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.     

A joined model for solar dynamo and differential rotation

A model for the solar dynamo, consistent in global flow and numerical method employed with the differential rotation model, is developed. The magnetic turbulent diffusivity is expressed in terms of the entropy gradient, which is controlled by the model equations. The magnetic Prandtl number and latitudinal profile of the alpha-effect are specified by fitting the computed period of the activity cycle and the equatorial symmetry of magnetic fields to observations. Then, the instants of polar field reversals and time-latitude diagrams of the fields also come into agreement with observations. The poloidal field has a maximum amplitude of about 10 Gs in the polar regions. The toroidal field of several thousand Gauss concentrates near the base of the convection zone and is transported towards the equator by the meridional flow. The model predicts a value of about 10³⁷ erg for the total magnetic energy of large-scale fields in the solar convection zone.     

The mean energy of a decaying magnetic field in MHD-turbulence

A magnetic field which is contained in a turbulent, electrically conducting fluid is supposed to decay. The decay time of the mean energy of the fluctuating part of the magnetic field – in contrast with the energy of the mean field – is shown to be determined by the molecular magnetic diffusivity and the correlation length of the turbulence. The investigation is carried out for weak magnetic fields in homogeneous isotropic mirror-symmetric turbulence and in the framework of the second-order correlations approximation.     

The helicity constraint in turbulent dynamos with shear

The evolution of magnetic fields is studied using simulations of forced helical turbulence with strong imposed shear. After some initial exponential growth, the magnetic field develops a large-scale travelling wave pattern. The resulting field structure possesses magnetic helicity, which is conserved in a periodic box by the ideal magnetohydrodynamics equations and can hence only change on a resistive time-scale. This strongly constrains the growth time of the large-scale magnetic field, but less strongly constrains the length of the cycle period. Comparing this with the case without shear, the time-scale for large-scale field amplification is shortened by a factor Q , which depends on the relative importance of shear and helical turbulence, and which also controls the ratio of toroidal to poloidal field. The results of the simulations can be reproduced qualitatively and quantitatively with a mean-field α Ω-dynamo model with alpha-effect and turbulent magnetic diffusivity coefficients that are less strongly quenched than in the corresponding α ²-dynamo.     

Hydrodynamic Source of the 11-year Solar Variations

In the traditional axisymmetric models of the 11-year solar cycle, oscillations of the magnetic fields appear in the background of nonoscillating (over time scale considered) turbulent velocity fields and differential rotation. In this paper, an alternative approach is developed: The excitation of magnetic oscillations with the 22-year period is the consequence of hydrodynamic oscillations with the 11-year period. In the excitation of hydrodynamic oscillations, two processes taking place in high latitudes near the interface between the convective and radiative zones play a key role. One is forcing of the westerly zonal flow, the conditions for which are due to deformation of the interfacial surface. The other process is the excitation of a shear instability of zonal flow as a consequence of a strong radial gradient of angular velocity. The development of a shear instability at some stage brings about the disruption of the forcing of differential rotation. In the first (hydrodynamic) part of the paper, the dynamics of axisymmetric flows near the bottom of the convection zone is numerically simulated. Forcing of differential rotation having velocity shear in latitude and the existence of solutions in the form of torsional waves with the 11-year oscillation period are shown. In the second part the dynamics of the magnetic field is studied. The most pronounced peculiarities of the solutions are the existence of forced oscillations with the 22-year period and the drift of the toroidal magnetic field component from the mid latitudes to the equator. In high and low latitudes after cycle maximum, the toroidal component is of opposite sign in accordance with observations. In the third part, the transport of momentum from the bottom of the convection zone to the outer surface by virtue of diffusivity is considered. The existence of some sources of differential rotation in the convection zone is not implied. A qualitative correspondence of the differential rotation profile in the bulk of the convection zone and on its outer surface to experimental data is shown. The time correspondence between torsional and magnetic oscillations is also in accordance with observations.     

The dual role of shear in large‐scale dynamos

The role of shear in alleviating catastrophic quenching by shedding small‐scale magnetic helicity through fluxes along contours of constant shear is discussed. The level of quenching of the dynamo effect depends on the quenched value of the turbulent magnetic diffusivity. Earlier estimates that might have suffered from the force‐free degeneracy of Beltrami fields are now confirmed for shear flows where this degeneracy is lifted. For a dynamo that is saturated near equipartition field strength those estimates result in a 5‐fold decrease of the magnetic diffusivity as the magnetic Reynolds number based on the wavenumber of the energy‐carrying eddies is increased from 2 to 600. Finally, the role of shear in driving turbulence and large‐scale fields by the magneto‐rotational instability is emphasized. New simulations are presented and the 3π /4 phase shift between poloidal and toroidal fields is confirmed. It is suggested that this phase shift might be a useful diagnostic tool in identifying mean‐field dynamo action in simulations and to distinguish this from other scenarios invoking magnetic buoyancy as a means to explain migration away from the midplane. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The dynamics of a toroidal magnetic ring

