deletions | additions       diff --git a/section_Magneto_Gravity_Waves_In__.tex b/section_Magneto_Gravity_Waves_In__.tex  index e6f13f1..b511cd8 100644  --- a/section_Magneto_Gravity_Waves_In__.tex  +++ b/section_Magneto_Gravity_Waves_In__.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_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 For one, the  higher multipole magneto-gravity waves have shorter wavelengths and damp out more quickly than dipole waves. Most More  importantly, the tunneling integral of equation 8 is much larger for higher $\ell$  waves of larger ℓ, are trapped within the radiative core  due to both a thicker evanescent region (see equation \ref{eqn:integral})  separating  the larger value of g-wave cavity in  the integrand and core from  the larger width of acoustic wave cavity in  the evanescent region (because of a higher Lamb frequency for high ℓ modes). envelope.  Therefore, any wave energy with ℓ≳3 $\ell \gtrsim 3$  will be completely trapped within the radiative core. The 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.} Hence, the  initially dipolar wave will be become  trapped in the core until it dissipates, unable to tunnel back toward the surface to create an observable signature.This is the essence of the magnetic greenhouse effect.  A similar effect occurs for magneto-gravity waves We see  thatare reflected at rMG rather than coupling with Alfven waves. The location of rMG is a function of latitude, because  the magnetic field cannot be spherically symmetric (since ∇⋅B=0). Even in the simplest case of a purely dipolar field, greenhouse effect arises not from  the waves will scatter into a broad spectrum alteration  ofℓ (see (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. The  incomingℓ=1  wave is thus inevitably scattered into higher ℓ waves. As described above, these waves cannot couple back frequencies, but rather due  to acoustic modes in modification of the wave angular structure. Such angular modification originates from  the envelope, and remain trapped within inherently non-spherical structure (since $\nabla \cdot {\bf B} = 0$) of even  the radiative core until they dissipate. simplest magnetic field configurations.  However, 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 greenhouse effect arises not from field cannot be spherically symmetric. Even in  the alteration simplest case  of incoming wave frequencies, but rather due to modification 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 angular structure. Such angular modification originates from is thus inevitably scattered into higher $\ell$ waves in  the inherently non-spherical structure presence  of even a strong magnetic field. As described above, these waves cannot couple back to acoustic modes in  the simplest field configurations. envelope, and remain trapped within the radiative core until they dissipate.  We have shown that, due to the magnetic greenhouse effect, waves that tunnel from the envelope into the radiative region cannot escape from a strongly magnetized stellar core. Their degree of suppression is therefore controlled by the degree of reflection at the bottom of the acoustic cavity (as determined by the tunneling integral through the evanescent region, see Eq. 8). This quantity can be used, as a first approximation, to predict the visibility of the ℓ=1,2 modes (Fig. undefined).