Non-ideal Magnetohydrodynamic Effects Research Articles

Effect of Magnetic Diffusion in the Chromosphere on the Solar Wind

We investigate nonideal magnetohydrodynamical (MHD) effects in the chromosphere on the solar wind by performing MHD simulations for Alfvén-wave-driven winds, explicitly including ohmic and ambipolar diffusion. We find that MHD waves are significantly damped in the chromosphere by ambipolar diffusion so that the Alfvénic Poynting flux that reaches the corona is substantially reduced. As a result, the coronal temperature and the mass-loss rate of the solar wind are considerably reduced, compared with those obtained from an ideal MHD case, which is indicative of the great importance of the nonideal MHD effects in the solar atmosphere. However, the temperature and the mass-loss rate are recovered by a small increase in the convection-originated velocity perturbation at the photosphere because of the sensitive dependence of the ambipolar diffusion and reflection of Alfvén waves on the physical properties of the chromosphere. We also find that density perturbations in the corona are reduced by the ambipolar diffusion of Alfvén waves in the chromosphere because the nonlinear generation of compressible perturbations is suppressed.

Hall-magnetohydrodynamic simulations of X-ray photoevaporative protoplanetary disc winds

ABSTRACT Understanding the evolution and dispersal via energetic stellar radiation of protoplanetary discs (PPDs) is a prominent challenge in astrophysics. It has been established that X-ray luminosity from the central protostar can significantly heat the surface of the disc, causing powerful photoevaporative winds that eject a considerable fraction of the disc’s mass. Recent work in the field has moreover shown the importance of global PPD simulations that simultaneously take into account non-ideal magnetohydrodynamic (MHD) effects and detailed thermochemistry. In this paper, we combine these two aspects and figure out how they interact. Focus is put on the Hall Effect (HE) and the impact it has on the overall field topology and mass-loss/accretion rates. Utilizing a novel X-ray temperature parametrization, we perform 2D-axisymmetric MHD simulations with the nirvana-iii fluid code, covering all non-ideal effects. We find that, in the aligned orientation, the HE causes prominent inward displacement of the poloidal field lines that increase the accretion rate through a laminar Maxwell stress. We find that outflows are mainly driven by photoevaporation – unless the magnetic field strength is considerable (i.e. βp ≤ 103) or the X-ray luminosity low enough (i.e. log LX ≤ 29.3). Inferred mass-loss rate are in the range of the expected values 10−8–$10^{-7}\, M_\odot \, {\rm yr}^{-1}$. Moreover, we performed pure hydrodynamic (HD) runs and compared them with the equivalent MHD runs. We concluded that the magnetic field does indeed contribute to the mass-loss rate, albeit only discernibly so for low enough LX (i.e. log LX ≤ 30.8).

Protoplanetary Disk Size under Nonideal Magnetohydrodynamics: A General Formalism with Inclined Magnetic Field

Many mechanisms have been proposed to alleviate the magnetic catastrophe, which prevents the Keplerian disk from forming inside a collapsing magnetized core. Such propositions include inclined field and nonideal magnetohydrodynamics effects, and have been supported with numerical experiments. Models have been formulated for typical disk sizes when a field threads the rotating disk, parallel to the rotation axis, while observations at the core scales do not seem to show evident correlation between the directions of angular momentum and the magnetic field. In the present study, we propose a new model that considers both vertical and horizontal fields and discuss their effects on the protoplanetary disk size.

