Numerical simulations of rotating axisymmetric sunspots

A numerical model of axisymmetric convection in the presence of a vertical magnetic flux bundle and rotation about the axis is presented. The model contains a compressible plasma described by the non-linear MHD equations, with density and temperature gradients simulating the upper layer of the Sun's convection zone. The solutions exhibit a central magnetic flux tube in a cylindrical numerical domain, with convection cells forming collar flows around the tube. When the numerical domain is rotated with a constant angular velocity, the plasma forms a Rankine vortex, with the plasma rotating as a rigid body where the magnetic field is strong, as in the flux tube, while experiencing sheared azimuthal flow in the surrounding convection cells, forming a free vortex. As a result, the azimuthal velocity component has its maximum value close to the outer edge of the flux tube. The azimuthal flow inside the magnetic flux tube and the vortex flow is prograde relative to the rotating cylindrical reference frame. A retrograde flow appears at the outer wall. The most significant convection cell outside the flux tube is the location for the maximum value of the azimuthal magnetic field component. The azimuthal flow and magnetic structure are not generated spontaneously, but decay exponentially in the absence of any imposed rotation of the cylindrical domain.     

Subsonic flows in the coronal-interplanetary regions of closed field lines

In the coronal-interplanetary space the plasma motion, in a reference frame corotating with the Sun, is aligned with the magnetic field. Just like the solar wind, which is the supersonically expanding flow along open field lines, the flow along closed field lines is mainly driven by the pressure gradient. The flow in the regions of closed field lines is subsonic, being determined by the conditions at the two footpoints of the magnetic flux tube.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Analysis and results of cooperative magnetographic measurements. I. Correction,comparison and discussion of measurements

On June 24, 1983 cooperative magnetographic measurements were made with the vectormagnetographs of the Sayan Observatory (Irkutsk) and the Potsdam Solar Observatory “Einsteinturm” and with the longitudinal magnetograph of the Ondrejov Observatory. Additionally, the maximum field strengths in the sunspot were measured with the photographic method. Using the photographically measured field strengths as reference values we provide a method to eliminate the influences of stray light in a time series of vectormagnetograms. A comparison of nearly simultaneous magnetograms shows a good correspondence in general. The deviations of the zero points as well as the scales are comparable with the results of other authors. Regarding the magnetic field distribution the magnetograms reflect a substantial nonsymmetric structure in the spot under study. The magnetic field lines tend to concentrate into several flux tube clusters. In one region of the penumbra we find a magnetic field with nearly longitudinal character in close neighbourhood of a strong, nearly horizontal flux tube bundle. This indicates a strong nonradial horizontal field gradient in the penumbra.     

Guided waves in magnetospheric tubes of enhanced density

Properties of a guided MHD-wave propagating in a magnetic field tube with the plasma density differing from the ambient density are studied. Like the Alvén wave this wave propagates along the magnetic field and is connected with the field-aligned currents flowing at the periphery of the oscillating tube. The guided wave is accompanied by the magnetic field compression, nevertheless the wave moves without attenuation. The guided wave velocity is between the Alvén velocities inside and outside the oscillating tube. In a tube of elliptical cross-section the propagation velocity depends on the polarization of the wave.     

On the possibility of investigation of the magnetic field structure in subphotospheric layers of the Sun according to observations of sunspot torsional oscillations

A method of investigation of the magnetic field structure in subphotospheric layers of the Sun has been developed. The method is based on observations of the torisonal oscillations of single sunspots. Characteristics of the torsional oscillations have been obtained from observations of the longitudinal magnetic field and radial velocities of seven single sunspots in the photospheric line Fe I λ5253 Å. The parameters of the torsional oscillations and magnetic tubes in the deep layers have been determined. The radius of the cross section of a magnetic flux tube forming a sunspot is greatest near the Sun’s surface and is approximately equal to the radius of a sunspot umbra. Down to the deeper layers, it decreases quite quickly. The longitudinal electric current appearing in the magnetic tube changes direction. The typical time of the current changes is determined by the period of the torsional oscillations. The intensity of the longitudinal magnetic field in the tube increases with depth. The Alfven wave velocity averaged over the length of a magnetic tube is tens or hundreds of times less than this velocity in a sunspot umbra. It decreases with an increase in the period of oscillations. A decrease in the Alfven wave velocity leads to an increase in the twisting angle of magnetic field lines.     

