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1. Non adiabatic oscillations of rapidly rotating stars :

Daniel Reese has just developped a non-adiabatic version of the TOP code. This offers the unique possibility to study the effect of rapid rotation on mode excitation. First results have been shown Reese (2015 arXiv:1502.01696) but a lot of work is needed to cover the different pulsating stars of the upper HR diagram. This task requires deformed 2D models like ESTER.

Deliverable : Report on the effect of rotation on mode excitation mechanism and new theoretical instability domain in the HR diagram

2. Patterns in gravito-inertial mode spectra

Summary : The evolution of the asymptotic regular frequency patterns with rotation has been well studied for p-modes and tools to detect them in observed spectra have been developped (see WP2). Some clues of regular patterns have been reported in the low-frequency part of the spectrum among gravito-inertial modes in rapidly rotating polytropic stellar models (Ballot et al. 2012). Here our goal is (i) to confirm their existence for more realistic stellar models, (ii) to construct new pattern recognition tools because the g-mode period separation is no longer uniform for the rotation of typical upper-main sequence pulsation stars.

The numerical computation of p-modes in rotating stars with the TOP code together with a ray based asymptotic theory has demonstrated that regular frequency patterns also exist in rapidly rotating stars. One of the predicted uniform frequency spacing, the large frequency separation, has been recently confirmed by seismic analysis of CoRoT and Kepler targets. In WP2, we describe our strategy to detect regular patterns among p-modes and infer constrains on stellar properties from them. Here, we want to follow the same path for the gravito-inertial modes, that consitute the low-frequency part of the oscillation spectrum. That is, we shall compute gravito-inertial modes in rotating stars with TOP, look for regular patterns and then construct tools to detect them in an observed spectrum. The TOP numerical exploration of the evolution of gravity modes with rotation has already been performed in the case of a polytropic model of star (Ballot et al, 2010). The main regularity of the gravity mode spectrum in a non-rotating star is the asymptotic uniform period spacing of modes of the same degree and consecutive radial order. As rotation increases, it is found that for most of the modes, this spacing is no longer uniform and becomes a function of the ratio 2 Omega/omega and of the azimuthal number m. It is encouraging that some sort of regular behaviour persists when rotational effects are fully taken into account. However, this regularity is not as simple as uniform period spacing. Thus, specific analysis tools must be developed to extract this period spacing from the observed oscillation spectra. In addition, Ballot et al. (2012) found that an approximate separable form of the equations governing the perturbations, the so-called Traditional Approximation, well reproduces the period spacing computed with TOP. This property could help as the Traditional Approximation is easier to use than the complete set of equations. However, contrary to the polytropic model used in Ballot et al. (2012), the internal structure of massive and intermediate-mass stars include a convective core. This property could significantly alter the validity of the Traditional Approximation to reproduce period spacings because subinertial waves (omega < 2 Omega) can propagate in convective region while this is forbbiden by the Traditional Approximation. We shall therefore compute gravitio-inertial modes for a realistic stellar model of main sequence hot star to test the validity of the Traditional Approximation. The open source ESTER code will be used to compute 2D stellar models at different rotation rates (see WP3 for details on the ESTER code). As in Ballot et al. (2010), the TOP code will be used to compute low degree gravity modes and follow their evolution with the rotation of the star. We note that some modes called rosette modes for their peculiar geometry do not show a regular spacing, but they are limited in numbers and they are unlikely to be observed because their small wavelength induces a small visibility in disk-integrated photometric observations.

Deliverable : Report on the existence of patterns in the gravito-inertial mode spectra and on analysis tools to detect them in observed spectra.

3. First 3D DNS of non-linear saturation of non-radial modes

Summary : We shall build on the pioneer numerical simulations of the non-linear saturation of radial modes excited by kappa mechanism (Gastine & Dintrans 2008a,b) and extend it to non-radial modes. Rather than trying to model the observed mode amplitudes, these first 2D Direct Numerical Simulations will be designed to better understand the physical mechanisms (by comparison with the weakly non-linear theory developped by Buchler et al. 1984) and to derive useful constraints on mode amplitudes.

Theoretical advances in taking into account the effects of rotation on oscillation modes has lifted an important obstacle towards the seismology of intermediate-mass and massive stars. The most evident success is the organization of the p-mode spectrum and its predicted regular patterns that can be at least partly detected in observed spectra (See WP2). Further progresses on the gravito-inertial mode spectrum (Task 2) and on the excitation mechanisms (Task1) are expected from the exploitation of the TOP code and its new non-adiabatic version. Nevertheless, we are still unable to predict the amplitude of individual modes. This has two important consequences : first, this observable is not exploited. Second, with a lack of constrain on the expected mode amplitude, the identification of individual modes is much more difficult if not impossible.

While the excitation mechanism is well described by the linearized form of the equations that governs the evolution of perturbations, the saturation of the unstable modes involves non-linear mechanisms that are significantly more complex to model. The saturation process is expected to be weakly non-linear because the ratio between the mode growth rate and the mode frequency is small. In this regime, non-linear interactions involving only two or three modes are possible and the corresponding equations for the amplitudes and phases of these modes have been derived (Buchler et al. 1984). However, in practice, these formalism is too complex for all the potential couplings between modes to be taken into account. One possible way forwards is to solve the full non-linear equations through Direct Numerical Simulations of the perturbations equations. This approach has been only attempted for radial modes excited through the kappa mechanism (Gastine & Dintrans 2008a,b) and our goal is to extend this pioneer work to non-radial modes.

Because of the prohibitive numerical constrains associated with the full 3D simulations in spherical geometry, we shall first consider (as Gastine & Dintrans) 2D DNS in cartesian geometry of the non-linear kappa mechanism. Our first goal is to study the general phenomelogy of the mode coupling that is basically unknow today. For example, Dziembowski (1982) has pointed out an important non-linear saturation mechanism whereby an unstable p-mode couples with a pair of resonnant g-modes of about half its frequency, but we still don't know whether the unstable p-mode will interact with one or various pairs of g-mode and how this will be modified when rotation increases the density of potentially resonnant modes (Nowakowski 2005). The configuration considered in (Gastine & Dintrans 2008a,b) will be tuned to investigate this type of problem. With this pionner work, we do not expect to predict individual mode amplitude for a given star. We rather intent to reach a qualitative understanding of the mode amplitude that translates into constrains on mode identification.

Deliverable : report on the constraints on mode amplitudes from numerical simulations of the non-linear saturation process