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.     

External and internal solutions for the twisted,flux-tube,prominence model

In this paper the twisted flux-tube model for the support of a prominence sheet with constant axial current density, given by Ridgway, Priest, and Amari (1991), is considered.The model is extended in Section 2 to incorporate a current sheet of finite height. The sheet is supported in a constant current density force-free field in the configuration of a twisted flux tube. The mass of the prominence sheet, using a typical height and field strength, is computed. Outside the flux tube the background magnetic field is assumed to be potential but the matching of the flux tube onto this background field is not considered here.Instead our attention is focussed, in Section 3, on the interior of the prominence. An expanded scale is used to stretch the prominence sheet to a finite width. We analytically select solutions for the internal magnetic field in this region which match smoothly onto the external force-free solutions at the prominence edge.The force balance equation applied inside the prominence then yields expressions for the pressure and density and a corresponding temperature may be computed.     

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.     

A New Model for Propagating Parts of EIT Waves: A Current Shell in a CME

EIT waves are observed in EUV as bright fronts. Some of these bright fronts propagate across the solar disk. EIT waves are all associated with a flare and a CME and are commonly interpreted as fast-mode magnetosonic waves. Propagating EIT waves could also be the direct signature of the gradual opening of magnetic field lines during a CME. We quantitatively addressed this alternative interpretation. Using two independent 3D MHD codes, we performed nondimensional numerical simulations of a slowly rotating magnetic bipole, which progressively result in the formation of a twisted magnetic flux tube and its fast expansion, as during a CME. We analyse the origins, the development, and the observability in EUV of the narrow electric currents sheets that appear in the simulations. Both codes give similar results, which we confront with two well-known SOHO/EIT observations of propagating EIT waves (7 April and 12 May 1997), by scaling the vertical magnetic field components of the simulated bipole to the line of sight magnetic field observed by SOHO/MDI and the sign of helicity to the orientation of the soft X-ray sigmoids observed by Yohkoh/SXT. A large-scale and narrow current shell appears around the twisted flux tube in the dynamic phase of its expansion. This current shell is formed by the return currents of the system, which separate the twisted flux tube from the surrounding fields. It intensifies as the flux tube accelerates and it is co-spatial with weak plasma compression. The current density integrated over the altitude has the shape of an ellipse, which expands and rotates when viewed from above, reproducing the generic properties of propagating EIT waves. The timing, orientation, and location of bright and faint patches observed in the two EIT waves are remarkably well reproduced. We conjecture that propagating EIT waves are the observational signature of Joule heating in electric current shells, which separate expanding flux tubes from their surrounding fields during CMEs or plasma compression inside this current shell. We also conjecture that the bright edges of halo CMEs show the plasma compression in these current shells.     

Sweeping-magnetic-twist mechanism for the acceleration of jets in the solar atmosphere

A magnetodynamic mechanism for the acceleration of jets in the solar atmosphere (surges, Brueckner's EUV jets, and so on) is proposed, and a 2.5-dimensional MHD simulation is performed to show how this mechanism operates in the situation of the chromosphere-corona region of the solar atmosphere. It is seen from the result of simulation that together with the release of the magnetic twist, e.g., into a reconnected open flux tube, the mass in the high density twisted loop is driven out into the open flux tube due both to the pinch effect progressing with the packet of the magnetic twist into the open flux tube, and to the j × B force at the front of the packet of the unwinding twist in the off-axis part of the tube. The former, the progressing pinch, is accompanied by an accelerated hot blob, while the latter, the unwinding front of the magnetic twist, drives a cool cylindrical flow, both with velocities of the order of the local Alfvén velocity. One of the characteristic properties of the jet in our model is that the jet, consisting of hot core and cool sheath, has a helical velocity field in it, explaining the thus-far unexplained observed feature.The sudden release of the magnetic twist into an open flux tube is most likely to be due to the reconnection between a twisted loop and the open flux tube. The mass is driven out in the relaxation process of the magnetic twist from the twisted loop to the open flux tube.     

Convective intensification of magnetic fields in the quiet Sun

Kilogauss-strength magnetic fields are often observed in intergranular lanes at the photosphere in the quiet Sun. Such fields are stronger than the equipartition field B _e, corresponding to a magnetic energy density that matches the kinetic energy density of photospheric convection, and comparable with the field B _p that exerts a magnetic pressure equal to the ambient gas pressure. We present an idealized numerical model of three-dimensional compressible magnetoconvection at the photosphere, for a range of values of the magnetic Reynolds number. In the absence of a magnetic field, the convection is highly supercritical and characterized by a pattern of vigorous, time-dependent, 'granular' motions. When a weak magnetic field is imposed upon the convection, magnetic flux is swept into the convective downflows where it forms localized concentrations. Unless this process is significantly inhibited by magnetic diffusion, the resulting fields are often much greater than B _e and the high magnetic pressure in these flux elements leads to their being partially evacuated. Some of these flux elements contains ultraintense magnetic fields that are significantly greater than B _p. Such fields are contained by a combination of the thermal pressure of the gas and the dynamic pressure of the convective motion, and they are constantly evolving. These ultraintense fields develop owing to non-linear interactions between magnetic fields and convection; they cannot be explained in terms of 'convective collapse' within a thin flux tube that remains in overall pressure equilibrium with its surroundings.     

