% $Id: paper.tex,v 1.28 2020/07/22 09:17:32 brandenb Exp $
\NeedsTeXFormat{LaTeX2e}
\documentclass[preprint2,numberedappendix]{emulateapj-rtx4}
\usepackage{graphicx,natbib,bm,url,color,mathtools}
\newcommand{\fsl}[1]{\ensuremath{\mathrlap{\!\not{\phantom{#1}}}#1}}
\graphicspath{{./fig/}{./png/}}
\thispagestyle{empty}
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%             Customized Commands
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%  mathematical abbreviations
%  =========================

% math defs
\newcommand{\EQ}{\begin{equation}}
\newcommand{\EN}{\end{equation}}
\newcommand{\EQA}{\begin{eqnarray}}
\newcommand{\ENA}{\end{eqnarray}}
\newcommand{\eq}[1]{(\ref{#1})}
\newcommand{\EEq}[1]{Equation~(\ref{#1})}
%\newcommand{\Eq}[1]{Eq.~(\ref{#1})}
%\newcommand{\Eqs}[2]{Eqs.~(\ref{#1}) and~(\ref{#2})}
%\newcommand{\Eqss}[2]{Eqs.~(\ref{#1})--(\ref{#2})}
\newcommand{\Eq}[1]{Equation~(\ref{#1})}
\newcommand{\Eqs}[2]{Equations~(\ref{#1}) and~(\ref{#2})}
\newcommand{\EEqs}[2]{Equations~(\ref{#1}) and~(\ref{#2})}
\newcommand{\Eqss}[2]{Equations~(\ref{#1})--(\ref{#2})}
\newcommand{\eqs}[2]{(\ref{#1}) and~(\ref{#2})}
\newcommand{\App}[1]{Appendix~\ref{#1}}

\newcommand{\Sec}[1]{Sect.~\ref{#1}}
\newcommand{\Secs}[2]{Sects.~\ref{#1} and \ref{#2}}

%\newcommand{\Sec}[1]{Section~\ref{#1}}
%\newcommand{\Secs}[2]{Sections~\ref{#1} and \ref{#2}}

%\newcommand{\Fig}[1]{Fig.~\ref{#1}}
%\newcommand{\FFig}[1]{Figure~\ref{#1}}
%\newcommand{\Figp}[2]{Fig.~\ref{#1}({#2})}
%\newcommand{\Figsp}[3]{Figs.~\ref{#1}({#2}) and ({#3})}
%\newcommand{\Figssp}[3]{Figs.~\ref{#1}({#2})--({#3})}
%\newcommand{\Figs}[2]{Figs.~\ref{#1} and \ref{#2}}
%\newcommand{\Figss}[2]{Figs.~\ref{#1}--\ref{#2}}
%\newcommand{\Tab}[1]{Table~\ref{#1}}
%\newcommand{\Tabs}[2]{Tables~\ref{#1} and \ref{#2}}
%
\newcommand{\Fig}[1]{Figure~\ref{#1}}
\newcommand{\FFig}[1]{Figure~\ref{#1}}
\newcommand{\Figp}[2]{Figure~\ref{#1}({#2})}
\newcommand{\Figsp}[3]{Figures~\ref{#1}({#2}) and ({#3})}
\newcommand{\Figssp}[3]{Figures~\ref{#1}({#2})--({#3})}
\newcommand{\Figs}[2]{Figures~\ref{#1} and \ref{#2}}
\newcommand{\Figss}[2]{Figures~\ref{#1}--\ref{#2}}
\newcommand{\Tab}[1]{Table~\ref{#1}}
\newcommand{\Tabs}[2]{Tables~\ref{#1} and \ref{#2}}

%swap these lines in/out only for purpose of editing
%\newcommand{\Fig}[1]{\textcolor{red}{Fig.~\ref{#1}}}
%\newcommand{\FFig}[1]{\textcolor{red}{Figure~\ref{#1}}}
%\newcommand{\Figs}[2]{\textcolor{red}{Figs.~\ref{#1} and \ref{#2}}}
%\newcommand{\Figss}[2]{\textcolor{red}{Figs.~\ref{#1}--\ref{#2}}}
%\newcommand{\Figp}[2]{\textcolor{red}{Fig.~\ref{#1}({#2})}}
%\newcommand{\Figsp}[3]{\textcolor{red}{Figs.~\ref{#1}({#2}) and ({#3})}}
%\newcommand{\Figssp}[3]{\textcolor{red}{Figs.~\ref{#1}({#2})--({#3})}}
%\newcommand{\Tab}[1]{\textcolor{magenta}{Table~\ref{#1}}}
%\newcommand{\Tabs}[2]{\textcolor{red}{Tables~\ref{#1} and \ref{#2}}}

