  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.