Dynamical behavior of thermal protons in the mid-latitude ionosphere and magnetosphere

Satellite and other observations have shown that H⁺ densities in the mid-latitude topside ionosphere are greatly reduced during magnetic storms when the plasmapause and magnetic field convection move to relatively low L-values. In the recovery phase of the magnetic storm the convection region moves to higher L-values and replenishment of H⁺ in the empty magnetospheric field tubes begins. The upwards flow of H⁺, which arises from O⁺—H charge exchange, is initially supersonic. However, as the field tubes fill with plasma, a shock front moves downwards towards the ionosphere, eventually converting the upwards flow to subsonic speeds. The duration of this supersonic recovery depends strongly on the volume of the field tube; for example calculations indicate that for L = 5 the time is approximately 22 hours. The subsonic flow continues until diffusive equilibrium is reached or a new magnetic storm begins. Calculations of the density and flux profiles expected during the subsonic phase of the recovery show that diffusive equilibrium is still not reached after an elapsed time of 10 days and correspondingly there is still a net loss of plasma from the ionosphere to the magnetosphere at that time. This slow recovery of the H⁺ density and flux patterns, following magnetic storms, indicates that the mid-latitude topside ionosphere may be in a continual dynamic state if the storms occur sufficiently often.     

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.     

Heating of coronal loops by fast mode MHD waves

A possible mechanism for the formation and heating of coronal loops through the propagation and damping of fast mode waves is proposed and studied in detail. Loop-like field structures are represented by a dipole field with the point dipole at a given distance below the solar surface. The density of the medium is determined by hydrostatic equilibrium along the field lines in an isothermal atmosphere. The fast mode waves propagating outward from the coronal base are refracted into regions with a low Alfvén speed and suffer collisionless damping when the gas pressure becomes comparable to the magnetic pressure. The propagation and damping of these waves are studied for three different cases: a uniform density at the coronal base, a density depletion within a given flux tube, and a density enhancement within a given flux tube. The fast mode waves are found to be important in the formation and heating of the loops if the wave energy flux density is of the order 10⁵ ergs cm^-2 s^-1 at the coronal base.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

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...     

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.     

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.     

Topology and current ribbons: A model for current,reconnection and flaring in a complex,evolving corona

Magnetic field enters the corona from the interior of the Sun through isolated magnetic features on the solar surface. These features correspond to the tops of submerged magnetic flux tubes, and coronal field lines often connect one flux tube to another, defining a pattern of inter-linkage. Using a model field, in which flux tubes are represented as point magnetic charges, it is possible to quantify this inter-linkage. If the coronal field were current-free then motions of the magnetic features would change the inter-linkage through implicit (vacuum) magnetic reconnection. Without reconnection the conductive corona develops currents to avoid changing the flux linkage. This current forms singular layers (ribbons) flowing along topologically significant field lines called separators. Current ribbons store magnetic energy as internal stress in the field: the amount of energy stored is a function of the flux tube displacement. To explore this process we develop a model called the minimum-current corona (MCC) which approximates the current arising on a separator in response to displacement of photospheric flux. This permits a model of the quasi-static evolution of the corona above a complex active region. We also introduce flaring to rapidly change the flux inter-linkage between magnetic features when the internal stress on a separator becomes too large. This eliminates the separator current and releases the energy stored by it. Implementation of the MCC in two examples reveals repeated flaring during the evolution of simple active regions, releasing anywhere from 10²⁷–10²⁹ ergs, at intervals of hours. Combining the energy and frequency gives a general expression for heat deposition due to flaring (i.e., reconnection).     

Is Null-Point Reconnection Important for Solar Flux Emergence?

The role of null-point reconnection in a three-dimensional numerical magnetohydrodynamic (MHD) model of solar emerging flux is investigated. The model consists of a twisted magnetic flux tube rising through a stratified convection zone and atmosphere to interact and reconnect with a horizontal overlying magnetic field in the atmosphere. Null points appear as the reconnection begins and persist throughout the rest of the emergence, where they can be found mostly in the model photosphere and transition region, forming two loose clusters on either side of the emerging flux tube. Up to 26 nulls are present at any one time, and tracking in time shows that there is a total of 305 overall, despite the initial simplicity of the magnetic field configuration. We find evidence for the reality of the nulls in terms of their methods of creation and destruction, their balance of signs, their long lifetimes, and their geometrical stability. We then show that due to the low parallel electric fields associated with the nulls, null-point reconnection is not the main type of magnetic reconnection involved in the interaction of the newly emerged flux with the overlying field. However, the large number of nulls implies that the topological structure of the magnetic field must be very complex and the importance of reconnection along separators or separatrix surfaces for flux emergence cannot be ruled out.     

