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1.
M. S. Wheatland 《Solar physics》2007,245(2):251-262
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. 相似文献
2.
Reliable measurements of the solar magnetic field are restricted to the level of the photosphere. For about half a century attempts have been made to calculate the field in the layers above the photosphere, i.e. in the chromosphere and in the corona, from the measured photospheric field. The procedure is known as magnetic field extrapolation. In the superphotospheric parts of active regions the magnetic field is approximately force-free, i.e. electric currents are aligned with the magnetic field. The practical application to solar active regions has been largely confined to constant-α or linear force-free fields, with a spatially constant ratio, α, between the electric current and the magnetic field. We review results obtained from extrapolations with constant-α force-free fields, in particular on magnetic topologies favourable for flares and on magnetic and current helicities. Presently, different methods are being developed to calculate non-constant-α or nonlinear force-free fields from photospheric vector magnetograms. We also briefly discuss these methods and present a comparison of a linear and a nonlinear force-free magnetic field extrapolation applied to the same photospheric boundary data. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
3.
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. 相似文献
4.
We develop an approach to deriving the three-dimensional non-force-free coronal magnetic field from vector magnetograms. Based
on the principle of minimum dissipation rate, a general non-force-free magnetic field is expressed as the superposition of
one potential field and two constant-α (linear) force-free fields. Each is extrapolated from its bottom boundary data, providing the normal component only. The
constant-α parameters are distinct and determined by minimizing the deviations between the numerically computed and measured transverse
magnetic field at the bottom boundary. The boundary conditions required are at least two layers of vector magnetograms, one
at the photospheric level and the other at the chromospheric level, presumably. We apply our approach to a few analytic test
cases, especially to two nonlinear force-free cases examined by Schrijver et al. (Solar Phys.
235, 161, 2006). We find that for one case with small α parameters, the quantitative measures of the quality of our result are better than the median values of those from a set
of nonlinear force-free methods. The reconstructed magnetic-field configuration is valid up to a vertical height of the transverse
scale. For the other cases, the results remain valid to a lower vertical height owing to the limitations of the linear force-free-field
solver. Because our method is based on the fast-Fourier-transform algorithm, it is much faster and easy to implement. We discuss
the potential usefulness of our method and its limitations. 相似文献
5.
We developed a code for the reconstruction of nonlinear force-free and non-force-free coronal magnetic fields. The 3D magnetic
field is computed numerically with the help of an optimization principle. The force-free and non-force-free codes are compiled
in one program. The force-free approach needs photospheric vector magnetograms as input. The non-force-free code additionally
requires the line-of-sight integrated coronal density distribution in combination with a tomographic inversion code. Previously
the optimization approach has been used to compute magnetic fields using all six boundaries of a computational box. Here we
extend this method and show how the coronal magnetic field can be reconstructed only from the bottom boundary, where the boundary
conditions are measured with vector magnetographs. The program is planed for use within the Stereo mission. 相似文献
6.
Force-free magnetic fields can be computed by making use of a new numerical technique, in which the fields are represented
by a boundary integral equation based on a specific Green's function. Vector magnetic fields observed on the photospheric
surface can be taken as the boundary conditions of this equation. In this numerical computation, the following two points
are emphasized: (1) A new method for data reduction is proposed, for removing uncertainties in boundary data and determining
the parameter in this Green's function, which is important for solving the boundary integral equation. In this method, the
transverse components of the observed boundary field are calibrated with a linear force-free field model without changing
their azimuth. (2) The computed 3-D fields satisfy the divergence-free and force-free conditions with high precision. The
alignment of these field lines is mostly in agreement with structures in Hα and Yohkoh soft X-ray images. Since the boundary data are calibrated with a linear force-free field model, the computed 3-D magnetic
field can be regarded as a quasi-linear force-free field approximation. The reconstruction of 3-D magnetic field in active
region NOAA 7321 was taken as an example to quantitatively exhibit the capability of our new numerical technique. 相似文献
7.
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. 相似文献
8.
