Impact of magnetic field gradients on the development of the magneto-rotational instability

Context. The magneto-rotational instability (MRI) plays a crucial role in accretion disk modelling, driving magnetohydrodynamic turbulence and facilitating enhanced angular momentum transport. Notably, MRI is believed to be pivotal in the development of large-scale poloidal magnetic fields during binary neutron star mergers. However, the few numerical simulations that start from weak seed magnetic fields and capture its growth until saturation show the effects of small-scale turbulence and winding but lack convincing evidence of MRI activity. Aims. We investigated how the axisymmetric MRI is impacted by magnetic fields with the realistic, complex topologies of the post-merger phase, where field gradients cannot be neglected. This analysis aims to improve our understanding of the conditions under which MRI develops in more realistic astrophysical scenarios. Methods. We performed a linear analysis of axisymmetric MRI under these extended conditions and studied the resulting generalized dispersion relation. After deriving the extended MRI criteria, we first applied them to simple analytical disk models. Finally, we analysed the results obtained from a numerical relativity simulation of a long-lived neutron star merger remnant. Results. We find that radial magnetic field gradients can significantly impact the instability, slowing it and suppressing it entirely if sufficiently large. Specifically, we derived modified expressions for the growth rate and wavelength of the fastest-growing mode. Evaluating them in the context of binary neutron star merger remnants, we find that conditions that potentially allow the instability to be active and grow on short enough timescales might be reached only in small portions of the post-merger remnant, and only at late times, t ≳ 100 ms after the merger. Conclusions. Our results indicate that the role of the axisymmetric MRI in amplifying the poloidal magnetic field in the post-merger environment during the first 𝒪(100) ms is likely limited.