Weak Initial Magnetic Field Research Articles

Influence of the magnetic field on the formation of protostellar disks

Abstract We numerically model the collapse of magnetic rotating protostellar clouds with mass of 10 M ⊙ {M}_{\odot } . The simulations are carried out with the help of 2D MHD code Enlil. The structure of the cloud at the isothermal stage of the collapse is investigated for the cases of weak, moderate, and strong initial magnetic field. Simulations reveal the universal hierarchical structure of collapsing protostellar clouds, consisting of the flattened envelope with the qausi-magnetostatc disk inside and the first core in its center. The size of the primary disk increases with the initial magnetic energy of the cloud. The magnetic braking efficiently transports the angular momentum from the primary disk into the envelope in the case, when the initial magnetic energy of the cloud is more than 20% of its gravitational energy. The intensity of the outflows launched from the region near the boundary of the first core increases with initial magnetic energy. The “dead” zone with small ionization fraction, x < 1 0 − 11 x\lt 1{0}^{-11} , forms inside the first hydrostatic core and at the base of the outflow. Ohmic dissipation and ambipolar diffusion determine conditions for further formation of the protostellar disk in this region.

Turbulent Dynamo in a Weakly Ionized Medium

Abstract The small-scale turbulent dynamo is an important process contributing to the cosmic magnetization. In partially ionized astrophysical plasmas, the dynamo growth of magnetic energy strongly depends on the coupling state between ions and neutrals and the ion-neutral collisional damping effect. A new damping stage of turbulent dynamo in a weakly ionized medium was theoretically predicted by Xu & Lazarian. By carrying out a 3D two-fluid dynamo simulation, we have for the first time numerically confirmed the physical conditions and the linear-in-time growth of the magnetic field strength of the damping stage of a dynamo. The dynamo-amplified magnetic field has a characteristic length as the damping scale, which increases with time and can reach the injection scale of turbulence after around eight turnover times of the largest eddy, given sufficiently low ionization fraction and a weak initial magnetic field. Due to the weak coupling between ions and neutrals, most turbulent energy carried by neutrals cannot be converted to magnetic energy, resulting in a relatively weak magnetic field at the end of the dynamo. This result has important implications for the growth of magnetic fields in the partially ionized interstellar medium and shock acceleration of Galactic cosmic rays.

How to form a millisecond magnetar? Magnetic field amplification in protoneutron stars

AbstractExtremely strong magnetic fields of the order of 1015G are required to explain the properties of magnetars, the most magnetic neutron stars. Such a strong magnetic field is expected to play an important role for the dynamics of core-collapse supernovae, and in the presence of rapid rotation may power superluminous supernovae and hypernovae associated to long gamma-ray bursts. The origin of these strong magnetic fields remains, however, obscure and most likely requires an amplification over many orders of magnitude in the protoneutron star. One of the most promising agents is the magnetorotational instability (MRI), which can in principle amplify exponentially fast a weak initial magnetic field to a dynamically relevant strength. We describe our current understanding of the MRI in protoneutron stars and show recent results on its dependence on physical conditions specific to protoneutron stars such as neutrino radiation, strong buoyancy effects and large magnetic Prandtl number.

