On the oscillations of a composite stellar model with a weak magnetic field

The unrestricted second-order virial tensor formalism has been used to calculate the characteristic frequencies of linear adiabatic oscillations of a composite stellar model having an isothermal core and a polytropic envelope in presence of a weak poloidal magnetic field. The frequencies of the transverse shear mode and the nonradial pulsation mode for both a radiative and a convective envelope (corresponding to polytropic index 3 and 1.5, respectively) alongwith that of the toroidal mode for the radiative envelope get increased in presence of the magnetic field. However, the frequency of the toroidal mode for the convective envelope registers a decrease in presence of the field. The corrections to the various frequencies decrease with increasing values of the parameter characterizing the lowering of the core temperature in presence of the magnetic field.     

Non-radial oscillations and convective instability of a polytrope with a toroidal magnetic field

We examine the non-radial modes of oscillation, belonging to spherical harmonics of ordersl=1 andl=3, of a gaseous polytrope with a toroidal magnetic field. We find that a toroidal magnetic field increases the growth rate of convective instability for deformations belonging to the spherical harmonicl=1 whereas it decreases the growth rate of convective instability for deformations belonging to the harmonicsl=2 andl=3. The frequencies of the ‘acoustic’ mode and the ‘Kelvin’ mode are decreased by the presence of the toroidal magnetic field.     

The periods of stellar models with linked poloidal-toroidal magnetic fields

First-order perturbation theory results for the changes in pulsation frequencies of a Cowling model star containing a magnetic field with both poloidal and toroidal components are presented. A toroidal field large enough to stabilize the poloidal field may reverse the sign of the frequency change caused by a purely poloidal field for some modes, including the fundamental radial mode.     

The equilibrium and the stability of the Jeans spheroids with toroidal magnetic fields

We examine the effect of a toroidal magnetic field on the equilibrium and stability of homogeneous masses distorted by the tidal effects of a secondary (of massM at a distanceR). It is shown that if the toroidal magnetic field is assumed to be axisymmetric about the direction of the line joining the centres of mass of the primary and the secondary, then the equilibrium configuration is a prolate spheroid. Also determined are the characteristic frequencies of the various modes of oscillations belonging to the second harmonics. It is found that the magnetic field shifts these frequencies to higher values than the ones which prevail in the absence of a magnetic field.     

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.     

Magnetically distorted polytropes

The oscillations of a gaseous polytrope with a magnetic field having both a toroidal and a poloidal component are examined using the second-order tensor virial equations on the assumption that the magnetic energy is small compared with the gravitational energy. The frequencies of oscillation of the transverse shear, the toroidal and the coupled pulsation modes are tabulated for polytropic indicesn=1, 1.5, 2, 3 and 3.5. It is found that the magnetic field decreases the frequency of oscillation of (i) the transverse shear mode and (ii) the mode which starts as a radial pulsation in the absence of a magnetic field while it increases the frequency of oscillation of (i) the toroidal mode and (ii) the Kelvin mode. In all cases the shift in frequency decreases with increasingn.     

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.     

Large-Scale Magnetic Field of the Sun and Evolution of Sunspot Activity

The evolution of the large-scale magnetic field of the Sun has been studied using an algorithm of tomographic inversion. By analyzing line-of-sight magnetograms, we mapped the radial and toroidal components of the Sun??s large-scale magnetic field. The evolution of the radial and toroidal magnetic field components in the 11-year solar cycle has been studied in a time?Clatitude aspect. It is shown that the toroidal magnetic field of the Sun is causally related to sunspot activity; i.e., the sunspot formation zones drift in latitude and follow the toroidal magnetic fields. The results of our analysis support the idea that the high-latitude toroidal magnetic fields can serve as precursors of sunspot activity. The toroidal fields in the current cycle are anomalously weak and also show a barely noticeable equatorward drift. This behavior of the toroidal magnetic field suggests low activity levels in the current cycle and in the foreseeable future.     

Toroidal magnetic field of a superfluid neutron star in a postnewtonian approximation

The effect of proton superconductivity on the generation of a toroidal magnetic field inside a neutron star is examined. It is shown that including the entrainment of superconducting protons by superfluid neutrons does not change the previously obtained results. Proton superconductivity does influence the structure of the generated magnetic field since, over a time on the order of 10⁴–10⁵ years, the magnetic field increases linearly with time and can exceed the first critical field for proton superconductivity. The distribution of the stationary toroidal magnetic field inside a neutron star is also found.     

