Impact of magnetic field gradients on the development of the MRI: Applications to binary neutron star mergers and proto-planetary disks

TL;DR

This work develops a generalized axisymmetric MRI framework that accounts for realistic magnetic-field gradients, deriving a gradient-inclusive dispersion relation and extended instability criteria. By applying the theory to analytical disk models and to a high-resolution binary neutron star remnant simulation, the authors show radial magnetic-field gradients can slow, restrict, or even suppress MRI growth, limiting poloidal-field amplification to narrow regions and late times. The study finds that the fastest-growing extended MRI modes often have shorter wavelengths (∼10–100 m, occasionally ∼1–10 m in central zones) and growth times around ∼1 ms or longer, raising significant numerical and physical constraints on MRI-driven turbulence in post-merger environments. These results suggest MRI may play a more limited role in the early post-merger evolution than previously assumed, with important implications for modeling jet formation and magnetic-field amplification, and they motivate further work on non-axisymmetric effects and fully global, high-resolution simulations.

Abstract

The magneto-rotational instability (MRI) is widely believed to play a central role in generating large-scale, poloidal magnetic fields during binary neutron star mergers. However, the few simulations that begin with a weak seed magnetic field and capture its growth until saturation predominantly show the effects of small-scale turbulence and winding, but lack convincing evidence of MRI activity. In this work, we investigate how the MRI is affected by the complex magnetic field topologies characteristic of the post-merger phase, aiming to assess the actual feasibility of MRI in such environments. We first derive the MRI instability criterion, as well as expressions for the characteristic wavelength and growth timescale of the fastest-growing modes, under conditions that include significant magnetic field gradients. Our analysis reveals that strong radial magnetic field gradients can impact significantly on the MRI, slowing its growth or suppressing it entirely if large enough. We then apply this extended framework to both idealized analytical disk models and data from a numerical relativity simulation of a long-lived neutron star merger remnant. We find that conditions favourable to MRI growth on astrophysically relevant timescales may occur only in limited regions of the post-merger disk, and only at late times $t\gtrsim 100$ ms after the merger. These results suggest that the MRI plays a limited role in amplifying poloidal magnetic fields in post-merger environments during the first $\mathcal{O}(100)$ms.

Figures (12)

• Figure 1: Extended MRI analysis for the analytical model given by a vertical magnetic field with radial power-law decay in \ref{['eq:analyticBverticalpowerlaw']}. Left: Generalized instability criterion. Middle: Comparison between the generalized and standard MRI for the wavelength of the fastest-growing mode. Right: Timescale of the instability. We have considered the values with $n=2,\,4$ and $C=0.01, \,0.025,\,0.1$ in the model defined by \ref{['eq:analyticBverticalpowerlaw']}. The wavelength and timescale are shown only when the instability is active according to the generalized criterion, a transition that is marked with black dots.
• Figure 2: Extended MRI analysis for the analytical model given by a vertical and toroidal magnetic field.From left to right: Analytic instability window, ratio of the wavelength of the fastest-growing mode, and ratio of the timescale of the instability in the generalized case vs the standard MRI result. The wavelength and timescale of the fastest-growing mode are shown only where the instability is active according to the generalized criterion.
• Figure 3: 2D maps of the azimuthally averaged magnetic field magnitudes at $t\simeq 40$ ms after merger.From left to right: 2D maps of the magnitude of the radial, azimuthal, and vertical component of the magnetic field. We see the different components present large gradients, which we take into account in the generalized MRI criterion.
• Figure 4: Instability window calculation at $t\simeq40$ ms after merger. From the angular frequency ($\Omega$) we computed its gradient ($s_0$) and mask those points where $s_0>0$, namely those for which the standard MRI would be inactive. Then we evaluated the updated instability criterion, considering at once the two cases $f>0$ and $f<0$. In the right panel, blue points are unstable according to the updated MRI criterion, and red are those for which the magnetic gradients are large enough to switch off the instability; all coloured points are MRI-unstable according to the standard criterion.
• Figure 5: Comparison of the wavelength of the fastest-growing mode in the standard vs generalized case, at $t\simeq40$ ms after merger. Top: Standard MRI case, all coloured points are unstable to the MRI. Bottom: Generalized case. We mask the points where the instability condition is not met. Where the generalized instability is active, we observe $\lambda_{\text{eMRI}}$ to reach values close to but slightly smaller than the standard case. Most interestingly, $\lambda_{max}$ is of the order of $100$ m or smaller at small radii, $r\lesssim 12$ km.