Thermal equilibria of coronal magnetic arcades

A coronal magnetic arcade can be thought of as consisting of an assembly of coronal loops. By solving equations of thermal equilibrium along each loop and assuming a base temperature of 2 × 10⁴ K, the thermal structure of the arcade can be found. By assuming a form for the plasma pressure in the arcade, the possible thermal structures can be shown to depend on three parameters. Arcades can contain hot loops with summits hotter than 400 000 K, cool loops at temperatures less than 80 000 K along their lengths, hot-cool loops with cool summits and cool footpoints but hotter intermediate portions, and warm loops, cooler than 80 000 K along most of their lengths but with summits as hot as 400 000 K. For certain arcades, there exist regions where more than one kind of loop is possible. If the parameters describing the arcade are varied, it is possible for non-equilibrium to occur when a type of solution ceases to exist. For example, hot or warm loops can cease to exist so that only cool solutions are possible when the arcade size or pressure is decreased, while warm or cool loops may give way to hot-cool loops when the heating is reduced or the pressure is increased.     

Thermal equilibria of isobaric coronal magnetic arcades

A coronal magnetic arcade can be thought of as consisting of an assembly of coronal loops. By solving equations of isobaric thermal equilibrium along each loop and assuming a base temperature of 2 × 10⁴ K, the thermal structure of the arcade can be found. The possible thermal equilibria can be shown to depend on two parameters L ^* p ^* and h ^*/p ^* representing the ratios of cooling (radiation) to condu and heating to cooling, respectively. Arcades can contain four types of loops: hot loops with summits hotter than 400000 K; cool loops at temperatures less than 80000 K along their lengths; hot-cool loops with cool summits and cool footpoints but hotter intermediate portions; and warm loops, cooler than 80000 K along most of their lengths but with summits as hot as 400000 K. Two possibilities for coronal heating are considered, namely a heating that is independent of magnetic field and a heating that is proportional to the square of the local magnetic field. When the arcade is sheared the thermal structure of the arcade may change, leading in some cases to non-equilibrium or in other cases to the formation of a cool core.     

The structure of coronal arcades and the formation of solar prominences

The temperature and density are obtained for coronal plasma in thermal and hydrostatic equilibrium and located in a force-free magnetic arcade. The isotherms are found to be inclined to the magnetic field lines and so care should be taken in inferring the magnetic structure from observed emission.When the coronal pressure becomes too great, the equilibrium ceases to exist and the material cools to form a quiescent prominence. The same process can be initiated at low heating rates when the width or shear of the arcade exceeds a critical value.We suggest that the prominence should be modelled as a dynamic structure with plasma always draining downwards. Material is continually sucked up along field lines of the ambient arcade and into the region lacking a hot equilibrium, where it cools to form new prominence material.     

Magnetic instability of coronal arcades as the origin of two-ribbon flares

The generally accepted scenario for the events leading up to a two-ribbon flare is that a magnetic arcade (supporting a plage filament) responds to the slow photospheric motions of its footpoints by evolving passively through a series of (largely) force-free equilibria. At some critical amount of shear the configuration becomes unstable and erupts outwards. Subsequently, the field closes back down in the manner modelled by Kopp and Pneuman (1976); but the main problem has been to explain the eruptive instability.The present paper analyses the magnetohydrodynamic stability of several possible arcade configurations, including the dominant stabilizing effect of line-tying at the photospheric footpoints. One low-lying force-free structure is found to be stable regardless of the shear; also some of the arcades that lie on the upper branch of the equilibrium curves are shown to be stable. However, another force-free configuration appears more likely to represent the preflare structure. It consists of a large flux tube, anchored at its ends and surrounded by an arcade, so that the field transverse to the arcade axis contains a magnetic island. Such a configuration is found to become unstable when either the length of the structure, the twist of the flux tube, or the height of the island becomes too great; the higher the tube is situated, the smaller is the twist required for instability.     

Effect of Flow on Resonant Absorption of Slow MHD Waves in Coronal Arcades

Resonant absorption of slow MHD waves is studied numerically by using the SGH method and is applied to a model of a coronal arcade in the presence of equilibrium plasma flows. The arcade is approximated by a 1D horizontal magnetic slab that is non-uniform along the vertical direction and which is surrounded by two homogeneous media. While propagating from the photosphere upwards into the corona, the magneto-acoustic waves can be resonantly absorbed in the inhomogeneous region of the arcade. Computational results show that the resonant absorption of the impinging waves strongly depends on the equilibrium model and on the characteristics of the driving wave. The results also indicate that the presence of an equilibrium plasma flow along the magnetic field of the arcade reduces the resonant absorption for the flow speed parameters considered.     

