Three-dimensional model for generation of the mean solar magnetic field

A three-dimensional (non-axisymmetric) model for the solar mean magnetic field generation is studied. The sources of generation are the differential rotation and mean helicity in the convective shell. The system is described by two equations of the first order in time and the fourth order in space coordinates. The solution is sought for in the form of expansion over the spherical function Y_n^m. The modes of different m are separated. A finite-difference scheme similar to the Peaceman-Rachford scheme is constructed so to find coefficients of the expansion depending on the time and radial coordinates. It is shown that a mode with a smaller azimuthal number m is primarily excited. The axisymmetric mode m = o describes the 22 year solar cycle oscillations. The modes of m o have no such periodicity, the oscillate with a period of rotation of the low boundary of the solar convective shell, The solutions which are symmetric relative to the equator plane are excited more easily compared with the antisymmetrical ones. The results obtained are confronted to the observational picture of the non-axisymmetric large-scale solar magnetic fields.     

Magnetic helicity in stellar dynamos: new numerical experiments

The theory of large scale dynamos is reviewed with particular emphasis on the magnetic helicity constraint in the presence of closed and open boundaries. In the presence of closed or periodic boundaries, helical dynamos respond to the helicity constraint by developing small scale separation in the kinematic regime, and by showing long time scales in the nonlinear regime where the scale separation has grown to the maximum possible value. A resistively limited evolution towards saturation is also found at intermediate scales before the largest scale of the system is reached. Larger aspect ratios can give rise to different structures of the mean field which are obtained at early times, but the final saturation field strength is still decreasing with decreasing resistivity. In the presence of shear, cyclic magnetic fields are found whose period is increasing with decreasing resistivity, but the saturation energy of the mean field is in strong super‐equipartition with the turbulent energy. It is shown that artificially induced losses of small scale field of opposite sign of magnetic helicity as the large scale field can, at least in principle, accelerate the production of large scale (poloidal) field. Based on mean field models with an outer potential field boundary condition in spherical geometry, we verify that the sign of the magnetic helicity flux from the large scale field agrees with the sign of α. For solar parameters, typical magnetic helicity fluxes lie around 10⁴⁷ Mx² per cycle.     

Secondary α-effect in the nearly symmetric dynamo

An attempt has been made to examine the role of the secondary α-effect in the nearly symmetric dynamo. The tensor form of the effect has been found and it is concluded that this effect only plays a role under special assumption of the disturbance velocity field. The secondary alfa effect can amplify the generation of the azimuthal field from the poloidal field in the α²-dynamo rather than in the αω-dynamo.     

Unrestricted second-order tensor virial equations for linear oscillations of magnetic configurations

This paper deals with the second-order tensor virial equations for the linear oscillations of a gaseous mass in the presence of a magnetic field. It is shown that the commonly used linearized versions of the tensor virial equations are restricted integral equations that incorporate the linearized equation of motion but not the boundary condition. These restricted equations only allow trial functions that fulfil the boundary condition and are of limited practical value.The unrestricted variational principle for the linear oscillations of a magnetic configuration is used to derive a more general formulation of the second-order tensor virial equations so that the linear trial function _i =X _ij x _j can be used to study the oscillations of a configuration with a magnetic field that extends in the exterior vacuum. The unrestricted virial equations have been applied to Ferraro's model and approximate results for the eigenfrequencies and eigenfunctions have been obtained for nine oscillation modes.     

Oscillatory motions in sunspots

We observe vertical velocity oscillations in some sunspot umbrae with periods of about 180 s and peak to peak amplitudes up to 1 km s^–1. These oscillations are not visible in either the line depth, line width or the continuum intensity. No correlation seems to exist between the occurence of these oscillations and the presence of the chromospheric umbral flashes (Solar Phys. 7, 351, 1069). In the spot penumbra there is an indication of a long period oscillation, the period increasing from about 300 s in the inner penumbra to nearly 1000 s at the penumbra-photosphere boundary. An attempt has been made to interpret these oscillations in terms of gravity or acoustic waves, travelling along the magnetic field lines, taking into account the variation of scale height and magnetic field direction across the sunspot.     

