Coronal magnetic field equilibrium with discrete flux sources

A model of the equilibrium structure of the coronal magnetic field is developed, taking account of the fact that field lines are rooted in the photosphere, where field is concentrated into isolated flux tubes. The field is force-free, described by ×B = B, with constant; this field has special physical significance, being the state of mininum energy after small-scale reconnections, and is also mathematically convenient in that the principle of superposition can be used to construct complex geometries. First a model of a single loop is presented, with a flux source and sink pair at the photosphere; both point flux tubes and finite radius flux tubes are considered. Then more complex topologies with multiple sources and sinks are investigated. It is shown that significant topology changes arise for different values of, indicating the possibility that there can be energy changes through magnetic reconnection if the field evolves ideally and then relaxes to a linear state.     

Exact static equilibrium of vertically oriented magnetic flux tubes

A method is prescribed for generating exact solutions of magnetostatic equilibrium describing a cylindrically symmetric magnetic flux tube oriented vertically in a stratified medium. Given the geometric shape of the field lines, compact formulae are presented for the direct calculation of all the possible distributions of pressure, density, temperature and magnetic field strength compatible with these field lines under the condition of static equlibrium. The plasma satisfies the ideal gas law and gravity is uniform in space. A particular solution is obtained by this method for a medium sized sunspot whose magnetic field obeys the similarity law of Schlüter and Temesváry (1958). With this solution, it is possible for the first time to illustrate explicitly the confinement of the magnetic field of the cool sunspot by the hotter external plasma in an exact relationship involving both magnetic pressure and field tension as well as the support of the weight of the plasma by pressure gradients. It is found that the cool region of the sunspot is not likely to extend much more than a few density scale heights below the photosphere. The sunspot field approaches being potential in the neighbourhood of the photosphere so that the Lorentz force exerting on the photosphere is less than what the magnetic pressure would suggest. This accounts for how the sunspot field can be confined in the photosphere where its magnetic pressure is often observed to even exceed the normal photospheric pressure. The energy mechanism operating in the sunspot and the question of mechanical stability are not treated in this paper.Normally at Lau Kuei Huat (Singapore) Private Limited, 55 Shipyard Road, Singapore 22, Singapore.     

Stability of magnetic flux tubes in close binary stars

Photometric and Doppler imaging observations of active binaries indicate the existence of starspots at preferred longitudes (position angles with respect to the companion star). We investigate the stability of magnetic flux tubes in the convection zone of close, fast‐rotating binary stars and explore whether the observed preferred longitudes could be caused by tidal forces and the deformation of the active star. We assume a synchronized binary system with spin axes perpendicular to the orbital plane and a rotation period of a few days. The tidal force and the deviation from spherical structure are considered in lowest‐order perturbation theory. The magnetic field is in the form of toroidal magnetic flux rings, which are stored in mechanical equilibrium within the stably stratified overshoot region beneath the convection zone until the field has grown sufficiently strong for the undulatory instability to initiate the formation of rising loops. Frequencies and geometry of stable as well as growth rates of unstable eigenmodes are determined by linear stability analysis. Particular consideration is given to the question whether the effects of tidal forces and perturbations of the stellar structure can force a rising flux loop to enter the convection zone at specific longitudes.     

The equilibrium shape of slender flux tubes in a linear force-free magnetic field

In this paper we extend previous work of Browning and Priest (1984, 1986) by studying the equilibrium path of twisted and untwisted thin flux tubes in a stratified, isothermal atmosphere using as the ambient field a linear force-free field. When an untwisted flux tube is considered, we find that shearing the magnetic arcade provides a different form to change the parameter which characterizes the external atmosphere, but at the same time this introduces a limitation in the width allowed for the external arcade. Also, the critical width found for the different analytical cases considered is always greater than one arch of the ambient arcade which prevents an eruption inside the arcade. In the case of twisted flux tubes, an analytical solution can be found for the critical _c , which separates regimes of strong and weak gravity, and the shape of the flux tube is now dependent on , a parameter which represents the magnetic field enhancement of the loop at the photosphere.     

