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\subsection{The red giant KIC 8366239}  We have chosen to model the red giant $\KIC$, for which \citet{Beck:2012} have observed a minimum rotational splitting of mixed modes of $\delta\nu_{n,\ell} = 0.135\pm0.008 \mu$Hz, arising from a p-dominated mode (mostly probing the slowly rotating stellar envelope). On the other hand the observed maximum values for the g-dominated mixed modes (mostly living in the stellar core) is on the order $0.2\mu$Hz. This confirmed the theoretical expectation that the core of this red giant is rotating faster than its envelope. However the inferred ratio is not very large, implying that the core is rotating only about 10 times more rapidly than the core. This is at odd with the expectations coming from models that only include transport of angular momentum from rotational instabilities and circulations, which assuming typical initial rotational velocities\footnote{Models with very slow initial rotation of $1\kms$ can not reproduce the observation either} predict a ratio of order $10^3$ between the angular velocity of the core and that of the envelope \citep[][ this work]{Eggenberger:2012}. Similar to \citet{Eggenberger:2012} we adopt an initial mass of $1.5\mso$ and calculate models assuming different physics for angular momentum transport. We select the background structure of the different calculations by matching the global asteroseismic properties of $\KIC$ (frequency of maximum oscillation power, $\nu_{\rm max}$, and large frequency separation $\Delta \nu$), as derived by \citet{Beck:2012}. The MESA background structure and rotational profiles are then used in ADIPLS to calculate the splitting of mixed modes for the different assumptions on the angular momentum transport. We show in Fig.~\ref{kernels} an example of the background rotational profile calculated by MESA, together with the radial integrals of the rotational kernels $K_{n,\ell}$ for $\ell=1,2$ calculated using ADIPLS. This figure shows how the p-dominated modes mostly probe the envelope of the star, where the angular velocity is quite low. The gravity dominated modes on the other hand tend to probe the radiative region below the H-burning shell, in which the angular velocity is higher. Note that higher $\ell$ modes have higher Lamb frequencies, implying a larger tunneling zone between the acoustic cavity in the envelope and the gravity modes region in the core. As a consequence a the modes become ``less mixed'', with $\ell=2$ p-modes (g-modes) being more p-like (g-like) than their $\ell=1$ analogue.   Our calculations confirm the results of \citet{Eggenberger:2012}. Models that only include angular momentum transport due to rotational instabilities and circulations fail to reproduce the observed splittings. The resulting cores are so rapidly rotating that the the perturbative approach to the splitting calculation is no more justified. Even models with an extremely slow initial rotation of $1 \kms$ result in rotational splittings one order of magnitude larger than the observed ones, which clearly shows this class of models can not explain the observations. This is in perfect agreement with the calculations of \citet{Eggenberger:2012}, even if the implementation of the physics of rotation is quite different in the GENEVA code compared to MESA \citep[See Sec.6 in ][and references therein]{Paxton:2013}.     We found that models including also angular momentum transport due to Spruit-Tayler magnetic fields \citep[which are generated by differential rotation in radiative regions][]{Spruit:2002} do a much better job, but still result in rotational splittings on the order of $1\mu$Hz. We explored if an increase in the efficiency of the ST mechanism could reconcile the models with the observations. However even increasing the efficiency of the diffusion coefficient resulting from the magnetic torques by a factor 100 result in a very small change in the overall coupling. This is due to the self-regulating nature of the Spruit-Tayler dynamo. The poloidal component of the magnetic field $B_r$ is generated by the Tayler instability that occurs in the toroidal component $B_{\phi}$. However the toroidal component is amplified by the differential rotation, which is in turn suppressed by the torque $\propto B_r B_{\phi}$. It is not too surprising then to observe that the system tend to relax around some average differential rotation state which is weakly dependent on a possible increase of the efficiency of the Tayler-Spruit dynamo loop.     We recall that, while the physics of the Tayler instability is solid, the existence of the Tayler-Spruit dynamo loop is currently theoretically debated \citep{Braithwaite:2006,Zahn:2007}. From the observational point of view observations of the spin rates of compact objects (WD and NS) are in much better agreement with models including this angular momentum transport mechanism \citep{Heger:2005,larends_Yoon_Heger_Herwig_2008}, which has also been discussed in the context of the rigid rotation of the solar core \citep{Eggenberger:2005}, but see also \citet{Denissenkov:2010}