The solar dynamo

The basic features of the solar activity mechanism are explained in terms of the dynamo theory of mean magnetic fields. The field generation sources are the differential rotation and the mean helicity of turbulent motions in the convective zone. A nonlinear effect of the magnetic field upon the mean helicity results in stabilizing the amplitude of the 22-year oscillations and forming a basic limiting cycle. When two magnetic modes (with dipole and quadrupole symmetry) are excited nonlinear beats appear, which may be related to the secular cycle modulation.The torsional waves observed may be explained as a result of the magnetic field effect upon rotation. The magnetic field evokes also meriodional flows.Adctual variations of the solar activity are nonperiodic since there are recurrent random periods of low activity of the Maunder minimum type. A regime of such a magnetic hydrodynamic chaos may be revealed even in rather simple nonlinear solar dynamo models.The solar dynamo gives rise also to three-dimensional, non-axisymmetric magnetic fields which may be related to a sector structure of the solar field.     

A thin disc model of the galactic dynamo

An axisymmetric αω-dynamo model of the Galactic disc is investigated. The disc is thin and its width, 2h^*, varies slowly with distance, ωT^*, from the axis of symmetry. The strength of the α^*, varies linearly with axialdistance, z^*, while differential rotation, dΩ^*/dωT^*, remains constant. Otherwise α^* and dΩ^*/dωT^* are arbitrary functions of ωT^*. The results are applied to the special case of the oblate spheroid first investigated by STIX (1976) and later by WHITE (1977). In the limit of small aspect ratio the new results agree with those obtained by WHITE and confirm that the dynamo numbers computed by STIX are too small.     

The quasi-biennial cycle of solar activity and dynamo theory

We have investigated the behavior of dynamo waves within the framework of a nonlinear αω-dynamo by taking into account the thickness of the convective zone, turbulent diffusivity, and meridional circulation. We show that there exists a regime in the model where short and longer patterns are simultaneously observed in the butterfly diagrams for the magnetic field. This regime is similar to the mixed cycle of the Sun, when fast quasi-biennial oscillations are observed against the background of the 22-year cycle.     

HALF-WIDTH OF A SOLAR DYNAMO WAVE IN PARKER'S MIGRATORY DYNAMO

A kinematic -dynamo model of magnetic field generation in a thin convection shell with nonuniform helicity for large dynamo numbers is considered in the framework of Parker's migratory dynamo. The asymptotic solution obtained of equations governing the magnetic field has the form of an anharmonic travelling dynamo wave. This wave propagates over most latitudes of the solar hemisphere from high latitudes to the equator, and the amplitude of the magnetic field first increases and then decreases with propagation. Over the subpolar latitudes, the dynamo wave reverses; there the dynamo wave propagates polewards and decays with latitude. The half-width of the maximum of the magnetic field localisation and the phase velocity of the dynamo wave are calculated. Butterfly diagrams are plotted and analysed and these show that even a simple model may reveal some properties of the solar magnetic fields.     

Nonhelical mean‐field dynamos in a sheared turbulence

Mechanisms of nonhelical large‐scale dynamos (shear‐current dynamo and effect of homogeneous kinetic helicity fluctuations with zero mean) in a homogeneous turbulence with large‐scale shear are discussed. We have found that the shearcurrent dynamo can act even in random flows with small Reynolds numbers. However, in this case mean‐field dynamo requires small magnetic Prandtl numbers (i.e., when Pm < Pm^cr < 1). The threshold in the magnetic Prandtl number, Pm^cr = 0.24, is determined using second order correlation approximation (or first‐order smoothing approximation) for a background random flow with a scale‐dependent viscous correlation time τ_c = (νk 2)^–1 (where ν is the kinematic viscosity of the fluid and k is the wave number). For turbulent flows with large Reynolds numbers shear‐current dynamo occurs for arbitrary magnetic Prandtl numbers. This dynamo effect represents a very generic mechanism for generating large‐scale magnetic fields in a broad class of astrophysical turbulent systems with large‐scale shear. On the other hand, mean‐field dynamo due to homogeneous kinetic helicity fluctuations alone in a sheared turbulence is not realistic for a broad class of astrophysical systems because it requires a very specific random forcing of kinetic helicity fluctuations that contains, e.g., low‐frequency oscillations. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the oscillatory behaviour of kinematic mean-field dynamos

The paper considers the question after the circumstances in which kinematic mean-field dynamos can have oscillatory magnetic field modes. The conducting fluid body is allowed to be of almost arbitrary shape; its surroundings are vacuum. A general relation for the frequency of oscillation is derived. This relation is discussed more closely for models with pure α²-mechanism. Proof is given that no oscillations can occur for constant α. The investigations published so far on spherical models with pure α²-mechanism call up the question whether there is a chance for axisymmetric modes to be oscillatory. For both spherical models and disk models the possibility of oscillatory axisymmetric modes is demonstrated by examples.     

