The equilibrium of a differentially rotating disc containing a poloidal magnetic field

Equilibrium models of accretion discs containing magnetic fields are constructed. The model system consists of a perfectly conducting, non-self-gravitating fluid in differential rotation about a massive central object. The fluid contains a purely poloidal magnetic field which bends as it passes through the disc, enforcing isorotation on magnetic surfaces. The general problem of constructing an equilibrium is first defined. Two methods are then used to obtain special solutions which require only the solution of ordinary differential equations. First, the asymptotic solutions for thin discs are derived. Two families of solutions are found: 'weakly magnetized' discs, in which gravitational, pressure and magnetic forces all contribute to the vertical equilibrium, and 'strongly magnetized' discs, in which the dominant vertical balance is between pressure and magnetic forces only. In the former case, the angular velocity deviates from the Keplerian value by an asymptotically small quantity, while the deviation is of order unity in the latter. Secondly, solutions are found by assuming self-similarity in the spherical radial coordinate, but without requiring the equilibria to be thin. Finally, the possibility of converting an equilibrium into a model of a self-consistent, wind-driven accretion disc is investigated, by including Ohmic resistivity, a meridional accretion flow and a toroidal magnetic field. In addition to being of intrinsic interest, the equilibria constructed here may be used as models for more realistic stability analyses than have been made previously.