Magnetorotational instability in a swirling partially ionized gas

ABSTRACT The magnetorotational instability (MRI) has been proposed as the method of angular momentum transport that enables accretion in astrophysical discs. However, for weakly ionized discs, such as protoplanetary discs, it remains unclear whether the combined non-ideal magnetohydrodynamic (MHD) effects of Ohmic resistivity, ambipolar diffusion, and the Hall effect make these discs MRI stable. While much effort has been made to simulate non-ideal MHD MRI, these simulations make simplifying assumptions and are not always in agreement with each other. Furthermore, it is difficult to directly observe the MRI astrophysically because it occurs on small scales. Here, we propose the concept of a swirling gas experiment of weakly ionized argon gas between two concentric cylinders threaded with an axial magnetic field that can be used to study non-ideal MHD MRI. For our proposed experiment, we derive the hydrodynamic equilibrium flow and a dispersion relation for MRI that includes the three non-ideal effects. We solve this dispersion relation numerically for the parameters of our proposed experiment. We find it should be possible to produce a non-ideal MRI in such an experiment because of the Hall effect, which increases the MRI growth rate when the vertical magnetic field is anti-aligned with the rotation axis. As a proof of concept, we also present experimental results for a hydrodynamic flow in an unmagnetized prototype. We find that our prototype has a small, but non-negligible, α-parameter that could serve as a baseline for comparison to our proposed magnetized experiment, which could be subject to additional turbulence from the MRI.

On the importance of disc chemistry in the formation of protoplanetary disc rings

ABSTRACT Radial substructures have now been observed in a wide range of protoplanetary discs (PPDs), from young to old systems; however, their formation is still an area of vigorous debate. Recent magnetohydrodynamic (MHD) simulations have shown that rings and gaps can form naturally in PPDs when non-ideal MHD effects are included. However, these simulations employ ad hoc approximations to the magnitudes of the magnetic diffusivities in order to facilitate ring growth. We replace the parametrization of these terms with a simple chemical network and grain distribution model to calculate the non-ideal effects in a more self-consistent way. We use a range of grain distributions to simulate grain formation for different disc conditions. Including ambipolar diffusion, we find that large grain populations (>1 $\mu$m), and those including a population of very small polyaromatic hydrocarbons (PAHs) facilitate the growth of periodic, stable rings, while intermediate-sized grains suppress ring formation. Including Ohmic diffusion removes the positive influence of PAHs, with only large grain populations still producing periodic ring and gap structures. These results relate closely to the degree of coupling between the magnetic field and the neutral disc material, quantified by the non-dimensional Elsasser number Λ (the ratio of magnetic forces to Coriolis force). For both the ambipolar-only and ambipolar-ohmic cases, if the total Elsasser number is initially of the order of unity along the disc mid-plane, ring and gap structures may develop.

Non-ideal MHD growth of current interchange tearing modes at plasma edge and response to externally-imposed flow

Current interchange tearing modes (CITMs) have been proposed to explain an intermittent eruption of electric current toward a divertor at an edge region of a tokamak device. Extended magnetohydrodynamic (MHD) simulations combined with the Scrape-Off-Layer diffusion model, which has been developed by the authors [Miura et al., Phys. Plasmas 24, 092111 (2017)], are carried out in order to study the growth of a CITM under influences of two-fluid and gyro-viscous effects, or non-ideal MHD effects. Numerical simulations both with and without an externally imposed azimuthal flow show that the growth of a CITM is observed for a relatively small flow velocity, and that a CITM can be a candidate mechanism for the intermittent eruption of the current in a tokamak, whether the growth is under influences of non-ideal MHD effects or not. The growth can be suppressed when an externally imposed azimuthal flow is sufficiently large to cause a finite radial displacement because such a displacement prevents a transition from an interchange to a tearing mode that is the key process of a CITM. Furthermore, it is also found that a stripe pattern similar to so-called streamers is formed in the course of the CITM growth. The basic nature of this streamer-like structure is also presented.

Impact of turbulence intensity and fragmentation velocity on dust particle size evolution and non-ideal magnetohydrodynamics effects

