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Magnetorotational instability (MRI) is one of the fundamental processes in astrophysics, driving angular momentum transport and mass accretion in a wide variety of cosmic objects. Despite much theoretical/numerical and experimental efforts over the last decades, its saturation mechanism and amplitude, which sets the angular momentum transport rate, remains not well understood, especially in the limit of high resistivity, or small magnetic Prandtl numbers typical to interiors (dead zones) of protoplanetary disks, liquid cores of planets and liquid metals in laboratory. Using direct numerical simulations, in this paper we investigate the nonlinear development and saturation properties of the helical magnetorotational instability (HMRI) -- a relative of the standard MRI -- in a magnetized Taylor-Couette flow at very low magnetic Prandtl number (correspondingly at low magnetic Reynolds number) relevant to liquid metals. For simplicity, the ratio of azimuthal field to axial field is kept fixed. From the linear theory of HMRI, it is known that the Elsasser number, or interaction parameter determines its growth rate and plays a special role in the dynamics. We show that this parameter is also important in the nonlinear problem. By increasing its value, a sudden transition from weakly nonlinear, where the system is slightly above the linear stability threshold, to strongly nonlinear, or turbulent regime occurs. We calculate the azimuthal and axial energy spectra corresponding to these two regimes and show that they differ qualitatively. Remarkably, the nonlinear state remains in all cases nearly axisymmetric suggesting that HMRI turbulence is quasi two-dimensional in nature. Although the contribution of non-axisymmetric modes increases moderately with the Elsasser number, their total energy remains much smaller than that of the axisymmetric ones., 26 pages, 9 figures, accepted for publication in New Journal of Physics

We compiled a sample of 98 radio-quiet active galactic nuclei observed by ASCA, Chandra, XMM-Newton, INTEGRAL and Swift with the aim of testing the formation of hot corona and the magnetic shear stress operating in a disk-corona system. We found a strong correlation between the hard X-ray luminosity, bolometric luminosity L-Bol and Eddington luminosity LEdd, in the sense that the fraction f of hard X-ray to the bolometfic luminosity is inversely proportional to the Eddington ratio. This correlation favors the shear stress tensor being of the form of tr phi proportional to P-gas, with which the disk-corona structure is stable.

The screw instability of the magnetic field is discussed based on its poloidal configuration generated by a single toroidal electric current flowing in the equatorial plane of a Kerr black hole (BH). The rotation of the BH relative to the disc induces an electromotive force, which in turn results in a poloidal electric current. By using Ampere's law, we calculate the toroidal component of the magnetic field and derive a criterion for the screw instability of the magnetic field connecting the rotating BH with its surrounding disc. It is determined that the screw instability is related to two parameters: the radius of the disc and the BH spin. The occurrence of screw instability is depicted in a parameter space. In addition, we discuss the effect of the screw instability on magnetic extraction of energy from the rotating BH.

Using the new model of anomalous viscosity, we investigate the magnetic instability in accretion discs and give the dispersion formula. On the basis of the dispersion relation obtained, it is numerically shown that the instability condition of the viscous accretion disc is well consistent with that of the ideal accretion disc, namely there would be magneto-rotational instability in the presence of a vertical weak magnetic field. For a given distance R from the centre of the disc, the growth rate in the anomalous case deviates from the ideal case more greatly when the vertical magnetic field is smaller. The large viscosity limits the instability. In the two cases, the distributions of growth rate with wavenumber k approach each other when the magnetic field increases. This greatly represses the effect of viscosity.

The self-electric field effect on the magnetorotational instability (MRI) of non-neutral rotating electron-ion plasmas, related to axisymmetric disks, is investigated by using a cylindrical model. Describing the equilibrium state of the rotational system in the presence of a self-electric field, the fluid-Maxwell equations as well as the linear perturbation theory are employed to derive the dispersion relation (DR) of the excited modes in the system. The obtained DR analysis leads to analytical stability conditions, the growth rate of the instability, and the saturation amount of the pure instability excited in the system, depending on the equilibrium velocity difference between the involved components. Moreover, the numerical analysis of the obtained DR shows that the self-electric field has an important role for the MRI in the region where the angular frequency due to drift, , dominates over all the frequencies, in the outer part of non-neutral disks. In this region, the growth rate of the MRI increases dramatically with the ratio of the ion density to electron density until it saturates. On the other hand, in the inner region of the disk in which the gravitational frequency overcomes the others, the growth rate of the instability is found to be much smaller than that of the outer region. These results can be applicable to understand the MRI growth rate of the differentially rotating non-neutral plasma systems existing in the astrophysical environments, particularly in the magnetosphere of a neutron star.

