deletions | additions       diff --git a/sectionAcknowledgmen.tex b/sectionAcknowledgmen.tex  index d14e6f7..8a0c9aa 100644  --- a/sectionAcknowledgmen.tex  +++ b/sectionAcknowledgmen.tex 
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\section*{Acknowledgments}   JF acknowledges partial support from NSF under grant no. AST-1205732 and through a Lee DuBridge Fellowship at Caltech. This research was supported by the National Science Foundation under grant No. NSF PHY11- 25915 and by NASA under TCAN grant No. NNX14AB53G  \begin{appendix} 
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Our stellar models are constructed using the MESA stellar evolution code \citep{paxton:11,paxton:13}, version 6794. The models are run using the following inlist controls file.  %\begin{multicols}{2}  \begin{verbatim}  \& star_job 
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/ ! end of controls namelist  \end{verbatim}  %\end{multicols}  The most important feature of this model is that it contains significant convective overshoot, especially above convective zones. It is non-rotating, thus there is no rotational mixing. Just before core O-burning, we change to a 201-isotope reaction network:  \begin{verbatim}  change_net = .true.   new_net_name = 'mesa_201.net'  \end{verbatim}  Although our choices affectthe precise  details of the model (e.g., He core mass), the general features of our model are robust. For instance, we find that a 14 $M_\odot$ model with less overshoot produces very similar results. Our low-mass models explode It always explodes  as a  red supergiants, and they undergo supergiant. It always undergoes  convective core C-burning, followed by shell C-burning, core O/Ne-burning, shell O-burning, core Si-burning, shell-Si burning, and then CC. The approximate convective properties (as described by MLT) are not strongly affected by model parameters. Since these properties are most important for IGW AM transport, we argue that the general features of IGWs described in this work are fairly robust against uncertain parameters in our massive star models.
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\label{eqn:spectrum}  \frac{d \dot{E}_0(\omega)}{d\omega} \sim \frac{\dot{E}_0}{\omega_c} \left(\frac{\omega}{\omega_c}\right)^{-a},  \end{equation}  where $a$ is the slope of the frequency spectrum, which is somewhat uncertain. As in F14, we expect a spectrum slope in the range $3 \lesssim a \lesssim 7$, and we use a value of $a=4.5$ in our calculations. Lower frequency waves have shorter wavelengths and slower group velocities, making them more prone to both radiative and non-linear damping. Thus, as waves propagate inwards, low frequency waves may damp out, and  increasing the wave frequency $\omega_*$ of waves that dominate AM transport. F14 show that radiative damping leads to   \begin{equation} 
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\label{eqn:nl}  | k_r \xi_r | = \bigg[ \frac{\lambda^{3/2} N \dot{E} }{\rho r^5 \omega^4} \bigg]^{1/2} \sim 1 \, .  \end{equation}  In the absence of damping, $\dot{E}$ is a conserved quantity. Therefore, waves become more non-linear as they propagate into regions with larger $N$, lower density, or smaller radius. In our problem, the geometrical focusing (i.e., the $r$-dependence) $r$-depdendence)  is the most important feature of equation \ref{eqn:nl}, and causes waves to non-linearly break as they propagate inward. Note also the $\omega^{-2}$ dependence of equation \ref{eqn:nl}, which causes low frequency waves to preferentially damp. Equation \ref{eqn:nl} entails there is a maximum energy flux that can be carried by waves of frequency $\omega$,   \begin{equation} 
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Solving equation \ref{eqn:efreq} yields the wave frequency which dominates energy transport,  \begin{equation}  \label{eqn:omstarnl}  \omega_* \sim {\rm max} \omega_c  \bigg[1, \, \, \bigg(  \frac{ A^2 \rho r^5 \omega_c^4}{\lambda^{3/2} N \dot{E}_0} \bigg)^{-1/(a+3)} \bigg] \omega_c \bigg]^{-1/(a+3)}  \, . \end{equation}  We expect frequency spectra with slopes somewhere near $3 \lesssim a \lesssim 7$. Therefore the exponent in equation \ref{eqn:omstarnl} is quite small, and in most cases, $\omega_*$ does not increase to values much larger than $\omega_c$.   Substituting equation \ref{eqn:omstarnl} back into equation \ref{eqn:emax} allows us to solve for the energy and AM flux as a function of radius due to non-linear attenuation. The result is  \begin{equation}  \label{eqn:jstarnl}  \dot{J}_* \sim \bigg[ \frac{\omega_*(r)}{\omega_c} \bigg]^{-a} \dot{J}_0 \sim {\rm min} \bigg[1, \, \, \bigg( \bigg[  \frac{A^2 \rho r^5 \omega_c^4}{\lambda^{3/2} N \dot{E}_0} \bigg)^{a/(a+3)} \bigg] \bigg]^{a/(a+3)}  \dot{J}_0 \, . \end{equation}  During some burning phases, waves can damp via C-shell burning, we find that  radiative diffusion damps waves  near the shell burning  convective shell, zone,  while non-linear breaking damps waves near the center of the star. In this case, we first calculate $\omega_*$ and its corresponding energy flux $\dot{E}_*$ via equations \ref{eqn:spectrum} \ref{eqn:omstar2}  and \ref{eqn:omstar2}. \ref{eqn:spectrum}.  We then substitute the value of $\dot{E}_*$ for $\dot{E}_0$ in equation \ref{eqn:omstarnl}. The appropriate value of $\omega_*$ is then $\omega_* = {\rm max} \big[ {\rm Eqn.}$ \ref{eqn:omstar2}$, {\rm Eqn.}$ \ref{eqn:omstarnl}$ Eqn.}$\ref{eqn:omstarnl}$  \big]$. The corresponding AM flux is $\dot{J}_* \sim \Big[ \frac{\omega_*(r)}{\omega_c} \Big]^{-a} \dot{J}_0$. The cores of massive stars nearing death cool primarily through neutrino emission, so it is not unreasonable to think that waves may be damped via neutrino emission. We calculate neutrino energy loss rates in the same manner as \cite{murphy:04}. We find that neutrino damping time scales are always longer than the wave crossing timescale  \begin{equation}