Distributed flames and Damk\"ohler ``small-scale turbulence'' in type Ia supernovae

High-resolution three-dimensional simulations of carbon-burning flames in type Ia supernovae are used to examine the distributed burning regime (high Karlovitz numbers), or the ``small-scale turbulence'' regime as it was referred to by Damk\"ohler (1940). He predicted that the turbulent flame speed and width are determined by the nuclear burning time scale and a diffusion coefficient prescribed by the turbulence. The scaling predicts its own breakdown when the turbulent Damk\"ohler number reaches unity. We demonstrate that the simulations are in agreement with the scaling predictions, and propose a method for predicting the limiting flame speed and width based on small-scale simulations. This flame speed can then be used to construct a turbulent flame model to study very large scale distributed supernova flames.