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Register a new userMagnetic self-organisation in Hall-dominated magnetorotational turbulence
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Matthew W. Kunz
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The magnetorotational instability (MRI) is the most promising mechanism by which angular momentum is efficiently transported outwards in astrophysical discs. However, its application to protoplanetary discs remains problematic. These discs are so poorly ionised that they may not support magnetorotational turbulence in regions referred to as `dead zones. It has recently been suggested that the Hall effect, a non-ideal magnetohydrodynamic (MHD) effect, could revive these dead zones by enhancing the magnetically active column density by an order of magnitude or more. We investigate this idea by performing local, three-dimensional, resistive Hall-MHD simulations of the MRI in situations where the Hall effect dominates over Ohmic dissipation. As expected from linear stability analysis, we find an exponentially growing instability in regimes otherwise linearly stable in resistive MHD. However, instead of vigorous and sustained magnetorotational turbulence, we find that the MRI saturates by producing large-scale, long-lived, axisymmetric structures in the magnetic and velocity fields. We refer to these structures as zonal fields and zonal flows, respectively. Their emergence causes a steep reduction in turbulent transport by at least two orders of magnitude from extrapolations based upon resistive MHD, a result that calls into question contemporary models of layered accretion. We construct a rigorous mean-field theory to explain this new behaviour and to predict when it should occur. Implications for protoplanetary disc structure and evolution, as well as for theories of planet formation, are briefly discussed.
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The rich structure that we observe in molecular clouds is due to the interplay between strong magnetic fields and supersonic (turbulent) velocity fluctuations. The velocity fluctuations interact with the magnetic field, causing it too to fluctuate. Using numerical simulations, we explore the nature of such magnetic field fluctuations, $vec{delta B}$, over a wide range of turbulent Mach numbers, $mathcal{M} = 2 - 20$ (i.e., from weak to strong compressibility), and Alfven Mach numbers, $mathcal{M}_{text{A}0} = 0.1 - 100$ (i.e., from strong to weak magnetic mean fields, $B_0$). We derive a compressible quasi-static fluctuation model from the magnetohydrodynamical (MHD) equations and show that velocity gradients parallel to the mean magnetic field give rise to compressible modes in sub-Alfvenic flows, which prevents the flow from becoming two-dimensional, as is the case in incompressible MHD turbulence. We then generalise an analytical model for the magnitude of the magnetic fluctuations to include $mathcal{M}$, and find $|vec{delta B}| = delta B = c_ssqrt{pirho_0}mathcal{M}mathcal{M}_{text{A}0}$, where $c_s$ is the sound speed and $rho_0$ is the mean density of gas. This new relation fits well in the strong $B$-field regime. We go on to study the anisotropy between the perpendicular ($ B_{perp}$) and parallel ($ B_{parallel}$) fluctuations and the mean-normalised fluctuations, which we find follow universal scaling relations, invariant of $mathcal{M}$. We provide a detailed analysis of the morphology for the $delta B_{perp}$ and $delta B_{parallel}$ probability density functions and find that eddies aligned with $B_0$ cause parallel fluctuations that reduce $B_{parallel}$ in the most anisotropic simulations. We discuss broadly the implications of our fluctuation models for magnetised gases in the interstellar medium.
Magnetorotational instability (MRI) has a potential to generate the vigorous turbulence in protoplanetary disks, although its turbulence strength and accretion stress remains debatable because of the uncertainty of MRI with low ionization fraction. We focus on the heating of electrons by strong electric fields which amplifies nonideal magnetohydrodynamic effects. The heated electrons frequently collide with and stick to dust grains, which in turn decreases the ionization fraction and is expected to weaken the turbulent motion driven by MRI. In order to quantitatively investigate the nonlinear evolution of MRI including the electron heating, we perform magnetohydrodynamical simulation with the unstratified shearing box. We introduce a simple analytic resistivity model depending on the current density by mimicking resistivity given by the calculation of ionization. Our simulation confirms that the electron heating suppresses magnetic turbulence when the electron heating occurs with low current density. We find a clear correlation between magnetic stress and its current density, which means that the magnetic stress is proportional to the squared current density. When the turbulent motion is completely suppressed, laminar accretion flow is caused by ordered magnetic field. We give an analytical description of the laminar state by using a solution of linear perturbation equations with resistivity. We also propose a formula that successfully predicts the accretion stress in the presence of the electron heating.