Dynamical stability of a self-gravitating rotating cylinder in a helical magnetic field

The effect of a helical magnetic field on the oscillations and the stability of a homogeneous self-gravitating rotating cylinder is investigated. The axial field has a tendency to stabilise long wave numbers and to destabilise small wave numbers so that maximum instability occurs for a finite wave number. If the toroidal and the axial component of the field have the same sign, the instability associated with the toroidal field can be removed by the rotation or by the axial field. Rotational instability is reduced but cannot removed by the field. If the components of the field have the opposite sign, rotational instability is increased. The maximum growth rate of the magnetic instability is reduced by a small axial field and tends to a finite value for large axial fields.     

On the quasi-periodic oscillations in magnetars

We study torsional Alfvén oscillations of magnetars, that is neutron stars with a strong magnetic field. We consider the poloidal and toroidal components of the magnetic field and a wide range of equilibrium stellar models. We use a new coordinate system  ( X , Y )  , where     and     and a ₁ is the radial component of the magnetic field. In this coordinate system, the one+two-dimensional evolution equation describing the quasi-periodic oscillations (QPOs), see Sotani et al., is reduced to a one+one-dimensional equation where the perturbations propagate only along the y -axis. We solve the one+one-dimensional equation for different boundary conditions and the open magnetic field lines, that is magnetic field lines that reach the surface and there match up with the exterior dipole magnetic field as well as closed magnetic lines, i.e. magnetic lines that never reach the stellar surface. For the open field lines, we find two families of QPO frequencies: a family of 'lower' QPO frequencies which is located near the x -axis and a family of 'upper' frequencies located near the y -axis. According to Levin, the fundamental frequencies of these two families can be interpreted as the turning point of the continuous spectrum. We find that the upper frequencies are multiples of the lower ones by a constant equalling  2 n + 1  . For the closed lines, the corresponding factor is   n + 1  . By using these relations, we can explain both the lower and the higher observed frequencies in SGR 1806−20 and SGR 1900+14.     

Thermal generation of magnetic fields with radiation pressure in rotating stars

It has been suggested by Biermann that in rotating stars the electron partial pressure could generate a toroidal magnetic field of a considerable strength. However, Mestel and Roxburgh have shown recently that the generation of such a toroidal magnetic field could almost completely be suppressed when a weak primodial poloidal magnetic field exists in the star. In this paper it is shown that a toroidal magnetic field of a moderate strength could be generated even in the presence of a primodial poloidal magnetic field, if the effect of radiation pressure is taken into consideration. This considered mechanism is effective for moderately massive stars, and numerical estimate indicates that in A type stars a toroidal magnetic field of the order of a thousand gauss can be generated near the surface within the time scale of the evolution of the star.Visiting Scientist to the High Altitude Observatory on leave of absence from the Department of Astronomy, University of Tokyo, Japan.     

Magnetic fields of Ap stars as a result of the Tayler instability

Ap star magnetism is often attributed to fossil magnetic fields which have not changed much since the pre‐main‐sequence epoch of the stars. Stable magnetic field configurations are known which could persist probably for the entire mainsequence life of the star, but they may not show the complexity and diversity exhibited by the Ap stars observed. We suggest that the Ap star magnetism is not a result of stable configurations, but is the result of an instability based on strong toroidal magnetic fields buried in the stars. The highly nonaxisymmetric remainders of the instability are reminiscent of the diversity of fields seen on Ap stars. The strengths of these remnant magnetic fields are actually between a few per cent up to considerable fractions of the internal toroidal field; this means field strengths of the order of kGauss being compatible with what is observed. The magnetic fields emerge at the surface rather quickly; rough estimates deliver time‐scales of the order of a few years. Since rotation stabilizes the instability, normal A stars may still host considerable, invisible toroidal magnetic fields (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-ideal evolution of non-axisymmetric, force-free magnetic fields in a magnetar

Recent numerical magnetohydrodynamic calculations by Braithwaite and collaborators support the 'fossil field' hypothesis regarding the origin of magnetic fields in compact stars and suggest that the resistive evolution of the fossil field can explain the reorganization and decay of magnetar magnetic fields. Here, these findings are modelled analytically by allowing the stellar magnetic field to relax through a quasi-static sequence of non-axisymmetric, force-free states, by analogy with spheromak relaxation experiments, starting from a random field. Under the hypothesis that the force-free modes approach energy equipartition in the absence of resistivity, the output of the numerical calculations is semiquantitatively recovered: the field settles down to a linked poloidal–toroidal configuration, which inflates and becomes more toroidal as time passes. A qualitatively similar (but not identical) end state is reached if the magnetic field evolves by exchanging helicity between small and large scales according to an α-dynamo-like, mean-field mechanism, arising from the fluctuating electromotive force produced by the initial random field. The impossibility of matching a force-free internal field to a potential exterior field is discussed in the magnetar context.     

