Resistive relaxation of a magnetically confined mountain on an accreting neutron star

Three-dimensional numerical magnetohydrodynamic (MHD) simulations are performed to investigate how a magnetically confined mountain on an accreting neutron star relaxes resistively. No evidence is found for non-ideal MHD instabilities on a short time-scale, such as the resistive ballooning mode or the tearing mode. Instead, the mountain relaxes gradually as matter is transported across magnetic surfaces on the diffusion time-scale, which evaluates to  τ_I∼ 10⁵–10⁸ yr  (depending on the conductivity of the neutron star crust) for an accreted mass of   M _a= 1.2 × 10⁻⁴ M_⊙  . The magnetic dipole moment simultaneously re-emerges as the screening currents dissipate over  τ_I  . For non-axisymmetric mountains, ohmic dissipation tends to restore axisymmetry by magnetic reconnection at a filamentary neutral sheet in the equatorial plane. Ideal-MHD oscillations on the Alfvén time-scale, which can be excited by external influences, such as variations in the accretion torque, compress the magnetic field and hence decrease  τ_I  by one order of magnitude relative to its standard value (as computed for the static configuration). The implications of long-lived mountains for gravitational wave emission from low-mass X-ray binaries are briefly explored.     

The influence of Hall drift on the magnetic fields of neutron stars

We consider the evolution of magnetic fields under the influence of Hall drift and Ohmic decay. The governing equation is solved numerically, in a spherical shell with   r _i / r _o = 0.75  . Starting with simple free-decay modes as initial conditions, we then consider the subsequent evolution. The Hall effect induces so-called helicoidal oscillations, in which energy is redistributed among the different modes. We find that the amplitude of these oscillations can be quite substantial, with some of the higher harmonics becoming comparable with the original field. Nevertheless, this transfer of energy to the higher harmonics is not sufficient to accelerate significantly the decay of the original field, at least not at the   R _B = O (100)  parameter values accessible to us, where this Hall parameter   R _B   measures the ratio of the Ohmic time-scale to the Hall time-scale. We do find clear evidence though of increasingly fine structures developing for increasingly large   R _B   , suggesting that perhaps this Hall-induced cascade to ever-shorter length-scales is eventually sufficiently vigorous to enhance the decay of the original field. Finally, the implications for the evolution of neutron star magnetic fields are discussed.     

The stability of accretion discs with inflow driven purely by magnetic winds

A model is presented for an accretion disc in which the inflow is driven purely by the angular momentum removed in a centrifugally accelerated magnetic wind. Turbulent discs around compact stars are considered, with the required magnetic field being generated in the disc by a simple dynamo. The turbulent magnetic Prandtl number, N _p, measures the ratio of turbulent viscosity to turbulent magnetic diffusivity. Formally, the hypothetical limit   N _p→ 0  corresponds to the magnetic wind torque dominating the viscous torque, but in practice the inflow is magnetically controlled for   N _p≲ 0.1  .
The suggestion by previous authors that purely magnetic wind-driven discs may be unstable is investigated. A detailed steady solution is found which allows perturbations to the thermal balance and vertical equilibrium to be calculated, and hence the effect of perturbations to the magnetic diffusivity, η, to be assessed. For a standard parametrized form of η, the wind-driven angular momentum balance is found to be linearly unstable. An increase in the inflow rate leads to increased bending of the poloidal magnetic field and an enhanced wind mass loss rate. This increases the angular momentum loss rate which drives further inflow. There is a resultant increase in η, due to the temperature perturbation, but this does not relieve field bending sufficiently to prevent the instability.     

耀斑大尺度电流片的湍流耗散和能谱分析

磁重联在宇宙的许多动力学现象中都是非常核心的过程.磁流体动力学(MHD)数值模拟是研究磁重联过程以及相应物理图像的一种很有效的手段.通过不同的参数组合,来研究MHD数值模拟中磁雷诺数和空间分辨率对磁重联率、数值耗散和能谱分布的影响.对得到的数据进行分析后,发现磁雷诺数对磁重联率和能谱分布有一定的影响.磁雷诺数越大,磁重联过程进入非线性阶段所需的特征时间越短,磁重联率就越早发生跃升.磁雷诺数R_m对耗散开始发挥作用的Kolmogorov微观尺度l_ko有明显影响:R_m越大,l_ko就越小.研究了磁重联过程中包括数值耗散在内的额外耗散对重联过程的影响.结果表明,撕裂模不稳定性开始之前的额外耗散以纯数值耗散为主,撕裂模不稳定性出现之后,额外耗散出现同步跃升,说明不稳定性导致的湍流明显增强了耗散的效果,相当于在局部湍流区引入了超电阻.能谱分析进一步表明,大尺度电流片的l_ko完全可能出现在宏观的MHD尺度上.     

