Nonlinear Force-Free Reconstruction of the Global Solar Magnetic Field: Methodology

We present a novel numerical method that allows the calculation of nonlinear force-free magnetostatic solutions above a boundary surface on which only the distribution of the normal magnetic field component is given. The method relies on the theory of force-free electrodynamics and applies directly to the reconstruction of the solar coronal magnetic field for a given distribution of the photospheric radial field component. The method works as follows: we start with any initial magnetostatic global field configuration (e.g. zero, dipole), and along the boundary surface we create an evolving distribution of tangential (horizontal) electric fields that, via Faraday’s equation, give rise to a respective normal-field distribution approaching asymptotically the target distribution. At the same time, these electric fields are used as boundary condition to numerically evolve the resulting electromagnetic field above the boundary surface, modeled as a thin ideal plasma with non-reflecting, perfectly absorbing outer boundaries. The simulation relaxes to a nonlinear force-free configuration that satisfies the given normal-field distribution on the boundary. This is different from existing methods relying on a fixed boundary condition – the boundary evolves toward the a priori given one, at the same time evolving the three-dimensional field solution above it. Moreover, this is the first time that a nonlinear force-free solution is reached by using only the normal field component on the boundary. This solution is not unique, but it depends on the initial magnetic field configuration and on the evolutionary course along the boundary surface. To our knowledge, this is the first time that the formalism of force-free electrodynamics, used very successfully in other astrophysical contexts, is applied to the global solar magnetic field.     

A Fluid Dynamics Approach for the Computation of Non—linear Force—Free Magnetic Field

Inspired by the analogy between the magnetic field and velocity field of incompressible fluid flow, we propose a fluid dynamics approach for computing nonlinear force-free magnetic fields. This method has the advantage that the divergence-free condition is automatically satisfied, which is a sticky issue for many other algorithms, and we can take advantage of modern high resolution algorithms to process the force-free magnetic field. Several tests have been made based on the well-known analytic solution proposed by Low & Lou. The numerical results are in satisfactory agreement with the analytic ones. It is suggested that the newly proposed method is promising in extrapolating the active region or the whole sun magnetic fields in the solar atmosphere based on the observed vector magnetic field on the photosphere.     

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 Energy of Force-Free Fields with Detached Field Lines

Using an axisymmetrical ideal MHD model in spherical coordinates, we present a numerical study of magnetic configurations characterized by a levitating flux rope embedded in a bipolar background field whose normal field at the solar surface is the same or very close to that of a central dipole. The characteristic plasma β (the ratio between gas pressure and magnetic pressure) is taken to be sosmall (β= 10^-4) that the magnetic field is close to being force-free. The system as a whole is then let evolve quasi-statically with a slow increase of either the annular magnetic flux or the axial magnetic flux of the rope, and the total magneticenergy of the system grows accordingly. It is found that there exists an energy threshold: the flux rope sticks to the solar surface in equilibrium if the magneticenergy of the system is below the threshold, whereas it loses equilibrium if the threshold is exceeded. The energy threshold is found to be larger than that of thecorresponding fully-open magnetic field by a factor of nearly 1.08 irrespective as towhether the background field is completely closed or partly open, or whether the magnetic energy is enhanced by an increase of annular or axial flux of the rope.This gives an example showing that a force-free magnetic field may have an energy larger than the corresponding open field energy if part of the field lines is allowed tobe detached from the solar surface. The implication of such a conclusion in coronal mass ejections is briefly discussed and some comments are made on the maximum energy of force-free magnetic fields.     