ABSTRACT We investigate the influence of dust particle size evolution on non-ideal magnetohydrodynamic (MHD) effects during the collapsing phase of star-forming cores, taking both the turbulence intensity in the collapsing cloud core and the fragmentation velocity of dust particles as parameters. When the turbulence intensity is small, the dust particles do not grow significantly, and the non-ideal MHD effects work efficiently in high-density regions. The dust particles rapidly grow in a strongly turbulent environment, while the efficiency of non-ideal MHD effects in such an environment depends on the fragmentation velocity of the dust particles. When the fragmentation velocity is small, turbulence promotes coagulation growth and collisional fragmentation of dust particles, producing small dust particles. In this case, the adsorption of charged particles on the dust particle surfaces becomes efficient and the abundance of charged particles decreases, making non-ideal MHD effects effective at high densities. On the other hand, when the fragmentation velocity is high, dust particles are less likely to fragment, even if the turbulence is strong. In this case, the production of small dust particles becomes inefficient and non-ideal MHD effects become less effective. We also investigate the effect of the dust composition on the star and disc formation processes. We constrain the turbulence intensity of a collapsing core and the fragmentation velocity of dust for circumstellar disc formation due to the dissipation of the magnetic field.

Non-ideal magnetohydrodynamics of self-gravitating filaments

Context. Filaments have been studied in detail through observations and simulations. A range of numerical works have separately investigated how chemistry and diffusion effects, as well as magnetic fields and their structure impact the gas dynamics of the filament. However, non-ideal effects have hardly been explored thus far. Aims. We investigate how non-ideal magnetohydrodynamic (MHD) effects, combined with a simplified chemical model affect the evolution and accretion of a star-forming filament. Methods. We modeled an accreting self-gravitating turbulent filament using LEMONGRAB, a one-dimensional (1D) non-ideal MHD code that includes chemistry. We explore the influence of non-ideal MHD, the orientation and strength of the magnetic field, and the cosmic ray ionization rate, on the evolution of the filament, with particular focus on the width and accretion rate. Results. We find that the filament width and the accretion rate are determined by the magnetic field properties, including the initial strength, the coupling with the gas controlled by the cosmic ray ionization rate, and the orientation of the magnetic field with respect to the accretion flow direction. Increasing the cosmic-ray ionization rate leads to a behavior closer to that of ideal MHD, reducing the magnetic pressure support and, hence, damping the accretion efficiency with a consequent broadening of the filament width. For the same reason, we obtained a narrower width and a larger accretion rate when we reduced the initial magnetic field strength. Overall, while these factors affect the final results by approximately a factor of 2, removing the non-ideal MHD effects results in a much greater variation (up to a factor of 7). Conclusions. The inclusion of non-ideal MHD effects and the cosmic-ray ionization is crucial for the study of self-gravitating filaments and in determining critical observable quantities, such as the filament width and accretion rate.

Fast methods for tracking grain coagulation and ionization

Dust grains influence many aspects of star formation, including planet formation and the opacities for radiative transfer, chemistry, and the magnetic field via Ohmic, Hall, as well as ambipolar diffusion. The size distribution of the dust grains is the primary characteristic influencing all these aspects. Grain size increases by coagulation throughout the star formation process. In this work, we describe numerical simulations of protostellar collapse using methods described in earlier papers of this series. We compute the evolution of the grain size distribution from coagulation and the non-ideal magnetohydrodynamics effects self-consistently and at low numerical cost. We find that the coagulation efficiency is mostly affected by the time spent in high-density regions. Starting from sub-micron radii, grain sizes reach more than 100 µm in an inner protoplanetary disk that is only 1000 yr old. We also show that the growth of grains significantly affects the resistivities, while also having an indirect effect on the dynamics and angular momentum of the disk.

Magnetically regulated collapse in the B335 protostar?