Corrugation Instability of a Coronal Arcade

We analyse the behaviour of linear magnetohydrodynamic perturbations of a coronal arcade modelled by a half-cylinder with an azimuthal magnetic field and non-uniform radial profiles of the plasma pressure, temperature, and the field. Attention is paid to the perturbations with short longitudinal (in the direction along the arcade) wavelengths. The radial structure of the perturbations, either oscillatory or evanescent, is prescribed by the radial profiles of the equilibrium quantities. Conditions for the corrugation instability of the arcade are determined. It is established that the instability growth rate increases with decreases in the longitudinal wavelength and the radial wave number. In the unstable mode, the radial perturbations of the magnetic field are stronger than the longitudinal perturbations, creating an almost circularly corrugated rippling of the arcade in the longitudinal direction. For coronal conditions, the growth time of the instability is shorter than one minute, decreasing with an increase in the temperature. Implications of the developed theory for the dynamics of coronal active regions are discussed.     

Formation of solar prominences by photospheric shearing motions

A numerical simulation is performed to investigate the prominence formation in a magnetic arcade by photospheric shearing motions. A two-and-a-half-dimensional magnetohydrodynamic (MHD) code is used, in which the gravitational force, radiative cooling, thermal conduction and a simplified form of coronal heating are included. It is found that a footpoint shear induces an expansion of the magnetic arcade and cooling of the plasma in it. Simultaneously the denser material from the lower part of the arcade is pulled up by the expanding field lines. A local enhancement of radiative cooling is thus effected, which leads to the onset of thermal instability and the condensation of coronal plasma. The condensed material grows vertically to form a sheet-like structure making dips on field lines, leading to the formation of the Kippenhahn- Schlüter type prominence. The mass of the prominence is found to be supplied not only by the condensation of the material in the vicinity but also by the siphon-type upflows. The upward growth of the vertical sheet-structure of the prominence is saturated at a certain stage and the newly condensed material is found to slide down from above the prominence along magnetic field lines. This drainage of material leads to the formation of an arc-shaped cavity of low density and low pressure around the prominence. The problem of force and heat balance is addressed and the prominence is found to be not in a static equilibrium but in a dynamic interaction with its environment.     

Orbits near critical inclination,including lunisolar perturbations

An improved theory is presented of long period perigee motion for orbits near the critical inclinations 63.4° and 116.6°. Inclusion of lunisolar perturbations andall measured zonal harmonic coefficients from a recent Earth model are significant improvements over existing theories. Phase portraits are used to depict the interaction between eccentricity magnitude and argument of perigee. The Hamiltonian constant can be chosen as the parameter to display a family of phase plane trajectories consisting of libration, circulation, and asymptotic motion along separatrices near equilibrium points. A two parameter family of phase portraits is defined by the other two integrals, the average semimajor axis and component of angular momentum resolved along the Earth's polar axis. There are regions of the parameter space where the stability and total number of equilibria can change, or two separatrices can coalesce. These phenomena signal large qualitative changes in phase portrait topology. Numerical studies show that lunisolar perturbations control stability of equilibria for orbits with semimajor axes exceeding 1.4 Earth radii. Moreover, a theory which includes lunisolar perturbations predicts larger maximum fluctuations in eccentricity and faster oscillations near stable equilibria compared to a theory which models only the zonal harmonics.     

The Asymmetrical Eruption of a Quiescent Filament and Associated Halo CME

We will present detailed observations of the asymmetrical eruption of a large quiescent filament on 24 November 2002, which was followed by a two-ribbon flare, three coronal dimmings, endpoint brightenings, and a very fast halo-type coronal mass ejection (CME). Before the eruption, the filament lay along the main neutral line (MNL) underneath a single-arcade helmet streamer with a simple bipolar configuration. However, photospheric magnetic fields on both sides of the filament showed an asymmetrical distribution, and the filament and MNL were not located just at the center of the streamer base but were closer to the eastern leg of the streamer arcade. Therefore, instead of erupting along the streamer’s symmetrical axis, the filament showed a nonradial and asymmetrical eruption. It lifted from the eastern flank of the streamer arcade to impact the western leg directly, leading to an asymmetrical CME that expanded westward; eventually the streamer was disrupted significantly. Accordingly, the opposite-polarity coronal dimmings at both sides of the filament forming in the eruption also showed an asymmetrical area distribution. We thus assume that the streamer arcade could guide the filament at the early eruption phase but failed to restrain it later. Consistent with previous results, these observations suggest that the global background magnetic field can impose additional action on the initial eruption of the filament and CME, as well as the dimming configuration.     