Cross Helicity and Turbulent Magnetic Diffusivity in the Solar Convection Zone

In a density-stratified turbulent medium, the cross helicity 〈u′⋅B′〉 is considered as a result of the interaction of the velocity fluctuations and a large-scale magnetic field. By means of a quasilinear theory and by numerical simulations, we find the cross helicity and the mean vertical magnetic field to be anti-correlated. In the high-conductivity limit the ratio of the helicity and the mean magnetic field equals the ratio of the magnetic eddy diffusivity and the (known) density scale height. The result can be used to predict that the cross helicity at the solar surface will exceed the value of 1 gauss km s⁻¹. Its sign is anti-correlated to that of the radial mean magnetic field. Alternatively, we can use our result to determine the value of the turbulent magnetic diffusivity from observations of the cross helicity.     

Galactic dynamo with helicity fluxes

We construct an approximation for the magnetic field of galaxies that takes into account the magnetic helicity conservation law. In our calculations, we use the fact that the galactic disk is fairly thin and, therefore, the magnetic field component perpendicular to the galactic disk can be neglected (the so-called no-z approximation). However, an averaging of the magnetic field over the entire galaxy, as was done in previous works, is not performed. Our results are compared both with the approximation that disregards the helicity flux and with the results obtained in models with helicity fluxes but without averaging. We show that, compared to the classical model, there are a number of new effects (for example, magnetic field oscillations) and, compared to the model with averaging, the behavior of the magnetic field “softens”: its stationary value is reached more slowly and the oscillation amplitude decreases. This is because the dissipative processes changing the magnetic field growth rate are taken into account in our model. In contrast to the model with averaging, here it becomes possible to construct the dependence of the magnetic field and helicity on the distance from the galactic center.     

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.     

Is the Long-Term Persistency of Circular Polarisation due to the Constant Helicity of the Magnetic Fields in Rotating Quasar Engines?

Many compact radio sources like quasars, blazars, radio galaxies, and micro-quasars emit circular polarisation (CP) with surprising temporal persistent handedness. We propose that the CP is caused by Faraday conversion (FC) of linear polarisation (LP) synchrotron light which propagates along a line-of-sight (LOS) through helical magnetic fields. Jet outflows from radio galaxies should have the required magnetic helicity in the emission region due to the magnetic torque of the accretion disc. Also advection dominated accretion flow (ADAF)should contain magnetic fields with the same helicity. However, a jetregion seems to be the more plausible origin of CP. The proposed scenario requires Faraday rotation (FR) to be insignificant in the emission region. The proposed mechanism works in electron-positron(e^±) as well as electron-proton (e/p) plasma. In the latter case, the emission region should consist of individual flux tubes with independent polarities in order to suppress too strong FR– as it was already proposed for FR based CP generation models. The predominant CP is expected to mostly counter-rotate (rotation is measured here in sky-projection) with respect to the central engine in all cases (jet or ADAF, e^± or e/p plasma) and therefore allows to measure the sense of rotation of quasar engines. The engine of SgrA^* is expected – in this scenario – to rotate clockwise and therefore counter-Galactic, as do the young hot stars in its vicinity, which are thought to feed SgrA^* by their winds. Generally, sources with Stokes-V<0 (V>0) are expected to rotate clockwise(counter-clockwise).     

Spatial variations in parameters of quasi-hourly sunspot fragment oscillations and a singular penumbra oscillator

We obtained three-dimensional interpolated portraits for the radial and torsional oscillations of fragments of 12 sunspots in the form of deviations of their polar coordinates from drift as functions of the time and distance from the sunspot center. We performed a wavelet analysis of the two orthogonal components and determined the dominant oscillation modes; the period varies between 40 and 100 min for different sunspots. We revealed two types of dominant modes, one is associated with the sunspot and the other is associated with its surrounding pores: the central-mode frequency depends on the maximum field strength of the sunspot and decreases from its center toward the boundary, while the peripheral-mode frequency depends on the heliographic latitude and decreases toward the sunspot boundary from the far periphery. We revealed radial variations in frequency and amplitude with a spatial period of 0.8 sunspot radius. The types of dominant modes and the radial variations in oscillation parameters are linked with the subphotospheric structure of an active region—with two types of spiral waves and concentric magnetic-field waves. We estimated the mean pore oscillation energy to be ～10³⁰ erg and found a singular oscillator with a mean energy of ～10³¹ erg in the penumbra at a distance of 0.8 sunspot radius. We argue that the singular penumbra oscillator is the source of solar flares.     

