The heavy-ion approximation for ambipolar diffusion calculations for weakly ionized plasmas

Ambipolar diffusion redistributes magnetic flux in weakly ionized plasmas and plays a critical role in star formation. Simulations of ambipolar diffusion using explicit MHD codes are prohibitively expensive for the level of ionization observed in molecular clouds (less than or similar to 10(-6)), since an enormous number of time steps is required to represent the dynamics of the dominant neutral component with a time step determined by the trace ion component. Here, we show that ambipolar diffusion calculations can be significantly accelerated by the heavy-ion approximation,'' in which the mass density of the ions is increased and the collisional coupling constant with the neutrals decreased such that the product remains constant. In this approximation the ambipolar diffusion time and the ambipolar magnetic Reynolds number remain unchanged. We present three tests of the heavy-ion approximation: C-type shocks, the Wardle instability, and the one-dimensional collapse of a magnetized slab. We show that this approximation is quite accurate provided that (1) the square of the Alfven Mach number is small compared to the ambipolar diffusion Reynolds number for dynamical problems, and that (2) the ion mass density is negligible for quasi-static problems; a specific criterion is given for the magnetized slab problem. The first condition can be very stringent for turbulent flows with large density fluctuations.