By analyzing the fifth-order dispersion relation, the influence of coupled magnetic field and self-gravity on the pulsational instability of gas pressure dominated accretion disk is investigated. Our main results are that the viscous modes become more stable with the increase of self-gravity, magnetized field and viscosity, while they enhance the instabilities of acoustic modes. The effect of self-gravity to the instability is much greater than that of magnetic field in transition zone of the accretion disk. Especially, the self-gravity affects the thermal-modes and acoustic modes strongly. Finally, we discuss our results.

The radial-azimuthal instability of an isothermal magnetized accretion disk is examined in this paper. We find that the azimuthal perturbation enhances the instability of the magneto-acoustic and non-axisymmetric modes. The magnetic field induces and enhances the instability of magneto-acoustic modes, but stabilizes the non-axisymmetric modes in the outer disk. The viscous modes are always stable. The instability of magneto-acoustic modes are dominant in this disk.

A mechanism is presented for generating high speed particles in Herbig-Haro flow coming from accretion disks associated with protostars. The disks are threaded with weak magnetic field lines, in which the magnetorotational instability results in magnetohydrodynamic turbulence. Then the turbulent waves accelerate the thermal particles out of the accretion disks to a few hundred kilometres per second, forming the high speed particles of optical jets in star-forming regions.

The magnetodynamic instability of a rotating self-gravitating fluid layer of finite thickness embedded in a fluid of different density is developed. A general eigenvalue relation valid for all perturbations is derived based on the linear perturbation technique. In absence of rotation and electromagnetic forces, the model is gravitationally unstable if the resultant dimensionless wavenumber is less than 0.64 and vice versa and if it is equal to 0.64 the marginal stability arises. The magnetic field has a strong stabilizing effect that extends the region in which stable waves occur below 0.64 but the gravitational instability is not completely suppressed. For high values of the angular velocity ?, the rotating force is stabilizing and can suppress the gravitating instability for symmetric perturbations in the absence of the magnetic field. Considering the combined effect of rotation and electromagnetic force simultaneously, it is found that the magnetic field would be able to suppress the instability influence of rotation and it appears that the stability criterion is unaffected by the rotation in this case.

The conjugate influence of current relaxation and of current-vortex sheet formation on the magnetorotational instability is reviewed as firstly derived in [13]. It is shown that the relative amplification of the magnetic viscosity from marginal stability to the instability determined by the maximum growth rate is around 924% when resistive effects dominate, while the corresponding quantity is around 220% in the ideal limit. This means that the conjugate influence is much more efficient to amplify the magnetic viscosity than just the effect due to the standard magnetic tension. The results presented here may contribute to the understanding of the various processes that play a significant role in the mechanism of anomalous viscosity observed in Keplerian disks. It is argued that the new effect shall be most relevant in thin accretion disks.

Resistive effects are investigated on the magnetorotational instability. The main modifications with respect to the approach to the problem in the ideal limit are the introduction of the diffusive term in the induction equation and of the decay time T of the velocity field perpendicular to the magnetic field in the dispersion relation. As a result, when resistive effects are strong, it is shown that the instability condition in the ideal limit is corrected by the coefficient κ2T2, where κ denotes the epicyclic frequency. The proposed formulation is then applied to the description of the instability for a Keplerian disk and it is found that the characteristic frequency of the unstable modes saturates asymptotically to a finite value which does not depend on the distance from the central object and is fully determined by the resistivity of the disk.

The instability of a vertically oscillated layer of fluid is a classical problem whose history dates back to Faraday in the 19th century. We consider the stability of a shallow layer for which the oscillation, as expressed through the effective gravitational acceleration, is a random function of time. Using both theoretical linear stability analysis and high-resolution numerical simulations, including both individual realizations and ensemble calculations, of the nonlinear system of equations, we find that two different stochastic modes of instability exist. Both modes find their expression in finite amplitude oscillations of the free surface that exhibit sharp crests and broad troughs, or in other words, that resemble the classical Stokes wave. We demonstrate that a necessary condition for the first type of instability is snapshot, or instantaneous, instability. The subdominant instability resembles classical parametric resonance that can exist in a harmonically oscillated layer of fluid, and occurs even when the flow is always snapshot stable (or the gravitational acceleration is non-negative).

The magnetorotational instability (MRI) is investigated using a two-fluid model. Electrons and singly charged ions are supposed to have the same angular velocity to eliminate the equilibrium current. The linear dispersion relation governing local MRI is derived. The instability criteria in the non-magnetized and weakly magnetized cases differ remarkably from those predicted by the one-fluid model. Based on the general magnetized case, we present the critical conditions for the occurrence of instability. Gyroeffects significantly alter the instability criterion and introduce four unstable regions, one of which is reduced to the magnetohydrodynamic result when the angular frequency is much less than the ion gyrofrequency. When the angular frequency is much less than the electron gyrofrequency, the ion gyroeffect contributes to the instability criterion and induces the Hall term.