Context. Whether or not magnetic fields play a key role in dynamically shaping the products of the star formation process is still largely debated. For example, in magnetized protostellar formation models, magnetic braking plays a major role in the regulation of the angular momentum transported from large envelope scales to the inner envelope, and is expected to be responsible for the resulting protostellar disk sizes. However, non-ideal magnetohydrodynamic effects that rule the coupling of the magnetic field to the gas also depend heavily on the local physical conditions, such as the ionization fraction of the gas. Aims. The purpose of this work is to observationally characterize the level of ionization of the gas at small envelope radii and to investigate its relation to the efficiency of the coupling between the star-forming gas and the magnetic field in the Class 0 protostar B335. Methods. We obtained molecular line emission maps of B335 with ALMA, which we use to measure the deuteration fraction of the gas, RD, its ionization fraction, χe, and the cosmic-ray ionization rate, ζCR, at envelope radii ≲1000 au. Results. We find large fractions of ionized gas, χe ≃ 1–8 × 10−6. Our observations also reveal an enhanced ionization that increases at small envelope radii, reaching values up to ζCR ≃ 10−14 s−1 at a few hundred astronomical units (au) from the central protostellar object. We show that this extreme ζCR can be attributed to the presence of cosmic rays accelerated close to the protostar. Conclusions. We report the first resolved map of ζCR at scales ≲1000 au in a solar-type Class 0 protostar, finding remarkably high values. Our observations suggest that local acceleration of cosmic rays, and not the penetration of interstellar Galactic cosmic rays, may be responsible for the gas ionization in the inner envelope, potentially down to disk-forming scales. If confirmed, our findings imply that protostellar disk properties may also be determined by local processes that set the coupling between the gas and the magnetic field, and not only by the amount of angular momentum available at large envelope scales and the magnetic field strength in protostellar cores. We stress that the gas ionization we find in B335 significantly stands out from the typical values routinely used in state-of-the-art models of protostellar formation and evolution. If the local processes of ionization uncovered in B335 are prototypical to low-mass protostars, our results call for a revision of the treatment of ionizing processes in magnetized models for star and disk formation.

Increasing Mass-to-flux Ratio from the Dense Core to the Protostellar Envelope around the Class 0 Protostar HH 211

To study the transportation of magnetic flux from large to small scales in protostellar sources, we analyzed the Nobeyama 45 m N2H+ (1–0), JCMT 850 μm polarization, and Atacama Large Millimeter/submillimeter Array (ALMA) C18O (2–1) and 1.3 and 0.8 mm (polarized) continuum data of the Class 0 protostar HH 211. The magnetic field strength in the dense core on a 0.1 pc scale was estimated with the single-dish line and polarization data using the Davis–Chandrasekhar–Fermi method, and that in the protostellar envelope on a 600 au scale was estimated from the force balance between the gravity and magnetic field tension by analyzing the gas kinematics and magnetic field structures with the ALMA data. Our analysis suggests that from 0.1 pc–600 au scales, the magnetic field strength increases from 40–107 μG to 0.3–1.2 mG with a scaling relation between the magnetic field strength and density of B ∝ ρ 0.36±0.08, and the mass-to-flux ratio increases from 1.2–3.7 to 9.1–32.3. The increase in the mass-to-flux ratio could suggest that the magnetic field is partially decoupled from the neutral matter between 0.1 pc and 600 au scales, and hint at efficient ambipolar diffusion in the infalling protostellar envelope in HH 211, which is the dominant nonideal magnetohydrodynamic effect considering the density on these scales. Thus, our results could support the scenario of efficient ambipolar diffusion enabling the formation of the 20 au Keplerian disk in HH 211.

Dust coagulation and fragmentation in a collapsing cloud core and their influence on non-ideal magnetohydrodynamic effects

ABSTRACT We determine the time-evolution of the dust particle size distribution during the collapse of a cloud core, accounting for both dust coagulation and dust fragmentation, to investigate the influence of dust growth on non-ideal magnetohydrodynamic (MHD) effects. The density evolution of the collapsing core is given by a one-zone model. We assume two types of dust model: dust composed only of silicate (silicate dust) and dust with a surface covered by H2O ice (H2O ice dust). When only considering collisional coagulation, the non-ideal MHD effects are not effective in the high-density region for both the silicate and H2O ice dust cases. This is because dust coagulation reduces the abundance of small dust particles, resulting in less efficient adsorption of charged particles on the dust surface. For the silicate dust case, when collisional fragmentation is included, the non-ideal MHD effects do apply at a high density of nH > 1012 cm−3 because of the abundant production of small dust particles. On the other hand, for the H2O ice dust case, the production of small dust particles due to fragmentation is not efficient. Therefore, for the H2O ice dust case, non-ideal magnetohydrodynamic effects apply only in the range nH ≳ 1014 cm−3, even when collisional fragmentation is considered. Our results suggest that it is necessary to consider both dust collisional coagulation and fragmentation to activate non-ideal magnetohydrodynamic effects, which should play a significant role in the star and disc formation processes.

