deletions | additions       diff --git a/subsection_Ray_Tracing_label_ray__.tex b/subsection_Ray_Tracing_label_ray__.tex  index 16708a1..c801139 100644  --- a/subsection_Ray_Tracing_label_ray__.tex  +++ b/subsection_Ray_Tracing_label_ray__.tex 
...
However, in the presence of a magnetic field, the last two terms of equation \ref{eqn:dkdt} break the spherical symmetry. Except in the unphysical case of a purely radial field or a constant field, these terms are non-zero, and therefore the angular component of the wave vector must change. At the radius $r_{\rm MG}$ where $v_A \sim \omega^2/(N k_\perp)$, each term in equation \ref{eqn:dkdt} is the same order of magnitude, assuming $|\nabla B|/B \sim 1/r$. Therefore, the rate of change in horizontal wavenumber is comparable to the rate of change in radial wavenumber at field strengths near $B_c$. Upon wave reflection or conversion into Alfven waves, the radial wavenumber will generally change by order unity, i.e., the change in radial wavenumber is $|\Delta k_r| \sim |k_r|$. We therefore expect a correspondingly large change in $k_\perp$, such that $|\Delta k_\perp| \sim |k_r|$. Hence, dipole waves will generally obtain high multipole moments when they propagate through strongly magnetized regions of the star.  {\bf Additional insight can be gained from a simple thought experiment. Consider a simplified model of a star with a purely radial field whose field strength increases with depth such that $d v_A/dr <0$, and with constant $N$. In this case the value of $k_\perp$ is conserved. For a downward propagating gravity wave (i.e., a fast wave with $k_r >0$), the wave will reach a turning point where $d r/dt =0$, which occurs where $B \approx B_c$. Examination of equation \ref{eqn:dkdt} shows that the value of $k_r$ increases steadily through the turning point, and $k_r$ continues to increase as the wave propagates back outward. When the wave propagates back outward, the increased value of $k_r$ causes the last term of equation \ref{eqn:dxdt} to dominate. In other words, the wave transitions into a slow Alfv\'enic wave as it reaches the turning point and propagates back outward.Although this scenario is highly simplified, there is a general tendency for waves to be converted into slow waves as they propagate into regions of increasing field strength. The conversion into slow waves }  Although this scenario is highly simplified, it demonstrates the tendency for waves to be converted into slow waves as they propagate into regions of increasing field strength. The conversion into slow waves is complete as long as the waves reach the magnetic turning point where $B \approx B_c$. If instead the waves reflect at the center of the star, the sign of $k_r$ is flipped upon reflection, and magnetic alterations previously made to the ingoing wave are undone as the wave propagates back outward.  The results of this thought experiment are threefold. First, fast (gravity-dominated) magneto-gravity waves are largely converted to slow (Alfv\'enic) waves if they encounter a radial field of field strength $B_c$. This conversion may be the primary agent for sapping the energy of the ingoing gravity waves 2. Waves which are not turned by the magnetic field (and instead are reflected at the center of the star) will not be converted to slow waves, and may lose much less of their energy. This may help explain the apparent dichotomy of observed dipole mode visibility in Figure \ref{fig:moneyplot}: the visibility is generally consistent with total wave energy loss in the core or zero wave energy loss in the core, corresponding to stars with $B>B_c$ and those with $BB_c$ would still exhibit depressed dipole modes, due to the conversion of ingoing waves into slow waves rather than the conversion to large angular degrees $\ell$. }