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.     

A two-dimensional model for a solar prominence: Effect of an external magnetic field

Using analytical approximations we study the effects of different external magnetic configurations on the half-width, mass, and internal magnetic structure of a quiescent solar prominence, modelled as a thin vertical sheet of cool plasma. Firstly, we build up a zeroth-order model and analyse the effects produced by a potential coronal field or a constant- force-free field. This model allows us to obtain the half-width and mass of the prominence for different values of the external field, pressure and shear angle. Secondly, the effects of these external magnetic configurations on a two-dimensional model proposed by Ballester and Priest (1987) are studied. The main effects are a change of the half-width with height, an increase of the mass, a decrease of the magnetic field strength with height and a change in the shape of the magnetic field lines.     

Helical structures around quiescent solar prominences computed from observable magnetic fields

We analyse the magnetic support of solar prominences in two-dimensional linear force-free fields. A line current is added to model a helical configuration, well suited to trap dense plasma in its bottom part. The prominence is modeled as a vertical mass-loaded current sheet in equilibrium between gravity and magnetic forces.We use a finite difference numerical technique which incorporates both vertical photospheric and horizontal prominence magnetic field measurements. The solution of this mixed boundary problem generally presents singularities at both the bottom and top of the model prominence. The removal of the singularities is achieved by superposition of solutions. Together with the line current equilibrium, these three conditions determine the amplitude of the magnetic field in the prominence, the flux below the prominence and the current intensity, for a given height of the line current. A numerical check of accuracy in the removal of singularities, is done by using known analytical solutions in the potential limit.We have investigated both bipolar and quadrupolar photospheric regions. In this mixed boundary problem the polarity of the field component orthogonal to the prominence is mainly fixed by the imposed height of the line current. For bipolar regions above (respectively below) a critical height the configuration is inverse (respectively normal). For quadrupolar regions the polarity is reversed if we refer the prominence polarity to the closest photospheric polarities. We introduce the polarity of the component parallel to the prominence axis with reference to a sheared arcade. Increasing the shear with fixed boundary conditions can increase or decrease the mass supported depending on the configuration.     

The vertical filamentary structures of quiescent prominences

We present a simple magnetostatic theory of the thin vertical filaments that make up the quiescent prominence plasma as revealed by fine spatial resolution H photographs. A class of exact equilibrium solutions is obtained describing a horizontal row of long vertical filaments whose weights are supported by bowed magnetic field lines. A free function is available to generate different assortments of filament sizes and spacings, as well as different density and temperature variations. The classic Kippenhahn-Schlüter solution for a long sheet without filamentary structures is a particular member of this class of solutions. The role of the magnetic field in supporting and thermally shielding the filament plasma is illustrated. It is found that the filament can have a sharp transition perpendicular to the local field, whereas the transition in the direction of the local field is necessarily diffuse. A consequence of the filamentary structure is that its support by the Lorentz force requires the electric current to have a component along the magnetic field. This electric current flowing into the rarefied region around the prominence can contain substantial energy stored in the form of force-free magnetic fields. This novel feature has implications for the heating and the disruption of prominences.     

A self-consistent model of a thermally balanced quiescent prominence in magnetostatic equilibrium in a uniform gravitational field

We present a theoretical model of quiescent prominences in the form of an infinite vertical sheet. Self-consistent solutions are obtained by integrating simultaneously the set of nonlinear equations of magnetostatic equilibrium and thermal balance. The basic features of the models are: (1) The prominence matter is confined to a sheet and supported against gravity by a bowed magnetic field. (2) The thermal flux is channelled along magnetic field lines. (3) The thermal flux is everywhere balanced by Low's (1975b) hypothetical heat sink which is proportional to the local density. (4) A constant component of the magnetic field along the length of the prominence shields the cool plasma from the hot surrounding. We assume that the prominence plasma emits more radiation than it absorbs from the radiation fields of the photosphere, chromosphere and corona, and we interpret the above hypothetical heat sink to represent the amount of radiative loss that must be balanced by a nonradiative energy input. Using a central density and temperature of 10¹¹ particles cm^–3 and 5000 K respectively, a magnetic field strength between 2 to 10 gauss and a thermal conductivity that varies linearly with temperature, we discuss the physical properties implied by the model. The analytic treatment can also be carried out for a class of more complex thermal conductivities. These models provide a useful starting point for investigating the combined requirements of magnetostatic equilibrium and thermal balance in the quiescent prominence.     