Solving the nonlinear partial differential equations of magnetohydrodynamics numerically, we examine (1) the time development of a purely toroidal magnetic field (a magnetic ring) and (2) the interaction of a magnetic ring with a poloidal magnetic field. Axisymmetry and incompressibility are assumed. Parameters are chosen to correspond to photospheric conditions. In case (1), the magnetic ring contracts to the axis and then splits in two with one ring travelling up along the axis and the other down. In case (2), a large toroidal velocity field is generated which has opposite direction of flow above and below the magnetic ring. The magnetic and flow patterns of case (2) may persist with little change for a relatively long time. We conjecture that toroidal magnetic fields may be involved in the bright rings of sunspots or in the dynamics of spicules.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Magnetic field evolution in simulations with Euler potentials

Using two- and three-dimensional hydromagnetic simulations for a range of different flows, including laminar and turbulent ones, it is shown that solutions expressing the field in terms of Euler potentials (EP) are in general incorrect if the EP are evolved with an artificial diffusion term. In three dimensions, standard methods using the magnetic vector potential are found to permit dynamo action when the EP give decaying solutions. With an imposed field, the EP method yields excessive power at small scales. This effect is more exaggerated in the dynamic case, suggesting an unrealistically reduced feedback from the Lorentz force. The EP approach agrees with standard methods only at early times when magnetic diffusivity did not have time to act. It is demonstrated that the usage of EP with even a small artificial magnetic diffusivity does not converge to a proper solution of hydromagnetic turbulence. The source of this disagreement is not connected with magnetic helicity or the three-dimensionality of the magnetic field, but is simply due to the fact that the non-linear representation of the magnetic field in terms of EP that depend on the same coordinates is incompatible with the linear diffusion operator in the induction equation.     

Influence of a dipole magnetic field on the topology of toroidal magnetic configurations around a gravitating body

The topological structure of the toroidal magnetic field, which is affected by a dipole magnetic field, is studied. It is shown, that a dipole magnetic field is able to split the initial toroidal configuration into two toroids and, at a certain critical value of the magnetic dipole, can completely destroy the toroidal configuration. The dependence of the total mass excess on the dipole magnetic field is found.     

Quantitative magnetospheric magnetic field modelling with toroidal and poloidal vector fields

Toroidal and poloidal vector fields allow divergence free magnetic field representations in regions where currents flow. We derive general magnetospheric magnetic fields using combinations of spherical harmonic expansions of the toroidal and poloidal fields. Adding restrictive conditions like the field line topology symmetry or the magnetic field measurements, more specific magnetospheric magnetic field models can be derived. Two examples of this technique are given : an axisymmetric model with a ring current in the equatorial region and a time-dependent model of the Earth's magnetosphere. Our results are compared with the Olson-Pfitzer model.     

Constraints on the magnetic field geometry of magnetars

We study the effect of the magnetic field geometry on the oscillation spectra of strongly magnetized stars. We construct a configuration of magnetic field where a toroidal component is added to the standard poloidal one. We consider a star with a type I superconductor core so that both components of the magnetic field are expelled from the core and confined in the crust. Our results show that the toroidal contribution does not influence significantly the torsional oscillations of the crust. On the contrary, the confinement of the magnetic field in the crust drastically affects the torsional oscillation spectrum. A comparison with estimations for the magnetic field strength, from observations, excludes the possibility that magnetars will have a magnetic field solely confined in the crust, that is, our results suggest that the magnetic field in whatever geometry has to permeate the whole star.     

A magnetic scaleheight: the effect of toroidal magnetic fields on the thickness of accretion discs

We consider accreting systems in which the central object interacts, via the agency of its magnetic field, with the disc that surrounds it. The disc is turbulent and, so, has a finite effective conductivity. The field sweeps across the face of the disc, thereby forming a current that is directed radially within the disc. In turn, this disc current creates a toroidal field, where the interaction between the disc current and the toroidal field produces a Lorentz force that compresses the disc. We investigate this compression, which creates a magnetic scaleheight of the disc that can be much smaller than the conventional scaleheight. We derive an analytic expression for the magnetic scaleheight and apply it to fully ionized discs.     

