Flame Acceleration in SN Ia Induced by the Burning of Local Unreacted Islands

The mechanism of type Ia supernova explosions is still unsolved due to the uncertainties in the processes of flame acceleration and the associated deflagration to detonation transition. In carbon-oxygen (C-O) white dwarfs, due to the well known Rayleigh-Taylor (R-T) instability and turbulent vortices in the R-T fingers, there exist unburned islands embedded in the downstream of the main flame front. The burning of such islands can be modeled by the spherical inward-propagating two-step C-O flame, which will accelerate while propagating inward due to the stretch effect and the Zel'Dovich mechanism. In turbulent flows under proper conditions, such acceleration can be further strengthened by turbulent mixing, possibly making the burning of these islands speed up to be supersonic. The supersonic burning of the embedded islands is then a source of heating that speeds up the main flame. Here we derive the relationship between the flame's spontaneous inward-propagating speed and the flame radius in consuming the unburned islands based on the Zel'Dovich mechanism in both laminar and turbulent flows. The calculation results show that, in suitable turbulent situation, i.e., for a C-O white dwarf with density 3.5 × 10⁷ g cm^-3, Reynolds number 1.0 × 10¹³, and integral scale 110 km, the island flame speed reaches the sound speed when the flame radius is 1.5 times the flame thickness, and then remains supersonic. Such supersonic combustion generates shock waves, which heat and speed up the main flame, and possibly lead to the initiation of global detonation.