Burial of the polar magnetic field of an accreting neutron star – II. Hydromagnetic stability of axisymmetric equilibria

The theory of polar magnetic burial in accreting neutron stars predicts that a mountain of accreted material accumulates at the magnetic poles of the star, and that, as the mountain spreads equatorward, it is confined by, and compresses, the equatorial magnetic field. Here, we extend previous, axisymmetric, Grad–Shafranov calculations of the hydromagnetic structure of a magnetic mountain up to accreted masses as high as   M _a= 6 × 10⁻⁴ M_⊙  , by importing the output from previous calculations (which were limited by numerical problems and the formation of closed bubbles to   M _a < 10⁻⁴ M_⊙  ) into the time-dependent, ideal-magnetohydrodynamic code zeus-3d and loading additional mass on to the star dynamically. The rise of buoyant magnetic bubbles through the accreted layer is observed in these experiments. We also investigate the stability of the resulting hydromagnetic equilibria by perturbing them in zeus-3d . Surprisingly, it is observed that the equilibria are marginally stable for all   M _a≤ 6 × 10⁻⁴ M_⊙  ; the mountain oscillates persistently when perturbed, in a combination of Alfvén and acoustic modes, without appreciable damping or growth, and is therefore not disrupted (apart from a transient Parker instability initially, which expels <1 per cent of the mass and magnetic flux).     

On the time dependence of differential rotation in young late-type stars

A model for the angular momentum transfer within the convection zone of a rapidly rotating star is introduced and applied to the analysis of recent observations of temporal fluctuations of the differential rotation on the young late-type stars AB Doradus (AB Dor) and LQ Hydrae (LQ Hya). Under the hypothesis that the mean magnetic field produced by the stellar dynamo rules the angular momentum exchanges and that the angular velocity depends only on the distance s from the rotation axis and the time, the minimum azimuthal Maxwell stress  | B_sB _φ|  , averaged over the convection zone, is found to range from ∼0.04 to  ∼0.14 T²  . If the poloidal mean magnetic field   B _s   is of the order of 0.01 T, as indicated by the Zeeman–Doppler imaging maps of those stars, then the azimuthal mean field   B _φ  can reach an intensity of several teslas, which significantly exceeds equipartition with the turbulent kinetic energy. Such strong fields can account also for the orbital period modulation observed in cataclysmic variables and RS Canum Venaticorum systems with a main-sequence secondary component. Moreover, the model allows us to compute the kinetic energy dissipation rate during the maintenance of the differential rotation. Only in the case of the largest surface shear observed on LQ Hya may the dissipated power exceed the stellar luminosity, but the lack of a sufficient statistic on the occurrence of such episodes of large shear does not allow us to estimate their impact on the energy budget of the convection zone.     

Continuous frequency spectrum of the global hydromagnetic oscillations of a magnetically confined mountain on an accreting neutron star

We compute the continuous part of the ideal-magnetohydrodynamic (ideal-MHD) frequency spectrum of a polar mountain produced by magnetic burial on an accreting neutron star. Applying the formalism developed by Hellsten & Spies, extended to include gravity, we solve the singular eigenvalue problem subject to line-tying boundary conditions. This spectrum divides into an Alfvén part and a cusp part. The eigenfunctions are chirped and anharmonic with an exponential envelope, and the eigenfrequencies cover the whole spectrum above a minimum ω_low. For equilibria with accreted mass  1.2 × 10⁻⁶≲ M _a/M_⊙≲ 1.7 × 10⁻⁴  and surface magnetic fields  10¹¹≲ B _*/G ≲ 10¹³, ω_low  is approximately independent of   B _*  , and increases with M _a. The results are consistent with the Alfvén spectrum excited in numerical simulations with the zeus-mp solver. The spectrum is modified substantially by the Coriolis force in neutron stars spinning faster than ∼100 Hz. The implications for gravitational-wave searches for low-mass X-ray binaries are considered briefly.     