Simulating the generation of the solar toroidal magnetic field by differential rotation

Within the kinematic dynamo theory, we construct a mathematical model for the evolution of the solar toroidal magnetic field, excited by the differential rotation of the convective zone in the presence of a poloidal field of a relic origin. We use a velocity profile obtained by decoding the data of helioseismological experiments. For the model of ideal magnetic hydrodynamics, we calculate the latitudinal profiles of the increasing-with-time toroidal field at different depths in the solar convection zone. It is found that, in the region of differential rotation, the excited toroidal field shows substantial fluctuations in magnitude with depth. Based on the simulations results, we propose an explanation for the “incorrect polarity” of magnetic bipolar sunspot groups in solar cycles.     

Double maxima of 11-year solar cycles

We propose a scenario to explain the observed phenomenon of double maxima of sunspot cycles, including the generation of a magnetic field near the bottom of the solar convection zone (SCZ) and the subsequent rise of the field from the deep layers to the surface in the royal zone. Five processes are involved in the restructuring of the magnetic field: the Ω-effect, magnetic buoyancy, macroscopic turbulent diamagnetism, rotary ?ρ-effect, and meridional circulation. It is found that the restructuring of magnetism develops differently in high-latitude and equatorial domains of the SCZ. A key role in the proposed mechanism of the double maxima is played by two waves of toroidal fields from the lower base of the SCZ to the solar surface in the equatorial domain. The deep toroidal fields are excited by the Ω-effect near the tachocline at the beginning of the cycle. Then these fields are transported to the surface due to the combined effect of magnetic buoyancy, macroscopic turbulent diamagnetism, and the rotary magnetic ?ρ-flux in the equatorial domain. After a while, these magnetic fragments can be observed as bipolar sunspot groups at the middle latitudes in the royal zone. This first, upward-directed wave of toroidal fields produces the main maximum of sunspot activity. However, the underlying toroidal fields in the high-latitude polar domains are blocked at the beginning of the cycle near the SCZ bottom by two antibuoyancy effects — the downward turbulent diamagnetic transfer and the magnetic ?ρ-pumping. In approximately 1 or 2 years, a deep equatorward meridional flow transfers these fields to low-latitude parts of the equatorial domain (where there are favorable conditions for magnetic buoyancy), and the belated magnetic fields (the second wave of toroidal fields) rise to the surface. When this second batch of toroidal fields comes to the solar surface at low latitudes, it leads to the second sunspot maximum.     

Turbulent dynamo near tachocline and reconstruction of azimuthal magnetic field in the solar convection zone

In order to extend the abilities of the αΩ dynamo model to explain the observed regularities and anomalies of the solar magnetic activity, the negative buoyancy phenomenon and the magnetic quenching of the α effect were included in the model, as well as newest helioseismically determined inner rotation of the Sun were used. Magnetic buoyancy constrains the magnitude of toroidal field produced by the Ω effect near the bottom of the solar convection zone (SCZ). Therefore, we examined two “antibuoyancy” effects: i) macroscopic turbulent diamagnetism and ii) magnetic advection caused by vertical inhomogeneity of fluid density in the SCZ, which we call the ∇ρ effect. The Sun's rotation substantially modifies the ∇ρ effect. The reconstruction of the toroidal field was examined assuming the balance between mean‐field magnetic buoyancy, turbulent diamagnetism and the rotationally modified ∇ρ effect. It is shown that at high latitudes antibuoyancy effects block the magnetic fields in the deep layers of the SCZ, and so the most likely these deep‐rooted fields could not become apparent at the surface as sunspots. In the near‐equatorial region, however, the upward ∇ρ effect can facilitate magnetic fields of about 3000 – 4000 G to emerge through the surface at the sunspot belt. Allowance for the radial inhomogeneity of turbulent velocity in derivations of the helicity parameter resulted in a change of sign of the α effect from positive to negative in the northern hemisphere near the bottom of the SCZ. The change of sign is very important for direction of the Parker's dynamo‐waves propagation and for parity of excited magnetic fields. The period of the dynamo‐wave calculated with allowance for the magnetic quenching is about seven years, that agrees by order of magnitude with the observed mean duration of the sunspot cycles. Using the modern helioseismology data to define dynamo‐parameters, we conclude that north‐south asymmetry should exist in the meridional field. At low latitudes in deep layers of the SCZ, the αΩ dynamo excites most efficiency the dipolar mode of the meridional field. Meanwhile, in high‐latitude regions a quadrupolar mode dominates in the meridional field. The obtained configuration of the net meridional field is likely to explain the magnetic anomaly of polar fields (the apparent magnetic “monopole”) observed near the maxima of solar cycles. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Toroidal magnetic field structure of the solar convective envelope inferred from the initial observations of the sunspots

