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Jim Fuller edited subsection_Joule_Damping_A_gravity__.tex
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\subsection{Rotation}
In principle, any physical effect that breaks the spherical symmetry of the star within the radiative core will give rise to the greenhouse effect described in Section \ref{greenhouse}. Other than magnetic fields, the only obvious candidate is rotation, which breaks spherical symmetry via the Coriolis and centrifugal forces. Here we briefly describe how rotation could lead to dipole mode suppression, while in Section \ref{norotation} we discuss why rotation is a less likely candidate in most stars.
It is well known that the effects of rotation on oscillation mode eigenfunctions become large when the Coriolis parameter is of comparable to the mode frequency, i.e., when $\nu \sim 2 \nu_s$, where $\nu_s$ is the spin frequency. It is also well known that each region of a stable (differentially rotating) star must spin below the breakup frequency, $\nu_K = \sqrt{G M(r)/r^3}/(2 \pi)$. For an ascending red-giant like the one shown in Figure \ref{Fig:Struc}, the surface breakup frequency is $\nu_K \sim 8 \, \mu{\rm Hz}$, while $\nu_{\rm max} \sim 100 \, \mu{\rm Hz}$. It is therefore impossible for the Coriolis force to strongly affect the acoustic oscillations near the surface of the star.
The centrifugal force can still affect acoustic modes by distorting the spherical symmetry of the background star. This can affect their visibility, as discussed in \cite{Reese_2013}. However, in this case we should expect all modes to be affected, not just the dipole modes. It would also require the surface rotation rate to be a significant fraction of breakup, which for the model in Figure \ref{Fig:Struc} corresponds to a rotation period of $P_K \sim 1.5 \, {\rm days}$.
%Neither of these effects seems to be observed in the majority of suppressed dipole stars.
However, rotation {\it can} strongly affect the modes in the core of a red giant. The breakup frequency evaluated near the H-burning shell of the model in Figure \ref{Fig:Struc} is $\nu_K {\rm (H-burn)} \sim 8 \times 10^3 \, \mu{\rm Hz} \gg \nu_{\rm max}$, and therefore the Coriolis force can have a strong influence on modes if the core of the star is rapidly rotating. Hence, rotation will only be important in the g-mode cavity of modes with frequencies near $\nu_{\rm max}$. This also requires very strong differential rotation between the core and envelope.
We can speculate on the effect of rotation on a dipole wave as it travels from a slowly rotating envelope to a rapidly rotating core. We will assume the rotation rate increases gradually such that a WKB analysis remains valid. In this case, the effects of rotation on the high order gravity waves is captured by the traditional approximation, in which the Coriolis force changes the gravity waves into Hough waves. The Hough waves are very similar to gravity waves except that their angular structure and radial wavelength are altered. When $2 \nu_s \gtrsim \nu$, the angular structure of the dipole waves will be strongly altered, and the waves will be composed of a broad spectrum of angular degrees $\ell$. This spectrum generally depends on the $m$ value of the incoming wave, i.e., whether the incoming wave was axisymmetric, prograde, or retrogade relative to the spin.
One key difference between the symmetry-breaking effects of rotation compared to a magnetic field is that rotation (in the WKB limit) will not create an evanescent region. Therefore, the waves will not be reflected by rapidly rotating layers (although prograde waves may be absorbed at critical layers), and the waves will continue to propagate toward the center of the star as Hough waves. When they reflect near the center of the star, our WKB analysis breaks down. At this point, the waves may be reflected onto the same branch of Hough waves, or energy may be transferred to other Hough wave branches, or to inertial waves or Rossby waves.
If the wave energy is reflected into inertial waves or Rossby waves, it will likely remain trapped in the core, because these waves can only exist in regions where $\nu < 2 \nu_s$. Moreover, these waves are generally composed of a broad spectrum of $\ell$, and will thus be trapped by the evanescent region between the core and envelope. It is therefore conceivable that rapid rotation can suppress the visibilities of dipole modes in a manner similar to the magnetic greenhouse effect.
\subsubsection{Problems with the Rotational Interpretation}
\label{norotation}
Although rotation can in principle lead to dipole mode suppression, we consider it to be less likely in most circumstances. First, it requires rapid core rotation rates
\begin{equation}
P_{\rm core} \lesssim 2\times 10^4 \bigg(\frac{\nu_{\rm max}}{100\,\mu{\rm Hz}}\bigg)^{-1}\,{\rm s}.
\end{equation}
This rotation period is approximately 2 orders of magnitude smaller than the RGB core rotation periods measured by \cite{Mosser_2012}.
Such rapid rotation could be generated by a stellar merger on the RGB, if some of the orbital angular momentum is deposited deep in the core of the red giant where $\nu_{\rm max} < \nu_K$. It is not clear whether this can commonly occur, and we speculate that stellar mergers cannot account for the suppressed fraction of $22 \%$ found in \cite{Mosser_2011}. Post-merger RGB stars will likely be rapidly rotating, and thus we may expect that suppressed dipole pulsators should always be rapidly rotating. In the case of the suppressed pulsator KIC 8561221 (\cite{Garcia_2014}), the envelope is not rapidly rotating. We therefore consider it unlikely that rapid core rotation can account for a large fraction of the suppressed pulsators.
%Although many suppressed dipole pulsators are rapidly rotating (Stello et al. 2015), many are not.
Rapid core rotation would also require a large amount of differential rotation to exist within the star. Although angular momentum transport is not well understood, the slow rotation rates observed in the cores of many RGB stars (\cite{Mosser_2012,Beck_2011,Deheuvels_2014}) suggests that such large degrees of differential rotation are difficult to sustain as a star evolves up the RGB.
Rotation may also cause different levels of suppression between axisymetric, prograde, and retrograde modes. However, no such differential suppression has been noted in observations. Nor do depressed pulsators show anomalous peaks due to very large and asymmetric amounts of rotational splitting (Dennis Stello, private communication). In fact, the oscillation spectrum of depressed pulsators is generally very simple: they simply exhibit less power in dipole modes. We therefore find a rotational explanation unlikely for most depressed pulsators.