Numerical Simulations of Torsional Alfvén Waves in Axisymmetric Solar Magnetic Flux Tubes

We numerically investigate Alfvén waves propagating along an axisymmetric and non-isothermal solar flux tube embedded in the solar atmosphere. The tube magnetic field is current-free and diverges with height, and the waves are excited by a periodic driver along the tube magnetic field lines. The main results are that the two wave variables, the velocity and magnetic field perturbations in the azimuthal direction, behave differently as a result of gradients of the physical parameters along the tube. To explain these differences in the wave behavior, the time evolution of the wave variables and the resulting cutoff period for each wave variable are calculated and used to determine regions in the solar chromosphere where strong wave reflection may occur.     

Nonaxisymmetric Oscillations of Thin Twisted Magnetic Tubes

In this paper we study nonaxisymmetric oscillations of thin twisted magnetic tubes taking the density variation along the tube into account. We use the approximation of the zero-beta plasma. The magnetic field outside the tube is straight and homogeneous; however, it is twisted inside the tube. We assume that the azimuthal component of the magnetic field is proportional to the distance from the tube axis and that the tube is only weakly twisted (i.e., the ratio of the azimuthal and axial components of the magnetic field is small). Using the asymptotic analysis we show that the eigenmodes and eigenfrequencies of the kink and fluting oscillations are described by a classical Sturm – Liouville problem for a second-order ordinary differential equation. The main result is that the twist does not affect the kink mode.     

Non-equilibrium of Magnetic Flux Tubes emerging into the Solar Corona

A lack of equilibrium of twisted magnetic flux tubes emerging from the photosphere into the corona is considered. Assuming mass and flux conservation in the tube and an isothermal tube that is in thermal equilibrium with the surrounding plasma, it is shown that a sufficently rapid temperature increase through the transition zone may lead to the loss of magnetohydrostatic equilibrium of the emerging flux tube due to the enhancement of the plasma pressure inside the tube. The non-equilibrium leads to a rapid expansion of the tube to reach a new equilibrium state. The rise and expansion of the tube before and after the non-equilibrium are accompanied by an increase in the twist of the magnetic field. This may lead to the field exceeding the threshold for the onset of the kink instability and a subsequent explosive release of magnetic energy.     

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.     

Dependence of the canopy height of solar magnetic flux tubes on its definition

By analyzing a canopy tube described by Athay, we point out the possibility that the different canopy heights given by different authors are caused by different definitions of canopy heights. For the same canopy tube, those heights where the magnetic field has horizontal components, where neighboring magnetic tubes meet each other and where the magnetic field becomes uniform or vertical range from near the solar photosphere to the solar corona.     

Evolution of emerging force-free flux tube to be a flare loop

Evolution of a filamentary magnetic flux tube emerging from the photosphere is investigated in the assumption that the magnetic field is force-free and unchanged during the evolution.If a characteristic radius of the flux tube is 3 km or less setting the field to 1000G, the temperature increases at first due to Joule heating up to about one million degree keeping the plasma density almost constant, and then the density decreases down to a critical value at which a current instability may occur. Thus, a.strong field-aligned electric field of 200 million volts or more is expected to be produced during the following anomalous Ohmic decay of the magnetic field as already shown by a numerical simulation.     

