Ambipolar diffusion and star formation: Formation and contraction of axisymmetric cloud cores. I. Formulation of the problem and method of solution

We formulate the problem of the formation and contraction of axially symmetric, isothermal, self-gravitating, molecular cloud fragments (or cores) due to ambipolar diffusion in magnetically supported parent clouds. The initial reference states are taken to be static and uniform with the magnetic field parallel to the axis of symmetry. The two-fluid MHD equations describing the evolution contain three dimensionless free parameters: α₀, the ratio of magnetic and thermal pressures in the initial state, v_ff,0, essentially the initial ratio of the free-fall and neutral-ion collision times, and the exponent k in the relation between the ion and neutral densities n_i ∝ n^k_n. A fourth free parameter, μ₀ < 1, enters through the initial conditions; it is the mass-to-flux ratio of the initial state in units of the critical value for gravitational collapse with a frozen-in magnetic field. In the absence of ambipolar diffusion, the magnetically subcritical fragments would contract toward, and oscillate about, magnetohydrostatic equilibrium states after a central density enhancement less than 10. If such equilibrium states are used as initial states, one of the free parameters is removed, since α_0,eq ∝ μ^-2₀ at tne center. The numerical method of solution involves a new, very general, fully adaptive, filtered mesh to accurately follow the quasistatic and dynamical phases of evolution. The mesh moves according to a prescription that provides increased spatial resolution where required by the variation of certain physical quantities. A global optimization is performed to ensure that the lengths of mesh zone sides vary smoothly from zone to zone, the zones never become too long and thin, and the angles between adjacent sides do not approach O or 180°. This filtering minimizes both the truncation errors of finite difference approximations to spatial derivatives and the spurious reflections of waves due to changes in mesh spacing. In a test run on a 30 × 30 grid with flux-freezing in the neutrals, we followed the evolution of a supercritical fragment to a central density enhancement of 10⁵ (e.g., from 3 × 10³ to 3 × 10⁸ cm^-3). The height and width of the innermost zone have decreased by factors of 180 and 45, respectively, and the mass in the central flux tube changed by less than 0.2%, which is a negligible amount compared to the change expected when ambipolar diffusion is included. In a subsequent paper, we present the results, including a parameter study, as they relate to the formation of cloud cores and protostars.