P – ω Dynamo in Spherical Geometry

Based on the fundamental P – ω dynamo equation, using spherical polar coordinates, we carry out a study of turbulent plasma wave dynamo effect. For various rotation laws, different analytical solutions are derived. In the cases of no rotation and rigid rotation, the dynamo generates poloidal field only, while with differential rotation, regardless the differential rotation is radial or latitudinal, poloidal and toroidal fields are all generated. We may think that the solutions are the analytical forms of the magnetic field in a turbulent source region of celestial bodies. This revised version was published online in July 2006 with corrections to the Cover Date.     

On axisymmetric hydromagnetic equilibria

The physical characteristics of possible axisymmetric equilibria are examined on the basis of the integrals of hydromagnetic equations. It is shown for nearly spherical configurations that a surface differential rotation is possible only in the absence of a meridional circulation with either purely toroidal or purely poloidal magnetic field. In the presence of a meridional circulation, it is shown that no surface rotation or constant rotation is possible if the magnetic field is purely toroidal, and that no rotation is possible if the magnetic field is purely poloidal. A brief discussion is given on the possible solutions including the case of stellar winds with force-free magnetic fields.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Observations of magnetic flux compression in jet impact experiments

The astrophysical jet experiment at Caltech generates a T=2–5 eV, n=10²¹–10²² m⁻³ plasma jet using coplanar disk electrodes linked by a poloidal magnetic field. A 100 kA current generates a toroidal magnetic field; the toroidal field pressure inflates the poloidal flux surface, magnetically driving the jet. The jet travels at up to 50 km/s for ∼20–25 cm before colliding with a cloud of initially neutral gas. We study the interaction of the jet and the cloud in analogy to an astrophysical jet impacting a molecular cloud. Diagnostics include magnetic probe arrays, a 12-channel spectroscopic system and a fast camera with optical filters. When a hydrogen plasma jet collides with an argon target cloud, magnetic measurements show the magnetic flux compressing as the plasma jet deforms. As the plasma jet front slows and the plasma piles up, the density of the frozen-in magnetic flux increases.     

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.     

Relativistic models of magnetars: the twisted torus magnetic field configuration

We find general relativistic solutions of equilibrium magnetic field configurations in magnetars, extending previous results of Colaiuda et al. Our method is based on the solution of the relativistic Grad–Shafranov equation, to which Maxwell's equations can be reduced. We obtain equilibrium solutions with the toroidal magnetic field component confined into a finite region inside the star, and the poloidal component extending to the exterior. These so-called twisted torus configurations have been found to be the final outcome of dynamical simulations in the framework of Newtonian gravity, and appear to be more stable than other configurations. The solutions include higher-order multipoles, which are coupled to the dominant dipolar field. We use arguments of minimal energy to constrain the ratio of the toroidal to the poloidal field.     

The Observed Long- and Short-Term Phase Relation between the Toroidal and Poloidal Magnetic Fields in Cycle 23

The observed phase relations between the weak background solar magnetic (poloidal) field and strong magnetic field associated with sunspots (toroidal field) measured at different latitudes are presented. For measurements of the solar magnetic field (SMF) the low-resolution images obtained from Wilcox Solar Observatory are used and the sunspot magnetic field was taken from the Solar Feature Catalogues utilizing the SOHO/MDI full-disk magnetograms. The quasi-3D latitudinal distributions of sunspot areas and magnetic fields obtained for 30 latitudinal bands (15 in the northern hemisphere and 15 in the southern hemisphere) within fixed longitudinal strips are correlated with those of the background SMF. The sunspot areas in all latitudinal zones (averaged with a sliding one-year filter) reveal a strong positive correlation with the absolute SMF in the same zone appearing first with a zero time lag and repeating with a two- to three-year lag through the whole period of observations. The residuals of the sunspot areas averaged over one year and those over four years are also shown to have a well defined periodic structure visible in every two – three years close to one-quarter cycle with the maxima occurring at − 40° and + 40° and drifts during this period either toward the equator or the poles depending on the latitude of sunspot occurrence. This phase relation between poloidal and toroidal field throughout the whole cycle is discussed in association with both the symmetric and asymmetric components of the background SMF and relevant predictions by the solar dynamo models.     

Magnetic fields in axisymmetric neutron stars

We derive general equations for axisymmetric Newtonian magnetohydrodynamics and use these as the basis of a code for calculating equilibrium configurations of rotating magnetized neutron stars in a stationary state. We investigate the field configurations that result from our formalism, which include purely poloidal, purely toroidal and mixed fields. For the mixed-field formalism, the toroidal component appears to be bounded at less than 7 per cent. We calculate distortions induced both by magnetic fields and by rotation. From our non-linear work, we are able to look at the realm of validity of perturbative work: we find for our results that perturbative-regime formulae for magnetic distortions agree to within 10 per cent of the non-linear results if the ellipticity is less than 0.15 or the average field strength is less than 10¹⁷ G. We also consider how magnetized equilibrium structures vary for different polytropic indices.     