A 2-D model for the support of a polar-crown solar prominence

We present a 2-D potential-field model for the magnetic structure in the environment of a typical quiescent polar-crown prominence. The field is computed using the general method of Titov (1992) in which a curved current sheet, representing the prominence, is supported in equilibrium by upwardly directed Lorentz forces to balance the prominence weight. The mass density of the prominence sheet is computed in this solution using a simple force balance and observed values of the photospheric and prominence magnetic field. This calculation gives a mass density of the correct order of magnitude. The prominence sheet is surrounded by an inverse-polarity field configuration adjacent to a region of vertical, open polar field in agreement with observations.A perturbation analysis provides a method for studying the evolution of the current sheet as the parameters of the system are varied together with an examination of the splitting of an X-type neutral point into a current sheet.Program Systems Institute of the Russian Academy of Sciences, Pereslavl-Zalessky 152140, Russia.     

无力场模型黑子上空的纵向磁场梯度

本文按常α无力场模型计算了1980年10月23日Boulder 2744活动区前导黑子的纵向磁场随高度的变化,并与用CIV 1548谱线观测得到的色球一日冕过渡区的磁场资料相结合,求得CIV 1548发射区的有效高度。这些结果与文献[4]中对同一黑子用势场模型推求的结果有很大差别。从而表明,势场和无力场在某些方面导致的结果是极不相同的。鉴于观测已表明活动区上空存在电流的事实,在活动区磁场的模拟中,特别是在强扭曲活动区磁场的计算中,应当避免采用势场,而尽可能采用无力场模型。     

Magnetic field evolution during prominence eruptions and two-ribbon flares

Simple models for the MHD eruption of a solar prominence are presented, in which the prominence is treated as a twisted magnetic flux tube that is being repelled from the solar surface by magnetic pressure forces. The effects of different physical assumptions to deal with this magneto-hydrodynamically complex phenomenon are evaluated, such as holding constant the prominence current, radius, flux or twist or modelling the prominence as a current sheet. Including a background magnetic field allows the prominence to be in equilibrium initially with an Inverse Polarity and then to erupt due to magnetic non-equilibrium when the background magnetic field is too small or the prominence twist is too great. The electric field at the neutral point below the prominence rapidly increases to a maximum value and then declines. Including the effect of gravity also allows an equilibrium with Normal Polarity to exist. Finally, an ideal MHD solution is found which incorporates self-consistently a current sheet below the prominence and which implies that a prominence will still erupt and form a current sheet even if no reconnection occurs. When reconnection is allowed it is, therefore, driven by the eruption.     

Force-free Magneitic Field Extrapolation for the Complex Sunspot Group of August 1972

Quantitative data on the magnetic field structure at all levels of the solar atmosphere are of basic importance for our understanding of physical processes in active regions in general and in flares in particular. Because photospheric longitudinal magnetograms are the most reliable data available, one has to look for a method of theoretical extrapolation of such data to higher levels. Such a method has been developed for force-free magnetic fields, i.e., ∇ X B = αB , with α = constant, satisfying more realistic boundary conditions as compared with earlier papers; e.g., the net magneitic flux through the magnetogram area is not required to be zero. The method has been used to calculate the magnetic field vector and lines of force in the flare-active region of August 1972. Calculated fields are compared with other observations such as structures in H α . Results of August 3rd show that the loop prominence systems observed during the flares of August 2nd and 4th are represented by a force-free field with positive α rather than by a current-free field ( α = 0) The extractable energy supply of this force-free field is of the order of magnitude of maximum flare demand (10³² erg); the height dependence of the magnetic field strengths agrees with that from radio and X-ray estimates. Similar results are obtained for the August 7th magnetic field structure.     

Numerical models of quiescent normal polarity prominences

We present 2-D numerical models of quiescent solar prominences with normal magnetic polarity. These models represent an extension to the classical Kippenhahn-Schlüter model in that the prominence is treated as having finite width and height and the external coronal field is matched smoothly to the internal prominence field so that there are no current sheets at the prominence sides. Using typical prominence and coronal values we find solutions to the generalised Grad-Shafranov equation which illustrate the necessary magnetic support. We also discuss some extensions to the basic model.     