Explosive reconnection in magnetars

The X-ray activity of anomalous X-ray pulsars and soft γ-ray repeaters may result from the heating of their magnetic corona by direct currents dissipated by magnetic reconnection. We investigate the possibility that X-ray flares and bursts observed from anomalous X-ray pulsars and soft γ-ray repeaters result from magnetospheric reconnection events initiated by development of the tearing mode in magnetically dominated relativistic plasma. We formulate equations of resistive force-free electrodynamics, discuss the relation of the latter to ideal electrodynamics, and give examples of both ideal and resistive equilibria. Resistive force-free current layers are unstable towards the development of small-scale current sheets where resistive effects become important. Thin current sheets are found to be unstable due to the development of the resistive force-free tearing mode. The growth rate of the tearing mode is intermediate between the short Alfvén time-scale  τ_A  and a long resistive time-scale  τ_R: Γ∼ 1/(τ_Rτ_A)^1/2  , similar to the case of non-relativistic non-force-free plasma. We propose that growth of the tearing mode is related to the typical rise time of flares, ∼10 ms. Finally, we discuss how reconnection may explain other magnetar phenomena and ways to test the model.     

A model of magnetically induced disc–corona for black hole binaries

We propose a model of magnetic connection (MC) of a black hole with its surrounding accretion disc based on large-scale magnetic field. The MC gives rise to transport of energy and angular momentum between the black hole and the disc, and the closed field lines pipe the hot matter evaporated from the disc, and shape it in the corona above the disc to form a magnetically induced disc–corona system, in which the corona has the same configuration as the large-scale magnetic field. We numerically solve the dynamic equations in the context of the Kerr metric, in which the large-scale magnetic field is determined by dynamo process and equipartition between magnetic pressure and gas pressure. Thus we can obtain a global solution rather than assuming the distribution of large-scale magnetic field beforehand. The main MC effects lie in three aspects. (1) The rotational energy of a fast-spinning black hole can be extracted, enhancing the dissipation in the accretion disc, (2) the closed field lines provide a natural channel for corona matter escaping from disc and finally falling into black hole and (3) the scope of the corona can be bounded by the conservation of magnetic flux. We simulate the high-energy spectra of this system by using Monte Carlo method, and find that the relative hardness of the spectra decreases as accretion rate or black hole spin a _* increases. We fit the typical X-ray spectra of three black hole binaries  (GRO J1655−40, XTE 1118+480 and GX 339−4)  in the low/hard or very high state.     

The response of the Fe Kα line to changes in the X-ray illumination of accretion discs

X-ray reflection spectra from photoionized accretion discs in active galaxies are presented for a wide range of illumination conditions. The energy, equivalent width (EW) and flux of the Fe K α line are shown to depend strongly on the ratio of illuminating flux to disc flux,   F _x/ F _disc  , the photon index of the irradiating power law, Γ, and the incidence angle of the radiation, i . When   F _x/ F _disc≤2  a neutral Fe K α line is prominent for all but the largest values of Γ. At higher illuminating fluxes an He-like Fe K α line at 6.7 keV dominates the line complex. With a high-energy cut-off of 100 keV, the thermal ionization instability seems to suppress the ionized Fe K α line when  Γ≤1.6  . The Fe K α line flux correlates with   F _x/ F _disc  , but the dependence weakens as iron becomes fully ionized. The EW is roughly constant when   F _x/ F _disc  is low and a neutral line dominates, but then declines as the line progresses through higher ionization stages. There is a strong positive correlation between the Fe K α EW and Γ when the line energy is at 6.7 keV, and a slight negative one when it is at 6.4 keV. This is a potential observational diagnostic of the ionization state of the disc. Observations of the broad Fe K α line, which take into account any narrow component, would be able to test these predictions. Ionized Fe K α lines at 6.7 keV are predicted to be common in a simple magnetic flare geometry. A model that includes multiple ionization gradients on the disc is postulated to reconcile the results with observations.     