Upper limits in the toroidal component of force-free magnetic fields in stellar atmospheres in the context of solar and stellar flares

Doing numerical calculations of axially symmetric force-free fields, Milsom and Wright (1976) have noticed that there seem to be no solutions if the toroidal component of the field exceeds a certain limit. In the present paper this problem is reexamined in the approximation of a plane stellar surface using a very simple analytic approximation. The results of Milsom and Wright (1976) are confirmed but, in contrast to their interpretation, it is shown that these limitations do not indicate the possibility of sudden changes of the topology of the magnetic field. This is because in a stellar atmosphere the toroidal component of the surface magnetic field is no independent quantity but is produced by shearing motions in the star which will prevent the toroidal magnetic field from exceeding its maximum value. To study the possibility of sudden changes in the magnetic field, which could cause stellar flares, the calculations are re-done prescribing the motion of the magnetic footpoints (shear in the stellar surface) instead of the toroidal component of the surface field. Using the same mathematical formalism it is found that no sudden changes can occur for configurations where all field lines connect to the stellar surface but that sudden changes may be possible for a more complicated field topology.     

Relation between the toroidal field in the solar convective layer and the BI-polar field of sunspots

In the solar convective layer, there is a strong toroidal field and a vertical gradient in the turbulent magnetic diffusivity. As a fluid blob rises through magnetic buoyancy, a steep gradient in the turbulent magnetic diffusivity across the surface of the blob is generated. This will perturb the toroidal field, resulting in the formation of a magnetic ring around the blob. An attempt is made to account for the concentration of the bipolar sunspot field in terms of this ring.     

Equilibrium of self gravitating polytropic cylinders with a magnetic field

The equations of hydromagnetics for a self-gravitating fluid of infinite conductivity are examined in the axisymmetric case in terms of toroidal and poloidal scalars. The stationary state with non-zero poloidal velocity scalar admits and analytical solution for polytropic cylinder of infinite length with a prevalent toroidal magnetic field. The case when the poloidal velocity scalar is zero is also considered.     

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.     

Constraints on the Solar Internal Magnetic Field from a Buoyancy Driven Solar Dynamo

We report here results from a dynamo model developed on the lines of the Babcock-Leighton idea that the poloidal field is generated at the surface of the Sun from the decay of active regions. In this model magnetic buoyancy is handled with a realistic recipe – wherein toroidal flux is made to erupt from the overshoot layer wherever it exceeds a specified critical field B _c (10⁵ G). The erupted toroidal field is then acted upon by the α-effect near the surface to give rise to the poloidal field. In this paper we study the effect of buoyancy on the dynamo generated magnetic fields. Specifically, we show that the mechanism of buoyant eruption and the subsequent depletion of the toroidal field inside the overshoot layer, is capable of constraining the magnitude and distribution of the magnetic field there. We also believe that a critical study of this mechanism may give us new information regarding the solar interior and end with an example, where we propose a method for estimating an upper limit of the difusivity within the overshoot layer. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stability of the solar tachocline with magnetic fields

The stability of magnetic fields in the solar tachocline is investigated. We present stability limits for higher azimuthal wave numbers and results on the dependence of the stability on the location of toroidal magnetic fields in latitude. While the dependence of the wave number with the largest growth rate on the magnetic field strength and the magnetic Prandtl number is small, the dependence on the magnetic Reynolds number Rm indicates that lowest azimuthal modes are excited for very high Rm. Upon varying the latitudinal position of the magnetic field belts, we find slightly lower stability limits for high latitudes, and very large stability limits at latitudes below 10°, with little dependence on latitude in between. An increase of the maximum possible field was achieved by adding a poloidal field. The upper limit for the toroidal field which can be stored in the radiative tachocline is then 1000 G, compared to about 100 G for a purely toroidal field as was found in an earlier work. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)