Line-tied Conditions Research Articles

Gravitational instability of solar prominence threads

Context. Prominence threads are dense and cold structures lying on curved magnetic fields that can be suspended in the solar atmosphere against gravity. Aims. The gravitational stability of threads, in the absence of non-ideal effects, is comprehensively investigated in the present work by means of an elementary but effective model. Methods. Based on purely hydrodynamic equations in one spatial dimension and applying line-tying conditions at the footpoints of the magnetic field lines, we derive analytical expressions for the different feasible equilibria (se) and the corresponding frequencies of oscillation (ω). Results. We find that the system allows for stable and unstable equilibrium solutions subject to the initial position of the thread (s0), its density contrast (ρt) and length (lt), and the total length of the magnetic field lines (L). The transition between the two types of solutions is produced at specific bifurcation points that have been determined analytically in some particular cases. When the thread is initially at the top of the concave magnetic field, that is at the apex, we find a supercritical pitchfork bifurcation, while for a shifted initial thread position with respect to this point the symmetry is broken and the system is characterised by an S-shaped bifurcation. Conclusions. The plain results presented in this paper shed new light on the behaviour of threads in curved magnetic fields under the presence of gravity and help to interpret more complex numerical magnetohydrodynamics simulations about similar structures.

Critical decay index for eruptions of ‘short’ filaments

ABSTRACT Model of a partial current-carrying torus loop anchored to the photosphere is analysed. Conditions of the catastrophic loss of equilibrium are considered and corresponding value of the critical decay index of external magnetic field is found. Taking into account line-tying conditions leads to non-monotonous dependence of the critical decay index on the height of the apex and length of the flux rope (its endpoints separation). For relatively short flux ropes, the critical decay index is significantly lower than unity, which is in contrast to widespread models with the typical critical decay index above unity. The steep decrease of the critical index with height at low heights is due to the sharp increase of the curvature of the flux-rope axis that transforms from a nearly straight line to a crescent.

Parametric study on kink instabilities of twisted magnetic flux ropes in the solar atmosphere

Aims. Twisted magnetic flux ropes (MFRs) in the solar atmosphere have been researched extensively because of their close connection to many solar eruptive phenomena, such as flares, filaments, and coronal mass ejections (CMEs). In this work, we performed a set of 3D isothermal magnetohydrodynamic (MHD) numerical simulations, which use analytical twisted MFR models and study dynamical processes parametrically inside and around current-carrying twisted loops. We aim to generalize earlier findings by applying finite plasma β conditions. Methods. Inside the MFR, approximate internal equilibrium is obtained by pressure from gas and toroidal magnetic fields to maintain balance with the poloidal magnetic field. We selected parameter values to isolate best either internal or external kink instability before studying complex evolutions with mixed characteristics. We studied kink instabilities and magnetic reconnection in MFRs with low and high twists. Results. The curvature of MFRs is responsible for a tire tube force due to its internal plasma pressure, which tends to expand the MFR. The curvature effect of toroidal field inside the MFR leads to a downward movement toward the photosphere. We obtain an approximate internal equilibrium using the opposing characteristics of these two forces. A typical external kink instability totally dominates the evolution of MFR with infinite twist turns. Because of line-tied conditions and the curvature, the central MFR region loses its external equilibrium and erupts outward. We emphasize the possible role of two different kink instabilities during the MFR evolution: internal and external kink. The external kink is due to the violation of the Kruskal-Shafranov condition, while the internal kink requires a safety factor q = 1 surface inside the MFR. We show that in mixed scenarios, where both instabilities compete, complex evolutions occur owing to reconnections around and within the MFR. The S-shaped structures in current distributions appear naturally without invoking flux emergence. Magnetic reconfigurations common to eruptive MFRs and flare loop systems are found in our simulations.