The emergence of magnetic flux

This paper first summarizes the morphology and dynamics of emerging flux regions and arch filament systems and then discusses detailed observations of a particular active region with emerging magnetic flux.The central part of the growing active region shows abnormal granulation and a weak magnetic field that, locally, is transverse. In the border zone, strong downward flows occur in the chromopshere and photosphere (small features with strong magnetic fields (faculae, pores) are formed here.) Near the leading and following edge, sunspots are formed by the coalescence of such small magnetic elements.The observational data are interpreted by means of a heuristic model of an emergent magnetic loop-shaped bundle consisting of many flux tubes. In this model we incorporate the theory of convective collapse and the buoyancy of flux tubes. The observed complexity in the structure and dynamics, including strong transverse fields and velocity shear, is attributed to the emergence of several flux regions within the active region at different orientations.     

The photospheric magnetic flux budget

The ensemble of bipolar regions and the magnetic network both contain a substantial and strongly variable part of the photospheric magnetic flux at any phase in the solar cycle. The time-dependent distribution of the magnetic flux over and within these components reflects the action of the dynamo operating in the solar interior. We perform a quantitative comparison of the flux emerging in the ensemble of magnetic bipoles with the observed flux content of the solar photosphere. We discuss the photospheric flux budget in terms of flux appearance and disappearance, and argue that a nonlinear dependence exists between the flux present in the photosphere and the rate of flux appearance and disappearance. In this context, we discuss the problem of making quantitative statements about dynamos in cool stars other than the Sun. This paper evolved out of a more comprehensive version which appeared in Harvey (1993).     

Dissipation of magnetic flux and magnetic energy in partially ionized gases

Macroscopic equations of motion are used to derive several forms of the generalized Ohm's law for partially ionized ternary gases in magnetic fields, and a conductivity σ is defined that is independent of the magnetic field. A flux theorem is derived using a velocityu _H that can be defined to be the velocity of magnetic field lines;u _H is only slightly different from the velocity of the electron component of the gas. It is shown that σ is the conductivity relevant to the decay of magnetic flux through any surface moving everywhere with velocityu _H . The rate of increase of the thermal energy density of the gas arising through collisions between particles of different species can be resolved into Joule heating at the ratej ²/σ, wherej is the current density, and heating associated with ambipolar drift. The latter, contrary to what has been claimed by some authors, is not necessarily fully compensated by a decrease in the energy of the electromagnetic field. In many applications such compensation does occur, but it may not in interstellar clouds where large amounts of gravitational energy can be made available by collapse, and then both heating and an increase in electromagnetic field energy may occur.     

The equilibrium of coronal flux tubes under toroidal forces

The properties of slender isolated flux tubes, taking into account curvature effects, were investigated by Parker (1975, 1979) and Spruit (1981), and many studies have been made concerning the equilibrium of slender flux tubes in the solar corona. In this paper we use a different approach considering the coronal loop as a part of a circular torus and studying the position of its top when the loop is in equilibrium under toroidal forces. Toroidal forces were considered by Shafranov (1966) for toroidal pinches and the equilibrium can be studied for different values of the toroidal current intensity and external magnetic field. The results show that it is possible to have a coronal flux tube in equilibrium without considering gravity and external magnetic field. Furthermore, the total twist of the flux tube and its variation with the toroidal intensity has been studied.     

The Mg/Ne abundance ratio in a recently emerged flux region observed by CDS

Evidence for the existence of the FIP-effect in the transition region is presented here based on recent observations from the Coronal Diagnostic Spectrometer (CDS) on-board the Solar and Heliospheric Observatory (SOHO). Observations of an emerging flux region in lines of Mgv–vii and Nevi–vii reveal differences in the relative Mg/Ne abundance of a factor of 9.2 between two transition region brightenings separated by less than 1 arc min on the Sun. The lower abundance ratio is approximately equal to the photospheric Mg/Ne value and is associated with a small loop-like feature in the central, hottest part of the active region. The higher abundance ratio is found in spike-like structures at the edge of the active region. A density diagnostic of Oiv is used to derive an electron number density of 10^11.3 cm^-3 for the low Mg/Ne brightening, while a Mgvii diagnostic gives a density of 10^9.2 cm for the high Mg/Ne brightening.     