Simple analytic models for the passive evolution of arcade-like magnetic fields through a series of force-free equilibria are presented. At the photospheric boundary, the normal magnetic field component is prescribed together with either the longitudinal field component or the photospheric shear. Analytic progress is made by considering either cylindrically symmetric solutions or using the separation of variables technique.
Two distinct cylindrically symmetric force-free fields are obtained that possess the same normal field component and photospheric shear. The scond field contains a magnetic bubble. As the shear increases beyond a critical value, so the magnetic energy of the first configuration exceeds that of the second. The possibility is therefore suggested of an eruption of the first field outwards towards the second. Such an eruptive instability is proposed as the origin of a two-ribbon solar flare.A new analytic solution to the force-free field equations, of separable form, is discovered and it is pointed out that the existence of shear in a magnetic field does not preclude it from being potential.Now at AWRE, Aldermaston, Reading, Berkshire. 相似文献
9.
Photospheric motion shears or twists solar magnetic fields to increase magnetic energy in the corona, because this process may change a current-free state of a coronal field to force-free states which carry electric current. This paper analyzes both linear and nonlinear two-dimensional force-free magnetic field models and derives relations of magnetic energy buildup with photospheric velocity field. When realistic data of solar magnetic field (B
0 103 G) and photospheric velocity field (v
max 1 km s–1) are used, it is found that 3–4 hours are needed to create an amount of free magnetic energy which is of the order of the current-free field energy. Furthermore, the paper studies situations in which finite magnetic diffusivities in photospheric plasma are introduced. The shearing motion increases coronal magnetic energy, while the photospheric diffusion reduces the energy. The variation of magnetic energy in the coronal region, then, depends on which process dominates. 相似文献
10.
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. 相似文献
11.
Magnetic field extrapolation is an alternative method to study chromospheric and coronal magnetic fields. In this paper, two
semi-analytical solutions of force-free fields (Low and Lou in Astrophys. J. 352:343, 1990) have been used to study the errors of nonlinear force-free (NLFF) fields based on force-free factor α. Three NLFF fields are extrapolated by approximate vertical integration (AVI) Song et al. (Astrophys. J. 649:1084, 2006), boundary integral equation (BIE) Yan and Sakurai (Sol. Phys. 195:89, 2000) and optimization (Opt.) Wiegelmann (Sol. Phys. 219:87, 2004) methods. Compared with the first semi-analytical field, it is found that the mean values of absolute relative standard deviations
(RSD) of α along field lines are about 0.96–1.19, 0.63–1.07 and 0.43–0.72 for AVI, BIE and Opt. fields, respectively. While for the
second semi-analytical field, they are about 0.80–1.02, 0.67–1.34 and 0.33–0.55 for AVI, BIE and Opt. fields, respectively.
As for the analytical field, the calculation error of 〈|RSD|〉 is about 0.1∼0.2. It is also found that RSD does not apparently depend on the length of field line. These provide the basic
estimation on the deviation of extrapolated field obtained by proposed methods from the real force-free field. 相似文献
12.
We present a careful investigation of the magnetofrictional relaxation and extrapolation technique applied to the reconstruction
of two test fields. These fields are taken from the family of nonlinear force-free magnetic equilibria constructed by Low
and Lou (Astrophys. J.
352, 343, 1990), which have emerged as standard tests for extrapolation techniques in recent years. For the practically relevant case that
only the field values in the bottom plane of the considered volume (vector magnetogram) are used as input information (i.e., not including the knowledge about the test field at the side and top boundaries), the test field is reconstructed to a higher
accuracy than obtained previously. Detailed diagnostics of the reconstruction accuracy show that the implementation of fourth-order
spatial discretization was essential to reach this accuracy for the given test fields and to achieve near machine precision
in satisfying the solenoidal condition. Different variants of boundary conditions are tested, which all yield comparable accuracy.
In its present implementation, the technique yields a scaling of computing time with total number of grid points only slightly
below N
5/3, which is too steep for applications to large (≥10242) magnetograms, except on supercomputers. Directions for improvement are outlined. 相似文献
13.
