deletions | additions       diff --git a/subsection_Magnetic_Greenhouse_Effect_label__.tex b/subsection_Magnetic_Greenhouse_Effect_label__.tex  index 188801d..8b69004 100644  --- a/subsection_Magnetic_Greenhouse_Effect_label__.tex  +++ b/subsection_Magnetic_Greenhouse_Effect_label__.tex 
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In RGB cores, the reflection/transmission process modifies the waves such that they will become trapped in the radiative zone. 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. 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 core. 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.   %\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.  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.