The magnetorotational instability (MRI) drives vigorous turbulence in a region of protoplanetary disks where the ionization fraction is sufficiently high. It has recently been shown that the electric field induced by the MRI can heat up electrons and thereby affect the ionization balance in the gas. In particular, in a disk where abundant dust grains are present, the electron heating causes a reduction of the electron abundance, thereby preventing further growth of the MRI. By using the nonlinear Ohms law that takes into account electron heating, we investigate where in protoplanetary disks this negative feedback between the MRI and ionization chemistry becomes important. We find that the e-heating zone, the region where the electron heating limits the saturation of the MRI, extends out up to 80 AU in the minimum-mass solar nebula with abundant submicron-sized grains. This region is considerably larger than the conventional dead zone whose radial extent is $sim20$ AU in the same disk model. Scaling arguments show that the MRI turbulence in the e-heating zone should have a significantly lower saturation level. Submicron-sized grains in the e-heating zone are so negatively charged that their collisional growth is unlikely to occur. Our present model neglects ambipolar and Hall diffusion, but our estimate shows that ambipolar diffusion would also affect the MRI in the e-heating zone.
Helical and azimuthal magnetorotational instabilities operate in rotating magnetized flows with relatively steep negative or extremely steep positive shear. The corresponding lower and upper Liu limits of the shear, which determine the threshold of modal growth of these instabilities, are continuously connected when some axial electrical current is allowed to pass through the rotating fluid. We investigate the nonmodal dynamics of these instabilities arising from the non-normality of shear flow in the local approximation, generalizing the results of the modal approach. It is demonstrated that moderate transient/nonmodal amplification of both types of magnetorotational instability occurs within the Liu limits, where the system is stable according to modal analysis. We show that for the helical magnetorotational instability this magnetohydrodynamic behavior is closely connected with the nonmodal growth of the underlying purely hydrodynamic problem.
Accretion disks are likely threaded by external vertical magnetic flux, which enhances the level of turbulence via the magnetorotational instability (MRI). Using shearing-box simulations, we find that such external magnetic flux also strongly enhances the amplitude of banded radial density variations known as zonal flows. Moreover, we report that vertical magnetic flux is strongly concentrated toward low-density regions of the zonal flow. Mean vertical magnetic field can be more than doubled in low-density regions, and reduced to nearly zero in high density regions in some cases. In ideal MHD, the scale on which magnetic flux concentrates can reach a few disk scale heights. In the non-ideal MHD regime with strong ambipolar diffusion, magnetic flux is concentrated into thin axisymmetric shells at some enhanced level, whose size is typically less than half a scale height. We show that magnetic flux concentration is closely related to the fact that the magnetic diffusivity of the MRI turbulence is anisotropic. In addition to a conventional Ohmic-like turbulent resistivity, we find that there is a correlation between the vertical velocity and horizontal magnetic field fluctuations that produces a mean electric field that acts to anti-diffuse the vertical magnetic flux. The anisotropic turbulent diffusivity has analogies to the Hall effect, and may have important implications for magnetic flux transport in accretion disks. The physical origin of magnetic flux concentration may be related to the development of channel flows followed by magnetic reconnection, which acts to decrease the mass-to-flux ratio in localized regions. The association of enhanced zonal flows with magnetic flux concentration may lead to global pressure bumps in protoplanetary disks that helps trap dust particles and facilitates planet formation.
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