deletions | additions       diff --git a/subsection_Magnetic_Greenhouse_Effect_label__.tex b/subsection_Magnetic_Greenhouse_Effect_label__.tex  index 5b5b14c..92374a6 100644  --- a/subsection_Magnetic_Greenhouse_Effect_label__.tex  +++ b/subsection_Magnetic_Greenhouse_Effect_label__.tex 
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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, any oscillation modes created by the waves will contain 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.   Additionally, magneto-gravity waves reflected at $r_{\rm MG}$ will dissipate much faster than normal dipole waves. First, they may be reflected onto the slow branch of magneto-gravity waves, which occursfor reflected waves  in the solar atmosphere (\cite{Newington_2009,Newington_2011}). In our case, as the slow waves propagate back outward into regions with weaker magnetic fields, their wavelength decreases rapidly (see supplementary material). The slow waves will thus dissipate rapidly via radiative diffusion. A less dramatic version of this effect also occurs for waves reflected back onto the fast branch, as higher $\ell$ gravity waves have shorter wavelengths and damp out more quickly than dipole waves. 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.