On the Shape of Force-Free Field Lines in the Solar Corona

This paper studies the shape parameters of looped field lines in a linear force-free magnetic field. Loop structures with a sufficient amount of kinking are generally seen to form S or inverse S (Z) shapes in the corona (as viewed in projection). For a single field line, we can ask how much the field line is kinked (as measured by the writhe), and how much neighbouring flux twists about the line (as measured by the twist number). The magnetic helicity of a flux element surrounding the field line can be decomposed into these two quantities. We find that the twist helicity contribution dominates the writhe helicity contribution, for field lines of significant aspect ratio, even when their structure is highly kinked. These calculations shed light on some popular assumptions of the field. First, we show that the writhe of field lines of significant aspect ratio (the apex height divided by the footpoint width) can sometimes be of opposite sign to the helicity. Secondly, we demonstrate the possibility of field line structures which could be interpreted as Z-shaped, but which have a helicity value sign expected of an S-shaped structure. These results suggest that caution should be exercised in using two-dimensional images to draw conclusions on the helicity value of field lines and flux tubes.     

太阳线性无力场评注和快速傅氏分析法的应用

首先对30年来用无力场计算太阳活动区磁场方法作了一些评注，特别研究了线性无力场解中的几个误区，提出了准线性无力场方法并探讨了快速傅氏变换在线性无力场计算中的应用，以2000年9月15日NOAA 9165活动区磁场为例，应用3种不同的方法计算出磁力线走向，然后同紫金山天文台赣榆观测站观测的精细Hα图形对比。     

Preprocessing of Vector Magnetograph Data for a Nonlinear Force-Free Magnetic Field Reconstruction

Knowledge regarding the coronal magnetic field is important for the understanding of many phenomena, like flares and coronal mass ejections. Because of the low plasma beta in the solar corona, the coronal magnetic field is often assumed to be force-free and we use photospheric vector magnetograph data to extrapolate the magnetic field into the corona with the help of a nonlinear force-free optimization code. Unfortunately, the measurements of the photospheric magnetic field contain inconsistencies and noise. In particular, the transversal components (say B _x and B _y) of current vector magnetographs have their uncertainties. Furthermore, the magnetic field in the photosphere is not necessarily force free and often not consistent with the assumption of a force-free field above the magnetogram. We develop a preprocessing procedure to drive the observed non–force-free data towards suitable boundary conditions for a force-free extrapolation. As a result, we get a data set which is as close as possible to the measured data and consistent with the force-free assumption.     

Intermittency of the Solar Magnetic Field and Solar Magnetic Activity Cycle

Small-scale solar magnetic fields demonstrate features of fractal intermittent behavior, which requires quantification. For this purpose we investigate how the observational estimate of the solar magnetic flux density \(B\) depends on resolution \(D\) in order to obtain the scaling \(\ln B_{D} = - k \ln D +a\) in a reasonably wide range. The quantity \(k\) demonstrates cyclic variations typical of a solar activity cycle. In addition, \(k\) depends on the magnetic flux density, i.e. the ratio of the magnetic flux to the area over which the flux is calculated, at a given instant. The quantity \(a\) demonstrates some cyclic variation, but it is much weaker than in the case of \(k\). The scaling obtained generalizes previous scalings found for the particular cycle phases. The scaling is typical of fractal structures. In our opinion, the results obtained trace small-scale action in the solar convective zone and its coexistence with the conventional large-scale solar dynamo based on differential rotation and mirror-asymmetric convection.     

非线性无力场模型的线性稳定性

本文从能量原理出发，对无力场稳定性进行了研究，给出了一般情形下无力场稳定性的充要判据。它可以把线性无力场稳定性判据作为特例包括在内。还对Ｋｒｕｇｅｒ给出的一个充分判据作了进一步的探讨，对无力场稳定性的物理图象也作了讨论。     

太阳磁场的观测

分析了太阳的观测磁图所涉及的磁场位形 ,以及磁场观测研究中的一些问题。     

无力磁场的稳定性

无力场被广泛用来模拟太阳活动区的强磁场,本文从Bernstein能量原理出发,导出了无力场能量原理的普遍形式,并给出了若干稳定性的充分条件,它们可方便而有效地对无力场进行稳定性判断。     

