Distributed Flames in Type Ia Supernovae

At a density near a few ×10 g cm, the subsonic burning in a Type Ia supernova enters the distributed regime (high Karlovitz number). In this regime, turbulence disrupts the internal structure of the flame, and so the idea of laminar burning propagated by conduction is no longer valid. The nature of the burning in this distributed regime depends on the turbulent Damköhler number (DaT ), which steadily declines from much greater than one to less that one as the density decreases to a few ×10 g cm. Classical scaling arguments predict that the turbulent flame speed sT , normalized by the turbulent intensity ǔ, follows sT/ǔ = Da 1/2 T for DaT <∼1. The flame in this regime is a single turbulentlybroadened structure that moves at a steady speed, and has a width larger than the integral scale of the turbulence. The scaling is predicted to break down at DaT ≈ 1, and the flame burns as a turbulently-broadened effective unity Lewis number flame. This flame burns locally with speed sλ and width lλ, and we refer to this kind of flame as a λ-flame. The burning becomes a collection of λ-flames spread over a region approximately the size of the integral scale. While the total burning rate continues to have a well-defined average, sT ∼ ǔ, the burning is unsteady. We present a theoretical framework, supported by both 1D and 3D numerical simulations, for the burning in these two regimes. Our results indicate that the average value of sT can actually be roughly twice ǔ for DaT >∼1, and that localized excursions to as much as five times ǔ can occur. We also explore the properties of the individual flames, which could be sites for a transition to detonation when DaT ∼ 1. The λ-flame speed and width can be predicted based on the turbulence in the star (specifically the energy dissipation rate ε) and the turbulent nuclear burning time scale of the fuel τ nuc . We propose a practical method for measuring sλ and lλ based on the scaling relations and small-scale computationally-inexpensive simulations. This suggests that a simple turbulent flame model can be easily constructed suitable for large-scale distributed supernovae flames. These results will be useful both for characterizing the deflagration Lawrence Berkeley National Laboratory, 1 Cyclotron Road, MS 50A-1148, Berkeley, CA 94720 Department of Astronomy and Astrophysics, University of California at Santa Cruz, Santa Cruz, CA 95064