures/TwoDc/spec_mag.ps} \\
% %
%    \includegraphics[width=0.3\textwidth]{./figures/TwoDa/spec_kin.ps}
%    \includegraphics[width=0.3\textwidth]{./figures/TwoDb/spec_kin.ps}
%    \includegraphics[width=0.3\textwidth]{./figures/TwoDc/spec_kin.ps}
\caption{Details of runs C16, C18, and C20.}
\end{figure}




\subsection{Chiral dynamos and at different frequencies of CMWs}
\label{sec_CMW_freq}


In run series C, we explore the dependence on the frequency of CMWs.
In all runs the damping rate of CMWs $\gamma_\mathrm{CMW}$ is much
less than the wave frequency $\omega_\mathrm{CMW}$.




\subsection{Generation of $\mu_5$ by chiral magnetic waves at different Reynolds numbers}

In run series D we explore the dependence on the magnetic Reynolds number. 
While decreasing the magnetic value of $\eta$ to increase $\Rm$, we keep 
the relative contributions of the non-linear term in (\ref{mu5-DNS_CSE}) and 
the CSE term constant. 
This implies that all runs in series D have the same production efficiency 
(\ref{eq_mu5Bumusat}). In practice, we achieve this by decreasing $B_\mathrm{ex}$ 
together with $\eta$.

% In Fig.~\ref{fig_highRm} the same analysis as in Fig.~\ref{fig_lowRm} is presented, but
% for a run with 100 times lower diffusivity (D5).

% Here, a much larger $\mu_5$ is generated.
% This leads to a growth of magnetic energy on much larger wavenumbers, since
% the peak of the magnetic energy spectrum $k_\mathrm{p}$ follows roughly
% $\mu_5/2$.




% % \begin{figure}[h!]
% % %
% % \begin{minipage}[t]{\textwidth}
% %   \includegraphics[width=0.245\textwidth]{./figures/RmRnin/ts.ps}
% %   \includegraphics[width=0.245\textwidth]{./figures/RmRnin/gamma_t.ps}
% %   \includegraphics[width=0.245\textwidth]{./figures/RmRnin/kmax_t.ps}
% %   \includegraphics[width=0.245\textwidth]{./figures/RmRnin/Rm_t.ps}\\
% %   \includegraphics[width=0.245\textwidth]{./figures/RmRnin/spec_muS2.ps}
% %   \includegraphics[width=0.245\textwidth]{./figures/RmRnin/spec_mu52.ps}
% %   \includegraphics[width=0.245\textwidth]{./figures/RmRnin/spec_mag.ps}
% %   \includegraphics[width=0.245\textwidth]{./figures/RmRnin/spec_kin.ps}
% % \end{minipage}
% % \caption{\textbf{[Run D5]} Same as Fig.~\ref{fig_lowRm} but for a run with $\Rm>1$.
% % [\textit{Caution: Still running.}]}
% % \label{fig_highRm}
% % \end{figure}


% We present the time evolution of all runs from series 3 in
% Fig.~\ref{fig_Rmtrends_ts}.
% The runs can be subdivided in to runs D1--5 which all have
% $\lambda_5=10^4$ but different $\eta$ ($\eta$ is decreasing
% from D1 to D5) and runs D1 plus D6--7 which all have the
% same value of $C_5 k_\mu B_{\mathrm{ex},x} /(\eta \lambda_5)$.
% In the second subseries, $\lambda_5$ is increased while $\eta$ is decreased
% which will result into similar $\mu_5 {\BB}^2/\mu$ according to
% Eq.~(\ref{eq_mu5Bumusat}).
% The estimate in Eq.~(\ref{eq_mu5Bumusat}) is describing
% our DNS well, as can be seen in the right panel of
% Fig.~\ref{fig_Rmtrends_ts}:
% At late times, the ratio of $\mu_{5,\mathrm{rms}} {\BB_\mathrm{rms}}^2/\mu_\mathrm{rms}$
% over $C_5 k_\mu B_{\mathrm{ex},x}/(\eta \lambda_5)$
% is close to $1$.

