deletions | additions       diff --git a/subsection_Magnetic_Greenhouse_Effect_label__.tex b/subsection_Magnetic_Greenhouse_Effect_label__.tex  index 1efdb87..2487e60 100644  --- a/subsection_Magnetic_Greenhouse_Effect_label__.tex  +++ b/subsection_Magnetic_Greenhouse_Effect_label__.tex 
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Let’s first consider waves that are transmitted into Alfven waves at $r_{\rm MG}$. The number of Alfven modes that can be excited is likely very high, due to the fact that the magnetic field has a large range of values and a non-trivial geometry in the region (stable magnetic equilibria require a mixture of toroidal and poloidal magnetic fields, (\cite{Braithwaite_2004,Braithwaite_2006,Duez_2010}). In fact the spectrum of Alfven modes is likely continuous (\cite{Reese_2004,Levin_2006}). An incoming dipolar ($\ell =1$, where $\ell$ is the spherical harmonic dependence of the angular structure of a wave) magneto-gravity wave is therefore transmitted into Alfven waves with a broad spectrum of $\ell$ values. The Alfven waves will travel along field lines and could eventually transmit their energy back into magneto-gravity waves. However, even if this occurs, the energy will be spread over a large number of $\ell$ values.   Once a dipolar wave has its energy spread to a broad spectrum of $\ell$, it is doomed to remain within the core. For one, the higher multipole magneto-gravity waves have shorter wavelengths and damp out more quickly than dipole waves. More importantly, The reason is that  higher $\ell$ waves are trapped within the radiative core due to by  a thicker evanescent region (see equation \ref{eqn:integral}) 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, the 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.