When the core hydrogen is exhausted during stellar evolution, the central region of a star contracts and the outer envelope expands and cools, giving rise to a red giant. Convection takes place over much of the star's radius. Conservation of angular momentum requires that the cores of these stars rotate faster than their envelopes; indirect evidence supports this 1,2 . Information about the angular-momentum distribution is inaccessible to direct observations, but it can be extracted from the effect of rotation on oscillation modes that probe the stellar interior. Here we report an increasing rotation rate from the surface of the star to the stellar core in the interiors of red giants, obtained using the rotational frequency splitting of recently detected 'mixed modes' 3,4 . By comparison with theoretical stellar models, we conclude that the core must rotate at least ten times faster than the surface. This observational result confirms the theoretical prediction of a steep gradient in the rotation profile towards the deep stellar interior 1,5,6 .The asteroseismic approach to studying stellar interiors exploits information from oscillation modes of different radial order n and angular degree l, which propagate in cavities extending at different depths 7 . Stellar rotation lifts the degeneracy of non-radial modes, producing a multiplet of (2l 1 1) frequency peaks in the power spectrum for each mode. The frequency separation between two mode components of a multiplet is related to the angular velocity and to the properties of the mode in its propagation region. More information on the exploitation of rotational splitting of modes may be found in the Supplementary Information. An important new tool comes from mixed modes that were recently identified in red giants 3,4 . Stochastically excited solar-like oscillations in evolved G and K giant stars 8 have been well studied in terms of theory [9][10][11][12] , and the main results are consistent with recent observations from space-based photometry 13,14 . Whereas pressure modes are completely trapped in the outer acoustic cavity, mixed modes also probe the central regions and carry additional information from the core region, which is probed by gravity modes. Mixed dipole modes (l 5 1) appear in the Fourier power spectrum as dense clusters of modes around those that are best trapped in the acoustic cavity. These clusters, the components of which contain varying amounts of influence from pressure and gravity modes, are referred to as 'dipole forests'.We present the Fourier spectra of the brightness variations of stars KIC 8366239 (Fig. 1a), KIC 5356201 ( Supplementary Fig. 3a) and KIC 12008916 ( Supplementary Fig. 5a), derived from observations with the Kepler spacecraft. The three spectra show split modes, the spherical degree of which we identify as l 5 1. These detected multiplets cannot have been caused by finite mode lifetime effects from mode damping, because that would not lead to a consistent multiplet appearance over several orders such as that shown in Fig. 1. ...

Abstract. We apply here the Time Dependent Convection (TDC) treatment presented in our earlier paper in this series to the study of δ Sct and γ Dor pulsating stars. Stabilization of the δ Sct p-modes at the red edge of the Instability Strip (IS) and the driving of the γ Dor g-modes are explained by our models. Theoretical IS obtained with different values of the Mixing Length (ML) parameter α are compared to observations and a good agreement is obtained for α between 1.8 and 2. The influence of each term of our TDC treatment (perturbation of convective flux, turbulent pressure, and turbulent kinetic energy dissipation) on the eigenfrequencies and on the driving and damping mechanisms is investigated. Finally, we show that our TDC models predict the likely existence of hybrid stars with both δ Sct p-modes and γ Dor g-modes oscillations.

Context. The CoRoT mission has provided thousands of red-giant light curves. The analysis of their solar-like oscillations allows us to characterize their stellar properties. Aims. Up to now, the global seismic parameters of the pressure modes have been unable to distinguish red-clump giants from members of the red-giant branch. As recently done with Kepler red giants, we intend to analyze and use the so-called mixed modes to determine the evolutionary status of the red giants observed with CoRoT. We also aim at deriving different seismic characteristics depending on evolution.Methods. The complete identification of the pressure eigenmodes provided by the red-giant universal oscillation pattern allows us to aim at the mixed modes surrounding the = 1 expected eigenfrequencies. A dedicated method based on the envelope autocorrelation function is proposed to analyze their period separation. Results. We have identified the mixed-mode signature separation thanks to their pattern that is compatible with the asymptotic law of gravity modes. We have shown that, independent of any modeling, the g-mode spacings help to distinguish the evolutionary status of a red-giant star. We then report the different seismic and fundamental properties of the stars, depending on their evolutionary status. In particular, we show that high-mass stars of the secondary clump present very specific seismic properties. We emphasize that stars belonging to the clump were affected by significant mass loss. We also note significant population and/or evolution differences in the different fields observed by CoRoT.

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Abstract. We present in details a time-dependent convection treatment in the frame of the Mixing-Length Theory (MLT). Following the original ideas by Unno (1967, PASJ, 19, 140), this theory has been developed by Gabriel et al. (1974, Bull. Ac. Roy. Belgique, Classe des Sciences, 60, 866) and Gabriel (1996, Bull. Astron. Soc. India, 24, 233). In this paper, we present it in a united form, we detail the basic derivations and approximations and give final improvements. A new perturbation of the energy closure equation is proposed for the first time, making it possible to avoid the occurrence of short wavelength spatial oscillations of the thermal eigenfunctions. This theory accounts for the perturbation of the convective flux, the turbulent Reynolds stress and the turbulent kinetic energy dissipation. It has been numerically implemented in a non-radial non-adiabatic pulsation code and the first results published in a Letter by Dupret et al. (2004a, A&A, 414, L17) indicate that the theory predicts the observed red border of the lower end of the instability strip and the driving mechanism of the recently discovered γ Dor stars.