Laboratory study of low-β forces in arched, line-tied magnetic flux ropes

The loss-of-equilibrium is a solar eruption mechanism whereby a sudden breakdown of the magnetohydrodynamic force balance in the Sun's corona ejects a massive burst of particles and energy into the heliosphere. Predicting a loss-of-equilibrium, which has more recently been formulated as the torus instability, relies on a detailed understanding of the various forces that hold the pre-eruption magnetic flux rope in equilibrium. Traditionally, idealized analytical force expressions are used to derive simplified eruption criteria that can be compared to solar observations and modeling. What is missing, however, is a validation that these idealized analytical force expressions can be applied to the line-tied, low-aspect-ratio conditions of the corona. In this paper, we address this shortcoming by using a laboratory experiment to study the forces that act on long-lived, arched, line-tied magnetic flux ropes. Three key force terms are evaluated over a wide range of experimental conditions: (1) the upward hoop force; (2) the downward strapping force; and (3) the downward toroidal field tension force. First, the laboratory force measurements show that, on average, the three aforementioned force terms cancel to produce a balanced line-tied equilibrium. This finding validates the laboratory force measurement techniques developed here, which were recently used to identify a dynamic toroidal field tension force that can prevent flux rope eruptions [Myers et al., Nature 528, 526 (2015)]. The verification of magnetic force balance also confirms the low-β assumption that the plasma thermal pressure is negligible in these experiments. Next, the measured force terms are directly compared to corresponding analytical expressions. While the measured and analytical forces are found to be well correlated, the low-aspect-ratio, line-tied conditions in the experiment are found to both reduce the measured hoop force and increase the measured tension force with respect to analytical expectations. These two co-directed effects combine to generate laboratory flux rope equilibria at lower altitudes than are predicted analytically. Such considerations are expected to modify the loss-of-equilibrium eruption criteria for analogous flux ropes in the solar corona.

Torsional Alfvén waves: magneto-seismology in static and dynamic coronal plasmas

Aims. We study the properties of torsional Alfven waves in coronal loops so that they may be exploited for coronal seismological applications. Methods. The governing equation is obtained for standing torsional Alfven waves of a dynamic, gravitationally stratified plasma. The footpoints are assumed to obey line-tying conditions necessary for standing oscillations. Solutions are found in a number of different but typical scenarios to demonstrate the possibilities for both temporal and spatial magneto-seismology exploitation of waveguides with the standing torsional Alfven oscillations. Results. It is found that the frequency of the standing Alfven oscillation increases as the stratification of the plasma increases. The ratio of the periods of the fundamental mode and the first overtone is also found to change as the stratification of the plasma increases. Further, the eigenfunctions of the higher overtones of the standing oscillations are found to experience a shift of their anti-nodes. The influence of a dynamic plasma on the amplitudes of the mode is also investigated. The amplitude of the torsional Alfven mode is found to increase as the plasma within the coronal loop experiences cooling.

Two-dimensional axisymmetric and three-dimensional helical equilibrium in the line-tied screw pinch

The line-tying condition at a conducting anode is shown to provide a localized modification to the well-understood 1-D screw pinch equilibrium in the presence of bulk plasma diamagnetism. Diamagnetic currents cannot flow near the conducting anode and are measured to disappear in a localized boundary layer, causing a weak mirror configuration that breaks 1-D equilibrium and causes large parallel pressure gradients suggestive of significant radial outflows. For sufficiently large plasma currents, the paramagnetic nature of parallel current drives the equilibrium to paramagnetism and destroys the mirror effect. At a critical plasma current, the axisymmetric equilibrium is found to transition to a long-lived, rotating, helical 3-D equilibrium state. Internal measurements of this state via multi-point correlation analysis techniques illustrate that it preserves the flux surfaces and pressure profile of the axisymmetric equilibrium. Measurements indicate that despite the fact that the flux surfaces wander at the anode, the line-tied boundary conditions are not necessarily violated.