% The early evolution where $\mu_{5,\mathrm{rms}}$ is
% generated is well-described by Eq.~(\ref{eq_mu5generation}),
% as can be seen in the left panel of Fig.~\ref{fig_Rmtrends_ts}.
% The time when the $\mu_5$ generation stops depends on
% $\mathrm{Re}_\mathrm{M}$.
% With increasing $\mathrm{Re}_\mathrm{M}$, the $\mu_5$
% grows linear in time for a longer time, which overall
% leads to a larger $\mu_5$.


% \begin{figure}[h!]
% \centering
%   \includegraphics[width=0.32\textwidth]{./figures/ts_theory_init.ps}
%   \includegraphics[width=0.32\textwidth]{./figures/ts_theory_sat.ps}
% %  \includegraphics[width=0.32\textwidth]{./figures/gammaBomegaCMW_t.ps}
% %
% %\caption{\textbf{[Runs D1-e + the same run series in 2D that is not listed in the table currently.]}
% \caption{\textbf{[Runs D1--g]}
% Characteristic properties of the simulations from run series 3.
% Solid lines represent runs D1--e which all have $\lambda_5=10^4$ but different $\eta$
% and dashed lines represent runs D6--g which have different $\eta$ but also different
% $\lambda_5$ so that the ratio $C_5 k_\mu B_{\mathrm{ex},x} \mu_\mathrm{rms}/(\eta \lambda_5)$
% is the same as in run D1.
% For better comparison between the runs, we normalize the time to the inverse frequency of the CMWs $\omega_\mathrm{CMW}$
% (in all other figures we used the resistive time for normalization).
% \textit{Left:} Growth of $\mu_{5,\mathrm{rms}}$ as a function time.
% The normalization of $\mu_{5,\mathrm{rms}}$ is the
% theoretical expectation $\mu_\mathrm{rms} C_5 B_{\mathrm{ex},x} k_x t $
% (see Eq.~\ref{eq_mu5generation})
% which works very well in the
% early stages of all runs.
% \textit{Middle:} The maximum value of $\mu_{5,\mathrm{rms}}B_{\mathrm{rms}}^2/\mu_\mathrm{rms}$ that should be reached when the generation
% term in the $\mu_{5}$ equation is of the same order as the destructive term.
% Here, we plot $\mu_{5,\mathrm{rms}}B_{\mathrm{rms}}^2/\mu_\mathrm{rms}$ normalized by its theoretically expected maximum value
% that is given in Eq.~(\ref{eq_mu5Bumusat}).
% \textit{Left:} Ratio of the measured growth rate of the magnetic field and $\omega_\mathrm{CMW}$.}
% \label{fig_Rmtrends_ts}
% \end{figure}



% In Fig.~\ref{fig_Rmtrends} we summarize the characteristic properties of
% the simulations from run series 3 as a function of the maximum
% $\mathrm{Re}_\mathrm{M}$.


% \begin{figure}[h!]
% \centering
%   \includegraphics[width=0.4\textwidth]{./figures/Rm_trends1.ps}
%   \includegraphics[width=0.4\textwidth]{./figures/Rm_trends2.ps}
% %
% %\caption{\textbf{[Runs D1-e + the same run series in 2D that is not listed in the table currently.]}
% \caption{\textbf{[Runs D1--7]} Characteristic properties of the simulations
% (maximum value $\mu_{5,\mathrm{rms}}$, difference between the logarithm of the
% maximum and minimum $B_{\mathrm{rms}}$, and the maximum normalized growth rate
% of the magnetic field measured) as a function of the maximum magnetic Reynolds
% number of each DNS.
% Filled diamonds represent the runs D1--e which all have $\lambda_5=10^4$ and open circles run D6--g which have a higher
% $\lambda_5$ to result in the same ratio $C_5 k_\mu B_{\mathrm{ex},x} \mu_\mathrm{S,rms}/(\eta \lambda_5)$ as in run D1.
% \textit{Left:} Reynolds number defined as $u_\mathrm{rms}/(k_1 \eta)$ where $k_1$ is the minimum wavenumber in the box.
% \textit{Right:} Reynolds number defined as $u_\mathrm{rms}/(k_\mathrm{p} \eta)$ where $k_\mathrm{p}$ is the wavenumber on which the magnetic energy spectrum has its peak.
% [\textit{Caution: Some data points will move a bit since some DNS are still running.}]}
% \label{fig_Rmtrends}
% \end{figure}