THERMAL STABILITY OF A THIN DISK WITH MAGNETICALLY DRIVEN WINDS

The absence of thermal instability in the high/soft state of black hole X-ray binaries, in disagreement with the standard thin disk theory, is a long-standing riddle for theoretical astronomers. We have tried to resolve this question by studying the thermal stability of a thin disk with magnetically driven winds in the $\dot{M}- \Sigma$ plane. It is found that disk winds can greatly decrease the disk temperature and thus help the disk become more stable at a given accretion rate. The critical accretion rate $\dot{M}_{\rm crit}$ corresponding to the thermal instability threshold is increased significantly in the presence of disk winds. For $\alpha=0.01$ and $B_{\rm \phi}=10B_{\rm _p}$, the disk is quite stable even for a very weak initial poloidal magnetic field [$\beta_{\rm p,0}\sim 2000, \beta_{\rm p}=(P_{\rm {gas}}+P_{\rm rad})/(B_{\rm p}^2/8\pi)$]. But when $B_{\rm \phi}=B_{\rm _p}$ or $B_{\rm \phi}=0.1B_{\rm _p}$, a somewhat stronger (but still weak) field ($\beta_{\rm p,0}\sim 200$ or $\beta_{\rm p,0}\sim 20$) is required to make the disk stable. Nevertheless, despite the great increase of $\dot{M}_{\rm crit}$, the luminosity threshold corresponding to instability remains almost constant or even decreases slowly with increasing $\dot{M}_{\rm crit}$ due to the decrease of gas temperature. The advection and diffusion timescales of the large-scale magnetic field threading the disk are also investigated in this work. We find that the advection timescale can be smaller than the diffusion timescale in a disk with winds, because the disk winds take away most of the gravitational energy released in the disk, resulting in the decrease of the magnetic diffusivity $\eta$ and the increase of the diffusion timescale.

MIXING IN MAGNETIZED TURBULENT MEDIA

Turbulent motions are essential to the mixing of entrained fluids and are also capable of amplifying weak initial magnetic fields by small-scale dynamo action. Here we perform a systematic study of turbulent mixing in magnetized media, using three-dimensional magnetohydrodynamic simulations that include a scalar concentration field. We focus on how mixing depends on the magnetic Prandtl number, Pm, from 1 to 4 and the Mach number, M}, from 0.3 to 2.4. For all subsonic flows, we find that the velocity power spectrum has a k^-5/3 slope in the early, kinematic phase, but steepens due to magnetic back reactions as the field saturates. The scalar power spectrum, on the other hand, flattens compared to k^-5/3 at late times, consistent with the Obukohov-Corrsin picture of mixing as a cascade process. At higher Mach numbers, the velocity power spectrum also steepens due to the presence of shocks, and the scalar power spectrum again flattens accordingly. Scalar structures are more intermittent than velocity structures in subsonic turbulence while for supersonic turbulence, velocity structures appear more intermittent than the scalars only in the kinematic phase. Independent of the Mach number of the flow, scalar structures are arranged in sheets in both the kinematic and saturated phases of the magnetic field evolution. For subsonic turbulence, scalar dissipation is hindered in the strong magnetic field regions, probably due to Lorentz forces suppressing the buildup of scalar gradients, while for supersonic turbulence, scalar dissipation increases monotonically with increasing magnetic field strength. At all Mach numbers, mixing is significantly slowed by the presence of dynamically-important small-scale magnetic fields, implying that mixing in the interstellar medium and in galaxy clusters is less efficient than modeled in hydrodynamic simulations.

Dynamical bar-mode instability in rotating and magnetized relativistic stars

We present three-dimensional simulations of the dynamical bar-mode instability in magnetized and differentially rotating stars in full general relativity. Our focus is on the effects that magnetic fields have on the dynamics and the onset of the instability. In particular, we perform ideal-magnetohydrodynamics simulations of neutron stars that are known to be either stable or unstable against the purely hydrodynamical instability, but to which a poloidal magnetic field in the range of $10^{14}$--$10^{16}$ G is superimposed initially. As expected, the differential rotation is responsible for the shearing of the poloidal field and the consequent linear growth in time of the toroidal magnetic field. The latter rapidly exceeds in strength the original poloidal one, leading to a magnetic-field amplification in the the stars. Weak initial magnetic fields, i.e. $ \lesssim 10^{15}$ G, have negligible effects on the development of the dynamical bar-mode instability, simply braking the stellar configuration via magnetic-field shearing, and over a timescale for which we derived a simple algebraic expression. On the other hand, strong magnetic fields, i.e. $\gtrsim 10^{16}$ G, can suppress the instability completely, with the precise threshold being dependent also on the amount of rotation. As a result, it is unlikely that very highly magnetized neutron stars can be considered as sources of gravitational waves via the dynamical bar-mode instability.

