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\begin{document}

%\title{Observed Extra Mixing Trends in Red Giants \evan{are Reproduced}{ Can be Parameterized} by the Reduced Density Ratio in Thermohaline Zones}
\title{Characterizing Observed Extra Mixing Trends in Red Giants using the \textit{Reduced Density Ratio} from Thermohaline Models}

%The Reduced density ratio in thermohaline zones... 

\author[0000-0003-4323-2082]{Adrian E. Fraser}
\altaffiliation{These authors contributed equally to this work}
\affiliation{Department of Applied Mathematics, Baskin School of Engineering, University of California, Santa Cruz, CA 95064, USA}
\affiliation{Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA}

\author[0000-0002-8717-127X]{Meridith Joyce}
\altaffiliation{These authors contributed equally to this work}
\affiliation{Lasker Fellow}
\affiliation{Space Telescope Science Institute,
3700 San Martin Drive,
Baltimore, MD 21218, USA}
\affiliation{Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA}
%\affiliation{\textdagger}

\author[0000-0002-3433-4733]{Evan H. Anders}
\altaffiliation{These authors contributed equally to this work}
\affiliation{CIERA, Northwestern University, Evanston IL 60201, USA}
\affiliation{Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA}

\author[0000-0002-4818-7885]{Jamie Tayar}
\altaffiliation{These authors contributed equally to this work}
\affiliation{NASA Hubble Fellow}
\affiliation{Department of Astronomy, University of Florida, Bryant Space Science Center, Stadium Road, Gainesville, FL 32611, USA }
\affiliation{Institute for Astronomy, University of Hawai‘i at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA}
\affiliation{Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA}

\author[0000-0001-5048-9973]{Matteo Cantiello}
\affiliation{Center for Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA}
\affiliation{Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA}
\affiliation{Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA}

\correspondingauthor{Adrian Fraser}
\email{adfraser@ucsc.edu}
\correspondingauthor{Meridith Joyce}
\email{mjoyce@stsci.edu}
\correspondingauthor{Evan Anders}
\email{evan.anders@northwestern.edu}
\correspondingauthor{Jamie Tayar}
\email{jtayar@ufl.edu}

\begin{abstract}
Observations show an almost ubiquitous presence of extra mixing in low-mass upper giant branch stars. The most commonly invoked explanation for this is the thermohaline instability. One-dimensional stellar evolution models include \textbf{various} prescriptions for thermohaline mixing, \textbf{but the use of observational data directly to discriminate \textit{between} thermohaline prescriptions has thus far been limited.}
%but our ability to make direct comparisons between models and observations has thus far been limited.
Here, we propose a new framework to facilitate direct comparison:
Using carbon to nitrogen measurements from the SDSS-IV APOGEE survey as a probe of mixing and a fluid parameter known as the \textit{reduced density ratio} from one dimensional stellar evolution programs, we compare the observed amount of extra mixing on the upper giant branch to predicted trends from three-dimensional fluid dynamics simulations. 
By applying this method, we are able to \textbf{empirically constrain how mixing efficiency should vary with the reduced density ratio.} %place empirical constraints on the efficiency of mixing across a range of masses and metallicities. 
We find that the observed amount of extra mixing is strongly correlated with the reduced density ratio and that trends between reduced density ratio and fundamental stellar parameters are robust across choices for modeling prescription. We show that stars with available mixing data tend to have relatively low density ratios, which should inform the regimes selected for future simulation efforts. Finally, we show that there is increased mixing at low values of the reduced density ratio, which is consistent with current hydrodynamical models of the thermohaline instability. 
The introduction of this framework sets a new standard for theoretical modeling efforts, as validation for not only the amount of extra mixing, but trends between the degree of extra mixing and fundamental stellar parameters is now possible.
%
\end{abstract}


\keywords{stellar evolution, stellar abundances, abundance ratios, stellar interiors, red giant branch, red giant bump, Dredge-up}

\section{Introduction} %1 
\label{sec:intro}
\setcounter{footnote}{0}
\input{sec_intro_jttest}

\section{Thermohaline Formalism } %2
\label{sec:formalism}
\input{sec_formalism}

\section{Parameterized Thermohaline Models } %3
\label{sec:parameterizations}
\input{sec_parameterizations}