The magnetic non-equilibrium of buoyant flux tubes in the solar corona

The equilibrium shape of a slender flux tube in the stratified solar atmosphere is studied. The path is determined by a balance between the downwards magnetic tension, which depends on the curvature of the loop, and the upwards buoyancy force. Previous results for untwisted slender tubes are extended to include twisted tubes embedded in an external magnetic field.The path of an untwisted tube in an atmosphere with an ambient magnetic field is calculated. For a given footpoint separation, the height of the tube is lowered by increasing the strength of the external magnetic field. If the footpoints are slowly moved apart, the tube rises, until a threshold separation is reached beyond which there is no possible equilibrium height. This threshold width does not depend on the strength of the external field.The effects of twisting up a curved loop are studied, using an extension of results obtained for slender curved tubes with a straight axis. It is shown that for a twisted tube of given width, there can be two possible values of the equilibrium height. If, however, the tube is twisted more than a certain amount or if the footpoints are too widely separated there is no equilibrium. The critical footpoint separation for non-equilibrium is smaller for a twisted tube that an untwisted one.Twisting a tube or moving its feet apart is thus likely to result in non-equilibrium, causing the tube to rise indefinitely under the influence of the unbalanced buoyant force. It is suggested that this phenomenon could be important in the preflare stage of a large two-ribbon solar flare, by causing the initial slow rise of an active region filament. As well as being involved in the onset of an erupting prominence, this non-equilibrium may also be relevant to the formation of coronal loop transients.     

Topological Departures from Translational Invariance along a Filament Observed by THEMIS

We study the topology of the 3D magnetic field in a filament channel to address the following questions: Is a filament always formed in a single flux tube? How does the photospheric magnetic field lead to filament interruptions and to feet formation? What is the relation between feet-related field lines and the parasitic polarities? What can topological analyses teach us about EUV filament channels? To do so, we consider a filament observed on 6 October 2004 with THEMIS/MTR, in Hα with the full line profile simultaneously and cospatially with its photospheric vector magnetic field. The coronal magnetic field was calculated from a “linear magnetohydrostatic” extrapolation of a composite THEMIS-MDI magnetogram. Its free parameters were adjusted to get the best match possible between the distribution of modeled plasma-supporting dips and the Hα filament morphology. The model results in moderate plasma β≤1 at low altitudes in the filament, in conjunction with non-negligible departures from force-freeness measured by various metrics. The filament here is formed by a split flux tube. One part of the flux tube is rooted in the photosphere aside an observed interruption in the filament. This splitted topology is due to strong network polarities on the edge of the filament channel, not to flux concentrations closer to the filament. We focus our study to the northwest portion of the filament. The related flux tube is highly fragmented at low altitudes. This fragmentation is due to small flux concentrations of two types. First, some locally distort the tube, leading to noticeable thickness variations along the filament body. Second, parasitic polarities, associated with filament feet, result in secondary dips above the related local inversion line. These dips belong to long field lines that pass below the flux tube. Many of these field lines are not rooted near the related foot. Finally, the present model shows that the coronal void interpretation cannot be ruled out to interpret the wideness of EUV filament channels.     

Field and flow perturbations outside the reconnected field line region in flux transfer events: Theory

We propose a simplified model of field draping in flux transfer events. To this end we investigate the perturbations in the magnetic field and plasma flow close to but outside a reconnected magnetic flux tube. Following Southwood (1985, in Physics of Ionosphere—Magnetosphere, Adv. Space Res.5, 4–7), we treat the case of incompressible flow and rigid flux tube boundary. We find that the external field perturbations are practically indistinguishable from those observed in the data and hitherto generally ascribed to actual penetration by the spacecraft of the open field line region. The magnetic signature of draping involves all three components of the magnetic field and is accompanied by changes in field strength. The perturbations in the plasma velocity are related to those in the magnetic field and there are concomitant pressure disturbances as well.The effect of magnetopause motion is to complicate the signature by introducing asymmetries and multiple extrema in the variations of the components.A discussion of observations corroborating the theory will appear separately.     

Magnetic flux tube in a stratified atmosphere under the influence of the vertical magnetic field

The magnetohydrostatic equilibrium of a magnetic flux tube in a homogeneous gravitational and vertical magnetic field is studied. Gas pressure and density are presented explicitly as a function of the external magnetic field. The influence of the external magnetic field on the magnetic and thermodynamic structures is illustrated by two exact solutions. The possible applications to sunspot and facular modeling are discussed.Work done at the Space Environment Laboratory, NOAA/ERL, Boulder, CO 80303, U.S.A.     