Corotation torques experienced by planets embedded in weakly magnetized turbulent discs

The migration of low-mass planets is driven by the differential Lindblad torque and the corotation torque in non-magnetic viscous models of protoplanetary discs. The corotation torque has recently received detailed attention as it may slow down, stall, or reverse migration. In laminar viscous disc models, the long-term evolution of the corotation torque is intimately related to viscous and thermal diffusion processes in the planet's horseshoe region. This paper examines the properties of the corotation torque in discs where MHD turbulence develops as a result of the magnetorotational instability, considering a weak initial toroidal magnetic field. We present results of 3D MHD simulations carried out with two different codes. Non-ideal MHD effects and the disc's vertical stratification are neglected, and locally isothermal disc models are considered. The running time-averaged torque exerted by the disc on a fixed planet is evaluated in three disc models. We first present results with an inner disc cavity (planet trap). As in viscous disc models, the planet is found to experience a positive running time-averaged torque over several hundred orbits, which highlights the existence of an unsaturated corotation torque maintained in the long term in MHD turbulent discs. Two disc models with initial power-law density and temperature profiles are also adopted, in which the time-averaged torque is found to be in decent agreement with its counterpart in laminar viscous disc models with similar viscosity at the planet location. Detailed analysis of the averaged torque density distributions indicates that the differential Lindblad torque takes very similar values in MHD turbulent and laminar viscous discs, and there exists an unsaturated corotation torque in MHD turbulent discs. This analysis also reveals the existence of an additional corotation torque in weakly magnetized discs.

Weak-field dynamo emerging in a rotating spherical shell with stress-free top and no-slip bottom boundaries

Numerical experiments are performed in order to investigate an MHD dynamo in a rotating spherical shell with stress-free top and no-slip bottom boundaries. The Ekman number, the Prandtl number, and the ratio of inner to outer radii are fixed as 10 −3, 1, and 0.35, respectively. The magnetic Prandtl number is varied from 5 to 50, and the modified Rayleigh number is increased from 1.5 to 10 times the critical Rayleigh number. The initial imposed magnetic field is either a weak or strong magnetic field, where the magnetic energy of the initial field is approximately two orders of magnitude smaller or larger than the kinetic energy of the quasi-steady state of non-magnetic thermal convection. For cases involving a weak initial magnetic field, self-sustained dynamo solutions are established when the magnetic Prandtl number is larger than or equal to 5, and the modified Rayleigh number is larger than or equal to 5 times the critical Rayleigh number. The solutions are categorized as a weak field-dynamo, where the mean magnetic energy is one order of magnitude smaller than the mean kinetic energy. The dynamo solutions are characterized by a radially two-layer spatial structure. The upper layer is dominated by a strong prograde zonal mean zonal flow with large-scale prograde propagating spiral vortices having a longitudinal wavenumber of 3. Toroidal kinetic energy is converted to toroidal magnetic energy through stretching of the field lines by large-scale prograde propagating spiral vortices. On the other hand, the lower layer contains small-scale retrograde propagating columnar convective vortices having a longitudinal wavenumber of 8. The magnetic field lines are not concentrated in the vortices, but rather wind around the vortices in each layer. Poloidal kinetic energy is converted to poloidal magnetic energy through winding of the field lines around small-scale retrograde propagating columnar convective vortices. For cases involving a strong initial magnetic field, self-sustained dynamo solutions are established when the magnetic Prandtl number is greater than or equal to 4 and the modified Rayleigh number is greater than or equal to 3 times the critical Rayleigh number. In contrast with the cases involving a weak initial magnetic field, all of the dynamo solutions are strong-field solutions. Bistability is observed when the magnetic Prandtl number is 5, and the modified Rayleigh number is approximately equal to 10 times the critical Rayleigh number, where weak-field and strong-field dynamo solutions coexist. However, transition between the weak- and strong-field solutions does not occur in this case.