\newcommand{\bra}[1]{\langle #1\rangle}
\newcommand{\bbra}[1]{\left\langle #1\right\rangle}
\newcommand{\mean}[1]{\overline #1}
\newcommand{\meanrho}{\overline{\rho}}
\newcommand{\meanPhi}{\overline{\Phi}}
\newcommand{\tildeFFFF}{\tilde{\mbox{\boldmath ${\cal F}$}}{}}{}
\newcommand{\hatFFFF}{\hat{\mbox{\boldmath ${\cal F}$}}{}}{}
\newcommand{\meanFFFF}{\overline{\mbox{\boldmath ${\cal F}$}}{}}{}
\newcommand{\meanemf}{\overline{\cal E} {}}
\newcommand{\meanAAAA}{\overline{\mbox{\boldmath ${\mathsf A}$}} {}}
\newcommand{\meanSSSS}{\overline{\mbox{\boldmath ${\mathsf S}$}} {}}
\newcommand{\meanAAA}{\overline{\mathsf{A}}}
\newcommand{\meanSSS}{\overline{\mathsf{S}}}
\newcommand{\meanuu}{\overline{\mbox{\boldmath $u$}}{}}{}
\newcommand{\meanoo}{\overline{\mbox{\boldmath $\omega$}}{}}{}
\newcommand{\meanEMF}{\overline{\mbox{\boldmath ${\cal E}$}}{}}{}
\newcommand{\meanEEEE}{\overline{\mbox{\boldmath ${\cal E}$}}{}}{}
\newcommand{\meanuxB}{\overline{\mbox{\boldmath $\delta u\times \delta B$}}{}}{}
\newcommand{\meanJB}{\overline{\mbox{\boldmath $J\cdot B$}}{}}{}
\newcommand{\meanAB}{\overline{\mbox{\boldmath $A\cdot B$}}{}}{}
\newcommand{\meanjb}{\overline{\mbox{\boldmath $j\cdot b$}}{}}{}
\newcommand{\meanEE}{\overline{\mbox{\boldmath $E$}}{}}{}
\newcommand{\meanFFf}{\overline{\mbox{\boldmath $F$}}_{\rm f}{}}{}
\newcommand{\meanFFm}{\overline{\mbox{\boldmath $F$}}_{\rm m}{}}{}
\newcommand{\hatFF}{\hat{\mbox{\boldmath $F$}}{}}{}
\newcommand{\meanGG}{\overline{\mbox{\boldmath $G$}}{}}{}
\newcommand{\meanHH}{\overline{\mbox{\boldmath $H$}}{}}{}
\newcommand{\tildeGG}{\tilde{\mbox{\boldmath $G$}}{}}{}
\newcommand{\meanCC}{\overline{\mbox{\boldmath $C$}}{}}{}
\newcommand{\meanKK}{\overline{\mbox{\boldmath $K$}}{}}{}
\newcommand{\meanUU}{\overline{\bm{U}}}
\newcommand{\meanVV}{\overline{\bm{V}}}
\newcommand{\meanWW}{\overline{\bm{W}}}
\newcommand{\meanQQ}{\overline{\mbox{\boldmath $Q$}}{}}{}
\newcommand{\meana}{\overline{a}}
\newcommand{\meanA}{\overline{A}}
\newcommand{\meanB}{\overline{B}}
\newcommand{\meanE}{\overline{E}}
\newcommand{\meanK}{\overline{K}}
\newcommand{\Qbar}{\overline{Q}}
\newcommand{\nnbar}{\overline{\bm{n}}}
\newcommand{\nnnbar}{\overline{\bm{n}}}
\newcommand{\meanb}{\tilde{b}}
\newcommand{\meanj}{\tilde{j}}
\newcommand{\meanC}{\overline{C}}
\newcommand{\meanD}{\overline{D}}
\newcommand{\meanF}{\overline{F}}
\newcommand{\meanFm}{\overline{F}_{\rm m}}
\newcommand{\meanFf}{\overline{F}_{\rm f}}
\newcommand{\meanh}{\overline{h}}
\newcommand{\meanhm}{\overline{h}_{\rm m}}
\newcommand{\meanhf}{\overline{h}_{\rm f}}
\newcommand{\meanG}{\overline{G}}
\newcommand{\meanGGG}{\overline{\cal G}}
\newcommand{\meanH}{\overline{H}}
\newcommand{\meanU}{\overline{U}}
\newcommand{\meanR}{\overline{R}}
\newcommand{\meanJ}{\overline{J}}
\newcommand{\means}{\overline{s}}
\newcommand{\meanp}{\overline{p}}
\newcommand{\meanP}{\overline{P}}
\newcommand{\meanS}{\overline{S}}
\newcommand{\meanT}{\overline{T}}
\newcommand{\meanW}{\overline{W}}
\newcommand{\meanQ}{\overline{Q}}
\newcommand{\meanEEE}{\overline{\cal E}}
\newcommand{\meanFFF}{\overline{\cal F}}
\newcommand{\meanHHH}{\overline{\cal H}}
\newcommand{\meanRRR}{\overline{\cal R}}
%\newcommand{\hatAA}{\hat{\mbox{\boldmath $A$}}{}}{}
%\newcommand{\hatBB}{\hat{\mbox{\boldmath $B$}}{}}{}
\newcommand{\hatg}{\hat{g}}
\newcommand{\hatk}{\hat{k}}
\newcommand{\hatkk}{\hat{\bm{k}}}
\newcommand{\hatAA}{\hat{\bm{A}}}
\newcommand{\hatBB}{\hat{\bm{B}}}
\newcommand{\hatJJ}{\hat{\mbox{\boldmath $J$}}{}}{}
%\newcommand{\hatOO}{\hat{\mbox{\boldmath $\Omega$}} {}}
\newcommand{\hatOO}{\hat{\bm{\Omega}}}
\newcommand{\hatEMF}{\hat{\mbox{\boldmath ${\cal E}$}}{}}{}
\newcommand{\Rgas}{{\cal R}}{}
\newcommand{\emf}{{\cal E}}{}
\newcommand{\hatA}{\hat{A}}
\newcommand{\hatB}{\hat{B}}
\newcommand{\hatJ}{\hat{J}}
\newcommand{\hatU}{\hat{U}}
\newcommand{\alphaK}{\alpha_{\rm K}}
\newcommand{\alphaM}{\alpha_{\rm M}}
\newcommand{\alphaSS}{\alpha_{\rm SS}}
%
% tilde
%
\newcommand{\tildexx}{\tilde{\bm{x}}}
\newcommand{\tildeBB}{\tilde{\bm{B}}}
\newcommand{\tildeJJ}{\tilde{\bm{J}}}
\newcommand{\tildeA}{\tilde{A}}