太阳磁场观测研究

简要回顾了近几年国际上太阳磁场研究的一些重要进展，包括耀斑与磁切和电流的关系，电流螺度和磁螺度，磁场拓扑性，三维磁场外推，色球磁场研究，日冕磁场研究，内网络磁元，磁流和振荡，极区磁场观测以及色球磁元观测等方面内容，同时也介绍了怀柔太阳观测站最近所取得的主要成果，自20世纪90年代以来，YOHKOH高分辨率的太阳X射线数据，SOHO的多波段大尺度观测，TRACE的高分辨太阳过渡区资料，为研究太阳磁场从内部到距离几十太阳半径处的大范围演化提供了依据，高效的空间资料结合长期的地面资料，将是正派推动太阳磁场研究的重要手段和必然趋势。     

确定太阳横向磁场方位角的综合方法

本文分析了现有几种常用的确定太阳磁场横向分量方位角的方法，如势场法，Ｋｒａｌｌ法、吴－艾法和＂方位角连续＂法，作者认为这些方法各有不同的适用范围，其中任何一种方法都不能单独确定太阳横向磁场。在此分析的基础上，提出了确定太阳横向磁场方位角的综合方法。该法的要点是：用势场法和Ｋｒａｌｌ法分别处理同一磁场观测资料，比较这两种方法所得到的横场分布图，找出它们的相同区域和有差别区域。从相同区域出发，利用＂无力因子相近＂假定，可以推断有差异区域的横场方位角。本文提供的应用实例初步显示了综合方法的有效性。     

Dipped Magnetic Field Configurations Associated with Filaments and Barbs

In this paper, three-dimensional linear force-free field configurations that can be associated with filaments are considered. It is assumed that the field configurations are suitable to represent filaments if they contain magnetic dips. With the photospheric flux distribution chosen to be an arcade with a dextral/sinistral axial component, it is found that dipped configurations exist only for large values of alpha (where, ×B=B). The dips always lie above the polarity inversion line in the centre of the channel between the flux regions. When the dips are viewed from above to a depth of 1 Mm they resemble closely the shape of filaments viewed in absorption on the solar disk. As the magnitude of alpha increases, the horizontal and vertical extent of the dips also increases, giving active-region filaments for low values of alpha and quiescient filaments for high values of alpha. Dextral filaments only form for negative values of alpha and sinistral filaments for positive values of alpha. The portion of the field line that is dipped is always of inverse polarity and the magnitude of the field in the dipped region increases with height, both of which are consistent with Leroy, Bommier, and Sahal-Bréchot (1983). Overlying the region of dips there are arcades of normal polarity which have the correct left-bearing/right-bearing orientation for dextral/sinistral filaments. When the hypothesis of barbs occurring in dipped field lines is used, barbs that branch out of the main axis and to the right/left for dextral/sinistral filaments can be formed around minority polarity elements on either side of the polarity inversion line. No barbs are found around normal polarity elements. The model reproduces many of the observed features of filament channels, filaments and their barbs.     

The Relationship Between Kinematics and Spatial Structure of the Large-Scale Solar Magnetic Field

The rotation of large-scale solar magnetic fields has been investigated by analysing a 20-yr series of synoptic maps of the radial magnetic field. For this purpose, a specially adapted method of spectral analysis was used. We calculated rotation spectra of the magnetic field as functions of the rotation period, heliographic latitude, and longitudinal wave number, k. These spectra reveal the existence of a number of discrete, rigidly rotating components (modes) of the magnetic field, whose rotation periods lie in the wide range from 26.5 to 30.5 days. The significant spectral maxima lie in the (rotation period–latitude) plane close to the curve that represents the differential rotation of small-scale magnetic features. For the first harmonic of the magnetic field (k=1) the properties of the rotation spectra are consistent with those reported by Antonucci, Hoeksema, and Scherrer (1990). However, the distribution of the rigidly rotating modes over rotation period and their latitudinal structure change systematically with the harmonic number k. As k increases, the mean distance P in rotation period between the modes decreases, from 1.2 days for k=1 to 0.3–0.5 days for k=4. This decreasing period separation is accompanied by a decrease of the characteristic latitude separation between the mode maxima. The latitudinal and longitudinal discrete spatial scales of the non-axisymmetric magnetic field appear to be connected with each other, as well as with the temporal scale P.     