Axisymmetric simulations of magneto-rotational core collapse: dynamics and gravitational wave signal

Aims. We have performed a comprehensive parameter study of the collapse of rotating, strongly magnetized stellar cores in axisymmetry to determine their gravitational wave signature based on the Einstein quadrupole formula. Methods. We use a Newtonian explicit magnetohydrodynamic Eulerian code based on the relaxing-TVD method for the solution of the ideal MHD equations, and apply the constraint-transport method to guarantee a divergence-free evolution of the magnetic field. We neglect effects due to neutrino transport and employ a simplified equation of state. The initial models are polytropes in rotational equilibrium with a prescribed degree of differential rotation and rotational energy. The initial magnetic fields are purely poloidal the field strength ranging from 1010 G to 10 13 G. The evolution of the core is followed until a few ten milliseconds past core bounce. Results. The initial magnetic fields are amplified mainly by the differential rotation of the core giving rise to a strong toroidal field component with an energy comparable to the rotational energy. The poloidal field component grows by compression during collapse, but does not change significantly after core bounce. In large parts of the simulated cores the growth time of the magneto-rotational instability (MRI) is of the order of a few milliseconds. The saturation field strengths that can be reached both via a pure 12 dynamo or the MRI are of the order of 10 15 G at the surface of the core. Sheet-like circulation flows which produce a strong poloidal field component transporting angular momentum outwards develop due to MRI, provided the initial field is not too weak. Weak initial magnetic fields (≤10 11 G) have no significant effect on the dynamics of the core and the gravitational wave signal. Strong initial fields (≤10 12 G) cause considerable angular momentum transport whereby rotational energy is extracted from the collapsed core which loses centrifugal support and enters a phase of secular contraction. The gravitational wave amplitude at bounce changes by up to a few ten percent compared to the corresponding non-magnetic model. If the angular momentum losses are large, the post-bounce model. If the angular momentum losses are large the post-bounce equilibrium state of the core changes from a centrifugally to a pressure supported one. This transition imprints in the gravitational wave signal a reduction of the amplitude of the large-scale oscillations characteristic of cores bouncing due to centrifugal forces. In some models the quasi-periodic large-scale oscillations are replaced by higher frequency irregular oscillations. This pattern defines a new signal type which we call a type IV gravitational wave signal. Collimated bipolar outflows give rise to a unique feature that may allow their detection by means of gravitational wave astronomy: a large positive quadrupole wave amplitude of similar size as that of the bounce signal.

On Magnetohydrodynamic Jet Production in the Collapsing and Rotating Envelope

We present results from axisymmetric, time-dependent hydrodynamic (HD) and magnetohydrodynamic (MHD) simulations of a gaseous envelope collapsing onto a black hole (BH). We consider gas with such small angular momentum that, after an initial transient, the flow in the HD case accretes directly onto a BH without forming a rotationally supported torus. However, in the MHD case, even with a very weak initial magnetic field, the flow settles into a configuration with four components: (1) an equatorial inflow, (2) a bipolar outflow, (3) polar funnel outflow, and (4) polar funnel inflow. We focus our analysis on the second flow component of the MHD flow, which represents a simple yet robust example of a well-organized inflow/outflow solution to the problem of MHD jet formation. The jet is heavy, highly magnetized, and driven by magnetic and centrifugal forces. A significant fraction of the total energy in the jet is carried out by a large-scale magnetic field. We review previous simulations, in which specific angular momentum was higher than that assumed here, and conclude that our bipolar outflow develops for a wide range of the properties of the flow near the equator and near the poles. Future work on such a simple inflow/outflow solution will help to pinpoint the key elements of real jets/outflows as well as help to interpret much more complex simulations aimed at studying jet formation and the collapse of magnetized envelopes.

