deletions | additions       diff --git a/IGW_are_generated_by.tex b/IGW_are_generated_by.tex  index 5b3c7ff..5c3c5a7 100644  --- a/IGW_are_generated_by.tex  +++ b/IGW_are_generated_by.tex 
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\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 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 a first check to see if IGW can have any affect on the spin of the star, we calculate the characteristic wave spin-up 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}  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.