A Nonlinear Force-Free Magnetic Field Approximation Suitable for Fast Forward-Fitting to Coronal Loops. I. Theory

We derive an analytical approximation of nonlinear force-free magnetic field solutions (NLFFF) that can efficiently be used for fast forward-fitting to solar magnetic data, constrained either by observed line-of-sight magnetograms and stereoscopically triangulated coronal loops, or by 3D vector-magnetograph data. The derived NLFFF solutions provide the magnetic field components B _x (x), B _y (x), B _z (x), the force-free parameter α(x), the electric current density j(x), and are accurate to second-order (of the nonlinear force-free α-parameter). The explicit expressions of a force-free field can easily be applied to modeling or forward-fitting of many coronal phenomena.     

太阳活动区磁场观测的一种新方法

在太阳活动区的物理研究中,特别是在二维动力学光谱分析中,迫切需要相应活动区的磁图资料。本文介绍了在太阳光谱仪的入射狭缝后安装一种新型偏振器进行活动区二维磁场观测的新方法。这种方法不仅能获得日面上任一点的磁场强度,且可快速获得活动区的纵向场磁图。除此之外,还可利用多条谱线的同时观测,获得有关磁力线管结构等方面的资料。     

Corrections to Solar Thermal Structure when a Turbulent Magnetic Field is Included

Correction of non-ideal effect due to a magnetic fluctuating tensor is derived from the ideal MHD equations. The inclusion of a magnetic turbulent field leads to modifications of the hydrostatic equilibrium equation and thermodynamical variables such as the temperature T, the adiabatic exponent γ, the adiabatic temperature gradient △↓ad and the temperature gradient △↓. In particular, the modifications in the adiabatic and radiative temperature gradients will result in a change in the Schwarzchild criterion, hence in the location of the base of the convective zone. Incorporating the modifications, we construct a modified thermodynamical equilibrium structure of the Sun.     

The Varying Multipolar Structure of the Sun's Magnetic Field and the Evolution of the Solar Magnetosphere Through the Solar Cycle

The Sun's magnetic field extends far from the photosphere, into the corona, defining a magnetically dominated region before being drawn out radially by the solar wind flow. This region, where the internal sources of the solar field dominate the plasma structures and the energetic particle movement, can be properly considered the solar magnetosphere. The magnetic field in this region can be approximately described by models that extrapolate photospheric magnetic field observations under some simplifying assumptions. In this paper we use a potential field model which describes the solar field up to a source surface at 3.25 R_s, where the field is constrained to become radial. We present the variation of the magnitude and inclination of the various multipolar components throughout the solar magnetic cycle that characterise the changes in the structure of the solar magnetosphere over a period of 22 years. We also present some 3-D images of the coronal magnetic structure to show the global evolution of the solar magnetosphere throughout the solar cycle and discuss the importance of taking this structure into account in order to relate interplanetary and solar features.     

太阳磁场理论外推研究进展

太阳磁场是研究太阳物理的关键。目前对太阳磁场的精确测量只限于光球层。对日冕磁场结构的了解,则多是以观测的光球磁场作为边界条件,在某种理论模型下进行外推。势场模型、线性无力场模型和非线性无力场模型是无力场假设下的三种理论外推模型。文章介绍了太阳磁场理论外推的基本方法和最新进展,和对三种模型中使用较多的外推方法,列举了它们在天文研究中的一些应用,同时也简略讨论了外推方法中存在的一些问题。