Latitudinal migration of sunspots based on the ESAI database

The latitudinal migration of sunspots toward the equator,which implies there is propagation of the toroidal magnetic flux wave at the base of the solar convection zone,is one of the crucial observational bases for the solar dynamo to generate a magnetic field by shearing of the pre-existing poloidal magnetic field through differential rotation.The Extended time series of Solar Activity Indices(ESAI)elongated the Greenwich observation record of sunspots by several decades in the past.In this study,ESAI's yearly mean latitude of sunspots in the northern and southern hemispheres during the years 1854 to 1985 is utilized to statistically test whether hemispherical latitudinal migration of sunspots in a solar cycle is linear or nonlinear.It is found that a quadratic function is statistically significantly better at describing hemispherical latitudinal migration of sunspots in a solar cycle than a linear function.In addition,the latitude migration velocity of sunspots in a solar cycle decreases as the cycle progresses,providing a particular constraint for solar dynamo models.Indeed,the butterfly wing pattern with a faster latitudinal migration rate should present stronger solar activity with a shorter cycle period,and it is located at higher latitudinal position,giving evidence to support the Babcock-Leighton dynamo mechanism.     

Solar cycle general magnetic fields of 1959–1974 and dynamical structure of the convection zone

The concept of the solar general magnetic field is extended from that of the polar fields to the concept of any axisymmetric fields of the whole Sun. The poloidal and toroidal general magnetic fields are defined and diagrams of their evolutionary patterns are drawn using the Mount Wilson magnetic synoptic chart data of Carrington rotation numbers from 1417 to 1620 covering approximately half of cycle 19 and cycle 20. After averaging over many rotations long-term regularities appear in the patterns. The diagrams of the patterns are compared with the Butterfly Diagram of sunspots of the same period. The diagram of the poloidal field shows that the Sun behaves like a magnetic quadrupole, each hemisphere having two branches of opposite polarities with mirror images on the other hemisphere. This was predicted by a solar cycle model driven by the dynamo action of the global convection by Yoshimura and could serve as a verification of the model. The diagram of the toriodal field is similar to the Butterfly Diagram of sunspots. The slight differences which do exist between the two diagrams seems to show that the fields responsible for the two may originate from different zones of the Sun. Common or different characteristics of the three diagrams are examined in terms of dynamical structure of the convection zone referring to the theoretical model of the solar cycle driven by the dynamo action of the global convection.     

A New Explanation for the Origin of Trans-equatorial Loops based on a Dynamo Model

Trans-equatorial loops (TLs) are one of the distinct magnetic structures in the solar corona and have a close relationship to solar activity. We present a systematic study of the origin of TLs linking with the Babcock?–?Leighton dynamo process based on the model of Chatterjee, Nandy, and Choudhuri (Astron. Astrophys. 427, 1019, 2004). We propose that TLs are visible signatures of poloidal field lines across the equator. The cycle variation of TL lengths obtained by the connectivities of poloidal field lines happens to be roughly in agreement with what one gets by considering the positions of sunspots. This explains why this cycle variation of TL lengths was found to be in conformity with Spörer’s law. The active regions always make the poloidal field configuration favorable to form TLs, which causes the conformity. The formation of TLs is a three-dimensional problem, which will require three-dimensional dynamo models for full investigation.     

Solar activity forecast with a dynamo model

Although systematic measurements of the Sun's polar magnetic field exist only from mid-1970s, other proxies can be used to infer the polar field at earlier times. The observational data indicate a strong correlation between the polar field at a sunspot minimum and the strength of the next cycle, although the strength of the cycle is not correlated well with the polar field produced at its end. This suggests that the Babcock–Leighton mechanism of poloidal field generation from decaying sunspots involves randomness, whereas the other aspects of the dynamo process must be reasonably ordered and deterministic. Only if the magnetic diffusivity within the convection zone is assumed to be high (of order  10¹² cm² s⁻¹  ), we can explain the correlation between the polar field at a minimum and the next cycle. We give several independent arguments that the diffusivity must be of this order. In a dynamo model with diffusivity like this, the poloidal field generated at the mid-latitudes is advected toward the poles by the meridional circulation and simultaneously diffuses towards the tachocline, where the toroidal field for the next cycle is produced. To model actual solar cycles with a dynamo model having such high diffusivity, we have to feed the observational data of the poloidal field at the minimum into the theoretical model. We develop a method of doing this in a systematic way. Our model predicts that cycle 24 will be a very weak cycle. Hemispheric asymmetry of solar activity is also calculated with our model and compared with observational data.     

Transport Effects in the Evolution of the Global Solar Magnetic Field

The axisymmetric component of the large-scale solar magnetic fields has a pronounced poleward branch at higher latitudes. In order to clarify the origin of this branch we construct an axisymmetric model of the passive transport of the mean poloidal magnetic field in the convective zone, including meridional circulation, anisotropic diffusivity, turbulent pumping and density pumping. For realistic values of the transport coefficients we find that diffusivity is prevalent, and the latitudinal distribution of the field at the surface simply reflects the conditions at the bottom of the convective zone. Pumping effects concentrate the field to the bottom of the convective zone; a significant part of this pumping occurs in a shallow subsurface layer, normally not resolved in dynamo models. The phase delay of the surface poloidal field relative to the bottom poloidal field is found to be small. These results support the double dynamo wave models, may be compatible with some form of a mixed transport scenario, and exclude the passive transport theory for the origin of the polar branch.     