\begin{figure}[h!]
\centering
   \includegraphics[width=0.19\textwidth]{./figures/ThreeDa/ts.ps}
   \includegraphics[width=0.19\textwidth]{./figures/ThreeDb/ts.ps}
   \includegraphics[width=0.19\textwidth]{./figures/ThreeDc/ts.ps}  \\
%
   \includegraphics[width=0.19\textwidth]{./figures/ThreeDa/gamma_t.ps}
   \includegraphics[width=0.19\textwidth]{./figures/ThreeDb/gamma_t.ps}
   \includegraphics[width=0.19\textwidth]{./figures/ThreeDc/gamma_t.ps} \\
%
   \includegraphics[width=0.19\textwidth]{./figures/ThreeDa/kmax_t.ps}
   \includegraphics[width=0.19\textwidth]{./figures/ThreeDb/kmax_t.ps} 
   \includegraphics[width=0.19\textwidth]{./figures/ThreeDc/kmax_t.ps} \\
%
   \includegraphics[width=0.19\textwidth]{./figures/ThreeDa/Rm_t.ps}
   \includegraphics[width=0.19\textwidth]{./figures/ThreeDb/Rm_t.ps}
   \includegraphics[width=0.19\textwidth]{./figures/ThreeDc/Rm_t.ps} \\
%
   \includegraphics[width=0.19\textwidth]{./figures/ThreeDa/spec_mu52.ps}
   \includegraphics[width=0.19\textwidth]{./figures/ThreeDb/spec_mu52.ps}
   \includegraphics[width=0.19\textwidth]{./figures/ThreeDc/spec_mu52.ps}  \\
%
   \includegraphics[width=0.19\textwidth]{./figures/ThreeDa/spec_mag.ps}
   \includegraphics[width=0.19\textwidth]{./figures/ThreeDb/spec_mag.ps}
   \includegraphics[width=0.19\textwidth]{./figures/ThreeDc/spec_mag.ps} \\
%
   \includegraphics[width=0.19\textwidth]{./figures/ThreeDa/spec_kin.ps}
   \includegraphics[width=0.19\textwidth]{./figures/ThreeDb/spec_kin.ps}
   \includegraphics[width=0.19\textwidth]{./figures/ThreeDc/spec_kin.ps}
\caption{Details of runs D1, D2, and D3.}
\end{figure}




%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec_conclusion}
%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Appendix}
\label{sec_appendix}
%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%