Magnetic field configurations associated with polarity intrusion in a solar active region

This paper presents a new class of exact solutions describing the non-linear force-free field above a spatially localized photospheric bipolar magnetic region. An essential feature is the variation in all three Cartesian directions and this could not be modelled adequately with previously known symmetric force-free fields. Sequences of force-free fields are constructed and analyzed to simulate the slow growth of a pair of spots on the photosphere. The axis connecting the spots executes rotational motion, distorting the photospheric neutral line separating fluxes of opposite signs. We show directly from the analytic solutions that the resulting reversal of the positions of the spots relative to the background field is associated with (i) the creation of magnetic free energy, (ii) the severe shearing of localized low-lying loops in the vicinity where the photospheric transverse field aligns with the photospheric neutral line, and (iii) the emergence and disappearance of flux from the photosphere at these highly stressed regions. The model relates theoretically for the first time these different magnetic field features that have been suggested by observation and theoretical considerations to be flare precursors. A general formula, based on the virial theorem, is also given for the free energy of a force-free field, strictly in terms of the field value at the photosphere. This formula has obvious practical application.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

A two-dimensional model for a solar prominence

A two-dimensional magnetohydrostatic model of a vertical prominence sheet is set up by allowing slow variations of the magnetic field and plasma properties with height. The width of the prominence is found to decrease with height and in many cases the field lines become less curved, while the strength of the horizontal magnetic field increases with height, in agreement with observations. Since we are only considering a local analysis, the model applies to a general prominence sheet, whether of Kippenhahn-Schlüter or Kuperus-Raadu type. The challenge in the future is to understand the detailed fine-scale microstructure which takes place in the mould formed by the present global macro-models.On leave from: Departament de Fisica, Universitat de les Illes Balears, E-07071, Palma de Mallorca, Spain.     

The structure of a prominence sheet

A one-dimensional analysis of Kippenhahn-Schlüter type is applied to a sheet of prominence material inclined at an angle, to the horizontal. It is found that the magnetic pressure across the prominence no longer has a symmetric profile, but is stronger on the lower side of the sheet. This excess in magnetic pressure is necessary to balance the component of prominence weight in that direction. A matching function is derived and allows for variations along the length of the sheet, enabling the internal prominence solution to be linked onto a given background potential field. In this way a curved prominence sheet in a potential field may be resolved. A smooth profile for the magnetic field and a continuous variation of plasma pressure across the prominence region is then possible. An example is given in which the analysis is applied to a polar-crown prominence configuration of inverse polarity and the basic properties of the prominence are determined.     

A class of force-free magnetic fields for modeling pre-flare coronal magnetic configurations

Three-dimensional, quasi-static evolutions of coronal magnetic fields driven by photospheric flux emergence are modeled by a class of analytic force-free magnetic fields. Our models relate commonly observed photospheric magnetic phenomena, such as the formation and growth of sunspots, the emergence of an X-type separator, and the collision and merging of sunspots, to the three-dimensional magnetic fields in the corona above. By tracking the evolution in terms of a continuous sequence of force-free states, we show that flux emergence and submergence along magnetic neutral lines in the photosphere are essential processes in all these photospheric phenomena. The analytic solutions we present have a parametric regime within which the magnetic energy attained by an evolving force-free field may be of the order of 10³⁰ ergs to several 10³¹ ergs, depending on the magnetic environment into which an emerging flux intrudes. The commonly used indicators of magnetic shear in magnetogram interpretation are discussed in terms of field connectivity in our models. It is demonstrated that the crossing angle of the photospheric transverse magnetic field with the neutral line may not be a reliable indicator of the magnetic shear in the coronal field above, due to the complexity of three-dimensionality. The poorly understood constraint of magnetic-helicity conservation on the availability of magnetic free energy for a flare is briefly discussed.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Calculating and Testing Nonlinear Force-Free Fields

Improvements to an existing method for calculating nonlinear force-free magnetic fields (Wheatland, Solar Phys. 238, 29, 2006) are described. In particular a solution of the 3-D Poisson equation using 2-D Fourier transforms is presented. The improved nonlinear force-free method is demonstrated in application to linear force-free test cases with localized nonzero values of the normal component of the field in the boundary. These fields provide suitable test cases for nonlinear force-free calculations because the boundary conditions involve localized nonzero values of the normal components of the field and of the current density, and because (being linear force-free fields) they have more direct numerical solutions. Despite their simplicity, fields of this kind have not been recognized as test cases for nonlinear methods before. The examples illustrate the treatment of the boundary conditions on current in the nonlinear force-free method, and in particular the limitations imposed by field lines that connect outside of the boundary region.     