Evolution of emerging force-free flux tube to be a flare loop

Solar Physics - Evolution of a filamentary magnetic flux tube emerging from the photosphere is investigated in the assumption that the magnetic field is force-free and unchanged during the...     

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.     

Numerical simulation of reconnection in an emerging magnetic flux region

The resistive MHD equations are numerically solved in two dimensions for an initial-boundary-value problem which simulates reconnection between an emerging magnetic flux region and an overlying coronal magnetic field. The emerging region is modelled by a cylindrical flux tube with a poloidal magnetic field lying in the same plane as the external, coronal field. The plasma betas of the emerging and coronal regions are 1.0 and 0.1, respectively, and the magnetic Reynolds number for the system is 2 × 10³. At the beginning of the simulation the tube starts to emerge through the base of the rectangular computational domain, and, when the tube is halfway into the computational domain, its position is held fixed so that no more flux of plasma enters through the base. Because the time-scale of the emergence is slower than the Alfvén time-scale, but faster than the reconnection time-scale, a region of closed loops forms at the base. These loops are gradually opened and reconnected with the overlying, external magnetic field as time proceeds.The evolution of the plasma can be divided into four phases as follows: First, an initial, quasi-steady phase during which most of the emergence is completed. During this phase, reconnection initially occurs at the slow rate predicted by the Sweet model of diffusive reconnection, but increases steadily until the fast rate predicted by the Petschek model of slow-shock reconnection is approached. Second, an impulsive phase with large-scale, super-magnetosonic flows. This phase appears to be triggered when the internal mechanical equilibrium inside the emerging flux tube is upset by reconnection acting on the outer layers of the flux tube. During the impulsive phase most of the flux tube pinches off from the base to form a cylindrical magnetic island, and temporarily the reconnection rate exceeds the steady-state Petschek rate. (At the time of the peak reconnection rate, the diffusion region at the X-line is not fully resolved, and so this may be a numerical artifact.) Third, a second quasi-steady phase during which the magnetic island created in the impulsive phase is slowly dissipated by continuing, but low-level, reconnection. And fourth, a static, non-evolving phase containing a potential, current-free field and virtually no flow.During the short time in the impulsive phase when the reconnection rate exceeds the steady-state Petschek rate, a pile-up of magnetic flux at the neutral line occurs. At the same time the existing Petschek-slow-mode shocks are shed and replaced by new ones; and, for a while, both new and old sets of slow shocks coexist.     

Current-driven plasma turbulence in a magnetic flux tube

The behaviour of a multi-component anisotropic plasma in a magnetic flux tube is studied in the presence of current-driven electrostatic ion-cyclotron turbulence. The plasma transport is considered in both parallel and perpendicular directions with respect to the given tube. As one of the sources of the parallel electric field, the anomalous resistivityof the plasma caused by the turbulence is taken into account. The acceleration and heating processes of the plasma are simulated numerically. It is found that at the upper boundary of the nightside auroral ionosphere, the resonant wave-particle interactions are most effective in the case of upward field-aligned currents with densities of a few 10^—6 A/m². The occurring anomalous resistivity maycause differences of the electric potential along the magnetic field lines of some kV. Further it is shown that the thickness of the magnetic flux tube and the intensity of the convection strongly influence the turbulent plasma heating.     

Radiative transfer across a magnetic flux tube

In the coming years, some solar telescopes will be able to yield the Stokes' parameters of polarized light with a resolution better than 1 arc sec (0.3 arc sec for THEMIS). We have simulated the Stokes' parameters of a solar magnetic flux tube as seen with such a resolution. We have shown that, observing with a line-of-sight not parallel to the axis of the flux tube (assumed vertical and axisymmetric), it is possible to see differences between different configurations of the magnetic field inside the flux tube (presence, and in what direction, of an azimuthal component of the field). Furthermore, along such a line-of-sight, the polarization profiles of any atomic line are strongly absorbed at the line center. We then suggest a strategy to infer the structure of the magnetic field from observations at high spatial resolution.