\newcommand{\tildeB}{\tilde{B}}
\newcommand{\tildeJ}{\tilde{J}}
\newcommand{\tildeemf}{\tilde{\cal E}}
\newcommand{\teps}{\tilde{\epsilon} {}}
\newcommand{\tkapz}{\tilde{\kappa_0}}
\newcommand{\Oh}{\hat{\Omega}}
\newcommand{\zh}{\hat{z}}
\newcommand{\SPARK}{{\tt SPARK}~}
\newcommand{\SPARKz}{{\tt SPARKz}}
\newcommand{\FLASH}{{\tt FLASH}~}
\newcommand{\FLASHz}{{\tt FLASH}}
\newcommand{\PC}{{\sc Pencil Code}~}
\newcommand{\PCz}{{\sc Pencil Code}}
%
%  unit vectors
%
\newcommand{\pphi}{\hat{\bm{\phi}}}
\newcommand{\ppom}{\bm{\hat{\varpi}}}
%\newcommand{\ppom}{\hat{\bm{\varpi}}}
\newcommand{\eee}{\hat{\mbox{\boldmath $e$}} {}}
%\newcommand{\nnn}{\hat{\mbox{\boldmath $n$}} {}}
\newcommand{\nnn}{\hat{\bm{n}}}
\newcommand{\rrr}{\hat{\mbox{\boldmath $r$}} {}}
\newcommand{\vvv}{\hat{\mbox{\boldmath $v$}} {}}
\newcommand{\xxx}{\hat{\mbox{\boldmath $x$}} {}}
\newcommand{\yyy}{\hat{\mbox{\boldmath $y$}} {}}
\newcommand{\zzz}{\hat{\mbox{\boldmath $z$}} {}}
\newcommand{\ttt}{\hat{\mbox{\boldmath $\theta$}} {}}
\newcommand{\OOO}{\hat{\mbox{\boldmath $\Omega$}} {}}
\newcommand{\ooo}{\hat{\mbox{\boldmath $\omega$}} {}}
%\newcommand{\ppom}{\hat{\mbox{\boldmath $\varpi$}} {}}
\newcommand{\BBBB}{\hat{\mbox{\boldmath $B$}} {}}
\newcommand{\BBhat}{\hat{\bm{B}}}
\newcommand{\meanAA}{{\overline{\bm{A}}}}
\newcommand{\meanBB}{{\overline{\bm{B}}}}
\newcommand{\meanJJ}{{\overline{\bm{J}}}}
\newcommand{\meanFF}{{\overline{\bm{F}}}}
\newcommand{\meanBBhat}{\hat{\overline{\bm{B}}}}
\newcommand{\meanJJhat}{\hat{\overline{\bm{J}}}}
\newcommand{\meanBhat}{\hat{\overline{B}}}
\newcommand{\meanJhat}{\hat{\overline{J}}}
\newcommand{\Bhat}{\hat{B}}
\newcommand{\Bh}{\hat{B}}
%
%  vectors
%
\newcommand{\nullvector}{{\bf0}}
\newcommand{\gggg}{\mbox{\boldmath $g$} {}}
\newcommand{\ddd}{\mbox{\boldmath $d$} {}}
\newcommand{\yy}{\mbox{\boldmath $y$} {}}
%\newcommand{\zz}{\mbox{\boldmath $z$} {}}
\newcommand{\vv}{\mbox{\boldmath $v$} {}}
\newcommand{\ww}{\mbox{\boldmath $w$} {}}
\newcommand{\mm}{\mbox{\boldmath $m$} {}}
\newcommand{\PP}{\mbox{\boldmath $P$} {}}
\newcommand{\bp}{\mbox{\boldmath $p$} {}}
\newcommand{\II}{\mbox{\boldmath $I$} {}}
\newcommand{\RR}{\mbox{\boldmath $R$} {}}
%\newcommand{\lll}{\bm{\ell}}
\newcommand{\kk}{\bm{k}}
\newcommand{\KK}{\bm{K}}
\newcommand{\pp}{\bm{p}}
\newcommand{\qq}{\bm{q}}
\newcommand{\xx}{\bm{x}}
\newcommand{\aaaa}{\bm{a}}
%\newcommand{\bb}{\bm{b}}
\newcommand{\jj}{\bm{j}}
\newcommand{\rr}{\bm{r}}
\newcommand{\grav}{\bm{g}}
\newcommand{\bb}{\bm{b}}
\newcommand{\BB}{\bm{B}}
\newcommand{\CC}{\bm{C}}
\newcommand{\EE}{\bm{E}}
\newcommand{\JJ}{\bm{J}}
\newcommand{\oo}{\bm{\omega}}
\newcommand{\eell}{\bm{\ell}}
\newcommand{\AAA}{\bm{A}}
\newcommand{\HH}{\bm{H}}
\newcommand{\MM}{\bm{M}}
\newcommand{\UU}{\bm{U}}
\newcommand{\FF}{\bm{F}}
\newcommand{\FFK}{\bm{F}_{\rm K}}
\newcommand{\FFM}{\bm{F}_{\rm M}}
\newcommand{\ffK}{\bm{f}_{\rm K}}
\newcommand{\ffM}{\bm{f}_{\rm M}}
\def\os{\omega_{\rm c}}
\def\oA{\omega_{\rm A}}
\def\oAy{\omega_{{\rm A}y}}
\def\oAz{\omega_{{\rm A}z}}
\newcommand{\vvA}{\bm{v}_{\rm A}}
\newcommand{\uu}{\bm{u}}
\newcommand{\uud}{\bm{u}_{\rm d}}
\newcommand{\rhod}{\rho_{\rm d}}
\newcommand{\sss}{\mbox{\boldmath $s$} {}}
\newcommand{\SSS}{\mbox{\boldmath $S$} {}}
\newcommand{\ee}{\mbox{\boldmath $e$} {}}
\newcommand{\nn}{\mbox{\boldmath $n$} {}}
\newcommand{\ff}{\mbox{\boldmath $f$} {}}
\newcommand{\hh}{\mbox{\boldmath $h$} {}}
\newcommand{\EEE}{\mbox{\boldmath ${\cal E}$} {}}
\newcommand{\FFF}{\mbox{\boldmath ${\cal F}$} {}}
\newcommand{\TT}{{\bm{T}}}
\newcommand{\nab}{{\bm{\nabla}}}
\newcommand{\GG}{\mbox{\boldmath $G$} {}}
\newcommand{\WW}{\mbox{\boldmath $W$} {}}
\newcommand{\QQ}{\mbox{\boldmath $Q$} {}}
%\newcommand{\ggg}{\mbox{\boldmath $g$} {}}
%\newcommand{\nab}{\mbox{\boldmath $\nabla$} {}}
\newcommand{\nabD}{\nabla_{\rm D}}
\newcommand{\nabel}{\nabla_{\rm el}}
\newcommand{\nabad}{\nabla_{\rm ad}}
\newcommand{\nabrad}{\nabla_{\rm rad}}
\newcommand{\nabk}{\mbox{\boldmath $\nabla$}_{\!\kk\,}}
%\newcommand{\OO}{\mbox{\boldmath $\Omega$} {}}
\newcommand{\OO}{\bm{\Omega}}
