  The thermonuclear explosion of a massive white dwarf in a Type Ia supernova
explosion is characterized by vastly disparate spatial and temporal scales. The
extreme dynamic range inherent to the problem prevents the use of direct
numerical simulation and forces modelers to resort to subgrid models to
describe physical processes taking place on unresolved scales.
  We consider the evolution of a model thermonuclear flame in a constant
gravitational field on a periodic domain. The gravitational acceleration is
aligned with the overall direction of the flame propagation, making the flame
surface subject to the Rayleigh-Taylor instability. The flame evolution is
followed through an extended initial transient phase well into the steady-state
regime. The properties of the evolution of flame surface are examined. We
confirm the form of the governing equation of the evolution suggested by
Khokhlov (1995). The mechanism of vorticity production and the interaction
between vortices and the flame surface are discussed. The results of our
investigation provide the bases for revising and extending previous
subgrid-scale model.
