deletions | additions       diff --git a/subsection_Magnetic_Greenhouse_Effect_label__.tex b/subsection_Magnetic_Greenhouse_Effect_label__.tex  index c90e1d9..bf0c864 100644  --- a/subsection_Magnetic_Greenhouse_Effect_label__.tex  +++ b/subsection_Magnetic_Greenhouse_Effect_label__.tex 
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Once a dipolar wave has its energy spread to a broad spectrum of $\ell$, it is doomed to remain within the core. The reason is that higher $\ell$ waves are trapped within the radiative core by a thicker evanescent region (see equation \ref{eqn:integral2}) separating the g-wave cavity in the core from the acoustic wave cavity in the envelope. Therefore, any wave energy with $\ell \gtrsim 3$ will be completely trapped within the radiative core.\footnote{For the same reason, mixed modes with $\ell \gtrsim 2$ are usually not observable in any red giants. Only the envelope modes can be seen, because the gravity-dominated modes in the core of the star are insulated by the thick evanescent evanescent region between core and envelope.} Moreover, higher multipole magneto-gravity waves have shorter wavelengths and damp out more quickly than dipole waves. Hence, an initially dipolar magnetically altered wave will become trapped in the core until it dissipates, unable to tunnel back toward the surface to create an observable signature.   We see that the magnetic greenhouse effect arises not from the alteration of incoming wave frequencies, but rather due to modification of the wave angular structure. Such angular modification originates from the inherently non-spherical structure (since $\nabla \cdot {\bf B} = 0$) of even the simplest magnetic field configurations.  The same effect occurs for magneto-gravity waves that are reflected at $r_{\rm MG}$ rather than coupling with Alfven waves. The location of $r_{\rm MG}$ is a function of latitude, because the magnetic field cannot be spherically symmetric. Even in the simplest case of a purely dipolar magnetic field, the waves will scatter into a broad spectrum of $\ell$ (\cite{Rincon_2003,Reese_2004}). In reality, purely poloidal fields are unstable, and the field will likely have a complex geometry containing both poloidal and toroidal components. An incoming $l=1$ wave is thus inevitably scattered into higher $\ell$ waves in the presence of a strong magnetic field. As described above, these waves cannot couple back to acoustic modes in the envelope, and remain trapped within the radiative core until they dissipate.  We emphasize that the magnetic greenhouse effect arises not from the alteration of incoming wave frequencies, but rather due to modification of the wave angular structure. Such angular modification originates from the inherently non-spherical structure (since $\nabla \cdot {\bf B} = 0$) of even the simplest magnetic field configurations.  Finally, we note that magneto-gravity waves which are reflected at $r_{\rm MG}$ may be reflected onto the slow branch of magneto-gravity waves, since this branch has the same wavenumber as the fast branch at $r=r_{\rm MG}$. This process is known to occur for reflected waves in the solar atmosphere (***REFS***). In our case, as the slow waves propagate back outward into regions with weaker magnetic fields, their wavenumber increases rapidly (see Figure \ref{fig:Prop}). The slow waves will thus dissipate very rapidly via radiative diffusion or non-linear effects, the end result being that they will not be observed as oscillations at the stellar surface.