\newcommand{\ppsi}{\mbox{\boldmath $\psi$} {}}
%\newcommand{\oo}{\mbox{\boldmath $\omega$} {}}
\newcommand{\pom}{\mbox{\boldmath $\varpi$} {}}
\newcommand{\ttau}{\mbox{\boldmath $\tau$} {}}
\newcommand{\LL}{\mbox{\boldmath $\Lambda$} {}}
\newcommand{\mmu}{\mbox{\boldmath $\mu$} {}}
\newcommand{\xxi}{\mbox{\boldmath $\xi$} {}}
\newcommand{\ddelta}{\mbox{\boldmath $\delta$} {}}
\newcommand{\ggamma}{\mbox{\boldmath $\gamma$} {}}
\newcommand{\kkappa}{\mbox{\boldmath $\kappa$} {}}
\newcommand{\llambda}{\mbox{\boldmath $\lambda$} {}}
\newcommand{\pomega}{\mbox{\boldmath $\varpi$} {}}
\newcommand{\PPsi}{\mbox{\boldmath $\Psi$} {}}
%
%  tensors
%
\newcommand{\DDDD}{\mbox{\boldmath ${\sf D}$} {}}
\newcommand{\IIII}{\mbox{\boldmath ${\sf I}$} {}}
\newcommand{\LLLL}{\mbox{\boldmath ${\sf L}$} {}}
\newcommand{\MMMM}{\mbox{\boldmath ${\sf M}$} {}}
\newcommand{\NNNN}{\mbox{\boldmath ${\sf N}$} {}}
\newcommand{\PPPP}{\mbox{\boldmath ${\sf P}$} {}}
\newcommand{\QQQQ}{\mbox{\boldmath ${\sf Q}$} {}}
\newcommand{\RRRR}{\mbox{\boldmath ${\sf R}$} {}}
\newcommand{\SSSS}{\mbox{\boldmath ${\sf S}$} {}}
\newcommand{\TTTT}{\mbox{\boldmath ${\sf T}$} {}}
\newcommand{\BBBBB}{\mbox{\boldmath ${\sf B}$} {}}
\newcommand{\tAAAA}{\tilde{\mbox{\boldmath ${\sf A}$}} {}}
\newcommand{\tDDDD}{\tilde{\mbox{\boldmath ${\sf D}$}} {}}
\newcommand{\tRRRR}{\tilde{\mbox{\boldmath ${\sf R}$}} {}}
\newcommand{\tQQQQ}{\tilde{\mbox{\boldmath ${\sf Q}$}} {}}
%
\newcommand{\AAAA}{\mbox{\boldmath ${\cal A}$} {}}
\newcommand{\BBB}{\mbox{\boldmath ${\cal B}$} {}}
\newcommand{\EMF}{\mbox{\boldmath ${\cal E}$} {}}
\newcommand{\GGG}{\mbox{\boldmath ${\cal G}$} {}}
%\newcommand{\HHH}{\mbox{\boldmath ${\cal H}$} {}}
%\newcommand{\QQQ}{\mbox{\boldmath ${\cal Q}$} {}}
\newcommand{\meanQQQ}{\overline{\mbox{\boldmath ${\cal Q}$}} {}}
%
\newcommand{\GGGG}{{\bf G} {}}
%
%  operators, subscripts, etc  (roman)
%
\newcommand{\ii}{{\rm i}}
\newcommand{\erf}{{\rm erf}}
\newcommand{\grad}{{\rm grad} \, {}}
\newcommand{\curl}{{\rm curl} \, {}}
\newcommand{\dive}{{\rm div}  \, {}}
\newcommand{\Dive}{{\rm Div}  \, {}}
\newcommand{\diag}{{\rm diag}  \, {}}
\newcommand{\asin}{{\rm asin}  \, {}}
\newcommand{\sgn}{{\rm sgn}  \, {}}
\newcommand{\meanDD}{{\overline{\rm D}} {}}
\newcommand{\DD}{{\rm D} {}}
\newcommand{\DDD}{{\cal D} {}}
\newcommand{\MMM}{{\cal M} {}}
\newcommand{\HHH}{{\cal H} {}}
\newcommand{\CCC}{{\cal C} {}}
\newcommand{\QQQ}{{\cal Q}}
\newcommand{\dd}{{\rm d} {}}
\newcommand{\dV}{\,{\rm d}V {}}
\newcommand{\dS}{\,{\rm d}{{\bm{S}}} {}}
\newcommand{\const}{{\rm const}  {}}
\newcommand{\CR}{{\rm CR}}
\def\degr{\hbox{$^\circ$}}
\def\la{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
$\displaystyle##$\hfil\cr<\cr\sim\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr<\cr\sim\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr<\cr\sim\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr<\cr\sim\cr}}}}}
\def\ga{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
$\displaystyle##$\hfil\cr>\cr\sim\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr>\cr\sim\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr>\cr\sim\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr>\cr\sim\cr}}}}}
%
%  numbers
%
\def\H{\mbox{\rm H}}
\def\pH{\mbox{\rm pH}}
\def\Ta{\mbox{\rm Ta}}
\def\Ra{\mbox{\rm Ra}}
\def\Rat{\mbox{\rm Ra}_{\rm turb}}
\def\Prt{\mbox{\rm Pr}_{\rm turb}}
\def\Ma{\mbox{\rm Ma}}
\def\Co{\mbox{\rm Co}}
\def\Ro{\mbox{\rm Ro}}
\def\Ri{\mbox{\rm Ri}}
\def\CoK{\mbox{\rm Co}_{\rm K}}
\def\CoM{\mbox{\rm Co}_{\rm M}}
\def\ShK{\mbox{\rm Sh}_{\rm K}}
\def\ShM{\mbox{\rm Sh}_{\rm M}}
\def\St{\mbox{\rm St}}
\def\Roo{\mbox{\rm Ro}^{-1}}
\def\Rooo{\mbox{\rm Ro}^{-2}}
\def\Pra{\mbox{\rm Pr}}
\def\Pran{\mbox{\rm Pr}}
\def\Sc{\mbox{\rm Sc}}
\def\Tr{\mbox{\rm Tr}}
\def\Da{\mbox{\rm Da}}
\def\Ka{\mbox{\rm Ka}}
\def\RM{\mbox{\rm RM}}
\def\Pm{\mbox{\rm Pr}_{\rm M}}
\def\Rm{\mbox{\rm Re}_{\rm M}}
%\def\Rm{R_{\rm m}}
\def\Dt{D_{\rm t}}
\def\DT{D_{\rm T}}
\def\Rmc{\mbox{\rm Re}_{\rm M}^{\rm crit}}
%\def\Pm{P_{\rm m}}
%\def\Rm{R_{\rm m}}
%\def\Rmc{R_{\rm m,{\rm crit}}}