The current sheet and Joule heating of a slender magnetic tube in the upper photosphere

Joule heating in a slender magnetic flux tube is investigated. The distribution of the magnetic field and electric sheet current encircling a vertical cylindrical magnetic tube is determined by equating the converging magnetic flux, which results from the converging and downward flow of the granulation, and the dissipative expanding magnetic flux due to Ohmic decay. Here, to ensure the mass flux conservation, an overshooting convective flow pattern resembling recent simulations was assumed. Even with the electrical resistivity from neutral hydrogen, the width of the current sheet was found to be 2 km, being much smaller than the tube diameter of 150 km, either from an exact or approximate (Gaussian) field distribution.The resultant energy flux density due to Joule heating averaged over the cylindrical cross sectional area, is 1 × 10⁹ erg cm^-2 s^-1 for an assumed photospheric magnetic field of 1500 G. This amount may supply enough energy to heat the temperature minimum region of the flux tube by T = 300 K in accord with observations, though our estimation of the excess radiation loss which should be supplied by the Joule heating to keep T = 300 K is rather uncertain.A possible role of the Joule heating on spicule formation is briefly discussed together with discussions on the slab geometry, general flow patterns, and non-constant field distributions inside the flux tube.     

A multiwavelength study of an M-class flare and the origin of an associated eruption from NOAA AR 11045

In this paper, we study multiwavelength observations of an M6.4 flare in Active Region NOAA 11045 on 7 February 2010. The space- and ground-based observations from STEREO, SoHO/MDI, EIT, and Nobeyama Radioheliograph were used for the study. This active region rapidly appeared at the north-eastern limb with an unusual emergence of a magnetic field. We find a unique observational signature of the magnetic field configuration at the flare site. Our observations show a change from dipolar to quadrapolar topology. This change in the magnetic field configuration results in its complexity and a build-up of the flare energy. We did not find any signature of magnetic flux cancellation during this process. We interpret the change in the magnetic field configuration as a consequence of the flux emergence and photospheric flows that have opposite vortices around the pair of opposite polarity spots. The negative-polarity spot rotating counterclockwise breaks the positive-polarity spot into two parts. The STEREO-A 195 Å and STEREO-B 171 Å coronal images during the flare reveal that a twisted flux tube expands and erupts resulting in a coronal mass ejection (CME). The formation of co-spatial bipolar radio contours at the same location also reveals the ongoing reconnection process above the flare site and thus the acceleration of non-thermal particles. The reconnection may also be responsible for the detachment of a ring-shaped twisted flux tube that further causes a CME eruption with a maximum speed of 446 km/s in the outer corona.     

Development of structure in pores and sunspots: flows around axisymmetric magnetic flux tubes

Flux elements, pores and sunspots form a family of magnetic features observed at the solar surface. As a first step towards developing a fully non-linear model of the structure of these features and of the dynamics of their interaction with solar convection, we conduct numerical experiments on idealized axisymmetric flux tubes in a compressible convecting atmosphere in cylindrical boxes of radius up to 8 times their depth. We find that the magnetic field strength of the flux tubes is roughly independent of both distance from the centre and the total flux content of the flux tube, but that the angle of inclination from the vertical of the field at the edge of the tube increases with flux content. In all our calculations, fluid motion converges on the flux tube at the surface. The results compare favourably with observations of pores; in contrast, large sunspots lie at the centre of an out-flowing moat cell. We conjecture that there is an inflow hidden beneath the penumbrae of large spots, and that this inflow is responsible for the remarkable longevity of such features.     

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.     

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.     

The Importance of Photospheric Intense Flux Tubes for Coronal Heating

Dissipation of magnetic energy in the corona requires the creation of very fine scale-lengths because of the high magnetic Reynolds number of the plasma. The formation of current sheets is a natural possible solution to this problem and it is now known that a magnetic field that is stressed by continous photospheric motions through a series of equilibria can easily form such sheets. Furthermore, in a large class of 3D magnetic fields without null points there are locations, called quasi-separatrix layers (QSLs), where the field-line linkage changes drastically. They are the relevant generalisation of normal separatrices to configurations without nulls: along them concentrated electric currents are formed by smooth boundary motions and 3D magnetic reconnection takes place when the layers are thin enough. With a homogenous normal magnetic field component at the boundaries, the existence of thin enough QSL to dissipate magnetic energy rapidly requires that the field is formed by flux tubes that are twisted by a few turns. However, the photospheric field is not homogeneous but is fragmented into a large number of thin flux tubes. We show that such thin tubes imply the presence of a large number of very thin QSLs in the corona. The main parameter on which their presence depends is the ratio between the magnetic flux located outside the flux tubes to the flux inside. The thickness of the QSLs is approximately given by the distance between neighbouring flux tubes multiplied by the ratio of fluxes to a power between two and three (depending on the density of flux tubes). Because most of the photospheric magnetic flux is confined in thin flux tubes, very thin QSLs are present in the corona with a thickness much smaller than the flux tube size. We suggest that a turbulent resistivity is triggered in a QSL, which then rapidly evolves into a dynamic current sheet that releases energy by fast reconnection at a rate that we estimate to be sufficient to heat the corona. We conclude that the fragmentation of the photospheric magnetic field stimulates the dissipation of magnetic energy in the corona.