Rotating Magnetic White Dwarf Models

In this paper we apply the so-called complex-plane iterative technique (CIT) to the computation of classical white dwarf models distorted by rotation (either rigid, or differential) and toroidal magnetic field. We give emphasis on computing critical rotations and rotations with effects that, combined with the effects of the toroidal magnetic field, give configurations with almost spherical shape. This revised version was published online in July 2006 with corrections to the Cover Date.     

Toroidal versus poloidal magnetic fields in Sun-like stars: a rotation threshold

From a set of stellar spectropolarimetric observations, we report the detection of surface magnetic fields in a sample of four solar-type stars, namely HD 73350, HD 76151, HD 146233 (18 Sco) and HD 190771. Assuming that the observed variability of polarimetric signal is controlled by stellar rotation, we establish the rotation periods of our targets, with values ranging from 8.8 d (for HD 190771) to 22.7 d (for HD 146233). Apart from rotation, fundamental parameters of the selected objects are very close to the Sun's, making this sample a practical basis to investigate the specific impact of rotation on magnetic properties of Sun-like stars.
We reconstruct the large-scale magnetic geometry of the targets as a low-order  (ℓ < 10)  spherical harmonic expansion of the surface magnetic field. From the set of magnetic maps, we draw two main conclusions. (i) The magnetic energy of the large-scale field increases with rotation rate. The increase in chromospheric emission with the mean magnetic field is flatter than observed in the Sun. Since the chromospheric flux is also sensitive to magnetic elements smaller than those contributing to the polarimetric signal, this observation suggests that a larger fraction of the surface magnetic energy is stored in large scales as rotation increases. (ii) Whereas the magnetic field is mostly poloidal for low rotation rates, more rapid rotators host a large-scale toroidal component in their surface field. From our observations, we infer that a rotation period lower than ≈12 d is necessary for the toroidal magnetic energy to dominate over the poloidal component.     

Rotation of Weakly Collisional Plasmas in Tokamaks, Operated with Alfvén Waves

In this paper we consider rotation induced by kinetic Alfvén waves in weakly collisional plasma of slightly rippled tokamaks in plateau and banana regimes. Rippled magnetic field of tokamaks retards the plasma rotation in toroidal direction. Here, we are going to find the quasistationary poloidal and toroidal plasma velocities and the radial electric field taking into account the complete form of ponderomotive forces.     

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.     

Axisymmetric magnetic fields in stars: relative strengths of poloidal and toroidal components

In this third paper in a series on stable magnetic equilibria in stars, I look at the stability of axisymmetric field configurations and, in particular, the relative strengths of the toroidal and poloidal components. Both toroidal and poloidal fields are unstable on their own, and stability is achieved by adding the two together in some ratio. I use Tayler's stability conditions for toroidal fields and other analytic tools to predict the range of stable ratios and then check these predictions by running numerical simulations. If the energy in the poloidal component as a fraction of the total magnetic energy is written as E_p / E , it is found that the stability condition is a ( E / U ) < E_p / E ≲ 0.8 where E /U is the ratio of magnetic to gravitational energy in the star and a is some dimensionless factor whose value is of order 10 in a main-sequence star and of order 10³ in a neutron star. In other words, whilst the poloidal component cannot be significantly stronger than the toroidal, the toroidal field can be very much stronger than the poloidal–given that in realistic stars we expect E / U < 10⁻⁶. The implications of this result are discussed in various contexts such as the emission of gravitational waves by neutron stars, free precession and a 'hidden' energy source for magnetars.     

Some recent developments in solar dynamo theory

We discuss the current status of solar dynamo theory and describe the dynamo model developed by our group. The toroidal magnetic field is generated in the tachocline by the strong differential rotation and rises to the solar surface due to magnetic buoyancy to create active regions. The decay of these active regions at the surface gives rise to the poloidal magnetic field by the Babcock-Leighton mechanism. This poloidal field is advected by the meridional circulation first to high latitudes and then down below to the tachocline. Dynamo models based on these ideas match different aspects of observational data reasonably well.     