\def\acos{\mbox{\rm acos}}
\def\Rey{\mbox{\rm Re}}
\def\Imag{\mbox{\rm Im}}
\def\Pe{\mbox{\rm Pe}}
\def\Co{\mbox{\rm Co}}
\def\Gr{\mbox{\rm Gr}}
\def\Lu{\mbox{\rm Lu}}
\def\tauc{\tau_{\rm c}}
\def\tauA{\tau_{\rm A}}
\def\taueff{\tau_{\rm eff}}
\def\Hp{H_{\!p}}
\def\HP{H_{\!P}}
\def\cP{c_{P}}
\def\cV{c_{V}}
\def\alpK{\alpha_{\rm K}}
\def\alpM{\alpha_{\rm M}}
\def\EEK{{\cal E}_{\rm K}}
\def\EEM{{\cal E}_{\rm M}}
\def\EEGW{{\cal E}_{\rm GW}}
\def\OmGW{{\Omega}_{\rm GW}}
\def\EEtot{{\cal E}_{\rm tot}}
\def\EErad{{\cal E}_{\rm rad}}
\def\EElam{{\cal E}_\lambda}
\def\EEcrit{{\cal E}_{\rm crit}}
\def\HHGW{{\cal H}_{\rm GW}}
\def\HHK{{\cal H}_{\rm K}}
\def\HHM{{\cal H}_{\rm M}}
\def\Ei{E_{\rm K}}
\def\EGW{E_{\rm GW}}
\def\HGW{H_{\rm GW}}
\def\EK{E_{\rm K}}
\def\EM{E_{\rm M}}
\def\EKM{E_{\rm K/M}}
\def\ET{E_{\rm T}}
\def\WL{W_{\rm L}}
\def\WC{W_{\rm C}}
\def\cmag{c_{\rm m}}
\def\cmso{c_{\rm ms}}
\def\csz{c_{\rm s0}}
\def\Teff{T_{\rm eff}}
\def\cp{c_{\rm p}}
\def\cv{c_{\rm v}}
\def\Rs{R_{\odot}}
\def\te{t_{\rm e}}
\def\cs{c_{\rm s}}
\def\xiM{\xi_{\rm M}}
\def\xiK{\xi_{\rm K}}
\def\xiKM{\xi_{\rm K/M}}
\def\Frad{F_{\rm rad}}
%\def\Hpz{H_{\rm p0}}
\def\Hpz{H_{\!\rm p0}}
\def\Hrz{H_{\rho 0}}
\def\qpz{q_{\rm p0}}
\def\qp{q_{\rm p}}
\def\betaK{\beta_{\rm K}}
\def\betaM{\beta_{\rm M}}
\def\betaKM{\beta_{\rm K/M}}
\def\betap{\beta_{\rm p}}
\def\betamin{\beta_{\min}}
\def\betastar{\beta_{\star}}
\def\Peff{{\cal P}_{\rm eff}}
\def\Pmin{{\cal P}_{\rm min}}
\def\pturbz{p_{\rm turb}^{(0)}}
\def\pturb{p_{\rm turb}}
\def\peff{p_{\rm eff}}
\def\qs{q_{\rm s}}
\def\qe{q_{\rm e}}
\def\qg{q_{\rm g}}
\def\qa{q_{\rm a}}
\def\pt{p_{\rm t}}
\def\ptz{p_{\rm t0}}
\def\vA{v_{\rm A}}
\def\vAy{v_{{\rm A}y}}
\def\vAZ{v_{{\rm A}z}}
\def\vAz{v_{\rm A0}}
\def\hf{h_{\rm f}}
\def\hm{h_{\rm m}}
\def\Sh{S_{h}}
\def\Ah{A_{h}}
\def\Shdot{S_{\dot{h}}}
\def\Ahdot{A_{\dot{h}}}
\def\hrms{h_{\rm rms}}
\def\kmean{k_{\rm m}}
\def\lf{l_{\rm f}}
\def\kone{k_1}
\def\kC{k_{\rm C}}
\def\kT{k_{\rm T}}
\def\kd{k_{\rm d}}
\def\kf{k_{\rm f}}
\def\tildekf{\tilde{k}_{\rm f}}
\def\kfeff{k_{\rm f}^{\rm eff}}
\def\kNy{k_{\rm Ny}}
\def\kp{k_{\rm p}}
\def\HC{H_{\rm C}}
\def\HM{H_{\rm M}}
\def\EM{E_{\rm M}}
\def\Ekin{E_{\rm kin}}
\def\kB{k_{\rm B}}
\def\sigmaSB{\sigma_{\rm SB}}
\def\kappaf{\kappa_{\rm f}}
\def\epsf{\epsilon_{\rm f}}
\def\epsfz{\epsilon_{\rm f0}}
\def\epsK{\epsilon_{\rm K}}
\def\epsM{\epsilon_{\rm M}}
\def\epsC{\epsilon_{\it C}}
\def\epsT{\epsilon_{\it T}}
\def\betK{\beta_{\it K}}
\def\betM{\beta_{\it M}}
\def\xid{\xi_{\rm D}}
\def\NA{N_{\rm A}}
\def\kB{k_{\rm B}}
\def\kK{k_{\rm K}}
\def\kM{k_{\rm M}}
\def\kD{k_{\rm D}}
\def\kmax{k_{\rm max}}
\def\kom{k_{\omega}}
\def\Ff{F_{\rm f}}
\def\Fm{F_{\rm m}}
\def\Hf{H_{\rm f}}
\def\Hm{H_{\rm m}}
\def\vArms{v_{\rm A,rms}}
\def\Brms{B_{\rm rms}}
\def\Bref{B_{\rm ref}}
\def\Jrms{J_{\rm rms}}
\def\Urms{U_{\rm rms}}
\def\urms{u_{\rm rms}}
\def\wrms{w_{\rm rms}}
\def\orms{\omega_{\rm rms}}
\def\omms{\omega_{\rm ms}}
\def\brms{b_{\rm rms}}
\def\uref{u_{\rm ref}}
\def\yN{y_{\rm N}}
\def\yL{y_{\rm L}}
\def\kappaBB{\kappa_{\rm BB}}
\def\kappaB{\kappa_{\rm B}}
\def\kappaOO{\kappa_{\Omega\Omega}}
\def\kappaO{\kappa_{\Omega}}
\def\kappah{\kappa_{\rm h}}
\def\kappat{\kappa_{\rm t}}
\def\kappaT{\kappa_{\rm T}}
\def\kappatz{\kappa_{\rm t0}}
\def\mut{\mu_{\rm t}}
\def\nut{\nu_{\rm t}}
\def\nuM{\nu_{\rm M}}
\def\nuT{\nu_{\rm T}}
\def\calKt{{\cal K}_{\rm t}}
\def\calKtz{{\cal K}_{\rm t0}}
\def\chit{\chi_{\rm t}}
\def\chitz{\chi_{\rm t0}}
\def\etat{\eta_{\rm t}}
\def\etatz{\eta_{\rm t0}}
\def\etatz{\eta_{\rm t0}}
\def\etaTz{\eta_{\rm T0}}
\def\nutz{\nu_{\rm t0}}
\def\etaT{\eta_{\rm T}}
\def\mut{\mu_{\rm t}}
\def\muT{\mu_{\rm T}}
\def\uT{\mu{\rm T}}
\def\BBeq{|\BB|/B_{\rm eq}}
\def\Beq{B_{\rm eq}}
\def\Beqz{B_{\rm eq0}}
\def\tautd{\tau_{\rm td}}
\def\tauto{\tau_{\rm to}}
%
%\newcommand{\etal}{{\em et al.}}
%\newcommand{\ea}{{\rm et al.\ }}
%\newcommand{\eaa}{{\rm et al.\ }}
\def\half{{\textstyle{1\over2}}}
\def\hhalf{{\scriptstyle{1\over2}}}
\def\threehalf{{\textstyle{3\over2}}}
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\begin{document}

