In this work we have explored the pulsational properties of detailed evolutionary models recently developed by Althaus et al. (2002). Attention has been focused on a ZZ Ceti model in the frame of linear, non-radial oscillations in the adiabatic approximation. White dwarf cooling has been computed in a self-consistent way with the evolution of the chemical abundances resulting from the various diffusion processes and nuclear burning. Element diffusion is based on a multicomponent gas treatment; so, the trace element approximation is avoided in our calculations. In addition, the evolutionary stages prior to the white dwarf formation have been considered. In particular, element diffusion causes near discontinuities in the chemical profile at the start of the cooling branch to be considerably smoothed out by the time the ZZ Ceti domain is reached.

An important aspect of this work has been to assess the role played by the internal chemical stratification of these new models in the behaviour of the eigenmodes, and the expectations for the full g-spectrum of periods. We have analyzed mainly the mode weight functions, which show the regions of the star that mostly contribute to the period formation. Our study suggests the existence of a much wider diversity of eigenmodes for periods shorter than $\approx$ 500-600 s than found in previous works (Brassard et al. 1992a,b). An important finding of this study is the effect of time-dependent element diffusion on the mode trapping properties in DA white dwarfs. We conclude that for periods longer than $\approx$ 500 -600 s all of the modes seem to be partially trapped in the hydrogen-rich envelope of the star. This conclusion, based on the fact that the weight functions of these modes show low amplitudes ( $WF \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$<$ }}}0.1$ ) below the hydrogen-helium transition (even lower as compared with the case of a simulated chemically homogeneous model), implies that the capability of mode selection due to mode trapping effects vanishes for high periods when account is made of white dwarf models with diffusively evolving stratifications. This conclusion is valid at least for massive hydrogen envelopes as predicted by our full evolutionary calculations. This behaviour is markedly different from that found in other studies based on the assumption of diffusive equilibrium in the trace element approximation. This assumption leads to a pronounced peak in the Ledoux term at the hydrogen-helium interface, which is responsible for the trapping of modes in the outer hydrogen-rich layers. We have verified this fact by performing additional pulsation calculations on a model in which the chemical profile at the hydrogen-helium transition is given by equilibrium diffusion in the trace element approximation. Finally, the prediction of both diffusion treatments for the node distribution of the eigenfunctions has been compared. We found that node distribution at the hydrogen-helium chemical interface is very sensitive to the treatment of the chemical profile at that interface.

$\begin{figure} \par\includegraphics[width=10.4cm,clip]{H3435F13.eps} \end{figure}$

Figure 13: The node distribution of eigenfunctions y₁ (filled dots) and y₂ (empty dots) at the hydrogen-helium transition region for modes with $\ell = 2$ , according to the diffusive equilibrium prediction in the trace element approximation.

$\begin{figure}\includegraphics[width=10.4cm,clip]{H3435F14.eps} \end{figure}$

Figure 14: Same as Fig. 13, but for the case of chemical profiles resulting from time dependent element diffusion. See text for details.

On the basis of these new results, we are forced to conclude that for high periods, trapping mechanism in massive hydrogen envelopes of stratified DA white dwarfs is not an appropriate one to explain the fact that all the modes expected from theoretical models are not observed in ZZ Ceti stars. Interestingly, a weaker trapping effect on the periodicities in DB white dwarfs has also been reported (Gautschy & Althaus 2002). We think that the results presented in this work deserves further exploration from the point of view of a non-adiabatic stability analysis. Work in this direction is in progress.

4 Conclusions