A Fast Current-Field Iteration Method for Calculating Nonlinear Force-Free Fields

Existing methods for calculating nonlinear force-free magnetic fields are slow, and are likely to be inadequate for reconstructing coronal magnetic fields based on high-resolution vector magnetic field data from a new generation of spectro-polarimetric instruments. In this paper a new implementation of the current-field iteration method is presented, which is simple, fast, and accurate. The time taken by the method scales as N ⁴, for a three-dimensional grid with N ³ points. The method solves the field-updating part of the iteration by exploiting a three-dimensional Fast Fourier Transform solution of Ampere’s law with a current density field constructed to satisfy the required boundary conditions, and uses field line tracing to solve the current-updating part of the iteration. The method is demonstrated in application to a known nonlinear force-free field and to a bipolar test case.     

Analysis of electric current helicity in active regions on the basis of vector magnetograms

The problem of (dc) magnetic field energy build up in the solar atmosphere is addressed. Although large-scale current generation may be due to large-scale shearing motions in the photosphere, recently a new approach was proposed: under the assumption that the magnetic field evolves through a sequence of force-free states, Seehafer (1994) found that the energy of small-scale fluctuations may be transferred into energy of large-scale currents in an AR (the α-effect). The necessary condition for the α-effect is revealed by the presence of a predominant sign of current helicity over the volume under consideration. We studied how frequently such a condition may occur in ARs. On the basis of vector magnetic field measurements we calculated the current helicity B _z · (▽ × B) _z in the photosphere over the whole AR area for 40 active regions and obtained the following results:

1. In 90% of cases there existed significant excess current helicity of some sign over the active region area. So one can suggest that the build up of large-scale currents in an active region due to small-scale fluctuations may be typical in ARs.
2. In 82.5% of cases, the excess current helicity in the northern (southern) hemisphere was negative (positive).

The method proposed can be applied to those ARs where the determination of the predominant sign of current helicity by traditional visual inspection of Hα-patterns is not reliable.     

Coronal activity from dynamos in astrophysical rotators

We show that a steady mean-field dynamo in astrophysical rotators leads to an outflow of relative magnetic helicity and thus magnetic energy available for particle and wind acceleration in a corona. The connection between energy and magnetic helicity arises because mean-field generation is linked to an inverse cascade of magnetic helicity. To maintain a steady state in large magnetic Reynolds number rotators, there must then be an escape of relative magnetic helicity associated with the mean field, accompanied by an equal and opposite contribution from the fluctuating field. From the helicity flow, a lower limit on the magnetic energy deposited in the corona can be estimated. Steady coronal activity including the dissipation of magnetic energy, and formation of multi-scale helical structures therefore necessarily accompanies an internal dynamo. This highlights the importance of boundary conditions which allow this to occur for non-linear astrophysical dynamo simulations. Our theoretical estimate of the power delivered by a mean-field dynamo is consistent with that inferred from observations to be delivered to the solar corona, the Galactic corona, and Seyfert 1 AGN coronae.     

Effects of external magnetic field on the outflows of an accretion disc

The aim of this work is to study the effects of an external magnetic field generated by a magnetized compact star on the outflows of its accretion disc. For this purpose, we solve a set of magneto-hydrodynamic (MHD) equations for an accretion disc in spherical coordinates to consider the disc structure along the θ-direction. We also consider the magnetic field of a compact star beyond its surface as a dipolar field, producing a toroidal magnetic field inside the disc. We convert the equations to a set of ordinary differential equations (ODEs) as a function of the θ only by applying self-similar assumptions in the radial direction. Then, this set of equations is solved under symmetrical boundary conditions in the equatorial plane to obtain the velocity field. The results are considered in the gas-pressure-dominated (GPD) region and radiation-pressure-dominated (RPD) region as well. The dipolar field of the compact stars can significantly enhance the speed of outflows. It also can change the structure of the disc. The results of this work would be useful in the study of X-ray binaries, the origin of ultra-relativistic outflows, and jet formation around the compact stars.