%%%%%%%%
\begin{table*}
\small
\centering
\caption{Summary of all runs with CSE presented in this paper.
In the column for $D_5 = D_\mu$, "L" stands for Laplacian diffusion and "H" for hyperdiffusion.
In the columns for $\BB(t=0)$, $\mu_5(t=0)$, and $\mu(t=0)$, "G" stands for Gaussian noise and "B" for Beltrami-x.
}
\begin{tabular}{| l | l  l | l l l l | l l l | }
 \hline
            & Setup:             &          & Parameters:\hspace{-2cm}   &            &                             &                    & Initial conditions:\hspace{-2cm}  &  & \\
                & Dim.   & Res.     & $\eta=\nu$         & $\lambda_5$        & $D_5 = D_\mu$               & $B_{\mathrm{ex},x}$  & $\mu_{5}(t=0)$      & $\mu(t=0)$   & $\BB(t=0)$  \\
%   Name            & MHD    & resolution   & $10^2\dfrac{ B_{\mathrm{rms},0}}{c_\mathrm{s}}$ & $\dfrac{k_{\mathrm{p},0}}{k_1}$ & $\dfrac{\mu_{5,0}}{k_1}$ & $\dfrac{B_{\mathrm{rms},0}}{\eta k_1}$ & $\dfrac{\lambda B_{\mathrm{rms},0}^2}{k_1 k_{\mathrm{p},0}}$  &   $\RmABmin$  &   $\Rmkpkone$ &  $\Rmmax$  \\
 \hline
 Series B & & & & & & & & & \\
%\newcommand{\initconRone}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/128_2D_eta1e-3_lambda51e4_diffmu5hyper245e-7_B0Gaussian1e-6_muSGaussian/postproc/}
   B1        & 2D     & $128^2$  & $10^{-3}$   & $10^{4}$    & $2.45\times10^{-7}$ (H)  & -                  & $0$                  & $300$  (G)           & $10^{-6}$  (G)    \\
%\newcommand{\initconRfiv}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/128_2D_eta1e-3_lambda51e4_diffmu5hyper245e-7_B0Belkx1ampl1e-9_muSGaussian_Bex/postproc/}
   B2        & 2D     & $128^2$  & $10^{-3}$   & $10^{4}$    & $2.45\times10^{-7}$ (H)  & -                  & $0$                  & $300$  (G)           & $10^{-9}$  (B)    \\
%\newcommand{\initconRsix}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/128_2D_eta1e-3_lambda51e4_diffmu5hyper245e-7_B0Gaussian1e-6_muSsin300kxmuS6_Bex/postproc/}
   B3        & 2D     & $128^2$  & $10^{-3}$   & $10^{4}$    & $2.45\times10^{-7}$ (H)  & -                  & $0$                  & $300~\mathrm{sin}(6 x)$           & $10^{-6}$  (G)    \\
%\newcommand{\initconRthr}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/128_2D_eta1e-3_lambda51e4_diffmu5hyper245e-7_B0Belkx1ampl1e-9_muSsin300kxmuS6/postproc/}
   B4        & 2D     & $128^2$  & $10^{-3}$   & $10^{4}$    & $2.45\times10^{-7}$ (H)  & -                  & $0$                  & $300~\mathrm{sin}(6 x)$           & $10^{-6}$  (B)    \\
%\newcommand{\initconRfou}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/128_2D_eta1e-3_lambda51e4_diffmu5hyper245e-7_B0Belkx1ampl1e-9_muSsin300kxmuS6_Bex/postproc/}
   B5        & 2D     & $128^2$  & $10^{-3}$   & $10^{4}$    & $2.45\times10^{-7}$ (H)  & $10^{-6}$   & $0$                  & $300~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
 \hline
%/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/postproc2D
   Series C &  & & & & & & & & \\
   C1       & 2D     & $224^2$  & $10^{-5}$   & $10^{2}$    & $7.97\times10^{-10}$ (H)  & $10^{-8}$   & $0$                  & $10~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C2       & 2D     & $224^2$  & $10^{-5}$   & $10^{3}$    & $7.97\times10^{-10}$ (H)  & $10^{-7}$   & $0$                  & $10~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C3       & 2D     & $224^2$  & $10^{-5}$   & $10^{4}$    & $7.97\times10^{-10}$ (H)  & $10^{-6}$   & $0$                  & $10~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C4       & 2D     & $224^2$  & $10^{-5}$   & $10^{5}$    & $7.97\times10^{-10}$ (H)  & $10^{-5}$   & $0$                  & $10~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C5       & 2D     & $224^2$  & $10^{-5}$   & $10^{6}$    & $7.97\times10^{-10}$ (H)  & $10^{-4}$   & $0$                  & $10~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C6       & 2D     & $224^2$  & $10^{-5}$   & $10^{2}$    & $7.97\times10^{-10}$ (H)  & $10^{-8}$   & $0$                  & $50~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C7       & 2D     & $224^2$  & $10^{-5}$   & $10^{3}$    & $7.97\times10^{-10}$ (H)  & $10^{-7}$   & $0$                  & $50~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C8       & 2D     & $224^2$  & $10^{-5}$   & $10^{4}$    & $7.97\times10^{-10}$ (H)  & $10^{-6}$   & $0$                  & $50~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C9       & 2D     & $224^2$  & $10^{-5}$   & $10^{5}$    & $7.97\times10^{-10}$ (H)  & $10^{-5}$   & $0$                  & $50~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C10       & 2D     & $224^2$  & $10^{-5}$   & $10^{6}$    & $7.97\times10^{-10}$ (H)  & $10^{-4}$   & $0$                  & $50~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C11       & 2D     & $224^2$  & $10^{-5}$   & $10^{2}$    & $7.97\times10^{-10}$ (H)  & $10^{-8}$   & $0$                  & $100~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C12       & 2D     & $224^2$  & $10^{-5}$   & $10^{3}$    & $7.97\times10^{-10}$ (H)  & $10^{-7}$   & $0$                  & $100~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C13       & 2D     & $224^2$  & $10^{-5}$   & $10^{4}$    & $7.97\times10^{-10}$ (H)  & $10^{-6}$   & $0$                  & $100~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C14       & 2D     & $224^2$  & $10^{-5}$   & $10^{5}$    & $7.97\times10^{-10}$ (H)  & $10^{-5}$   & $0$                  & $100~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C15       & 2D     & $224^2$  & $10^{-5}$   & $10^{6}$    & $7.97\times10^{-10}$ (H)  & $10^{-4}$   & $0$                  & $100~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C16       & 2D     & $448^2$  & $10^{-5}$   & $10^{2}$    & $1.99\times10^{-10}$ (H)  & $10^{-8}$   & $0$                  & $200~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C17       & 2D     & $448^2$  & $10^{-5}$   & $10^{3}$    & $1.99\times10^{-10}$ (H)  & $10^{-7}$   & $0$                  & $200~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C18       & 2D     & $448^2$  & $10^{-5}$   & $10^{4}$    & $1.99\times10^{-10}$ (H)  & $10^{-6}$   & $0$                  & $200~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C19       & 2D     & $448^2$  & $10^{-5}$   & $10^{5}$    & $1.99\times10^{-10}$ (H)  & $10^{-5}$   & $0$                  & $200~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C20       & 2D     & $448^2$  & $10^{-5}$   & $10^{6}$    & $1.99\times10^{-10}$ (H)  & $10^{-4}$   & $0$                  & $200~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C21       & 2D     & $448^2$  & $10^{-5}$   & $10^{2}$    & $1.99\times10^{-10}$ (H)  & $10^{-8}$   & $0$                  & $300~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C22       & 2D     & $448^2$  & $10^{-5}$   & $10^{3}$    & $1.99\times10^{-10}$ (H)  & $10^{-7}$   & $0$                  & $300~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C23       & 2D     & $448^2$  & $10^{-5}$   & $10^{4}$    & $1.99\times10^{-10}$ (H)  & $10^{-6}$   & $0$                  & $300~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C24       & 2D     & $448^2$  & $10^{-5}$   & $10^{5}$    & $1.99\times10^{-10}$ (H)  & $10^{-5}$   & $0$                  & $300~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
   C25       & 2D     & $448^2$  & $10^{-5}$   & $10^{6}$    & $1.99\times10^{-10}$ (H)  & $10^{-4}$   & $0$                  & $300~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
 \hline
  Series D & & & & & & & & &\\