The effects of the equilibrium density gradient on non-axisymmetric magnetorotational instability are investigated in a pure axial magnetic field for ideal incompressible plasmas. A second-order ordinary differential equation is employed to determine the magnetic field perturbation and the full dispersion relationship regarding the non-axisymmetric magnetohydrodynamic instability in the presence of gravitation and density gradient effects. By means of local linear analysis, the reduced dispersion relationship is derived with a small azimuthal wavenumber. Spatial variations in the radial field perturbation magnitude cannot be neglected in the calculation since this term has the same order of magnitude as LD, which is the scale length of radial density gradient. The analytical expression of the instability growth rate is presented. Our analysis shows that the instability criterion is modified by the density gradient which has a stabilizing effect when increasing outwards and conversely a destabilizing effect when decreasing outwards. The growth rate increases with LD when LD is small. For a sufficiently large LD, the growth rate decreases with increasing LD. The magnetic field exerts a similar effect on the growth rate and can totally quench the instability. The non-axisymmetric effect introduces a frequency shift and increases the growth rate but does not affect the instability criterion.

Black holes surrounded by self-gravitating tori are astrophysical systems which may naturally form following the core collapse of a massive star or the merger of two neutron stars. We present here results from fully general relativistic numerical simulations of such systems in order to assess the influence of the torus self-gravity on the onset of the so-called runaway instability. This instability, which might drive the rapid accretion of the disk on shorter timescales than those required to power a relativistic fireball, potentially challenges current models of gamma-ray bursts. Our simulations indicate that the self-gravity of the torus does not actually favour the onset of the instability.

Taking advantage of disparate spatio-temporal scales relevant to astrophysics and laboratory experiments, we derive asymptotically exact reduced partial differential equation models for the magnetorotational instability. These models extend recent single-mode formulations leading to saturation in the presence of weak dissipation, and are characterized by a back-reaction on the imposed shear. Numerical simulations performed for a broad class of initial conditions indicate an initial phase of growth dominated by the optimal (fastest growing) magnetorotational instability fingering mode, followed by a vertical coarsening to a box-filling mode.

Most regions of protoplanetary disks are magnetorotationally unstable. Only the inner part of the disk, which is the so-called 'dead zone', is stabilized by ohmic dissipation. The characteristics of dust grains affect the abundances of charged particles significantly and change the size of the dead zones. The outer radius is typically around 10 AU, but could be zero. The gas, dust grains and magnetic field in protoplanetary disks are closely correlated with each other and thus the evolution of these components should be solved consistently. Especially dust evolution in turbulent disks is an important process in planet formation theory.

The charged dust effect on stability of a magnetized rotating plasma is analysed using approximation of immobile dust. In the presence of the dust, a term with the electric field appears in the one-fluid equation of plasma motion. This electric field affects the equilibrium plasma rotation and also gives rise to a family of instabilities of the rotating plasma called dust-induced rotational instabilities (DRIs). The DRIs are related to the charge imbalance between the plasma ions and electrons because of the charged dust. In contrast to the well-known magnetorotational instability driven by the radially decreasing plasma rotation frequency, the DRI can appear for an arbitrary rotation frequency profile. A mathematical technique for the analysis of the magnetorotational phenomena is presented. It is based on the one-fluid magnetohydrodynamic approach developed for a pure plasma and generalized to include the immobile dust effects. The mode equation and local dispersion relation are derived in terms of the canonical parameters.

We present magnetohydrodynamic (MHD) models of stellar core collapse computed with our astrophysical MHD code GenASiS. Microphysics in the models include a realistic equation of state and a simple parametrization of deleptionization during collapse. We give a brief description of some numerical results, and discuss requirements for capturing the physics of the magnetorotational instability in the context of core collapse supernovae.

The authors investigate the stability of a space-bounded relativistic beam of electrons accelerated by a nonuniform electric field under conditions where the transverse dimension of the beam is significantly greater than the Debye radius and a collective approach is necessary for describing small oscillations. For currents exceeding some critical value, there may develop in such a system an instability analogous to the ‘slipping’ instability of non-relativistic electron beams with a nonuniform velocity profile. The instability conditions are established and the instability growth rates calculated both in the absence and in the presence of a strong longitudinal magnetic field. The energy spread of the electrons has a stabilizing effect on the beam instability in question. In the Annex it is shown that the slipping instability is characteristic of any non-uniform plasma flow. In particular, it may develop in an isotropic plasma if the flow velocities exceed the thermal velocity of the ions.

The effect of magnetic viscosity is investigated in two simple stability problems — Rayleigh-Taylor instability and Kelvin-Helmholtz instability of superposed fluids. In the Rayleigh-Taylor instability, perturbations perpendicular to the magnetic field are suppressed. In the Kelvin-Helmholtz instability, a magnetic field transverse to the direction of streaming velocities in general opposes the development of instability.