On the Properties of Low‐β Magnetohydrodynamic Waves in Curved Coronal Fields

The solar corona is a complex magnetic environment where several kinds of waves can propagate. In this work, the properties of fast, Alfven, and slow magnetohydrodynamic waves in a simple curved structure are investigated. We consider the linear regime, i.e., small-amplitude waves. We study the time evolution of impulsively generated waves in a coronal arcade by solving the ideal magnetohydrodynamic equations. We use a numerical code specially designed to solve these equations in the low-beta regime. The results of the simulations are compared with the eigenmodes of the arcade model. Fast modes propagate nearly isotropically through the whole arcade and are reflected at the photosphere, where line-tying conditions are imposed. On the other hand, Alfven and slow perturbations are very anisotropic and propagate along the magnetic field lines. Because of the different physical properties in different field lines, there is a continuous spectrum of Alfven and slow modes. Curvature can have a significant effect on the properties of the waves. Among other effects, it considerably changes the frequency of oscillation of the slow modes and enhances the possible dissipation of the Alfven modes due to phase mixing.

A Three‐dimensional Line‐tied Magnetic Field Model for Solar Eruptions

We introduce a three-dimensional analytical model of a coronal flux rope with its ends embedded in the solar surface. The model allows the flux rope to move in the corona while maintaining line-tied conditions at the solar surface. These conditions ensure that the normal component of the coronal magnetic field at the surface remains fixed during an eruption and that no magnetic energy enters the corona through the surface to drive the eruption. The model is based on the magnetic configuration of Titov & Demoulin, where a toroidal flux rope is held in equilibrium by an overlying magnetic arcade. We investigate the stability of this configuration to specific perturbations and show that it is subject to the torus instability when the flux rope length exceeds a critical value. A force analysis of the configuration shows that flux ropes are most likely to erupt in a localized region near the apex, while the regions near the surface remain relatively undisturbed. Thus, the flux rope will tend to form an aneurysm-like structure once it erupts. Our analysis also suggests how the flux rope rotation seen in some eruptions and simulations may be related to the observed orientation of the overlying arcade field. This model exhibits the potential for catastrophic loss of equilibrium as a possible trigger for eruptions, but further study is required to prove this property.

Equilibrium and observational properties of line-tied twisted flux tubes

We describe a new explicit three-dimensional magnetohydrodymanic code, which solves the standard zero-β MHD equations in Cartesian geometry, with line-tied conditions at the lower boundary and open conditions at the other ones. Using this code in the frame of solar active regions, we simulate the evolution of an initially potential and concentrated bipolar magnetic field, subject to various sub-Alfvénic photospheric twisting motions which preserve the initial photospheric vertical magnetic field. Both continuously driven and relaxation runs are performed. Within the numerical domain, a steep equilibrium curve is found for the altitude of the apex of the field line rooted in the vortex centers as a function of the twist. Its steepness strongly depends on the degree of twist in outer field lines rooted in weak field regions. This curve fits the analytical expression for the asymptotic behaviour of force-free fields of spherical axisymmetric dipoles subject to azimuthal shearing motions, as well as the curve derived for other line-tied twisted flux tubes reported in previous works. This suggests that it is a generic property of line-tied sheared/twisted arcades. However, contrary to other studies we never find a transition toward a non-equilibrium within the numerical domain, even for twists corresponding to steep regions of the equilibrium curve. The calculated configurations are analyzed in the frame of solar observations. We discuss which specific conditions are required for the steepness of the generic equilibrium curve to result in dynamics which are typical of both fast and slow CMEs observed below 3 . We provide natural interpretations for the existence of asymmetric and multiple concentrations of electric currents in homogeneoulsy twisted sunspots, due to the twisting of both short and long field lines. X-ray sigmoïds are reproduced by integrating the Joule heating term along the line-of-sight. These sigmoïds have inverse-S shapes associated with negative force-free parameters α, which is consistent with observed rules in the northern solar hemisphere. We show that our sigmoïds are not formed in the main twisted flux tube, but rather in an ensemble of low-lying sheared and weakly twisted field lines, which individually never trace the whole sigmoïd, and which barely show their distorded shapes when viewed in projection. We find that, for a given bipolar configuration and a given twist, neither the α nor the altitude of the lines whose envelope is a sigmoïd depends on the vortex size.

