Incorporation of ambipolar diffusion into the Zeus magnetohydrodynamics code

Magnetic fields tied to ions can diffuse through mostly neutral gas: this occurs in protostellar disks and in the cores of molecular clouds. We describe an algorithm that includes ambipolar diffusion in the astrophysical magnetohydrodynamics code ZEUS. We use the approximations that both electrons and ions have equal and constant temperature, that the ion inertia is negligible, and that the ion density is proportional to a power of the neutral density. Our algorithm is fully explicit, and treats the magnetic field using constrained transport and the method of characteristics. We test the algorithm by computing the gravitational collapse of a magnetically supported slab, and by comparing the computed solution for an oblique C-shock to a semi-analytic solution that we have derived. We then compute the development of the magnetorotational instability described by Balbus and Hawley in a magnetized accretion disk, including the effects of ambipolar diffusion. Our computation agrees with the published linear analysis of how diffusion prevents instability, and it allows us to describe the nonlinear development of the instability when diffusion is important but not dominant. We find that ambipolar diffusion indeed creates the sharp structures predicted by Brandenburg and Zweibel.