%\title{Numerical and physical magnetic helicity production during the kinematic stage of a dynamo}
\title{Numerical and physical magnetic helicity production in a kinematic dynamo}
\author{
Axel Brandenburg$^{1,2,3}$\thanks{E-mail:brandenb@nordita.org}
\& Evan O'Connor$^{2}$
}

\affil{
$^1$Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden\\
$^2$Department of Astronomy, AlbaNova University Center, Stockholm University, SE-10691 Stockholm, Sweden\\
$^3$McWilliams Center for Cosmology \& Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA
}

\date{\!$ \, $Revision: 1.28 $ $\!}
%\submitted{Astrophys. J. 889, 55 (2020)}
%\date{Received 2019 October 14; revised 2019 November 27; accepted 2019 December 2; published 2020 January 24}

\begin{abstract}
...
\end{abstract}

\keywords{
dynamo --- magnetic fields --- MHD --- turbulence --- methods: numerical
}

\section{Introduction}

Magnetic helicity is conserved by the magnetohydrodynamic (MHD) equations
in the limit of large magnetic Reynolds numbers.
This is a remarkable property of MHD, because 
something analogous does not exist for the kinetic helicity, which does
change---even when the Reynolds number is very large.
When the flow is forced with a helical forcing function,
it leads to an efficient dynamo process.
Since magnetic helicity is conserved, the magnetic field tends to become
bihelical with opposite signs of magnetic helicity at large and small
length scales, whose contributions cancel.

Numerical codes tend to obey magnetic helicity conservation reasonably
well when magnetic energy and helicity dissipation is accomplished
through Ohm's law by microphysical resistivity \citep{BS20}.
Magnetic energy dissipation through purely numerical means, on the other
hand, yields a magnetic helicity evolution that strongly depends on the details
of the numerical scheme.
With the eight-wave solver of the \FLASH code, for example, spurious
magnetic helicity is generated both over long and short time scales
\citep{BS20}.
By contrast, the \SPARK solver in \FLASH produces much less spurious
magnetic helicity, as will be demonstrated here in detail.

At finite magnetic Reynolds numbers, some level of magnetic helicity
is always generated.
There has been significant progress in characterizing magnetic helicity
production during the late evolution of the dynamo process in its
saturation phase \citep{Bra01,FB02,BB02}, but not much is known about
the early evolution before nonlinear effects have played a role.
An exception is the work of \cite{BDS02}, who found that for the ABC
flow, the normalized magnetic helicity decreases with magnetic Reynolds
number $\Rm$ like $\Rm^{-1/2}$; see their Figure~5.
Assuming $\Rm$ to be proportional to the number of mesh points $N$,
this suggests a relatively slow convergence with increasing resolution.
%For a fixed mesh magnetic Reynolds number, $\Rm=\urms\delta x/\eta$,
%where $\delta x$ is the mesh width and $\urms$ is the rms velocity,
%we expect $\Rm$ to increase linearly with the number of mesh points $N$.
It turns out, however, that the actual magnetic helicity produced
decreases faster---approximately like $N^{-3/2}$.
Investigating and understanding this in detail is the purpose of the
present work.

%\section{Preliminary considerations}
\section{Definition of the model}

The evolution of the magnetic field $\BB$ is governed by the induction
equation,
\EQ
{\dd\BB\over\dd t}=\nab\times\left(\UU\times\BB-\eta\JJ\right),
\EN
where $\UU(\xx,t)$ is the velocity field, which is in general time-dependent,
$\JJ=\nab\times\BB/\mu_0$ is the current density, with $\mu_0$ being the
vacuum permeability, and $\eta$ is the magnetic diffusivity.
The mean magnetic energy density is given by $\bra{\BB^2}/2\mu_0$,
where angle brackets denote volume averaging.
A more convenient quantity is the mean squared magnetic field,
\EQ
\MMM=\bra{\BB^2},
\EN
as well as the magnetic and normalized current helicities
\EQ
\HHH=\bra{\AAA\cdot\BB}\quad\mbox{and}\quad
\CCC=\nu_0\bra{\JJ\cdot\BB}.
\EN

