Stationary magnetic field configurations in accretion discs around neutron stars – the role of velocity and diffusivity

We report results from calculations investigating stationary magnetic field configurations in accretion discs around magnetised neutron stars. Our strategy is to start with a very simple model and then progressively improve it providing complementary insight into results obtained with large numerical simulations. In our first model, presented here, we work in the kinetic approximation and consider the stellar magnetic field as being a dipole aligned with the stellar rotation axis and perpendicular to the disc plane, while the flow in the disc is taken to be steady and axisymmetric. The behaviour in the radial direction is then independent of that in the azimuthal direction. We investigate the distortion of the field caused by interaction with the disc matter, working in full 2D and solving the induction equation numerically. The influence of turbulent diffusivity and fluid velocity on the poloidal field configuration is analysed, including discussion of outflows from the top and bottom of the disc. We find that the distortions increase with increasing magnetic Reynolds number Rm (calculated using the radial velocity). However, a single global parameter does not give an adequate description in different parts of the disc and we use instead a ‘magnetic distortion function’ Dm(r, θ) (a magnetic Reynolds number defined locally). Where Dm ≪ 1 (near to the inner edge of the disc) there is little distortion, but where Dm ≫ 1 (most of the rest of the disc), there is considerable distortion and the field becomes weaker than the dipole would have been. Between these two regions, there is a transition zone where the field is amplified and can have a local minimum and maximum. The location of this zone depends sensitively on the diffusivity. The results depend very little on the boundary conditions at the top of the disc.