Dissipation of hydromagnetic waves in the viscous polytropic zone of the solar wind including FLR corrections, ohmic diffusion, and the Hall effect

ABSTRACT In the polytropic zone of the solar wind, we have used the generalized polytrope pressure laws to investigate the dissipation of hydromagnetic waves and pressure-anisotropy-driven fluid instabilities in magnetized viscous plasmas, including finite Larmor radius (FLR) corrections and non-ideal magnetohydrodynamic (MHD) effects. The modified dispersion properties have been analysed in the MHD and Chew–Goldberger–Low (CGL) limits for typical conditions of the solar wind and corona. The theoretical results are found to be in good agreement with the observational data, which shows that the MHD and CGL waves are dissipated due to viscous and ohmic diffusion. The FLR and Hall parameters show destabilizing and stabilizing influences, respectively, for the strong magnetic fields in the solar corona, and reversed effects in the case of weak magnetic fields in the solar wind. In the solar corona, the CGL wave dissipation achieves the required damping rate in the minimum time than the dissipation of the MHD waves. The damping time is mainly associated with the considered parameters and was found to be larger for the MHD wave dissipation than the CGL wave dissipation. The theoretical results successfully demonstrate the role of the considered parameters on the reverse and forward shock waves and instabilities as observed in the solar wind parameters versus heliolatitude graph using Ulysses observations for r = 5.41 au. The results are helpful to explore the possibilities of MHD waves and pressure-anisotropy-driven fluid instabilities in the polytropic zone of the solar wind that will probably be observed by the Parker Solar Probe (PSP) mission.

Steady-state accretion in magnetized protoplanetary disks

Context. The transition between magnetorotational instability (MRI)-active and magnetically dead regions corresponds to a sharp change in the disk turbulence level, where pressure maxima may form, hence potentially trapping dust particles and explaining some of the observed disk substructures. Aims. We aim to provide the first building blocks toward a self-consistent approach to assess the dead zone outer edge as a viable location for dust trapping, under the framework of viscously driven accretion. Methods. We present a 1+1D global magnetically driven disk accretion model that captures the essence of the MRI-driven accretion, without resorting to 3D global nonideal magnetohydrodynamic (MHD) simulations. The gas dynamics is assumed to be solely controlled by the MRI and hydrodynamic instabilities. For given stellar and disk parameters, the Shakura–Sunyaev viscosity parameter, α, is determined self-consistently under the adopted framework from detailed considerations of the MRI with nonideal MHD effects (Ohmic resistivity and ambipolar diffusion), accounting for disk heating by stellar irradiation, nonthermal sources of ionization, and dust effects on the ionization chemistry. Additionally, the magnetic field strength is numerically constrained to maximize the MRI activity. Results. We demonstrate the use of our framework by investigating steady-state MRI-driven accretion in a fiducial protoplanetary disk model around a solar-type star. We find that the equilibrium solution displays no pressure maximum at the dead zone outer edge, except if a sufficient amount of dust particles has accumulated there before the disk reaches a steady-state accretion regime. Furthermore, the steady-state accretion solution describes a disk that displays a spatially extended long-lived inner disk gas reservoir (the dead zone) that accretes a few times 10−9 M⊙ yr−1. By conducting a detailed parameter study, we find that the extent to which the MRI can drive efficient accretion is primarily determined by the total disk gas mass, the representative grain size, the vertically integrated dust-to-gas mass ratio, and the stellar X-ray luminosity. Conclusions. A self-consistent time-dependent coupling between gas, dust, stellar evolution models, and our general framework on million-year timescales is required to fully understand the formation of dead zones and their potential to trap dust particles.

