Wind-driving protostellar accretion discs – II. Numerical method and illustrative solutions

We continue our study of weakly ionized protostellar accretion discs that are threaded by a large-scale magnetic field and power a centrifugally driven wind. It has been argued that there is already evidence in several protostellar systems that such a wind transports a significant fraction of the angular momentum from at least some part of the disc. We model this situation by considering a radially localized disc model in which the matter is everywhere well coupled to the field and the wind is the main repository of excess angular momentum. We consider stationary configurations in which magnetic diffusivity counters the shearing and advection of the magnetic field lines. In Wardle & Königl we analysed the disc structure in the hydrostatic approximation (vertical motions neglected inside the disc) and presented exact disc/wind solutions for the ambipolar diffusivity regime. In Königl, Salmeron & Wardle (Paper I) we generalized the hydrostatic analysis to the Hall and Ohm diffusivity domains and used it to identify the disc parameter sub-regimes in which viable solutions with distinct physical properties can be expected to occur. In this paper we test the results of Paper I by deriving full numerical solutions (integrated through the sonic critical surface) of the disc equations in the Hall domain. We confirm all the predictions of the hydrostatic analysis and demonstrate its usefulness for clarifying the behaviour of the derived solutions. We further show that the outflow solutions can be extended to larger scales (so that, in particular, they also cross the Alfvén critical surface) by matching the localized disc solutions to global 'cold' wind solutions of the type introduced by Blandford & Payne. To facilitate this matching, we construct a library of wind solutions for a wide range of wind model parameters; this library is made available to the community. The results presented in Wardle & Königl, Paper I and this work combine to form a comprehensive framework for the study of wind-driving accretion discs in protostellar and other astrophysical environments. This theoretical tool could be useful for interpreting observations and for guiding numerical simulations of such systems