We consider a periodic domain of size $L^3$, so the smallest wavenumber
is $k_1=2\pi/L$.
Because of periodicity, there are no magnetic helicity fluxes and the mean
magnetic helicity density $\bra{\AAA\cdot\BB}$ is governed by the equation
\EQ
{\dd\over\dd t}\bra{\AAA\cdot\BB}=-2\eta\bra{\JJ\cdot\BB}
\EN
During the kinematic dynamo phase of interest, the rms magnetic field
increases exponentially with the growth rate $\lambda$ and the mean squared magnetic
field $\bra{\BB^2}$ grows at a rate $2\lambda$.
It is then useful to introduce normalized quantities
\EQ
\epsM=k_1\bra{\AAA\cdot\BB}/\bra{\BB^2},\quad
\kC=\bra{\JJ\cdot\BB}/\bra{\BB^2},
\EN
where $\epsM$ is non-dimensional and characterizes the net
magnetic helicity in the system, and $\kC$ has the units of a
wavenumber that characterizes the magnetic energy spectrum.
We then have
\EQ
2\lambda \epsM=-2\eta\kC k_1\quad
\mbox{(kinematic phase)}.
\EN
Thus, we see that $\epsM<0$, when the magnetic field grows
($\lambda>0$).
It is conceivable that $\kC$ increases with $\Rm$ to some power $n$, so
we write $\kC=k_1\Rm^n$, where $n$ is an empirically determined parameter.
For Kolmogorov turbulence, for example, one expects $n=1/4$
\citep{BB02}.
In general, however, we have
\EQ
\epsM=-\eta\kC k_1^2/\lambda
=-\tilde{\lambda}^{-1}\tildekf^{-2}\Rm^{-(1-n)},
\EN
where $\tilde{\lambda}=\lambda/\urms\kf$ is the nondimensional growth rate
and $\tildekf=\kf/k_1$ is the nondimensional forcing wavenumber.
The findings of \cite{BDS02} are reproduced when $n=1/2$.
In an ideal code, we expect there to be an effective magnetic
Reynolds number proportional to $N$, so we define
\EQ
R_{\rm eff}=R_0 N,
\EN
where $R_0$ is the mesh Reynolds number.
We have dropped here the specification to the {\em magnetic}
Reynolds number, because for ideal codes one expects the
fluid and magnetic Reynolds numbers to be the same.
The empirical finding $\epsM\propto N^{-3/2}$ then
implies then for $n$ a negative value, $n=-1/2$, and thus
\EQ
\epsM=-\tilde{\lambda}^{-1} R_{\rm eff}^{-3/2}.
\EN
This allows us to compute the effective grid Reynolds number as
\EQ
R_0=-N^{-1}\,(\tilde{\lambda}\,\tildekf^2\epsM)^{-2/3}.
\label{R0eff}
\EN

%\begin{equation}
%{\cal H}(\xx,\yy)=\int \xx\curl^{-1}\yy\,\dd V
%\end{equation}

\begin{table*}[t!]\caption{
\FLASH (first three rows) and {\sc Pencil Code} results.
}\vspace{12pt}\centerline{\begin{tabular}{crccccrrrccc}
\hline
Run& $N$& $\kf$&$\urms$ &$\nu$ & $\eta$ &$\Pm$&$\Rey$ &$\Rm$& $-\epsM$ & $\kC/\kone$ &$\lambda$ \\%run
\hline
O& 16& 16.5& 0.405& 0& 0&--&--&--& 2.4e-02&-2.5e-01& 7.8e-02\\%data_initialbeta1e20_16cubed
O& 32& 16.5& 0.427& 0& 0&--&--&--& 8.7e-03& 4.8e-02& 1.2e-01\\%data_initialbeta1e20_32cubed
O& 64& 16.5& 0.408& 0& 0&--&--&--& 2.8e-03& 5.4e-02& 1.3e-01\\%data_initialbeta1e20_64cubed
\hline
I& 128& 16.5& 0.211& 5.0e-05& 5.0e-05&  1&  255.3&  255.3& 3.5e-04& 9.4e-01& 2.0e-01\\%Box2_5em5_128a
I& 256& 16.5& 0.212& 5.0e-05& 5.0e-05&  1&  256.5&  256.5& 3.1e-04& 1.1e+00& 1.9e-01\\%Box2_5em5_256a
I& 512& 16.5& 0.211& 5.0e-05& 5.0e-05&  1&  255.8&  255.8& 3.0e-04& 1.1e+00& 1.9e-01\\%Box2_5em5_512c
\hline
P& 256& 16.5& 0.206& 2.0e-04& 5.0e-06& 40&   62.5& 2501.7& 5.8e-05& 6.2e-01& 3.5e-01\\%Box2_5em6_256a
P& 512& 16.5& 0.206& 2.0e-04& 5.0e-06& 40&   62.3& 2493.6& 6.3e-05& 4.2e+00& 3.5e-01\\%Box2_5em6_512a
\hline
X&  64& 28.3& 0.085& 2.0e-03& 5.0e-05& 40&    1.5&   59.8& 4.3e-03& 1.2e+01& 1.5e-01\\%Box1_5em5_64a2
X& 128& 28.3& 0.085& 2.0e-03& 5.0e-05& 40&    1.5&   60.1& 4.1e-03& 1.3e+01& 1.6e-01\\%Box1_5em5_128a
X& 256& 28.3& 0.084& 2.0e-03& 5.0e-05& 40&    1.5&   59.6& 4.3e-03& 1.3e+01& 1.5e-01\\%Box1_5em5_256a
\hline
Y& 128& 28.3& 0.178& 4.0e-04& 1.0e-05& 40&   15.7&  629.0& 1.3e-04&-2.7e+00& 4.0e-01\\%Box1_5em5_128b
Y& 256& 28.3& 0.180& 4.0e-04& 1.0e-05& 40&   15.9&  635.6& 2.8e-04& 9.5e+00& 3.8e-01\\%Box1_5em5_256c2 (spec,sat)
Y& 512& 28.3& 0.180& 4.0e-04& 1.0e-05& 40&   15.8&  634.0& 2.7e-04& 1.0e+01& 3.8e-01\\%Box1_5em5_512b (no spec)
\hline
Z& 128& 28.3& 0.194& 2.0e-04& 5.0e-06& 40&   34.3& 1372.4& 8.1e-05&-7.7e+00& 1.4e+00\\%Box1_5em5_128c2 not ok
Z& 256& 28.3& 0.199& 2.0e-04& 5.0e-06& 40&   35.2& 1407.5& 8.0e-05& 5.1e-01& 4.9e-01\\%Box1_5em5_256b  not ok
Z& 512& 28.3& 0.199& 2.0e-04& 5.0e-06& 40&   35.2& 1407.7& 9.3e-05& 8.6e+00& 4.9e-01\\%Box1_5em6_512a
Z&1024& 28.3& 0.196& 2.0e-04& 5.0e-06& 40&   34.5& 1380.9& 9.5e-05& 9.3e+00& 4.9e-01\\%Box1_5em6_1024a (spec!)
\label{Tsummary}\end{tabular}}\end{table*}

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pabm3}
\end{center}\caption[]{
\FLASH.
}\label{pabm3}\end{figure}

\FFig{pabm3} shows magnetic helicity production with the \SPARK solver in \FLASHz.
The early exponential growth phase of the dynamo corresponds to the time
interval from 20 to 140 (in code units).
In the following, we average the instantaneous growth rate,
$\lambda(t)=\dd\ln\Brms/\dd t$ and various other normalized
quantities such as $\epsM$ and $\kC$, as well as $\urms$,
over this time interval.
\FFig{pabm_table_flash} shows that the magnetic helicity production decreases with
increasing resolution $N$ like $N^{-1.5}$.
(Not clear why.)