Non-ideal magnetohydrodynamic simulations of subcritical pre-stellar cores with non-equilibrium chemistry

ABSTRACT Non-ideal magnetohydrodynamic (MHD) effects are thought to be gravity’s closest ally in overcoming the support of magnetic fields and in forming stars. Here, we modify the publicly available version of the adaptive mesh refinement code flash (Fryxell et al. 2000; Dubey et al. 2008) to include a detailed treatment of non-ideal MHD and study such effects in collapsing pre-stellar cores. We implement two very extended non-equilibrium chemical networks, the largest of which is comprised of ∼ 300 species and includes a detailed description of deuterium chemistry. The ambipolar diffusion, Ohmic and Hall resistivities are then self-consistently calculated from the abundances of charged species. We present a series of 2D axisymmetric simulations where we vary the chemical model, cosmic ray ionization rate, and grain distribution. We benchmark our implementation against ideal MHD simulations and previously published results. We show that, at high densities ($n_{\rm {H_2}}\gt ~10^6~\rm {cm^{-3}}$), the ion that carries most of the perpendicular and parallel conductivities is not $\rm {H_3^+}$ as was previously thought, but is instead $\rm {D_3^+}$.

Global three-dimensional simulations of outer protoplanetary discs with ambipolar diffusion

ABSTRACT The structure and evolution of protoplanetary discs (PPDs) are largely governed by disc angular momentum transport, mediated by magnetic fields. In the most observable outer disc, ambipolar diffusion is the primary non-ideal magnetohydrodynamic (MHD) effect. In this work, we study the gas dynamics in outer PPDs by conducting a series of global three-dimensional non-ideal MHD simulations with ambipolar diffusion and net poloidal magnetic flux, using the Athena++ MHD code, with resolution comparable to local simulations. Our simulations demonstrate the coexistence of magnetized disc winds and turbulence driven by the magneto-rotational instability (MRI). While MHD winds dominate disc angular momentum transport, the MRI turbulence also contributes significantly. We observe that magnetic flux spontaneously concentrates into axisymmetric flux sheets, leading to radial variations in turbulence levels, stresses, and accretion rates. Annular substructures arise as a natural consequence of magnetic flux concentration. The flux concentration phenomena show diverse properties with different levels of disc magnetization and ambipolar diffusion. The disc generally loses magnetic flux over time, though flux sheets could prevent the leak of magnetic flux in some cases. Our results demonstrate the ubiquity of disc annular substructures in weakly MRI turbulent outer PPDs and imply a stochastic nature of disc evolution.

On the vertical shear instability in magnetized protoplanetary discs

ABSTRACT The vertical shear instability (VSI) is a robust phenomenon in irradiated protoplanetary discs (PPDs). While there is extensive literature on the VSI in the hydrodynamic limit, PPDs are expected to be magnetized and their extremely low ionization fractions imply that non-ideal magnetohydrodynamic effects should be properly considered. To this end, we present linear analyses of the VSI in magnetized discs with Ohmic resistivity. We primarily consider toroidal magnetic fields, which are likely to dominate the field geometry in PPDs. We perform vertically global and radially local analyses to capture characteristic VSI modes with extended vertical structures. To focus on the effect of magnetism, a locally isothermal equation of state is employed. We find that magnetism provides a stabilizing effect to dampen the VSI, with surface modes, rather than body modes, being the first to vanish with increasing magnetization. Subdued VSI modes can be revived by Ohmic resistivity, where sufficient magnetic diffusion overcomes magnetic stabilization, and hydrodynamic results are recovered. We also briefly consider poloidal magnetic fields to account for the magnetorotational instability (MRI), which may develop towards surface layers in the outer regions of PPDs. The MRI grows efficiently at small radial wavenumbers, in contrast to the VSI. When resistivity is considered, the VSI dominates over the MRI for Ohmic Elsässer numbers ≲0.09 at plasma beta parameter βZ ∼ 104.