% %\newcommand{\RmRsix}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/140_3D_eta1e-3_lambda51e4_diffmu5hyper204e-7_B0Belkx1ampl1e-9_muSsin300kxmuS6_Bex_rst/postproc/}
%    D1        & 3D     & $140^3$  & $10^{-3}$   & $10^{4}$    & $2.04\times10^{-7}$ (H)  & $10^{-6}$   & $0$                  & $300~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
% %\newcommand{\RmRsev}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/140_3D_eta5e-4_lambda51e4_diffmu5hyper102e-7_B0Belkx1ampl1e-9_muSsin300kxmuS6_Bex_rst/postproc/}
%    D2        & 3D     & $140^3$  & $5\times10^{-4}$   & $10^{4}$    & $1.02\times10^{-7}$ (H)  & $10^{-6}$   & $0$                  & $300~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
% %\newcommand{\RmReig}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/140_3D_eta2e-4_lambda51e4_diffmu5hyper408e-8_B0Belkx1ampl1e-9_muSsin300kxmuS6_Bex/postproc/}
%    D3        & 3D     & $140^3$  & $2\times10^{-4}$   & $10^{4}$    & $4.08\times10^{-8}$ (H)  & $10^{-6}$   & $0$                  & $300~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
% %\newcommand{\RmRten}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/224_3D_eta5e-5_lambda51e4_diffmu5hyper399e-9_B0Belkx1ampl1e-9_muSsin300kxmuS6_Bex/postproc/}
%    D4        & 3D     & $224^3$  & $5\times10^{-5}$   & $10^{4}$    & $3.99\times10^{-9}$ (H)  & $10^{-6}$   & $0$                  & $300~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
% %\newcommand{\RmRnin}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/224_3D_eta1e-5_lambda51e4_diffmu5hyper797e-10_B0Belkx1ampl1e-9_muSsin300kxmuS6_Bex/postproc/}
%    D5        & 3D     & $224^3$  & $10^{-5}$   & $10^{4}$    & $7.97\times10^{-10}$ (H)  & $10^{-6}$   & $0$                  & $300~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
% %\newcommand{\RmRnin}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/140_3D_eta2e-4_lambda55e4_diffmu5hyper408e-8_B0Belkx1ampl1e-9_muSsin300kxmuS6_Be/postproc/}
%    D6        & 3D     & $140^3$  & $2\times10^{-4}$   & $5\times10^{4}$    & $4.08\times10^{-8}$ (H)  & $10^{-6}$   & $0$                  & $300~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
% %\newcommand{\RmRnin}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/224_3D_eta1e-5_lambda51e6_diffmu5hyper797e-10_B0Belkx1ampl1e-9_muSsin300kxmuS6_Bex/postproc/}
%    D7        & 3D     & $224^3$  & $10^{-5}$   & $10^{6}$    & $7.97\times10^{-10}$ (H)  & $10^{-6}$   & $0$                  & $300~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
%\newcommand{\RmThreeDa}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/448_3D_eta1e-4_lambda51e6_diffmu5hyper199e-9_B0Belkx1ampl1e-9_muSsin200kxmuS6_Bex1e-3_2/postproc/}
   D1        & 3D     & $448^3$  & $10^{-3}$   & $10^{6}$    & $1.99\times10^{-9}$ (H)  & $10^{-3}$   & $0$                  & $200~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
%\newcommand{\RmThreeDb}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/448_3D_eta1e-5_lambda51e6_diffmu5hyper199e-10_B0Belkx1ampl1e-9_muSsin200kxmuS6_Bex1e-4_2/postproc/}
   D2        & 3D     & $448^3$  & $10^{-4}$   & $10^{6}$    & $1.99\times10^{-10}$ (H)  & $10^{-4}$   & $0$                  & $200~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
%\newcommand{\RmThreeDc}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/448_3D_eta1e-6_lambda51e6_diffmu5hyper199e-11_B0Belkx1ampl1e-9_muSsin200kxmuS6_Bex1e-5_2/postproc/}
   D3        & 3D     & $448^3$  & $10^{-5}$   & $10^{6}$    & $1.99\times10^{-11}$ (H)  & $10^{-5}$   & $0$                  & $200~\mathrm{sin}(6 x)$           & $10^{-9}$  (B)    \\
 \hline
\end{tabular}
\label{tab_DNSoverview_CSE}
\end{table*}
%%%%%%%%


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