deletions | additions       diff --git a/IGW_are_generated_by.tex b/IGW_are_generated_by.tex  index 1ded666..589fec5 100644  --- a/IGW_are_generated_by.tex  +++ b/IGW_are_generated_by.tex 
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where $\bar{m}$ and $\bar{\omega}$ are characteristic horizontal wave numbers and angular frequencies, respectively. For shell burning, we expect typical values of $\bar{m} \sim {\rm max}(r/H,1)$, and $\bar{\omega} \sim \omega_{\rm con}$. Here, $H$ and $r$ are the thickness and radial coordinate of the convective zone, $\omega_{\rm con} \sim v_{\rm con}/(2H)$ is the angular convective turnover frequency of the largest eddies, and $v_{\rm con}$ is the MLT convective velocity. In reality, the 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 characteristic AM flux of equation \ref{Jwaves} \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. Then we define a maximum core rotation rate 
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This maximum spin rate 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}, \ref{eqn:wavestar},  we calculate characteristic wave frequencies $\omega_*(r)$ and AM fluxes $J_*(r)$ entering the core during different phases of massive star evolution. During core He and shell C-burning, waves of lower frequency are somewhat attenuated by radiative photon diffusion. Neutrino damping is likely irrelevant at all times. During shell burning phases, the waves become highly non-linear as they approach the inner $\sim 0.5 M_\odot$, and we expect them to be mostly dissipated via non-linearly breaking before reflecting at the center of the star. For each phase, we calculate the characteristic IGW spin down time scale, $t_*(r)$, on which the waves can change the spin rate by an amount $\omega_*(r)$, \begin{equation}  \label{eqn:tstar}  t_*(r) = \frac{I(r) \omega_*(r)}{J_*(r)} \, ,