% \documentclass[10pt,a4paper]{article} %\documentclass[twocolumn]{article} % \setlength{\textwidth}{130mm} % \setlength{\oddsidemargin}{-0mm} % \setlength{\textheight}{200mm} % \setlength{\topmargin}{-0mm} \usepackage{subfigure} \usepackage{graphicx,natbib} \usepackage{bm} \usepackage{color} %\usepackage{mathtools} \usepackage{amssymb} %\usepackage{verbatim} %\usepackage{wrapfig} %\definecolor{light-gray}{gray}{0.85} %\usepackage{tcolorbox} %\usepackage[export]{adjustbox} \usepackage[margin=0.6in]{geometry} \newcommand{\AAA}{\bm{A}} \newcommand{\BB}{\bm{B}} \newcommand{\UU}{\bm{U}} \newcommand{\JJ}{\bm{J}} \newcommand{\GG}{\bm{G}} \newcommand{\EE}{\bm{E}} \newcommand{\SSSS}{\mbox{\boldmath ${\sf S}$} {}} \newcommand{\ff}{\bm{f}} \newcommand{\meanrho}{\overline{\rho}} \def\cs{c_{\rm s}} \newcommand{\nab}{\bm{\nabla}} \def\Rm{{\rm Re}_{_\mathrm{M}}} \newcommand{\bra}[1]{\langle #1\rangle} \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.} %/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/chiral_dynamo_modes/postproc \newcommand{\initasymRone}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/chiral_dynamo_modes/128_2D_eta1e-3_diffmu51e-3_lambda51e4_mu5const10/postproc/} \newcommand{\initasymRtwo}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/chiral_dynamo_modes/128_2D_eta1e-3_diffmu51e-3_lambda51e4_mu50sin_amplmu510_kxmu51/postproc/} \newcommand{\initasymRthr}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/chiral_dynamo_modes/128_2D_eta1e-3_diffmu51e-3_lambda51e4_mu50sin_amplmu510_kxmu54/postproc/} \newcommand{\initasymRfou}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/chiral_dynamo_modes/128_2D_eta1e-3_diffmu5hyper2245e-7_lambda51e4_mu50sin_amplmu510_kxmu54/postproc/} \newcommand{\initasymRfiv}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/chiral_dynamo_modes/128_2D_eta1e-3_diffmu5hyper2245e-7_lambda51e4_mu50doublesin_amplmu510_kxmu54/postproc/} \newcommand{\initasymRsix}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/chiral_dynamo_modes/128_2D_eta1e-3_diffmu5hyper2245e-7_lambda51e4_mu50triplesin_amplmu510_kxmu54/postproc/} \newcommand{\initasymRsev}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/chiral_dynamo_modes/128_2D_eta1e-3_diffmu5hyper2245e-7_lambda51e4_mu50gaussian_amplmu510/postproc/} \newcommand{\initasymReig}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/chiral_dynamo_modes/128_2D_eta1e-3_diffmu5hyper2245e-7_lambda51e4_mu50gaussian_amplmu550/postproc/} \newcommand{\initasymRnin}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/chiral_dynamo_modes/128_3D_eta1e-3_diffmu5hyper2245e-7_lambda51e4_mu50random_amplmu506_slopem0/postproc/} \newcommand{\initasymRten}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/chiral_dynamo_modes/128_3D_eta1e-3_diffmu5hyper2245e-7_lambda51e4_mu50random_amplmu51_slopem1/postproc/} \newcommand{\initasymRele}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/chiral_dynamo_modes/128_3D_eta1e-3_diffmu5hyper2245e-7_lambda51e4_mu50random_amplmu516_slopem3/postproc/} % \newcommand{\initasymRnin}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/chiral_dynamo_modes/128_2D_eta1e-3_diffmu5hyper2245e-7_lambda51e4_mu50gaussian_amplmu570/postproc/} % \newcommand{\initasymRten}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/chiral_dynamo_modes/128_2D_eta1e-3_diffmu5hyper2245e-7_lambda51e4_mu50sin_amplmu510_kxmu51/postproc/} \newcommand{\initasymRthirt}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/chiral_dynamo_modes/128_2D_eta1e-3_diffmu5hyper2245e-7_lambda51e4_mu50triplesin_amplmu510_kxmu52/postproc/} % \newcommand{\initasymRtwe}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/chiral_dynamo_modes/128_2D_eta1e-3_diffmu5hyper2245e-7_lambda51e4_mu50triplesin_amplmu510_kxmu51/postproc/} % \newcommand{\initasymRthirt}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/chiral_dynamo_modes/128_2D_eta1e-3_diffmu5hyper2245e-7_lambda51e4_mu50gaussian_amplmu530/postproc/} % \newcommand{\initasymRfourt}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/chiral_dynamo_modes/128_2D_eta1e-3_diffmu5hyper2245e-9_lambda51e4_mu50gaussian_amplmu530/postproc/} % \newcommand{\initasymRfivt}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/chiral_dynamo_modes/128_2D_eta1e-3_diffmu5hyper2245e-8_lambda51e4_mu50gaussian_amplmu530/postproc/} %/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/postproc_initcon \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/} \newcommand{\initconRtwo}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/128_2D_eta1e-3_lambda51e4_diffmu5hyper245e-7_B0Gaussian1e-6_muSGaussian_Bex/postproc/} \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/} \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/} \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/} \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/} \newcommand{\initconRsev}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/128_2D_eta1e-3_lambda51e4_diffmu5hyper245e-7_B0Belkx1ampl1e-9_muSsin300kxmuS6_Bex2/postproc/} \newcommand{\initconReig}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/128_2D_eta1e-3_lambda51e4_diffmu5hyper245e-7_B0Belkx1ampl1e-9_muSsin300kxmuS6_Bex3/postproc/} \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/} \newcommand{\initconRten}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/128_2D_eta5e-3_lambda51e4_diffmu5hyper122e-6_B0Belkx1ampl1e-9_muSsin300kxmuS6_Bex/postproc/} \newcommand{\initconRele}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/140_3D_eta1e-3_lambda51e4_diffmu5hyper245e-7_B0Belkx1ampl1e-9_muSsin300kxmuS6_Bex/postproc/} \newcommand{\initconRtwe}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/140_3D_eta5e-4_lambda51e4_diffmu5hyper122e-7_B0Belkx1ampl1e-9_muSsin300kxmuS6_Bex_rst/postproc/} \newcommand{\initconRthirt}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/128_2D_eta2e-3_lambda51e4_diffmu5hyper488e-7_B0Belkx1ampl1e-9_muSsin300kxmuS6_Bex/postproc/} \newcommand{\initconRfourt}{/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/128_2D_eta2e-4_lambda51e4_diffmu5hyper488e-8_B0Belkx1ampl1e-9_muSsin300kxmuS6_Bex_fidis/postproc/} %/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/postproc_2D \newcommand{\TwoDa}{/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_2/postproc/ts.ps} \newcommand{\TwoDb}{/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_2/postproc/ts.ps} \newcommand{\TwoDc}{/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_2/postproc/ts.ps} \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/} \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/} \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/} % %/Users/jennifer/Science/Coding/pencil-code/jenny/chiral_fluids/generate_mu5/external_B/postproc_Rm % \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/} % \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/} % \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/} % \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/} % \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/} \setlength\parindent{0pt} \newcommand\FramedBox[3]{% \setlength\fboxsep{0pt} \fbox{\parbox[t][#1][c]{#2}{\centering\huge #3}}} \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*} %%%%%%%% \bibliographystyle{plain} \bibliography{paper} \end{document}