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%             Customized Commands
%|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
%  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{\eqss}[2]{(\ref{#1})--(\ref{#2})}
\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{\tildeh}{\tilde{h}}
\newcommand{\tildeT}{\tilde{T}}
\newcommand{\tildehhh}{\tilde{\sf h}}
\newcommand{\tildeTTT}{\tilde{\sf T}}
%
\newcommand{\tildexx}{\tilde{\bm{x}}}
\newcommand{\tildeee}{\tilde{\mathbf{e}}}
\newcommand{\tildeAA}{\tilde{\mathbf{A}}}
\newcommand{\tildeBB}{\tilde{\mathbf{B}}}
\newcommand{\tildeEE}{\tilde{\mathbf{E}}}
\newcommand{\tildeJJ}{\tilde{\mathbf{J}}}
%\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{\ttheta}{\bm{\hat{\theta}}}
%\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{\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{\eeps}{\bm{\epsilon}}
\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}}
%AB--\newcommand{\lll}{\bm{l}}
\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}{\bm{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{\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&lt;\cr\sim\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr&lt;\cr\sim\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr&lt;\cr\sim\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr&lt;\cr\sim\cr}}}}}
\def\ga{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
$\displaystyle##$\hfil\cr&gt;\cr\sim\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr&gt;\cr\sim\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr&gt;\cr\sim\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr&gt;\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\Sp{\mbox{\rm Sp}}
\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\EET{{\cal E}_{\rm T}}
\def\EETz{{\cal E}_{\rm T0}}
\def\EEK{{\cal E}_{\rm K}}
\def\EEM{{\cal E}_{\rm M}}
\def\EEMz{{\cal E}_{\rm M0}}
\def\EEGW{{\cal E}_{\rm GW}}
\def\EEEM{{\cal E}_{\rm EM}}
\def\EEEl{{\cal E}_{\rm E}}
\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\rstar{r_{*}}
\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\omf{\omega_{\rm f}}
\def\tildekf{\tilde{k}_{\rm f}}
%\def\kfeff{k_{\rm f}^{\rm eff}}
\def\kfeff{k_{\rm f,eff}}
\def\kNy{k_{\rm Ny}}
\def\kp{k_{\rm p}}
%AB: italics for ApJ
%\def\kf{k_{\it f}}
\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_{\rm C}}
\def\epsT{\epsilon_{\rm 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\Arms{A_{\rm 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\nuTz{\nu_{\rm T0}}
\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}}
%AB: for ApJ they have italics t here.
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%
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%
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%
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%
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%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  journals
%
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\def\prl{PhRvL}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
</pre></body></html>