\section{Core Spin-down by Internal Gravity Waves}
\label{spindown}


The characteristic AM flux of equation \ref{eqn:Jwaves} is positive by definition, although we expect turbulent convection to generate prograde and retrograde waves in nearly equal quantities, so that the net AM flux is nearly zero. However, differential rotation can set up powerful wave filtration mechanisms (see F14), which filter out either prograde or retrograde waves. Consider a wave packet of angular frequency $\omega$ and azimuthal number $m$ that propagates across a radial region of thickness $\Delta r$, whose endpoints have angular spin frequencies that differ by an amount $\Delta \Omega$. If  $\Delta \Omega > \omega/m$, the waves will encounter a critical layer within the region, and will be absorbed. Only waves of opposing AM will penetrate through the layer, therefore, rapidly rotating regions will only permit influxes of negative AM, which will act to slow the rotation of the underlying layers.

IGW can therefore limit differential rotation to maximum of amplitudes $\Delta \Omega_{\rm max} \sim \omega/m$, provided the IGW AM flux is large enough to change the spin rate on time scales shorter than relevant stellar evolution times. In the case of a rapidly rotating core (which has contracted and spun-up) surrounded by a slowly rotating burning shell, we may expect a maximum core rotation rate of $\omega$, provided waves of this frequency can propagate into the core. This maximum rotation rate assumes $|m|=1$ waves dominate the AM flux, the actual rotation rate could be smaller if $|m|>1$ waves have a substantial impact. Thus we define a maximum core rotation rate
\begin{equation}
\label{eqn:Omegamax}
\Omega_{\rm max} \sim \omega_* \,,
\end{equation}
where $\omega_*$ is the characteristic frequency of waves able to penetrate into the core.

This maximum spin rate describe above can only be established if waves of frequency $\omega_*$ can change the spin rate of the core on short enough time scales. However, some of the waves generated by the convective zone may dissipate before reaching the core, and will therefore not be able to change its spin. Indeed, F14 showed that convectively excited IGW likely cannot change the spin rate of the cores of low mass red giants because most of the wave energy is damped out before the waves reach the core. Low frequency waves are particularly susceptible to damping because they have smaller radial wavelengths, causing them to damp out (via diffusive and/or non-linear processes) on shorter time scales. 

In Appendix \ref{wavestar}, we calculate characteristic wave frequencies $\omega_*(r)$ and AM fluxes $\dot{J}_*(r)$ entering the core during different phases of massive star evolution. During core He burning, waves of lower frequency are significantly attenuated by radiative photon diffusion. Neutrino damping is likely irrelevant at all times. During C/O/Si shell burning phases, the waves become highly non-linear as they approach the inner $\sim 0.3 M_\odot$, and we expect them to be mostly dissipated via non-linear breaking instead of reflecting at the center of the star. 

The AM deposited by IGW is only significant if it is larger than the amount of AM contained within the core of the star. A typical massive star has a zero-age main sequence equatorial rotation velocity of $v_{\rm rot} \sim 150 \,{\rm km \ s}^{-1}$ \citep{de_Mink_2013}, corresponding to a rotation period of $P_{\rm MS} \sim 1.5 \, {\rm d}$ for our stellar model. Using this rotation rate, we calculate the AM $J_0(M)$ contained within the mass coordinate $M(r)$, given rigid rotation on the main sequence. In the absence of AM transport, this AM is conserved, causing the core to spin-up as it contracts. Of course, magnetic torques may extract much of this AM and in the contracting core $J_0$ represents an upper limit to the AM contained within the mass coordinate $M(r)$. Both $J_0$ and the corresponding evolving rotation profiles are shown in Figure \ref{fig:MassiveIGWtime}. We also plot the approximate AM $J_{\rm NS}$ contained within a NS rotating at $P_{\rm NS} = 10 \, {\rm ms}$, which is more than two orders of magnitude smaller than the value of $J_0$ within the inner $1.4 M_\odot$. 

In the top panel of Figure \ref{fig:MassiveIGWtime}, we plot the amount of AM capable of being extracted by IGW during each burning phase, 
\begin{equation}
J_{\rm ex} = \dot{J}_*(r) T_{\rm shell} \, .
\label{eqn:Jex}
\end{equation}
We plot both a pessimistic and optimistic estimate, corresponding to the left and right-hand sides of equation \ref{eqn:Ewaves}, respectively. We find that the values of $J_{\rm ex}$ are comparable to $J_0$ for waves emitted during He core burning and C shell burning. This implies that IGW emitted during these phases may be able to significantly spin down the cores of massive stars. However, given the uncertainties, it is unclear whether IGW have a significant effect. During O and Si shell burning, we find that IGW most likely cannot remove the AM contained within the core, if the core retains its full AM from birth. This does not imply IGW have no effect, as the value of $J_{\rm ex}$ for O/Si burning is larger than $J_{\rm NS}$. Therefore, if the core has been spun down by IGW or magnetic torques during previous burning phases, IGW during late burning phases may be critical in modifying the core spin rate (see Section \ref{spinup}). 

If IGW are able to spin down the core during He core burning and C shell burning, this entails a minimum possible core rotation period $P_{\rm min} = 2 \pi/\omega_*(r)$ at the end of these phases. The bottom panel of Figure \ref{fig:MassiveIGWtime} plots the value of $P_{\rm min}$, in addition to the rotation profile $P_0$ corresponding to the AM profile $J_0$ that would occur in the absence of AM transport. If IGW are able to spin down the cores, they entail minimum rotation periods 10-100 times larger than those that would exist without AM transport. Thus, IGW may significantly spin down the cores of massive stars. Table 1 lists the values of $P_{\rm min}$ corresponding to He and C burning, as well as corresponding minimium spin periods for the pre-collapse iron core ($P_{\rm min,Fe}$) and for the neutron star remnant ($P_{\rm min,NS}$) given no subsequent AM transport. The minimum NS rotation period $P_{\rm min,NS}$ we calculate is on the order of milliseconds, which is shorter than that inferred for most newly born NSs. Therefore either IGW spin-down is significantly more effective than our conservative estimates, or (perhaps more likely) magnetic torques are the primary mechanism responsible for spinning down the cores of massive stars.