For different life spans, we measure the line of sight component of the magnetic field structure of the bipolar sunspots from the SOHO/MDI magnetograms during their initial appearance on the surface and toroidal component of the magnetic field structure is separated. Irrespective of their sizes, strength of the measured line of sight component of the magnetic field structure varies from ∼450 G for the life span of 2 days to ∼300 G for the life span of 12 days. Where as strength of the estimated surface toroidal component of the bipolar spots varies from ∼10 G for the life span of 2 days to ∼700 G for the life span of 12 days. We use rederived Parker’s (1955a) flux tube model in spherical coordinates and Hiremath’s (2002) life span anchoring depth information to infer the strength of line of sight and toroidal components of the magnetic field structures at different anchoring depths of the bipolar spots in the convective envelope and the important findings are: (i) both the line of sight and toroidal components of the magnetic field structures at the sites of sunspots’ different anchoring depths in the convective envelope have a similar radial variation and the strength (∼10⁴ G near base of the convective envelope to ∼100 G near the surface) and, (ii) rate of emergence of toroidal magnetic field structure near base of the convective envelope is estimated to be ∼100 times the rate of emergence of toroidal magnetic field structure near the surface.     

Non-linear magnetic diffusivity in mean-field electrodynamics

We consider non-linear transport and drift processes caused by an inhomogeneous magnetic field in a turbulent fluid. The coefficients of magnetic diffusivity and drift velocity are calculated by making use of the second-order correlation approximation. Transport processes in the presence of a sufficiently strong magnetic field become anisotropic with larger diffusion rate and turbulent electrical resistivity across the field than along the field. Non-linear effects also lead to a drift of the magnetic field away from the regions with a higher magnetic energy.     

The magnetic field transfer in the solar convective zone

The large-scale azimuth magnetic field is pumping to the bottom of the solar convective zone due to the diamagnetic action of turbulent conductive fluids. When the field at the bottom is of about 10³ G, an equilibrium is established between diamagnetic pumping and buoyancy.If, in addition to the density gradient, an additional anisotropy exists (for instance, due to rotation), another mechanism of the magnetic field transfer appears, the efficiency of which greatly depends on the magnitude of the anistropy parameter.     

On the X-ray image of the Crab nebula: comparison with Chandra observations

An axisymmetric model for the Crab nebula is constructed to examine the flow dynamics in the nebula. The model is based on that of Kennel & Coroniti, although we assume that the kinetic-energy-dominant wind is confined to an equatorial region. The evolution of the distribution function of the electron–positron plasma flowing out in the nebula is calculated. Given viewing angles, we reproduce an image of the nebula and compare it with the Chandra observation.
The reproduced image is not ring-like, but is rather 'lip-shaped'. It is found that the assumption of a toroidal field does not reproduce the Chandra image. We must assume that there is a disordered magnetic field with an amplitude as large as the mean toroidal field. In addition, the brightness contrast between the front and back sides of the ring cannot be reproduced if we assume that the magnetization parameter σ is as small as ∼10⁻³. The brightness profile along the semimajor axis of the torus is also examined. The non-dissipative, ideal-magnetohydrodynamic approximation in the nebula appears to break down.
We speculate that if the magnetic energy is released by some process that produces a turbulent field in the nebula flow and causes heating and acceleration – for example, by magnetic reconnection – then the present difficulties may be resolved (i.e. we can reproduce a ring image and a higher brightness contrast). Thus, the magnetization parameter σ can be larger than previously expected.     

Viscous-resistive ADAF with a general large-scale magnetic field

We have studied the structure of hot accretion flow bathed in a general large-scale magnetic field. We have considered magnetic parameters , where are the Alfvén sound speeds in three direction of cylindrical coordinate (r,φ,z). The dominant mechanism of energy dissipation is assumed to be the magnetic diffusivity due to turbulence and viscosity in the accretion flow. Also, we adopt a more realistic model for kinematic viscosity (ν=αc _s H), with both c _s and H as a function of magnetic field. As a result in our model, the kinematic viscosity and magnetic diffusivity (η=η ₀ c _s H) are not constant. In order to solve the integrated equations that govern the behavior of the accretion flow, a self-similar method is used. It is found that the existence of magnetic resistivity will increase the radial infall velocity as well as sound speed and vertical thickness of the disk. However the rotational velocity of the disk decreases by the increase of magnetic resistivity. Moreover, we study the effect of three components of global magnetic field on the structure of the disk. We found out that the radial velocity and sound speed are Sub-Keplerian for all values of magnetic field parameters, but the rotational velocity can be Super-Keplerian by the increase of toroidal magnetic field. Also, Our numerical results show that all components of magnetic field can be important and have a considerable effect on velocities and vertical thickness of the disk.