Magnetic Helicity as a Predictor of the Solar Cycle

It is well known that the polar magnetic field is at its maximum during solar minima, and that the behaviour during this time acts as a strong predictor of the strength of the following solar cycle. This relationship relies on the action of differential rotation (the Omega effect) on the poloidal field, which generates the toroidal flux observed in sunspots and active regions. We measure the helicity flux into both the northern and the southern hemispheres using a model that takes account of the Omega effect, which we apply to data sets covering a total of 60 years. We find that the helicity flux offers a strong prediction of solar activity up to five years in advance of the next solar cycle. We also hazard an early guess as to the strength of Solar Cycle 25, which we believe will be of similar amplitude and strength to Cycle 24.     

On the large-scale diffuse magnetic field of the Sun – II. The Contribution of Active Regions

Dikpati and Choudhuri (1994, 1995) developed a model for the poleward migration of the weak diffuse magnetic field on the Sun's surface. This field was identified with the poloidal component produced by the solar dynamo operating at the base of the convection zone, and its evolution was studied by considering the effects of meridional circulation and turbulent diffusion. The earlier model is extended in this paper by incorporating the flux from, the decay of tilted active regions near the solar surface as an additional source of the poloidal field. This extended model can now explain various low-latitude features in the time-latitude diagram of the weak diffuse fields. These low-latitude features could not be accounted for in the earlier model, which was very successful in modeling the behavior at high latitudes. The time-latitude diagrams show that regions of a particular polarity often have tongues of opposite polarity. Such tongues can be produced in the theoretical model by incorporating fluctuations in the source term arising out of the decaying active regions.     

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.     

On the Nature of Neutral-Line-Associated Radio Sources

A number of authors claimed that radio sources above the neutral line of the magnetic field in solar active regions are due to non-thermal emission. This study shows that the thermal mechanism explains the radio emission from such sources. Models similar to those used for interpreting cyclotron lines were used in this study. Such models account for a steep decline in the spectrum at high frequencies and a low degree of polarization. The magnetic field between the two sunspots with an anti-parallel magnetic field has a lower gradient than the field above the sunspots. This, combined with the possibly high temperature in coronal loops connecting the sunspots, leads to the following conclusions. The optical thickness of the gyroresonance layers is increased and leads to more effective radiation at a harmonic number of 4 or 5. The lower gradient of the field between the sunspots also results in more rapid growth of emission intensity with increasing wavelength in this region than in the regions immediately above the sunspots. Additionally, the spatial averaging of the source structure due to the antenna beam pattern leads to a decrease in the degree of polarization in the region between the sunspots.     

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.     

A Spatio-temporal Description of the Abrupt Changes in the Photospheric Magnetic and Lorentz-Force Vectors During the 15 February 2011 X2.2 Flare

The active region NOAA 11158 produced the first X-class flare of Solar Cycle 24, an X2.2 flare at 01:44 UT on 15 February 2011. The Helioseismic and Magnetic Imager (HMI) instrument on the Solar Dynamics Observatory (SDO) satellite produces 12-minute, 0.5′′ pixel^?1 vector magnetograms. Here we analyze a series of these data covering a 12-hour interval centered at the time of this flare. We describe the spatial distributions of the photospheric magnetic changes associated with the flare, including the abrupt changes in the field vector, vertical electric current and Lorentz-force vector acting on the solar interior. We also describe these parameters’ temporal evolution. The abrupt magnetic changes were concentrated near the neutral line and in two neighboring sunspots. Near the neutral line, the field vectors became stronger and more horizontal during the flare and the shear increased. This was due to an increase in strength of the horizontal field components near the neutral line, most significant in the horizontal component parallel to the neutral line but the perpendicular component also increased in strength. The vertical component did not show a significant, permanent overall change at the neutral line. The increase in field strength at the neutral line was accompanied by a compensating decrease in field strength in the surrounding volume. In the two sunspots near the neutral line the integrated azimuthal field abruptly decreased during the flare but this change was permanent in only one of the spots. There was a large, abrupt, downward vertical Lorentz-force change acting on the solar interior during the flare, consistent with results of past analyses and recent theoretical work. The horizontal Lorentz force acted in opposite directions along each side of neutral line, with the two sunspots at each end subject to abrupt torsional forces relaxing their magnetic twist. These shearing forces were consistent with a contraction of field and decrease of shear near the neutral line, whereas the field itself became more sheared as a result of the field collapsing towards the neutral line from the surrounding volume. The Lorentz forces acting on the atmospheric volume above the photosphere were equal and opposite.     