The interplay between ambipolar diffusion and Hall effect on magnetic field decoupling and protostellar disc formation

ABSTRACT Non-ideal magnetohydrodynamic (MHD) effects have been shown recently as a robust mechanism of averting the magnetic braking ‘catastrophe’ and promoting protostellar disc formation. However, the magnetic diffusivities that determine the efficiency of non-ideal MHD effects are highly sensitive to microphysics. We carry out non-ideal MHD simulations to explore the role of microphysics on disc formation and the interplay between ambipolar diffusion (AD) and Hall effect during the protostellar collapse. We find that removing the smallest grain population (≲10 nm) from the standard MRN size distribution is sufficient for enabling disc formation. Further varying the grain sizes can result in either a Hall-dominated or an AD-dominated collapse; both form discs of tens of au in size regardless of the magnetic field polarity. The direction of disc rotation is bimodal in the Hall-dominated collapse but unimodal in the AD-dominated collapse. We also find that AD and Hall effect can operate either with or against each other in both radial and azimuthal directions, yet the combined effect of AD and Hall is to move the magnetic field radially outward relative to the infalling envelope matter. In addition, microphysics and magnetic field polarity can leave profound imprints both on observables (e.g. outflow morphology, disc to stellar mass ratio) and on the magnetic field characteristics of protoplanetary discs. Including Hall effect relaxes the requirements on microphysics for disc formation, so that prestellar cores with cosmic ray ionization rate of ≲2–3 × 10−16 s−1 can still form small discs of ≲10 au radius. We conclude that disc formation should be relatively common for typical prestellar core conditions, and that microphysics in the protostellar envelope is essential to not only disc formation, but also protoplanetary disc evolution.

Growth of magnetorotational instability in circumstellar discs around class 0 protostars

ABSTRACT We investigate the possibility of the growth of magnetorotational instability (MRI) in discs around Class 0 protostars. We construct a disc model and calculate the chemical reactions of neutral and charged atoms, molecules, and dust grains to derive the abundance of each species and the ionization degree of the disc. Then, we estimate the diffusion coefficients of non-ideal magnetohydrodynamics effects such as ohmic dissipation, ambipolar diffusion, and the Hall effect. Finally, we evaluate the linear growth rate of MRI in each area of the disc. We investigate the effect of changes in the strength and direction of the magnetic field in our disc model and we adopt four different dust models to investigate the effect of dust size distribution on the diffusion coefficients. Our results indicate that an MRI-active region possibly exists with a weak magnetic field in a region far from the protostar where the Hall effect plays a role in the growth of MRI. On the other hand, in all models, the disc is stable against MRI in the region within <20 au from the protostar on the equatorial plane. Since the size of the discs in the early stage of star formation is limited to ≲10–20 au, it is difficult to develop MRI-driven turbulence in such discs.

Linear Stability Analysis of a Magnetic Rotating Disk with Ohmic Dissipation and Ambipolar Diffusion

Abstract We perform a linear analysis of the stability of isothermal, rotating, magnetic, self-gravitating sheets that are weakly ionized. The magnetic field and rotation axis are perpendicular to the sheet. We include a self-consistent treatment of thermal pressure, gravitational, rotational, and magnetic (pressure and tension) forces together with two nonideal magnetohydrodynamic (MHD) effects (ohmic dissipation and ambipolar diffusion) that are treated together for their influence on the properties of gravitational instability for a rotating sheetlike cloud or disk. Our results show that there is always a preferred length scale and associated minimum timescale for gravitational instability. We investigate their dependence on important dimensionless free parameters of the problem: the initial normalized mass-to-flux ratio μ 0, the rotational Toomre parameter Q, the dimensionless ohmic diffusivity η ˜ OD , 0 , and the dimensionless neutral–ion collision time τ ˜ ni , 0 , which is a measure of the ambipolar diffusivity. One consequence of η ˜ OD , 0 is that there is a maximum preferred length scale of instability that occurs in the transcritical (μ 0 ≳ 1) regime, qualitatively similar to the effect of τ ˜ ni , 0 , but with quantitative differences. The addition of rotation leads to a generalized Toomre criterion (that includes a magnetic dependence) and modified length scales and timescales for collapse. When nonideal MHD effects are also included, the Toomre criterion reverts back to the hydrodynamic value. We apply our results to protostellar disk properties in the early embedded phase and find that the preferred scale of instability can significantly exceed the thermal (Jeans) scale and the peak preferred fragmentation mass is likely to be ∼10–90 M Jup.