An investigation of magnetic field distortions in accretion discs around neutron stars. I. Analysis of the poloidal field component

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 kinematic 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, solving the induction equation numerically in full 2D. 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 R_m (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’ D_m(r,\theta) (a magnetic Reynolds number defined locally). Where D_m«1 (near to the inner edge of the disc) there is little distortion, but where D_m>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.

💡 Research Summary

The paper presents a systematic investigation of how the poloidal component of a neutron‑star’s magnetic field is distorted by its surrounding accretion disc. The authors adopt a deliberately simple, yet physically motivated, kinematic model as a first step toward interpreting the results of large‑scale magnetohydrodynamic (MHD) simulations. In this baseline model the stellar field is a pure dipole aligned with the rotation axis and perpendicular to the disc plane. The disc flow is assumed steady, axisymmetric, and described by radial (v_r) and vertical (v_θ) velocity components. Turbulent magnetic diffusivity η is allowed to vary spatially, being larger in the inner disc and decreasing outward, thereby mimicking realistic non‑uniform turbulence.

The governing equation is the steady‑state induction equation ∇×(v×B) = ∇×(η∇×B). By exploiting axisymmetry the azimuthal dependence drops out, reducing the problem to a two‑dimensional (r, θ) system. The authors solve this system numerically on a high‑resolution grid using finite‑difference discretisation and impose dipolar boundary conditions at large radii while applying free (Neumann) conditions at the disc surfaces.

A central methodological point is the critique of the traditional, globally defined magnetic Reynolds number R_m = v_r L/η. Because η varies strongly with radius and height, a single R_m cannot capture the local balance between advection and diffusion. The authors therefore introduce a “magnetic distortion function” D_m(r, θ) = v(r, θ) L/η(r, θ), which is essentially a local magnetic Reynolds number. When D_m ≪ 1, diffusion dominates and the field remains close to the vacuum dipole; when D_m ≫ 1, advection dominates and the field is strongly dragged and weakened by the disc material.

The numerical results reveal three distinct radial zones. Near the inner edge of the disc, η is high and v_r is low, giving D_m ≪ 1; the magnetic field there is essentially undistorted. Moving outward, η drops while v_r rises, so D_m exceeds unity over most of the disc. In this region the poloidal field is considerably weakened compared with the pure dipole, confirming the expectation that the disc “screens” the stellar field. Between the diffusion‑dominated inner zone and the advection‑dominated outer zone lies a transition layer where D_m≈1. In this layer the field can be locally amplified, producing a small maximum (or minimum) in the magnetic strength. The exact radial position and thickness of this transition layer are highly sensitive to the prescribed η profile: a steep decline in η pushes the layer inward, while a gentle decline spreads it over a larger radial extent.

The authors also explore the effect of outflows (winds) from the top and bottom disc surfaces. Adding modest vertical velocities does not appreciably alter the overall field geometry, indicating that the boundary conditions at the disc surfaces have limited impact on the interior distortion pattern.

Overall, the study demonstrates that a local magnetic Reynolds number (the distortion function D_m) provides a far more nuanced description of magnetic field behaviour in accretion discs than a single global R_m. The findings corroborate earlier full‑MHD simulations that show disc‑induced suppression of the stellar field, but they also highlight the existence of a narrow amplification zone whose location is dictated by the diffusivity profile. These insights suggest that future global simulations and analytic models should pay close attention to spatial variations in turbulent diffusivity and to the local balance of advection versus diffusion when predicting magnetosphere–disc coupling, torque transfer, and the conditions for launching magnetically driven outflows.