Tearing-mediated reconnection in magnetohydrodynamic poorly ionized plasmas. I. Onset and linear evolution

In high-Lundquist-number plasmas, reconnection proceeds via onset of tearing, followed by a nonlinear phase during which plasmoids continuously form, merge, and are ejected from the current sheet (CS). This process is understood in fully ionized, magnetohydrodynamic plasmas. However, many plasma environments, such as star-forming molecular clouds and the solar chromosphere, are poorly ionized. We use theory and computation to study tearing-mediated reconnection in such poorly ionized systems. In this paper, we focus on the onset and linear evolution of this process. In poorly ionized plasmas, magnetic nulls on scales below $v_{\rm A,n0}/\nu_{\rm ni0}$, with $v_{\rm A,n0}$ the neutral Alfv\'{e}n speed and $\nu_{\rm ni0}$ the neutral-ion collision frequency, will self-sharpen via ambipolar diffusion. This sharpening occurs at an increasing rate, inhibiting the onset of reconnection. Once the CS becomes thin enough, however, ions decouple from neutrals and thinning of the CS slows, allowing tearing to onset in a time of order $\nu_{\rm ni0}^{-1}$. We find that the wavelength and growth rate of the mode that first disrupts the forming sheet can be predicted from a poorly ionized tearing dispersion relation; as the plasma recombination rate increases and ionization fraction decreases, the growth rate becomes an increasing multiple of $\nu_{ni0}$ and the wavelength becomes a decreasing fraction of $v_{\rm A,n0}/\nu_{\rm ni0}$.