Frequency response of magnetic flux sheaths

We suggest to identify the elementary flare bursts with the excitation of the small kernels that occur in flare loops that are observed in soft X-ray pictures of flares. We stress the need of simultaneous observations of spatial structure and time variations of hard X-ray bursts sources in various wavelength regions.     

Motions in the field of two rotating magnetic dipoles. II. Stability of the equilibrium points

The stability of the equilibrium points found to exist (cf. Goudaset al., 1985, referred to henceforth as Paper I) in the problem of two parallel, or antiparallel, magnetic dipoles that rotate about the centre of mass of their carrier stars, is studied by computing the characteristic roots of their variational equations. The characteristic equation, a biquadratic, solved for many combinations of and showed that all equilibrium points of this problem are unstable.     

Transport of reconnected magnetic flux and magnetic energy due to magnetotail reconnection

The Ph.D. thesis summarized here was defended at the University of Graz, Austria, in February 2008. The Ph.D. supervisor was Prof. Helfried K. Biernat.     

Thermodynamical properties of unresolved magnetic flux tubes

We propose a diagnostic method, based on the observation of circular polarization signals in line pairs, to derive the thermodynamical properties of unresolved magnetic elements in the solar atmosphere. The concept of response function for the ratio of circular polarization signals in two lines is introduced and its main properties are analyzed. Some detailed calculations for suitably selected line pairs are presented.     

Topological stability of finite-length magnetic flux tubes

It has been suggested that the activity of cosmical magnetic fields is a consequence of a general topological nonequilibrium in the neighbourhood of magnetostatic equilibria. Evidence for this suggestion can be obtained from the Kolmogorov-Arnold-Moser theorem of classical mechanics, applied to the magnetic field line flow as a Hamiltonian system. A finite-length magnetic flux tube, however, always possesses two independent sets of flux surfaces - or, equivalently, the corresponding Hamiltonian system two independent first integrals - and is topologically stable if in the volume occupied by the tube there are no singular (null) points of the magnetic field and the normal field component does not change its sign on the end faces of the tube. Therefore, the concept of nonequilibrium due to flux surface destruction is not applicable to solar atmospheric loops with each end situated in the interior of one polarity of the photospheric normal field component. Further, it seems unlikely that the tearing-mode mechanism can play a role in such loops.     

The separation velocity of emerging magnetic flux

We measure the separation velocity of opposite poles from 24 new bipoles on the Sun. We find that the measured velocities range from about 0.2 to 1 km s^–1. The fluxes of the bipoles range over more than two orders of magnitude, and the mean field strength and the sizes range over one order of magnitude. The measured separation velocity is not correlated with the flux and the mean field strength of the bipole. The separation velocity predicted by the present theory of magnetic buoyancy is between 7.4Ba ^–1/4 cot and 13 cot km s^–1, where is the elevation angle of the flux tube at the photosphere (see Figure 9), B is the mean field strength, and a is the radius of the observed bipole. The rising velocity of the top of flux tubes predicted by the theory of magnetic buoyancy is between 3.7Ba ^–1/4 and 6.5 km s^–1. The predicted separation velocity is about one order of magnitude higher than those measured, or else the flux tubes are almost vertical at the photosphere. There is no correlation between the measured separation velocity and the theoretical value, 7.4Ba ^–1/4. The predicted rising velocity is also higher than the vertical velocity near the line of inversion in emerging flux regions observed by other authors.     

The rise of twisted magnetic flux tubes

Sunspots are caused by the eruption of magnetic flux tubes through the solar photosphere: current theories of the internal magnetic field of the Sun suggest that such tubes must rise relatively unscathed from the base of the convection zone. In order to understand how the structure of the magnetic field within a buoyant flux tube affects its stability as it rises, we have considered the quasi-two-dimensional rise of isolated magnetic flux tubes through an adiabatically stratified atmosphere. The magnetic field is initially helical; we have investigated a range of initial field configurations, varying the distribution and strength of the twist of the field.