
\subsection{Stellar Models}

We have used the Modules for Experiments in Stellar Evolution (MESA, release 7385) code \cite{Paxton_2010,Paxton_2013} to evolve low-mass stars with initial mass in the range 1-3.0$\,M_\odot$. Models have been evolved from the pre-main-sequence to the tip of the red giant branch. We chose a metallicity of $Z=0.02$ with a mixture taken from  \cite{2005ASPC..336...25A}; the plasma opacity is determined  using the OPAL opacity tables from \cite{Iglesias_1996}. Convective regions are calculated using the mixing-length theory (MLT) with $\alpha_{\rm MLT} = 2.0$. The boundaries of convective regions are determined using the Ledoux criterion. Overshooting is parameterized by an exponentially decaying diffusivity that decays over a distance $f_{\rm ov} H$ above the convective boundary \cite{2000A&A...360..952H}, with $f_{\rm ov}= 0.018$. We include in Section \ref{inlist} the inlist used for running the calculations.

To estimate plausible magnetic field strengths within the cores of red giants, we consider two scenarios: one based on observational evidence of surface {fi}elds, and one based on MHD modeling of magnetic {fi}elds in convective stellar cores. First, we extrapolate inward from a main sequence surface field of $B \! \sim \! 3\, {\rm kG}$, as appropriate for magnetic Ap stars \cite{Auri_re_2007}, assuming the field is a pure dipole such that the field strength scales as $B \! \propto \! r^{-3}$. Since the radius of the convective core is typically $r_c \! \sim \! R/10$ for low mass main sequence stars, field strengths of $B \! > \! 10^{5} \, {\rm G}$ are attainable near the core.

Second, we estimate field strengths produced by a magnetic dynamo that operates within the convective core of a star while it is on the main sequence. In this case, MHD simulations suggest equipartition (and even super-equipartition) magnetic field strengths may be generated \cite{Featherstone_2009}, i.e., magnetic fields whose energy density is comparable to that of the kinetic energy of convective flows such that

\begin{equation}
\label{eqn:beq}
\frac{B^2}{8 \pi} \sim \rho v_{\rm con}^2 \, .
\end{equation}
We calculate typical core convective velocities $v_{\rm con}$ using mixing length theory. In our stellar models, we evaluate Eqn. \ref{eqn:beq} to find that core magnetic fields of $B \! \sim \! 3 \! \times \! 10^5 \, {\rm G}$ could be generated during the main sequence.

To extrapolate to field strengths plausibly obtained within the radiative cores of red giants, we assume that the magnetic flux (calculated via the methods above) within the core is conserved as it contracts. This is a good approximation for stable magnetic equilibria discussed here because the timescale for the field to diffuse through the star (the Ohmic timescale) is longer than the main sequence timescale. At each mass shell within a red giant, the field strength is then approximated by

\begin{equation}
\label{eqn:BRG}
B_{\rm RG} = \bigg(\frac{r_{\rm MS}}{r_{\rm RG}}\bigg)^2 B_{\rm MS} \, ,
\end{equation}
where $B_{\rm RG}$ is the field strength while on the RGB, $r_{\rm RG}$ and $r_{\rm MS}$ are the radial coordinates of the shell on the RGB and MS, respectively, and $B_{\rm MS}$ is the MS field strength. Mass shells enclosing $M \sim 0.2 \, M_\odot$ (which are located just outside the MS core and near the H-burning shell on the lower RGB) typically contract by a factor of a few from the MS to the lower RGB. The magnetic field may therefore be amplified and field strengths in excess of $10^6 \, {\rm G}$ are quite plausible within the H-burning shells of RGB stars.

Fig. \ref{Fig:Struc} shows the density, mass, and magnetic field profiles of the $1.6 \, M_\odot$ stellar model used to generate Fig. \ref{fig:Prop}. To make this model, we extrapolate a dipole field inward from a surface value of $3 \! \times \! 10^{3} \, {\rm G}$ (as described above), with an artificial cap at a field strength of $5 \! \times \! 10^{4} \, {\rm G}$. We then calculate the corresponding RGB field profile using the flux conservation described above (for simplicity we set the field equal to zero in convective regions of the RGB model). This relatively conservative approach yields a field strength of $\sim \! \! 10^6 \, {\rm G}$ at the H-burning shell, sufficient for magnetic suppression of dipole oscillation modes. We note that field strengths of this magnitude are orders of magnitude below equipartition with the gas pressure, and therefore have a negligible influence on the stellar structure.