On the formation of current sheets in the solar corona

view Abstract Citations (42) References (14) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS On the Formation of Current Sheets in the Solar Corona Karpen, Judith T. ; Antiochos, Spiro K. ; DeVore, C. Richard Abstract Several theoretical studies have proposed that, in response to photospheric footpoint motions, current sheets can be generated in the solar corona without the presence of a null point in the initial potential magnetic field. A fundamental assumption in these analyses, commonly referred to as the line-tying assumption, is that all coronal field lines are anchored to a boundary surface representing the top of the dense, gas pressure-dominated photosphere. It is shown here that line-typing cannot be applied indiscriminately to dipped coronal fields, and that the conclusions of the line-tied models are incorrect. To support the theoretical arguments, the response of a dipped potential magnetic field in a hydrostatic-equilibrium atmosphere to shearing motions of the footpoints is studied, using a 2.5-dimensional MHD code. The results show that, in the absence of artificial line-tying conditions, a current sheet indeed does not form at the location of the dip. Rather, the dipped magnetic field rises, causing upflows of photospheric and chromospheric plasma. Publication: The Astrophysical Journal Pub Date: June 1990 DOI: 10.1086/185751 Bibcode: 1990ApJ...356L..67K Keywords: Current Sheets; Solar Corona; Solar Magnetic Field; Chromosphere; Magnetic Field Configurations; Mathematical Models; Photosphere; Solar Physics; HYDROMAGNETICS; SUN: CORONA; SUN: MAGNETIC FIELDS full text sources ADS |

The Evolution of a Sheared Potential Magnetic Field in the Solar Corona

Several theoretical studies have proposed that, in response to photospheric foot-point motions, current sheets can be generated in the solar corona without the presence of a null point in the initial potential magnetic field. In these analytic models, current sheets form wherever the coronal field dips down and is parallel to the photosphere. A fundamental assumption in these analyses — commonly referred to as the line-tying assumption — is that all coronal field lines are anchored to a boundary surface representing the top of the dense, gas-pressure-dominated photosphere. In theoretical arguments presented elsewhere (Karpen, Antiochos, and DeVore 1989), however, we show that line-tying is not valid for “dipped” coronal fields, and hence that the conclusions of the line-tied models are incorrect. We contend that current sheets will not form if the photosphere-corona interface is represented by a physically valid model. Here we summarize a numerical investigation of the response of a “dipped” potential magnetic field in a hydrostatic-equilibrium atmosphere to shearing motions of the foot points. Our results show that, in the absence of artificial line-tying conditions, a current sheet indeed does not form at the location of the dip. Rather, the dipped magnetic field rises, causing upflows of photospheric and chromospheric plasma.

The ideal magnetohydrodynamic stability of a line-tied coronal magnetohydrostatic equilibrium

An energy method is used to determine a condition for local instability of field lines in magnetohydrostatic equilibrium which are rooted in the photosphere. The particular equilibrium studied is isothermal and two-dimensional and may model a coronal arcade of loops where variations along the axis of the arcade are weak enough to be ignorable. If line tying conditions are modelled by perturbations that vanish on the photosphere, then, when the field is unsheared, the condition for stability is necessary and sufficient. However, when the axial field component is non-zero, so that the field is sheared, the stability condition is only sufficient. It is found that when β 0.34 a magnetic island is present and all the field lines inside the island are unstable as well as some beyond it. As β increases, the size of the island and the extent of unstable field lines increase. The effect of the instability is likely to be to create small-scale filamentation in the solar corona and to enhance the global transport coefficients.

Photospheric line-tying conditions for the MHD stability of coronal magnetic fields

Using the localised, ballooning ordering, the effect of a density stratification on the ideal MHD stability of magnetic fields is investigated. It is found that, when the photospheric density is very much greater than the coronal value, the line tying conditons are best simulated by assuming that all coronal disturbances vanish at the photospheric boundary. This is commonly known as the rigid wall conditions.