A magnetostatic sunspot model with ‘twisted’ field

It is shown that the internal structure of sunspots is affected only to a small degree by the presence of the azimuthal component of the sunspot magnetic field. The magnetostatic theory of sunspots developed by Schlüter and Temesvary (1958) is extended to include such an effect. In the present study, the radial dependence of the azimuthal component of the field is adopted from the observation of Stepanov (1965). A typical sunspot of an effective temperature of 4000K is considered and its detailed structure is presented.Publications of the Goethe Link Observatory, No. 115.     

Cooling of neutron stars with strong toroidal magnetic fields

We present models of temperature distribution in the crust of a neutron star in the presence of a strong toroidal component superposed to the poloidal component of the magnetic field. The presence of such a toroidal field hinders heat flow toward the surface in a large part of the crust. As a result, the neutron star surface presents two warm regions surrounded by extended cold regions and has a thermal luminosity much lower than in the case the magnetic field is purely poloidal. We apply these models to calculate the thermal evolution of such neutron stars and show that the lowered photon luminosity naturally extends their life-time as detectable thermal X-ray sources. Work partially supported by UNAM-DGAPA grant #IN119306.     

The decay of torque free toroidal magnetic fields confined beneath the solar convection zone

Calculation similar to those of Mestel and Moss (1983) are performed to investigate the decay of a toroidal field through configurations satisfying the torque free condition, imposed by the presence of a poloidal field of dipolar form confined beneath the solar convection zone. It is found that initially stable field configurations diffuse into unstable configurations on time-scales of order a few x 10⁸ yr. The results are similar to those of Tayler (1982) for a simpler field model without any dynamical constraints.     

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.     

The measurement of magnetic fields in the solar atmosphere above sunspots using gyroresonance emission

An analysis of the local sources (LS) structure of the S-component of solar radio emission confirms the presence of a core component which is characterized by strong circular polarization and a steep growing spectrum at shorter centimeter wavelengths. These details coincide in position with the sunspots' umbra and their height above the photosphere does not generally exceed about 2000 km. Gyroresonance emission of thermal electrons of the corona is generally accepted as being responsible for this type of emission. The spectral and polarization observations of LS made with RATAN-600 using high resolution in the wavelength range 2.0–4.0 cm, allow us to measure the maximum magnetic fields of the corresponding sunspots at the height of the chromosphere-corona transition region (CCTR). This method is based on determining the short wavelength limit of gyroresonance emission of the LS and relating it to the third harmonic of gyrofrequency.An analysis of a large number of sunspots and their LS (core component) has shown a good correlation between radio magnetic fields near the CCTR and optical photospheric ones. The magnetic field in CCTR above a sunspot is found only 10 to 20% lower than in the photosphere. The resulting gradient of the field strength is not less than 0.25 G km^–1. This result seems to contradict the lower values of magnetic fields generally found above sunspots using the chromospheric H line. Some possible ways of overcoming this difficulty are proposed.     

Large-scale photospheric magnetic field: The diffusion of active region fields

The large-scale photospheric magnetic field has been computed by allowing observed active region fields to diffuse and to be sheared by differential rotation in accordance with the Leighton (1969) magnetokinematic model of the solar cycle. The differential rotation of the computed field patterns as determined by autocorrelation curves is similar to that of the observed photospheric field, and poleward of 20° latitude both are significantly different from the differential rotation of the long-lived sunspots (Newton and Nunn, 1951) used as an input into the computations.Now at Department of Physics, Victoria University of Wellington, Wellington, New Zealand.     

Changes in measured vector magnetic fields when transformed into heliographic coordinates

In this paper we investigate the changes that occur in measured magnetic fields when they are transformed into a heliographic coordinate system. To carry out this investigation we took measurements of the vector magnetic field of an active region that was observed at 1/3 the solar radius from disk center and transformed the observed field into heliographic coordinates. We also examined differences in the calculated potential field that occur when the heliographic normal component of the field is used as the boundary condition rather than the observed line-of-sight component. The results of this analysis show (1) that the observed fields of sunspots more closely resemble the generally accepted picture of the distribution of umbral fields if they are displayed in heliographic coordinates, (2) that the differences in the potential calculations are less than 200 G in field strength and 20° in field azimuth outside sunspots, and (3) that differences in the two potential calculations in the sunspot areas are no more than 400 G in field strength but range from 60 to 80° in field azimuth in localized umbral areas.     

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.     

On the large-scale diffuse magnetic field of the sun

Although the sunspots migrate towards the equator, the large-scale weak diffuse magnetic fields of the Sun migrate poleward with the solar cycle, the polar field reversing at the time of the sunspot maxima. We apply the vector model of Dikpati and Choudhuri (1994, Paper I) to fit these observations. The dynamo layer at the base of the convection zone is taken to be the source of the diffuse field, which is then evolved in the convection zone subject to meridional circulation and turbulent diffusion. We find that the longitudinally averaged observational data can be fitted reasonably well both for positive and negative values of the-effect by adjusting the subsurface meridional flow suitably. The model will be extended in a future paper to include the decay of active regions as an extra source of the diffuse field, which may be necessary to explain the probable phase lag betweenB _r andB at lower latitudes.