deletions | additions       diff --git a/In_stars_with_field_strengths__.tex b/In_stars_with_field_strengths__.tex  index 950b27b..8e9fd36 100644  --- a/In_stars_with_field_strengths__.tex  +++ b/In_stars_with_field_strengths__.tex 
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In stars with field strengths exceeding $B_c$ (equation \ref{eqn:Bc}) somewhere in their core, incoming dipolar gravity waves will become evanescent at the radius $r_{\rm MG}$ where $B>B_c$. At this point, the waves must either reflect or be transmitted into the strongly magnetized region as Alfven waves. An analogous process occurs in the Sun's atmosphere, where outwardly propagating magneto-acoustic-gravity waves become magnetically dominated as they propagate upward into regions where the magnetic pressure is larger than the gas pressure. In this case, the reflection or transmission of the wave depends on the geometry of the magnetic field (\cite{Zhugzhda_1984}).   In RGB cores, the reflection/transmission process modifies the waves such that they will become trapped in the radiative zone (see Supplementary Material). Incoming $\ell=1$ magneto-gravity waves can transmit energy into a continuous spectrum (\cite{Reese_2004,Levin_2006}) of Alfven waves with a broad spectrum of $\ell$ values \cite{Rincon_2003}, even for simple field geometries. The same effect occurs for magneto-gravity reflected 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, any oscillation modes created by the waves will contain a broad spectrum of $\ell$ (\cite{Lee_2007,Lee_2010}). In reality,purely poloidal fields are unstable, and  the field will likely have a complex geometry containing both poloidal and toroidal components (\cite{Braithwaite_2004,Braithwaite_2006,Duez_2010}). An incoming $l=1$ wave is thus inevitably scattered into higher $\ell$ waves in the presence of a strong magnetic field. %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}). 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.  
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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-mode cavity in the core from the p-mode cavity in the envelope. 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. Even if the wave eventually does return to the surface to create an oscillation mode, the increased time spent in the core translates to a very large mode inertia, greatly reducing the mode visibility.   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 (because  $\nabla \cdot {\bf B} = 0$) of even the simplest magnetic field configurations.