
A full understanding of AM transport by IGW should include the combined effects of waves emitted from each convective zone. For simplicity, we focus on cases in which a convective shell overlies the radiative core, irradiating it with IGW. These convective shell phases typically occur after core burning phases and thus have the final impact for a given burning phase. We use mixing length theory (MLT), as described in F14, to calculate IGW frequencies and fluxes. Our MLT calculations yield convective velocities and Mach numbers that tend to be a factor of a few smaller than those seen in simulations (e.g., \citealt{Meakin_2006,meakinb:07,Meakin_2007,Arnett_2008}). This could be due to the larger mass of their stellar model or the inadequacy of the MLT approximation. We proceed with our MLT results, but caution that realistic wave frequencies and fluxes may differ from those used here by a factor of a few. 


Figure \ref{fig:MassiveIGWhist} shows a Kippenhahn diagram for our stellar model, and Figure \ref{fig:Massivestruc} shows the density ($\rho$), mass [$M(r)$], and Brunt-V\"{a}is\"{a}l\"{a} frequency ($N$) profiles of our model during important convective shell phases. The first convective shell phase occurs during He-core burning, at which point the star has evolved into a red supergiant. At this stage, IGW are generated at the base of the surface convection zone and propagate toward the He-burning core. We have also shown profiles during shell C-burning, O-burning, and Si-burning, when the radiative core contains a mass of $M_c \sim 1 M_\odot$ and is being irradiated by IGW generated from the overlying convective burning shell. The basic features of each of these phases is quite similar, the main difference is that more advanced burning stages are more vigorous but shorter in duration. We find that the characteristics of the convective burning shells (convective luminosities, turnover frequencies, mach numbers, and lifetimes) are similar to those listed in QS12 and SQ13, although the shell burning phases are generally more vigorous and shorter-lived than the core burning examined in SQ13. Table 1 lists some of the parameters of our convective zones.


\begin{table*}
\begin{center}
\begin{tabular}{ccccccccc}
\hline\
Burning Phase & $r_c$ (km) & $T_{\rm shell}$ (s) & $t_{\rm waves}$ (s) & $\mathcal{M}$ & $\omega_{\rm c}$ (rad/s) & $P_{\rm min}$ (s) & $P_{\rm min,Fe}$ (s) & $P_{\rm min,NS}$ (s) \\
\hline
He Core Burn & $ 1.6 \times 10^7 $ & $ 4 \times 10^{13}$ & $ 2 \times 10^{5}$ & $0.06$ & $ 3 \times 10^{-6}$ & $ 2 \times 10^{5}$ & $ 40$ & $ 2 \times 10^{-3} $  \\
\hline
C Shell Burn & $ 9.7 \times 10^3$ & $ 3 \times 10^8$ & $ 10^{6}$ & $0.002$ & $ 4 \times 10^{-3}$ & $ 2 \times 10^{3}$ & $ 50$ & $  2.5 \times 10^{-3} $ \\
\hline
O Shell Burn & $ 3.6 \times 10^3$ & $ 4 \times 10^6$ & $ 10^{5}$ & $0.004$ & $ 2 \times 10^{-2}$ & - & - & - \\
\hline
Si Shell Burn & $ 1.7 \times 10^3$ & $ 7 \times 10^3$ &  $ 2 \times 10^{3}$ & $0.02$ & $ 4 \times 10^{-1}$ & - & - & - \\
\hline
\end{tabular}
\end{center}
\label{tab:table} Properties of IGW AM transport during evolutionary stages corresponding to the stellar models shown in Figures \ref{fig:Massivestruc} and \ref{fig:MassiveIGWtime}. Here, $r_c$ is the radius at the base of the convection zone in consideration, $T_{\rm shell}$ is the duration of the burning phase, $t_{\rm waves}$ is a wave spin-up timescale (Equation \ref{eqn:twave}), and $\omega_{\rm c}$ is the angular convective turnover frequency. For He and C burning phases, IGW may be able to spin down the core, and $P_{\rm min}$ is the approximate minimum rotation period set by IGW during each phase. $P_{\rm min,Fe}$ is the minimum rotation period if AM is conserved until just before CC (when the iron core has a radius of 1500 km), and $P_{\rm min,NS}$ is the minimum rotation period if AM is conserved until NS birth. The listed values of $P_{\rm min}$ can only be achieved if IGW are able to extract sufficient AM from the core (see text), which is unlikely to occur during O/Si shell burning. 
\end{table*}


