Magneto-gravity waves cannot exist in regions with $B_r \! > \! B_c$ where magnetic tension overwhelms the buoyancy force, i.e., the stiff field lines cannot be bent by the placid gravity wave motion. Consequently, dipole magneto-gravity waves become evanescent when $\omega \! < \! \omega_{\rm MG}$,  where the magneto-gravity frequency $\omega_{\rm MG}$ is defined as

\begin{equation}
\label{eqn:maggrav}
\omega_{\rm MG} = \bigg[ \frac{2}{\pi} \frac{B_r^2 N^2}{\rho r^2} \bigg]^{1/4} \, .
\end{equation}
Fig. \ref{fig:Prop} shows a wave propagation diagram in which a strong internal magnetic field prevents magneto-gravity wave propagation in the core.

In red giants, $B_c$ is typically smallest at the peak in $N$ corresponding to the sharp density gradient within the hydrogen burning (H-burning) shell. Therefore, gravity waves are most susceptible to magnetic alteration in the H-burning shell, and the observation of a star with depressed dipole modes thus provides a {\it lower limit} to the radial field strength (Eqn. \ref{eqn:Bc}) evaluated in the H-burning shell. We refer to this field strength as $B_{c,{\rm min}}$. Magnetic suppression via horizontal fields can also occur, but in general requires much larger field strengths.

In stars with field strengths exceeding $B_c$ (Eqn. \ref{eqn:Bc}) somewhere in their core, incoming dipole gravity waves will become evanescent where $B_r \! > \! B_c$. At this point, the waves must either reflect or be transmitted into the strongly magnetized region as Alfv\'en waves. In either case, the reflection/transmission process modifies the angular structure of the waves such that their energy is spread over a broad spectrum of $\ell$ values (supplementary material). Once a dipole wave has its energy transferred to higher values of $\ell$, it will not substantially contribute to observable oscillations at the stellar surface, because higher $\ell$ waves are trapped within the radiative core by a thicker evanescent region (see Eqn. \ref{eqn:integral2} and Fig. \ref{fig:cartoon}) separating the core from the envelope. Even if some wave energy does eventually return to the surface to create an oscillation mode, the increased time spent in the core results in a very large mode inertia, greatly reducing the mode visibility. Additionally, high $\ell$ waves will not be detected in {\it Kepler} data due to the geometric cancellation which makes $\ell \! \gtrsim \! 3$ modes nearly invisible \cite{Bedding_2010}.

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 (because $\nabla \cdot {\bf B} = 0$) of even the simplest magnetic field configurations.

Dipole oscillation modes can be suppressed if the magnetic field strength exceeds $B_c$ (Eqn. \ref{eqn:Bc}) at some point within the core. We therefore posit that stars with depressed dipole oscillation modes have minimum core field strengths of $B_{c,{\rm min}}$. Stars with normal dipole oscillation modes cannot have radial field strengths in excess of $B_{c,{\rm min}}$ within their H-burning shells. However, they may contain larger fields away from the H-burning shell, or they may contain fields that are primarily horizontal (e.g., strong toroidal fields).
  
  