Magneto‐driven Shock Waves in Core‐Collapse Supernovae

We perform a series of two-dimensional magnetohydrodynamic simulations of the rotational core collapse of a magnetized massive star. We employ a realistic equation of state and take into account the neutrino cooling by the so-called leakage scheme. In this study we systematically investigate how the strong magnetic field and the rapid rotation affect the propagation of the shock waves. Our results show that in the case of the strong initial poloidal magnetic field, the toroidal magnetic field amplified by the differential rotation becomes strong enough to generate a tightly collimated shock wave along the rotational axis. On the other hand, in the case of the weak initial magnetic field, although the differential rotation amplifies the toroidal magnetic field over the long rotational period, the launched shock wave is weak and the shape of it becomes wider. The former case is expected to be accompanied by the formation of the so-called magnetar. Our models with rapid rotation and strong magnetic field can create a nozzle formed by the collimated shock wave. This might be the analogous situation to the collapsar that is plausible as the central engine of the gamma-ray bursts.

Two-Dimensional Simulation of the Dynamics of the Collapse of a Rotating Core with Formation of a Neutron Star on an Adaptive Triangular Grid in Lagrangian Coordinates

Results are presented from a two-dimensional numerical simulation of the collapse of a rotating core with formation of a neutron star that has strong differential rotation in its outer regions. A specially developed numerical method is used which is based on a fully conservative implicit operator difference scheme for gravitational gas dynamics problems in lagrangian coordinates on a variable-structure triangular grid. The recoil shock wave generated by the collapse causes ejection of a small amount of material. This cannot explain the explosion of type II supernovae. The strong differential rotation in the presence of even a weak initial magnetic field obtained in these calculations must lead to a rise in the magnetic pressure, formation of an MHD shock wave, and conversion of rotational energy into the energy of radial expansion (magnetorotational supernova explosion).

Alpha effect in helically forced turbulence

The effect on large-scale dynamics of small-scale helicity injection in three-dimensional resistive magnetohydrodynamic turbulence is investigated for a weak initial magnetic field. The magnetic configuration of α 2 dynamo prototype flows is such that, generally, the energy concentrates on one large-scale mode. However, we obtain that alpha effect is not limited to the dominant mode and that a non-local equation (in Fourier space) is more appropriate to describe it. It gives some insights into the non-local theory of Pouquet et al. [J. Fluid Mech. 77 (1976) 321] where the inverse cascade results from a competition between the helicity and Alfvén effects.

Global Three‐dimensional Magnetohydrodynamic Simulations of Accretion Disks and the Surrounding Magnetosphere

We present the results of global three-dimensional numerical simulations of magnetized accretion disks which have been performed in order to study the evolution of an accretion disk and its surrounding corona within the framework of ideal magnetohydrodynamics. A self-consistent study of this coupled system is a necessary step toward gaining an understanding of both the production and structure of outflows and the angular momentum transport in accretion disks. We study model disks that are initially threaded by a uniform, homogenous magnetic field parallel to the rotational axis. The evolution of these disks is examined under the assumption of an adiabatic equation of state. When the disk is threaded with a weak initial magnetic field, we observe new, intrinsically three-dimensional nonaxisymmetric phenomena that do not occur in the two-dimensional axisymmetric approximation. Our three-dimensional simulations show that in the weak field case MHD waves are generated at the surface of the disk and subsequently propagate into the corona, possibly contributing to coronal heating. To investigate the nature and the origin of these waves, we have performed a series of exploratory simulations with different rotation profiles of the corona and different density relations between disk and corona, as well as simulations with a purely toroidal initial magnetic field. When the simulations are computed with a strong initial field, we find that there is very little difference between two-dimensional and three-dimensional evolution. In both the weak field and strong field cases, in agreement with previous two-dimensional simulations, the disk collapses on orbital timescales. Disk collapse arises from angular momentum transport by the Balbus-Hawley instability in the weak field case and by the external torque of an outflow in the strong field case. We observe the generation of winds in all our simulations. In the weak field case, these winds are unsteady, and they do not collimate within the time frame of the simulation.