Magnetohydrodynamic behaviour of a finite amplitude disturbance in the solar magnetic field

In this paper, dynamic processes in the solar atmosphere are studied numerically from a complete set of MHD equations. Dynamic evolution of the non-linear magnetic field is produced by the finite amplitude of the azimuthai magnetic field at the base of the flux tube of the solar atmosphere. It is assumed that the initial configuration of the magnetic field is a force-free and potential field, the magnetic field is disturbed at the base, the plasma is driven and a part of the magnetic energy is transformed into the kinetic energy of the plasma.The compressed flow of the plasma has the features of fast MHD waves. The computation results give quantitatively the non-linear evolution of strong magnetic fields. These results could be used in an explanation of coronal transients, surge, spray and eruptive prominence events in the solar atmosphere, as well as in a modelling of plasma behaviour in high-β structure experiments in the laboratory.     

双重电流片和日冕瞬变

当背景磁场在日冕中存在零磁场线时,反向新磁通量的喷发将会产生双重电流片,包括零场区附近的磁场受到挤压而形成的横向电流片和新喷发场、原背景场之间形成的拱形电流片、本文用一对线偶极子来模拟背景场,用一对线磁荷来模拟反向喷发场,讨论了上述双重电流片的形成和演变过程。在电流片形成过程中,物质将向电流片集中。拱形电流片物质主要来自过渡层和光球层,并通过辐射损失进一步冷却,形成低温日珥环;横向电流片的物质则全部来自日冕,从而形成高温日冕环。以上结果可用来解释1984年4月14日观测到的日冕瞬变。     

A Twisted Flux Rope Model for Coronal Mass Ejections and Two-Ribbon Flares

We present a new approach to the theory of large-scale solar eruptive phenomena such as coronal mass ejections and two-ribbon flares, in which twisted flux tubes play a crucial role. We show that it is possible to create a highly nonlinear three-dimensional force-free configuration consisting of a twisted magnetic flux rope representing the magnetic structure of a prominence (surrounded by an overlaying, almost potential, arcade) and exhibiting an S-shaped structure, as observed in soft X-ray sigmoid structures. We also show that this magnetic configuration cannot stay in equilibrium and that a considerable amount of magnetic energy is released during its disruption. Unlike most previous models, the amount of magnetic energy stored in the configuration prior to its disruption is so large that it may become comparable to the energy of the open field.     

A self-consistent model of the jovian plasma sheet

An axially-symmetric, rapidly-rotating magnetosphere containing low-energy plasma is considered. The resulting plasma sheet is presumed isothermal and thin compared with the radius of the sheet. Solutions of the model equations are found which include the effects of centrifugal, pressure and electro-magnetic forces. These solutions show that the sheet has a constant thickness and that the pressure decays exponentially with distance from the equatorial plane. The calculated curves for the magnetic induction field are compared with the observed field of Jupiter.     

CAN WE EXTRAPOLATE A MAGNETIC FIELD WHEN ITS TOPOLOGY IS COMPLEX?

In order to understand various solar phenomena controlled by the magnetic field, such as X-ray bright points, flares and prominence eruptions, the structure of the coronal magnetic field must be known. This requires a precise extrapolation of the photospheric magnetic field. Presently, only potential or linear force-free field approximations can be used easily. A more realistic modelling of the field is still an active research area because of well-known difficulties related to the nonlinear mixed elliptic-hyperbolic nature of the equations. An additional difficulty arises due to the complexity of the magnetic field structure which is caused by a discrete partition of the photospheric magnetic field. This complexity is not limited to magnetic regions having magnetic nulls (and so separatrices) but also occurs in those containing thin elongated volumes (called Quasi-Separatrix Layers) where the photospheric field-line linkage changes rapidly. There is a wide range for the thickness of such layers, which is determined by the character (bipolar or quadrupolar) of the magnetic region, by the sizes of the photospheric field concentrations and by the intensity of the electric currents. The aim of this paper is to analyse the recent nonlinear force-free field extrapolation techniques for complex coronal magnetic fields.