The total energy flux carried by waves emitted from the bottom of the convective zone is of order
\begin{equation}
\label{eqn:Ewaves}
\mathcal{M} L_{\rm c} \lesssim \dot{E} \lesssim \mathcal{M}^{5/8} L_{\rm c}  
\end{equation}
\citep{Goldreich_1990,Kumar_1999}, where $\mathcal{M}$ is the convective Mach number (defined as the ratio of MLT convective velocity to sound speed, $v_{\rm c}/c_s$), and $L_{\rm c}$ is the luminosity carried by convection near the base of the convective zone. Many previous works have used the left-hand side of equation \ref{eqn:Ewaves} as an estimate for the IGW energy flux, although \cite{Lecoanet_2013} argue that a more accurate estimate may be $\dot{E} \sim \mathcal{M}^{5/8} L_{\rm c}$, which is larger by a factor of $\mathcal{M}^{-3/8}$. We consider the left-hand side of \ref{eqn:Ewaves} to be a lower limit for the wave flux, and the right-hand side to be an upper limit.


For shell burning, this energy flux is dominated by waves with horizontal wave numbers and angular frequencies near $\bar{m} \sim {\rm max}(r_c/H_c,1)$, and $\bar{\omega} \sim \omega_{\rm c}$, respectively. Here, $H_c$ and $r_c$ are the pressure scale height and radial coordinate near the base of the convective zone, and we define the angular convective turnover frequency as
\begin{equation}
\omega_{\rm c} = \frac{ \pi v_{\rm c}}{ \alpha H_c} \, ,
\end{equation}
where $\alpha H_c$ is the mixing length. Our models use $\alpha = 1.5$. For the convective shells we consider, $H_c \sim r_c$ and we expect waves of low angular degree to be most efficiently excited. The characteristic AM flux carried by these waves is 
\begin{equation}
\label{eqn:Jwaves}
\dot{J} \sim \frac{\bar{m}}{\bar{\omega}} \dot{E}.
\end{equation}
Turbulent convection generates waves with a spectrum of azimuthal numbers $m$ and angular frequencies $\omega$, the values given above are characteristic values which dominate the AM flux. The waves carrying the most AM flux sometimes damp before they reach the core, and might not be able to affect the spin of the core. Then the waves with $\bar{m}$ and $\bar{\omega}$ would not dominate the AM flux to the core; instead, other waves in the turbulent spectrum become important (see Appendix \ref{wavestar}). 

As a first check to see if IGW can have any affect on the spin of the core of the star, we assume all waves can propagate to the core. We suppose that IGW could be important for the spin evolution if they are able to carry an amount of AM comparable to that contained in a young NS, which contains $J_{NS} \approx 10^{48} \, g \, {\rm cm}^2 \, {\rm s}^{-1}$ for a rotation period of $P_{\rm NS} = 10 \, {\rm ms}$. Then the characteristic timescale on which waves could affect the AM of the core is
\begin{equation}
\label{eqn:twave}
t_{\rm waves} = \frac{J_{\rm NS}}{\dot{J}} \, .
\end{equation}
Table 1 lists shell burning stage lifetimes $T_{\rm shell}$ and wave spin-down time scales $t_{\rm waves}$, evaluated using $\dot{E} \propto \mathcal{M}$, $\bar{m}=1$ and $\bar{\omega}=\omega_c$. In all phases, $t_{\rm waves} \ll T_{\rm shell}$ for our model, indicating that waves may be able to have a substantial impact on the spin rate of the core. We examine this impact in the following sections.