\section{Stellar Evolution Models}%4
\label{sec:mesa_experiment}
\input{sec_mesa_experiment}

\section{Results from Numerical Experiments }
\label{sec:mesa_results}
\input{sec_mesa_results}

\section{Observed Mixing Signatures }
\label{sec:obs}
\input{sec_observations}

\section{Results} 
\label{sec:punchline}
\input{sec_punchline}

\section{Conclusions }
\label{sec:conclusions}
\input{sec_conclusions}


\begin{comment}
%FIGURE omgcomp---------------------------------------------------
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=9cm,clip=true, trim=0.5in 0in 0in 0in]{./Figs/protversusloggmodelpmmPYboth.eps}%[width=9cm, clip=true, trim=1in 1in 1in 1in]{./Figs/omgcomp.eps}
\caption{The measured core rotation rates for the stars in our sample as a function of gravity compared to the predictions of our solid body model (lue) and our moerately differentially convection zone (pink), showing that these models provide limits on the allowable amount of radial differential rotation in the surface convection zone.}
\label{Fig:bothmodels}
\end{center}
\end{figure}
%FIGURE omgcomp---------------------------------------------------
\end{comment}

\begin{acknowledgements}
The first four authors of this manuscript contributed equally to this work. 
\begin{itemize}
\item[] A.~Fraser contributed the majority of text and was responsible for Secs.~2 and 3.
%
\item[] M. Joyce wrote and tested the MESA modeling templates, including \texttt{inlists}, evolutionary and structural output, and \texttt{run\_star\_extras} functionality, wrote the data analysis and parameter extraction scripts in collaboration with E.H. Anders (Python), and contributed text.
%
\item[] E.H. Anders computed and analyzed stellar structure models, created Figs.~2 and 4-7, and contributed text.
%
\item[] J. Tayar was responsible for all analysis related to observations, constructed the initial manuscript template, and contributed text. 
%
\end{itemize}
Author order within this set was decided according to height in heels, descending.

The authors thank C. Hayes, A. Jermyn, and M. Pinsonneault for helpful discussions that contributed to this work and T. White for providing the script that translates between [Fe/H] and Z. The authors acknowledge the thermohaline working group at the KITP program ``Probes of transport in stars'' for fruitful discussion that led to the conception of this idea. The authors likewise thank the Space Telescope Science Institute for providing accommodation and infrastructure support during the week-long meeting during which the majority of this paper was written.
%
AF thanks Pascale Garaud for many discussions of thermohaline mixing which helped improve his understanding of the theory of that field. 
MJ thanks John Bourke for proofreading and helpful discussion of numerics.
EHA thanks MJ for her mentorship in teaching him how to run and interact with MESA models.
%
%
Support for this work was provided by NASA through the NASA Hubble Fellowship grant No.51424 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. JT also acknowledges support from 80NSSC20K0056. AF acknowledges support from NSF Grant No.~AST-1908338. 
MJ acknowledges the Space Telescope Science Institute's Lasker Data Science Fellowship, the Kavli Institute for Theoretical Physics at UC Santa Barbara, and the MESA developers team.
EHA acknowledges the support of a CIERA Postdoctoral fellowship.
 The Center for Computational Astrophysics at the Flatiron Institute is
supported by the Simons Foundation.
 %
 Computations were conducted with support from the NASA High End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center on Pleiades with allocation GID s2276 which is provided through NASA HTMS grant 80NSSC20K1280. 
This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958.


\end{acknowledgements}

\software{Astropy \citep{Astropy,Astropy_2018}, Matplotlib \citep{Matplotlib}, NumPy \citep{numpy}, SciPy \citep{2020SciPy-NMeth}}

\facilities{Du Pont (APOGEE), Sloan (APOGEE)} 

\bibliographystyle{aasjournal}
\bibliography{ms.bib, library.bib, library2.bib, thermohaline.bib, mesa.bib, Joyce_bibliography_4.12.22}


\appendix

\section{MESA}
\label{app:mesa}
\input{app_mesa}

\section{Resolution testing}
\label{app:resolution_test}
\input{app_resolution_test}

\section{Movie of thermohaline front evolution}
\label{app:movie}
\input{app_evolution_movie}

\end{document}