A dynamo model for axisymmetric and non-axisymmetric solar magnetic fields

More and more observations are showing a relatively weak, but persistent, non-axisymmetric magnetic field co-existing with the dominant axisymmetric field on the Sun. Its existence indicates that the non-axisymmetric magnetic field plays an important role in the origin of solar activity. A linear non-axisymmetric  α²– Ω  dynamo model is derived to explore the characteristics of the axisymmetric  ( m = 0)  and the first non-axisymmetric  ( m = 1)  modes and to provide a theoretical basis with which to explain the 'active longitude', 'flip-flop' and other non-axisymmetric phenomena. The model consists of an updated solar internal differential rotation, a turbulent diffusivity varying with depth, and an α-effect working at the tachocline in a rotating spherical system. The difference between the  α²–Ω  and the  α–Ω  models and the conditions that favour the non-axisymmetric modes under solar-like parameters are also presented.     

On the MHD effects on the force-free monopole outflow

The stationary axisymmetric outflow from a rotating sphere with a (split) monopole magnetic field is considered. The stream equation describing the outflow is linearized in terms of the Michel magnetization parameter σ⁻¹ ≪ 1, which allows a self-consistent analysis of the direct problem. It is shown that for a finite σ the fast magnetosonic surface is located at a finite distance ∼ σ^1/3 R _L ( R _L =  c /Ω_F is the light cylinder). We have also found that the particle energy at the fast surface is just equal to the Michel value γ ∼ 1/3σ. The particle acceleration and magnetic field collimation are shown to become ineffective outside the fast magnetosonic surface.     

Hydrostatic equilibrium in a magnetized, warped Galactic disc

Hydrostatic equilibrium of the multiphase interstellar medium in the solar vicinity is reconsidered, with the regular and turbulent magnetic fields treated separately. The regular magnetic field strength required to support the gas is consistent with independent estimates, provided that energy equipartition is maintained between turbulence and random magnetic fields. Our results indicate that a mid-plane value of B ₀=4 μG for the regular magnetic field near the Sun leads to more attractive models than B ₀=2 μG . The vertical profiles of both the regular and random magnetic fields contain disc and halo components, the parameters of which we have determined. The layer at 1≲| z |≲4 kpc can be overpressured and an outflow at a speed of about 50 km s⁻¹ may occur there, presumably associated with a Galactic fountain flow, if B ₀≃2 μG .
We show that hydrostatic equilibrium in a warped disc must produce asymmetric density distributions in z , in rough agreement with H  i observations in the outer Galaxy. This asymmetry may be a useful diagnostic of the details of the warping mechanism in the Milky Way and other galaxies. We find indications that gas and magnetic field pressures are different above and below the warped midplane in the outer Galaxy, and quantify the difference in terms of turbulent velocity and/or magnetic field strength.     

Variability associated with α accretion disc theory for standard and advection-dominated discs

The α turbulent viscosity formalism for accretion discs must be interpreted as a mean field theory, modelling a steady state only on spatial or time-scales greater than those of the turbulence. The extent of the scale separation determines the relative precision error (RPE) of the predicted luminosity L _ν. Turbulence and the use of α implies that (1) field line stretching gives a magnetic pressure  α²/6 of the total pressure generally, and a one-to-one relation between α and the pressure ratio for thin discs, and (2) large turbulent scales in advection-dominated accretion flows (ADAFs) predict a lower L _ν precision than thin discs for a given observation duration and central mass. The allowed variability (or RPE) at frequency ν increases with the size of the contributing region. For X-ray binary ADAFs, the RPE ∼ 5 per cent at R  ≤ 1000 Schwarzchild radii ( R _s) for averages over  1000 s. However, current data for galaxies like NGC 4258 and M87 give RPEs in L _ν of 50–100 per cent even at R  ≤ 100  R _S. More data are required, but systematic deviations from ADAF predictions are more significant than random deviations, and may constrain properties of the turbulence, the accretion mode, the assumption of a steady state or the accretion rate.     