Thermal equilibria of coronal magnetic loops

Equations of thermal equilibrium along coronal loops with footpoint temperatures of 2 × 10⁴ K are solved. Three fundamentally different categories of solution are found, namely hot loops with summit temperatures above about 4 × 10⁵ K, cool loops which are cooler than 8 × 10⁴ K along their whole length and hot-cool loops which have summit temperatures around 2 × 10⁴ K but much hotter parts at intermediate points between the summit and the footpoints. Hot loops correspond to the hot corona of the Sun. The cool loops are of relevance for fibrils, for the cool cores observed by Foukal and also for active-region prominences where the magnetic field is directed mainly along the prominence. Quiescent prominences consist of many cool threads inclined to the prominence axis, and each thread may be modelled as a hot-cool loop. In addition, it is possible for warm loops at intermediate summit temperatures (8 × 10⁴K to 4 × 10⁵ K) to exist, but the observed differential emission measure suggests that most of the plasma in the solar atmosphere is in either the hot phase or the cool phase. Thermal catastrophe may occur when the length or pressure of a loop is so small that the hot solution ceases to exist and there are only cool loop solutions. Many loops can be superimposed to form a coronal arcade which contains loops of several different types.     

Bounds on the ideal MHD stability of line-tied 2-D coronal magnetic fields

A tractable method for investigating the linear stability of line-tied 2-D coronal magnetic fields is introduced. It is based on the Bernstein et al. (1958) energy principle and can be applied to non-isothermal equilibria with gravity, having a translational invariance. The perturbed potential energy integral is manipulated to produce either necessary conditions for stability to localized modes or sufficient conditions for stability to global modes. Each condition only requires the solution of a set of ordinary differential equations, integrated along the magnetic field lines. The tests are employed to two different classes of equilibria. A linear force-free field is shown to be completely stable, regardless of the shear. The role of pressure gradients, footpoint displacements, line-tying and stratification on an isothermal magneto-hydrostatic equilibrium is assessed.     

Simple tests for the ideal MHD stability of line-tied coronal magnetic fields

Methods for investigating the stability of line-tied, cylindrically-symmetric magnetic fields are presented. The energy method is used and the perturbed potential energy integral is manipulated to produce simple tests that predict either stability to general coronal disturbances or instability to localized modes, both satisfying photospheric line-tying. Using these tests the maximum amount of magnetic energy, that can be stored in the coronal magnetic field prior to an instability, can be estimated. The tests are applied to four different classes of equilibria and results are obtained for both arcade and loop geometries.     

Stabilization due to gyroscopic coupling in dual-spin satellites subject to gravitational torques

The stability of attitude equilibria relative to gravitational torques for a rigid satellite in a circular orbit has been divided into three inertia regions, the Lagrange region of assured Liapunov stability, the Beletskii-Delp region which is often described as stabilized due to gyroscopic coupling, and an assured instability region. The generalization of these regions to the case of dual-spin or gyrostat satellites whose internal spin momentum is along a principal axis is treated here. The stability boundaries are obtained for all possible equilibrium orientations for such vehicles, and the variations of these boundaries corresponding to changes in the internal momentum magnitude, or to aligning the momentum with a different principal axis, are determined.Alexander von Humboldt Research Fellow at the Institut für Mechanik; on sabbatical leave from Columbia University, New York, U.S.A.     

A prominence model in a simple magnetic arcade

This paper offers an evolution scenario for a simple magnetic arcade where the frozen-in magnetic field decreases with the ascent of its arches together with the plasma. Uplift is produced by the movement of photospheric plasma with a frozen-in magnetic field, which is divergent with respect to a neutral line. A decrease in magnetic field leads to the appearance in the arcade of a height range of arches, with no high-temperature thermal equilibrium present, and to a variation of the nonequilibrium range with time. Uplift of the arcade is accompanied by the consecutive entry of new arches into this range. All arches entering the nonequilibrium range experience a transient process. Some of the earlier inquiries into the physics of this process were instrumental, in the first place, in identifying those arches which – through the production of an ascending plasma flow from the base of the arcade – are involved in the formation of a prominence (with magnetic dips appearing and evolving at the tops of these arches) and, secondly, in synthesizing a computational algorithm for the final state of the transient process, the quasi-steady-state dynamic structure of the prominence. The arcade evolution scenario, combined with the computational algorithm, constitutes a unified prominence model, a model for the transition from a simple static magnetic arcade to a quasi-steady dynamic prominence structure. The model has been used in numerical calculations of parameters of two classes of prominences: in and outside active regions. Results of the calculations are in good agreement with observations.     

