We revisit the problem of the formation of DB white dwarfs, as well as the origin of hydrogen in DBA stars, using a new set of envelope model calculations with stratified and mixed hydrogen/helium compositions. We first describe an approximate model to simulate the so-called convective dilution process, where a thin, superficial hydrogen radiative layer is gradually eroded by the underlying and more massive convective helium envelope, thus transforming a DA white dwarf into a DB star. We show that this convective dilution process is able to account for the large increase in the number of DB white dwarfs below Teff ~ 20,000 K, but that the residual hydrogen abundances expected from this process are still orders of magnitude lower than those observed in DBA white dwarfs. Scenarios involving the accretion of hydrogen from the interstellar medium or other external bodies have often been invoked to explain these overabundances of hydrogen. In this paper, we describe a new paradigm where hydrogen, initially diluted within the thick stellar envelope, is still present and slowly diffusing upward in the deeper layers of a Teff ~ 20,000 K white dwarf. When the convective dilution process occurs, the bottom of the mixed H/He convection zone sinks deep into the star, resulting in large amounts of hydrogen being dredged-up to the stellar surface, a phenomenon similar to that invoked in the context of DQ white dwarfs.
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