On large-scale solar convection

We examine the effects of rotation about a vertical axis on thermal convection with a simple model in which an inviscid, incompressible fluid of zero thermal conductivity and electrical resistivity is contained in a thin annulus of rectangular cross-section. The initial steady state assumed is one of no motion relative to the rotating frame with constant (unstable) vertical temperature gradient and uniform toroidal magnetic field. Small periodic disturbances are then introduced and the linearized perturbation equations solved. We also determine the second-order mean circulations and magnetic fields that are forced by non-zero Reynolds and thermal stresses and magnetic field transports.The solutions have several properties which are relevant to large-scale solar phenomena if giant long-lived convection cells exist on the sun. In particular, the convective cells are tilted in latitude in the same sense as bipolar magnetic regions, and induce vertical magnetic fields with the same tilt. They transport momentum across latitude circles through Reynolds stresses and induced meridional circulations thus setting up a differential rotation. Cells which grow slowly compared to the rotation rate and have comparable dimensions in latitude and longitude transport momentum toward the equator. The cells also form a poloidal magnetic field from initial toroidal field, in a manner similar to that put forth by Parker.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

The rotation of magnetic loop systems in the solar atmosphere

Eclipse photographs indicate that large regions of the inner solar corona are confined in various types of closed magnetic configurations and, as a result, do not participate in the general solar wind expansion. In this paper, the rotation of initially poloidal loop configurations of this type, as influenced by differential rotation of the footpoints, is investigated. The analysis is restricted to axially symmetric fields and it is assumed that the toroidal magnetic field induced by differential rotation is small as compared to the initial poloidal field. This restricts the validity of the analysis to times less than about one month.The most interesting physical situation is that of flux tubes existing in one solar hemisphere only, one end of the tube being fixed in the photosphere at a higher latitude than the other. As a consequence, the lower end of the tube rotates at a faster rate than the upper end. Solution of the pertinent equations reveals that the angular velocity measured along a field line increases monotonically from its value at the poleward footpoint to that at the lower footpoint. The variation of angular velocity along the field depends upon the field geometry only and is not directly related to the variation of angular velocity along the solar surface between the footpoints. Depending upon the field configuration, both outward radial increases and decreases are possible. Using the Newton and Nunn model for the surface differential rotation rate, the angular velocity distribution on two particularly simple types of closed magnetic loop systems is determined analytically. It is shown that the angular velocity increases outward in the polar regions but decreases outward near the equator - leading to a decrease in differential rotation with height.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Magnetorotational supernovae. Magnetorotational instability. Jet formation

2D numerical simulations of magnetorotational (MR) supernova mechanism are described. It is shown that magnetic field is amplified due to the differential rotation after core collapse. When magnetic pressure reaches some level, a compression wave starts to move outwards. Moving along steeply decreasing density profile the compression wave transforms quickly into fast MHD shock. The magnetorotational instability (MRI) was found in our simulations. MRI leads to the exponential growth of the components of the magnetic field. The MRI significantly reduces MR supernova explosion time. Configuration of the initial magnetic field qualitatively defines the shape of MR supernova explosion. For the quadrupole-like initial poloidal field the MR supernova explosion develops mainly along equatorial plane, the dipole-like initial field results in MR supernova developing as mildly collimated jet along axis of rotation. The explosion energy of MR supernova found in our simulations is ∼0.5–0.6×10⁵¹ erg.     

Toroidal magnetic field in pulsars

The differential rotation of plasma in the core of pulsars (Ω_s ≠ Ω_e) generates convective currents increasing with time which in turn generates the toroidal magnetic field. To avoid difficulties of physical interpretation inherent to the theory of general relativity we have adopted the tetrad approach to discuss the generation of the magnetic field in the core of the neutron stars. The results which we have obtained are in agreement with those obtained earlier. Published in Astrofizika, Vol. 49, No. 4, pp. 613–620 (August 2006).     

Natural MHD oscillations of a neutron star

Natural, low-frequency, hydromagnetic oscillations of an isolated, nonrotating neutron star, which are localized in the peripheral crust, the structure of which is determined by the electron-nuclear plasma (the Ae phase), are studied. The plasma medium of the outer crust is treated as a homogeneous, infinitely conducting, incompressible continuum, the motions of which are determined by the equations of magnetohydrodynamics. In the approximation of a constant magnetic field inside the crust (the magnetic field outside the star is assumed to have a dipole structure), the spectrum of normal poloidal and toroidal hydromagnetic oscillations, due to presumed residual fluctuations of flow and their associated fluctuations in magnetic field strength, is calculated. Numerical estimates given for the periods of MHD oscillations fall in the range of periods of radio pulsar emission, indicating a close connection between the residual hydromagnetic oscillations and the electromagnetic activity of neutron stars. Translated from Astrofizika, Vol. 40, No. 1, pp. 77–86, January–March, 1997.     

The effect of rotation on the luminosity of magnetic stars

The effect of rotation and a general magnetic field on the luminosity, radius, and effective temperature of the upper Main-Sequence stars has been investigated using a perturbation analysis. The magnetic field profile prevailing inside the star is assumed to have both poloidal and toroidal components. The case of constant as well as differential rotation is admitted. Model calculations indicate that these stellar parameters modify considerably as a result of coupling between rotation and the magnetic field.     

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)