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 < 0.34 the equilibrium is stable. When = 0.34 a magnetic neutral line appears at the photosphere and it is marginally stable. When > 0.34 a magnetic island is present and all the field lines inside the island are unstable as well as some beyond it. As 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.     

Energy equilibrium in a coronal magnetic loop

Temperature distribution in the cylindrically symmetric coronal magnetic loop, (i) with constant pressure and (ii) with the pressure varying along the radial distance, of the (a) hotter apex and (b) cooler apex than base is investigated analytically by considering the equilibrium between the heat conduction and radiation loss. If the temperature of the loop does not lie within one of the specified temperature ranges, then the distribution is calculated numerically.The effect of the inclusion of heating due to an external source is studied and found that it increases the length of the loop. On the basis of the observed phenomenon, that the magnetic field varies along the loop, the temperature distribution in the loop is investigated for the loop-geometries proposed by Antiochos and Sturrock (1976). It is concluded that for the larger compression in the area of cross section, the height of the loop decreases.Present investigation shows that no loop with equal apex and base temperatures can exist, but a small variation between the two temperatures supports the existence of the loop, which can be observed in nature.     

Two-dimensional steady-state pressure structure in a coronal loop

By expressing the magnetic field and fluid velocity in terms of two Chandrasekhar-Kendall functions (n = 0, m = 0; n = 1, m = 0) we investigated the steady-state pressure profile inside a solar coronal loop. For constant density loops, we found a two-dimensional (radial and axial) structure of pressure. This work is the modified version of the work of Krishan (1985). At the base of the loop, the pressure is found to increase steeply outwards along the radius, whereas at the apex it decreases slowly. The radial variation of pressure is found to be minimum around L/5, where L is the length of the loop measured from one foot to another one. But Krishan (1985) found that the rate of increase of pressure at the base was nearly equal to the rate of decrease of pressure at the apex, and the pressure was found nearly constant at L/4. For axial variation, we found that along the loop axis the pressure increases from the base up to z = 3L/8 and then decreases up to the apex, whereas at the surface, the pressure decreases from the base up to the apex. Krishan (1985), however, found the axial variation to be linear.     

MHD equilibrium and stability properties of a bipolar current loop

A study is made of equilibrium and stability properties of a semi-toroidal current loop imbedded in a high temperature plasma. The loop carries a toroidal current density J _t and poloidal current density J _p. By explicity including the global curvature of the loop, the net Lorentz and pressure forces acting along the major radius are calculated. Requirement of equilibrium force-balance gives rise to conditions that must be satisfied by the physical parameters and geometry. On the basis of these conditions, we deduce a class of equilibrium semi-toroidal current loops satisfying c ^#X2212;1 J × B ? ▽p = 0. It is found that the averge pressure inside the loop is less than the ambient coronal pressure in equilibrium. Furthermore, this class of equilibria is shown to be stable to a number of destructive MHD modes. The theoretical results are discussed in the context of solar bipolar current loops.     

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.     

Loss of equilibrium in coronal loops

The loss of equilibrium in coronal magnetic field structures is a possible source of energy for coronal heating and solar flares. We investigate whether such a loss of equilibrium occurs when a coronal loop is progressively twisted by photospheric motions. In studies of 2-D cylindrical equilibria, long loops have been found to be of constant cross-sectional area along most of their length, with axial variations being confined to narrow boundary layers. We use this information to develop a 1-D line-tied model, for a 2-D coronal loop. We specify the twist in terms of the azimuthal field and more physically, in terms of the photospheric footpoint displacement. In the former case we find a loss of equilibrium, but not in the latter. We also examine a twisted loop with a non-zero plasma pressure. The loss of equilibrium is only found at high-plasma . It is conjectured that such high- can occur in flare loops and prior to a prominence eruption. However, when the plasma evolves adiabatically, there is no loss of equilibrium.