Shell models of MHD turbulence and the problem of superequipartition

Within the framework of the shell model of turbulence, we consider the long-time evolution of the magnetic field in a turbulent flow of conducting fluid. A weak initial magnetic field rapidly increases up to magnitudes corresponding to the equipartition with kinetic energy, and then the stage of slow evolution begins. It is shown that in some cases, the magnetic energy can significantly exceed the kinetic energy of the turbulence at this stage. We discuss the possible observational evidences for such a superequipartition of the magnetic field. It is shown that the superequipartition is realized if the initial energy of the magnetic field exceeds the kinetic energy, although even in such cases the equipartition is reached in many realizations.

The effect of the neutron-star crust on the evolution of a core magnetic field

We consider the expulsion of the magnetic field from the super-conducting core of a neutron star and its subsequent decay in the crust. Particular attention is paid to a strong feedback of the distortion of magnetic field lines in the crust on the expulsion of the flux from the core. This causes a considerable delay in the core flux expulsion if the initial field strength is larger than 1011 G. It is shown that the hypothesis on the magnetic field expulsion induced by the neutron-star spin-down is adequate only for a relatively weak initial magnetic field B≈1011 G. The expulsion time-scale depends not only on the conductivity of the crust, but also on the initial magnetic field strength itself. Our model of the field evolution naturally explains the existence of the residual magnetic field of neutron stars. Its strength is correlated with the impurity concentration in neutron-star crusts and anticorrelated with the initial field strengths.

Magnetic field amplification and saturation in two-dimensional magnetohydrodynamic turbulence

Two-dimensional (2-D) magnetohydrodynamic turbulence is investigated for weak initial magnetic fields using numerical simulation. It is found that the magnetic field is amplified owing to the formation of flux sheets with saturation due either to resistive diffusion (kinematic regime) or to nonlinear effects (dynamic regime). In the kinematic regime, which corresponds to the problem of passive scalar convection by 2-D Navier–Stokes turbulence, the saturation value of the magnetic energy is observed to scale as EMmax∝η−0.8 in approximate agreement with a simple theoretical estimate, EMmax/EM(0)≂Rm, where Rm is the magnetic Reynolds number. Because of the strongly disparate kinetic and magnetic energy spectra in the kinematic regime, roughly EVk∼k−3, EMk∼k, dynamic interaction on small scales already occurs at very small global energy ratios EM/EV, giving rise to strongly enhanced kinetic energy dissipation. In the fully dynamic regime (reached for EM/EV‖t=0>R−1m) global magnetic and kinetic energies become tightly coupled, with EM/EV being approximately constant in time and the energy dissipation rates being independent of the collisional diffusion coefficients. Finally, the effect of the magnetic Prandtl number Pr=μ/η is discussed.

Confinement of a Hot Electron Plasma in a Mirror Machine, DECA 1

Plasma, produced by a hydrogenated titanium washer gun, was compressed between two pulsed magnetic mirrors (rise time : 150 μsec, decay time : 6 msec). For weak initial magnetic fields (below 150 G), and large compression ratios (above 100), the resulting plasma consists of very energetic electrons and of colder ions; it is confined during several milliseconds. Typically, with an initial field of 30 G antiparallel to a compression field of 10 kG, the compressed plasma has a diameter of 1.5 cm and a length equal to or less than 7 cm; the average hot electron density is (4 ± 2) × 1011 cm−3, with mean energy of 45 ± 10 keV; the density of the cold plasma, which might be present around the compressed plasma, is less than 1011 cm−3. Calculations show moreover that, in addition to the adiabatic heating, an important nonadiabatic acceleration occurs at the beginning of the compression. When the initial density is higher than 1013 cm−3, the energetic electrons do not appear. One observes, then, large radial leaks which might be caused by an instability. Such conditions correspond to a β which is higher than 1 to 3% for the compressed plasma.