Microlensing in dark matter haloes

Using eight dark matter haloes extracted from fully self-consistent cosmological N -body simulations, we perform microlensing experiments. A hypothetical observer is placed at a distance of 8.5 kpc from the centre of the halo measuring optical depths, event durations and event rates towards the direction of the Large Magellanic Cloud. We simulate 1600 microlensing experiments for each halo. Assuming that the whole halo consists of massive astronomical compact halo objects (MACHOs),   f = 1.0  , and a single MACHO mass is   m _M= 1.0 M_⊙  , the simulations yield mean values of  τ= 4.7^+5.0_−2.2× 10⁻⁷  and  Γ= 1.6^+1.3_−0.6× 10⁻⁶  events star⁻¹ yr⁻¹. We find that triaxiality and substructure can have major effects on the measured values so that τ and Γ values of up to three times the mean can be found. If we fit our values of τ and Γ to the MACHO collaboration observations, we find   f = 0.23^+0.15_−0.13  and   m _M= 0.44^+0.24_−0.16  . Five out of the eight haloes under investigation produce f and m _M values mainly concentrated within these bounds.     

Magnetic field dragging and the vertical structure of thin accretion discs

The presence of an imposed vertical magnetic field may drastically influence the structure of thin accretion discs. If the field is sufficiently strong, the rotation law can depart from the Keplerian one. We consider the structure of a disc for a given eddy magnetic diffusivity but neglect details of the energy transport. The magnetic field is assumed to be in balance with the internal energy of the accretion flow. The thickness of the disc as well as the turbulent magnetic Prandtl number and the viscosity, α , are the key parameters of our model. The calculations show that the radial velocity can reach the sound speed for a magnetic disc if the thickness is comparable to that of a non-magnetic one. This leads to a strong amplification of the accretion rate for a given surface density. The inclination angle of the magnetic field lines can exceed the critical value 30° (required to launch cold jets) even for a relatively small magnetic Prandtl number of order unity. The toroidal magnetic fields induced at the disc surface are smaller than predicted in previous studies.     

Whistler wave cascades in solar wind plasma

Non-linear, three-dimensional, time-dependent fluid simulations of whistler wave turbulence are performed to investigate role of whistler waves in solar wind plasma turbulence in which characteristic turbulent fluctuations are characterized typically by the frequency and length-scales that are, respectively, bigger than ion gyrofrequency and smaller than ion gyroradius. The electron inertial length is an intrinsic length-scale in whistler wave turbulence that distinguishably divides the high-frequency solar wind turbulent spectra into scales smaller and bigger than the electron inertial length. Our simulations find that the dispersive whistler modes evolve entirely differently in the two regimes. While the dispersive whistler wave effects are stronger in the large-scale regime, they do not influence the spectral cascades which are describable by a Kolmogorov-like   k ^−7/3  spectrum. By contrast, the small-scale turbulent fluctuations exhibit a Navier–Stokes-like evolution where characteristic turbulent eddies exhibit a typical   k ^−5/3  hydrodynamic turbulent spectrum. By virtue of equipartition between the wave velocity and magnetic fields, we quantify the role of whistler waves in the solar wind plasma fluctuations.     

Mean-field effects in the Galloway–Proctor flow

In the framework of mean-field electrodynamics the coefficients defining the mean electromotive force in Galloway–Proctor flows are determined. These flows show a two-dimensional pattern and are helical. The pattern wobbles in its plane. Apart from one exception a circularly polarized Galloway–Proctor flow, i.e. a circular motion of the flow pattern is assumed. This corresponds to one of the cases considered recently by Courvoisier, Hughes & Tobias. An analytic theory of the α effect and related effects in this flow is developed within the second-order correlation approximation and a corresponding fourth-order approximation. In the validity range of these approximations there is an α effect but no γ effect, or pumping effect. Numerical results obtained with the test-field method, which are independent of these approximations, confirm the results for α and show that γ is in general non-zero. Both α and γ show a complex dependency on the magnetic Reynolds number and other parameters that define the flow, that is, amplitude and frequency of the circular motion. Some results for the magnetic diffusivity  η_t  and a related quantity are given, too. Finally, a result for α in the case of a randomly varying linearly polarized Galloway–Proctor flow, without the aforementioned circular motion, is presented. The flows investigated show quite interesting effects. There is, however, no straightforward way to relate these flows to turbulence and to use them for studying properties of the α effect and associated effects under realistic conditions.