1 Introduction

Since photometric variations were detected in the white dwarf HL Tau 76 (Landolt 1968), astronomers have been observing multimode pulsations in an increasing number of these objects. Of particular interest are the variable white dwarfs characterized by hydrogen-rich atmospheres. These variable stars, known in the literature as ZZ Ceti or DAV stars, constitute the most numerous group amongst degenerate pulsators. Other class of pulsating white dwarfs are the DBV, with helium-rich atmospheres, and the pre-white dwarfs DOVs and PNNVs, which show spectroscopically pronounced carbon, oxygen and helium features (for reviews of the topic, see Winget 1988 and Kepler & Bradley 1995). In particular, ZZ Ceti stars are found in a narrow interval of effective temperature ( $T_{\rm eff}$) ranging from 12 500 K $\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$>$ }}}T_{\rm eff} \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$>$ }}}$ 10 700 K. Their brightness variations, which reach up to 0.30 mag, are interpreted as being caused by spheroidal, non-radial g(gravity)-modes of low degree ( $\ell \leq 2$) and low and intermediate overtones k (the number of zeros in the radial eigenfunction), with periods (P_k) between 2 and 20 min. Radial modes, although found overstables in a number of theoretical studies of pulsating DA white dwarfs (see, e.g. Saio et al. 1983), have been discarded as the cause of variability in such stars. This is so because the periods involved are shorter than 10 s. Observationally these high-frequency signatures have not been detected thus far. With regard to the mechanism that drives pulsations, the $\kappa-\gamma$ mechanism is the traditionally accepted one (Dolez & Vauclair 1981; Winget et al. 1982). Nonetheless, Brickhill (1991) proposed the convective driving mechanism as being responsible for the overstability of g-modes in DAVs (see also Goldreich & Wu 1999). Although both mechanisms predict roughly the observed blue edge of the instability strip, none of them are capable to yield the red edge, where pulsations of DA white dwarfs seemingly cease in a very abrupt way (Kanaan 1996).

A longstanding problem in the study of pulsating DA white dwarfs is to find the reason of why only a very reduced number of modes are observed, as compared with the richness of modes predicted by theoretical studies. Indeed, it has been long suspected that some filtering mechanism must be acting quite efficiently. The explanation commonly proposed is that of "mode trapping'' phenomenon (Winget et al. 1981; Brassard et al. 1992a; Bradley 1996). According to this mechanism, those modes (the trapped ones) having a local radial wavelength comparable with the thickness of the hydrogen envelope, require low kinetic energy to reach observable amplitudes. Then, most of the observed periods should correspond to trapped modes. Nevertheless, in a recent asteroseismological study of the ZZ Ceti star G117-B15A (Bradley 1998), the observed period of 215 s, which has the larger amplitude in the power spectrum, does not correspond to the trapped mode predicted by the best fitting model.

The exploration of these very important aspects requires the construction of detailed DA white dwarf evolutionary models, particularly regarding the treatment of the chemical abundance distribution. Work in this direction has recently begun to be undertaken. In fact, Althaus et al. (2002) have carried out full evolutionary calculations which take into account time dependent element diffusion, nuclear burning and the history of the white dwarf progenitor in a self-consistent way. Specifically, these authors have followed the evolution of an initially 3 $M_{\odot }$ stellar model all the way from the stages of hydrogen and helium burning in the core through the thermally pulsing and mass loss phases till the white dwarf state. Althaus et al. (2002) find that the shape of the Ledoux term (an important ingredient in the computation of the Brunt-Väisälä frequency; see Brassard et al. 1991, 1992a,b) is markedly different from that found in previous detailed studies of white dwarf pulsations. This is due partly to the effect of smoothness in the chemical abundance distribution caused by element diffusion, which gives rise to less pronounced peaks in the Ledoux term. This, in turn, leads to a substantially weaker mode trapping effect, as it has recently been found by Córsico et al. (2001). These authors have presented the first results regarding the trapping properties of the Althaus et al. (2002) evolutionary models.

The present work is designed to explore at some length the pulsation properties of the Althaus et al. (2002) evolutionary models and to compare our predictions with those of others investigators. In addition, the work is intended to bring some more insight to the phenomenon of mode trapping in the frame of these new evolutionary models. In particular, we shall restrict ourselves to analyse the same stellar model as that studied in Córsico et al. (2001). Specifically, the model, which belongs to the ZZ Ceti instability strip, is analyzed in the frame of linear, non-radial stellar pulsations in the adiabatic approximation. Emphasis will be placed on assessing the role played by the internal chemical stratification in the behaviour of eigenmodes, and the expectations for the full spectrum of periods. Specifically, we shall explore the effects of the chemical interfaces on the kinetic energy distribution of the modes and their ability to modify the properties of the g-mode propagation throughout the star interior. We want to mention that, because of the high computational demands involved in the evolutionary calculations in which white dwarf cooling is assessed in a self-consistent way with element diffusion and the history of the pre-white dwarf, we are forced to restrict our attention to only one value for the stellar mass.

The article is organized as follows. In Sect. 2, we briefly describe our evolutionary and pulsational codes. In this section we also discuss some aspects concerning the evolutionary properties of our models. In Sect. 3 we present in detail the pulsation results. Finally, Sect. 4 is devoted to summarizing our findings.