deletions | additions       diff --git a/IGW_are_generated_by.tex b/IGW_are_generated_by.tex  index d9dace0..c670615 100644  --- a/IGW_are_generated_by.tex  +++ b/IGW_are_generated_by.tex 
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\end{table*}  The total  energy flux carried by waves emitted from the bottom of the convective zone is \begin{equation}  \label{eqn:Ewaves}  \dot{E} \sim \mathcal{M} L_{\rm con} \, ,   \end{equation}  where $\mathcal{M}$ is the convective Mach number and $L_{\rm con}$ is the luminosity carried by convection near the base of the convective zone. For shell burning, this energy flux is dominated by waves with horizontal wave numbers and angular frequencies near $\bar{m} \sim {\rm max}(r/H,1)$, and $\bar{\omega} \sim \omega_{\rm con}$, respectively. 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.  The characteristic AM flux carried by these waves  is \begin{equation}  \label{eqn:Jwaves}  \dot{J} \sim \frac{\bar{m}}{\bar{\omega}} \dot{E}, \dot{E}.  \end{equation}  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 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. As 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}). However, as  a first check to see if IGW can have any affect on the spin of the star, we calculate assume all waves can propagate to the core. Then  the characteristic wave spin-down timescale for waves to change the spin of the core by an amount $\omega_c$, \begin{align}  \label{eqn:twave}  t_{\rm waves} &= \frac{I_c \omega_c}{\dot{J}} \\   &\approx \frac{I_c \omega_c^2}{\mathcal{M} L_{\rm con}} \, .  \end{align}  Here $I_c$ is the moment of inertia of the core.The second line follows from using $\bar{m}=1$ and $\bar{\omega}=\omega_c$ and provides a conservative estimate of $t_{\rm waves}$.  Table 1 lists shell burning stage lifetimes $T_{\rm shell}$ and wave spin-down time scales $t_{\rm waves}$. In each case, $t_{\rm waves} \omega/m$, 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.