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pabm_table_flash}
\end{center}\caption[]{
Results with \FLASH.
}\label{pabm_table_flash}\end{figure}

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pabm_pc}
\end{center}\caption[]{
Results with {\sc Pencil Code}.
}\label{pabm_pc}\end{figure}

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pabm_table}
\end{center}\caption[]{
Results with {\sc Pencil Code}.
}\label{pabm_table}\end{figure}

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pabm_table_comb}
\end{center}\caption[]{
Combined presentation of the results both with \FLASH and the
{\sc Pencil Code}.
}\label{pabm_table_comb}\end{figure}

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pabm_table_comb_Rm}
\end{center}\caption[]{
Combined presentation of the results both with \FLASH and the
{\sc Pencil Code} versus $\Rm$.
For the \FLASH results, we have assumed $\Rm=N^{7/3}/2000$.
}\label{pabm_table_comb_Rm}\end{figure}

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pabm_table_eff}
\end{center}\caption[]{
Effective grid Reynolds number.
(Dubious?)
}\label{pabm_table_eff}\end{figure}

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pabm_table_kC}
\end{center}\caption[]{
Dependence of $\kC$ on $N$.
}\label{pabm_table_kC}\end{figure}

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pabm_table_kC_Rm}
\end{center}\caption[]{
Dependence of $\kC$ on $\Rm$.
}\label{pabm_table_kC_Rm}\end{figure}

In \Fig{pabm_pc} we show the magnetic helicity evolution in DNS
for different values of $\Rm$; see also \Tab{Tsummary} for a summary
of quantities averaged over the linear growth phase.
The data of the DNS correspond to an inhomogeneous dataset because both
$\Pm$ and $\Rm$ vary.
We also note that in the DNS, $\urms$ is only about half the value
used in the \FLASH code results.
\FFig{pabm_table} shows that the magnetic helicity production
decreases with $\Rm$ like $\Rm^{-0.7}$, at least for the range
of values shown here.

In \Fig{pabm_table_comb} we combine the \FLASH and {\sc Pencil Code}
results.
For each value of $\Rm$, we show here separate lines with different
resolutions.

Using \Eq{R0eff}, we compute $R_0$, which is shown
in \Fig{pabm_table_eff} as a function of $N$.

Looking at \Tab{Tsummary}, we see that $\kC/\kf$
converges with $N$ more slowly than $\epsM$.

\section{Two-scale approach}

In a closed or periodic domain, magnetic helicity can only be produced
by the $\bra{\JJ\cdot\BB}$ term, even if $\bra{\AAA\cdot\BB}=0$ initially.
To understand this, we now compute $\bra{\JJ\cdot\BB}$ under the
two-scale approximation, i.e.,
\EQ
\bra{\JJ\cdot\BB}=\bra{\meanJJ\cdot\meanBB}+\bra{\jj\cdot\bb}
\EN
\EQ
\bra{\JJ\cdot\BB}=k_1^2\bra{\meanAA\cdot\meanBB}+\kf^2\bra{\aaaa\cdot\bb}
\EN
Making use of $\HHM=\bra{\meanAA\cdot\meanBB}+\bra{\aaaa\cdot\bb}$, we have
$\bra{\meanAA\cdot\meanBB}=\HHM-\bra{\aaaa\cdot\bb}$, so
\EQ
\bra{\JJ\cdot\BB}=k_1^2(\HHM-\bra{\aaaa\cdot\bb})+\kf^2\bra{\aaaa\cdot\bb}
\EN
\EQ
\bra{\JJ\cdot\BB}=k_1^2\HHM+(\kf^2-k_1^2)\bra{\aaaa\cdot\bb}
\EN
\EQ
\frac{\dd}{\dd t}\bra{\AAA\cdot\BB}=-2\eta\bra{\JJ\cdot\BB}
\EN
\EQ
\frac{\dd\HHM}{\dd t}=-2\eta\HHM-2\eta(\kf^2-k_1^2)\bra{\aaaa\cdot\bb}
\EN
Using $\kf^2\bra{\aaaa\cdot\bb}\approx\bra{\jj\cdot\bb}
\approx\mu_0\rho_0\bra{\oo\cdot\uu}$, we have
\EQ
\frac{\dd\HHM}{\dd t}=-2\eta\HHM-2\eta(1-k_1^2/\kf^2)\mu_0\rho_0\bra{\oo\cdot\uu}
\EN

\section{Conclusions so far}

The magnetic helicity produced during the early kinematic phase of a
dynamo is being reduced with increasing resolution.
Both \SPARK and the {\sc Pencil Code} reduce the magnetic helicity
production approximately like $N^{-1.5}$.

\begin{verbatim}
\end{verbatim}

%r e f
\begin{thebibliography}{}

\bibitem[Blackman \& Brandenburg(2002)]{BB02}
Blackman, E. G., \& Brandenburg, A.\yapj{2002}{579}{359}
%{373}{Dynamic nonlinearity in large scale dynamos with shear}

\bibitem[Brandenburg(2001)]{Bra01}
Brandenburg, A.\yapj{2001}{550}{824}
%{840}{The inverse cascade and nonlinear alpha-effect in simulations of isotropic helical hydromagnetic turbulence}

\bibitem[Brandenburg \& Scannapieco(2020)]{BS20}
Brandenburg, A., \& Scannapieco, E.\yapj{2020}{889}{55}
%{Magnetic helicity dissipation and production in an ideal MHD code}

\bibitem[Brandenburg et al.(2002)]{BDS02}
Brandenburg, A., Dobler, W., \& Subramanian, K.\yan{2002}{323}{99}
%{122}{Magnetic helicity in stellar dynamos: new numerical experiments}

\bibitem[Field \& Blackman(2002)]{FB02}
Field, G. B., \& Blackman, E. G.\yapj{2002}{572}{685}
%{692}{Dynamical quenching of the $\alpha^2$ dynamo}

\end{thebibliography}
\end{document}