Su Qing-Rui 《Solar physics》1982,75(1-2):229-236
In this paper, we extend B. C. Low's study on nonlinear force-free magnetic fields. Based on Low's mathematical method, a revised boundary-value problem of the two-dimensional nonlinear force-free magnetic field is solved analytically. The solution shows that higher magnetic loops evolve towards preflare loops when the gradient of longitudinal magnetic field at the photospheric level and the angle (shear) included between the magnetic field line and magnetic neutral line increase with time. The density, temperature and the current density are higher in the preflare loops than in the high-lying magnetic loops. We believe it is the loops that provide conditions for the eruption of the flare.The original has been published in the Acta Astronomica Sinica
21 (1980), 152, in Chinese. The present paper completes the discussion and revises some of the preliminary results. 相似文献
14.
In earlier works we introduced and tested a nonlinear force-free (NLFF) method designed to self-consistently calculate the coronal free magnetic energy and the relative magnetic helicity budgets of observed solar magnetic structures. In principle, the method requires only a single, photospheric or low-chromospheric, vector magnetogram of a quiet-Sun patch or an active region and performs calculations without three-dimensional magnetic and velocity-field information. In this work we strictly validate this method using three-dimensional coronal magnetic fields. Benchmarking employs both synthetic, three-dimensional magnetohydrodynamic simulations and nonlinear force-free field extrapolations of the active-region solar corona. Our time-efficient NLFF method provides budgets that differ from those of more demanding semi-analytical methods by a factor of approximately three, at most. This difference is expected to come from the physical concept and the construction of the method. Temporal correlations show more discrepancies that are, however, soundly improved for more complex, massive active regions, reaching correlation coefficients on the order of, or exceeding, 0.9. In conclusion, we argue that our NLFF method can be reliably used for a routine and fast calculation of the free magnetic energy and relative magnetic helicity budgets in targeted parts of the solar magnetized corona. As explained in this article and in previous works, this is an asset that can lead to valuable insight into the physics and triggering of solar eruptions. 相似文献
15.
T. Wiegelmann 《Solar physics》2007,240(2):227-239
We describe a newly developed code for the extrapolation of nonlinear force-free coronal magnetic fields in spherical coordinates.
The program uses measured vector magnetograms on the solar photosphere as input and solves the force-free equations in the
solar corona. The method is based on an optimization principle and the heritage of the newly developed code is a corresponding
method in Cartesian geometry. We test the newly developed code with the help of a semi-analytic solution and rate the quality
of our reconstruction qualitatively by magnetic field line plots and quantitatively with a number of comparison metrics. We
find that we can reconstruct the original test field with high accuracy. The method is fast if the computation is limited
to low co-latitudes (say 30°≤θ≤150°), but it becomes significantly slower if the polar regions are included. 相似文献
16.
S. Régnier 《Solar physics》2012,277(1):131-151
In the last decades, force-free-field modelling has been used extensively to describe the coronal magnetic field and to better
understand the physics of solar eruptions at different scales. Especially the evolution of active regions has been studied
by successive equilibria in which each computed magnetic configuration is subject to an evolving photospheric distribution
of magnetic field and/or electric-current density. This technique of successive equilibria has been successful in describing
the rate of change of the energetics for observed active regions. Nevertheless the change in magnetic configuration due to
the increase/decrease of electric current for different force-free models (potential, linear and nonlinear force-free fields)
has never been studied in detail before. Here we focus especially on the evolution of the free magnetic energy, the location
of the excess of energy, and the distribution of electric currents in the corona. For this purpose, we use an idealised active
region characterised by four main polarities and a satellite polarity, allowing us to specify a complex topology and sheared
arcades to the coronal magnetic field but no twisted flux bundles. We investigate the changes in the geometry and connectivity
of field lines, the magnetic energy and current-density content as well as the evolution of null points. Increasing the photospheric
current density in the magnetic configuration does not dramatically change the energy-storage processes within the active
region even if the magnetic topology is slightly modified. We conclude that for reasonable values of the photospheric current
density (the force-free parameter α<0.25 Mm−1), the magnetic configurations studied do change but not dramatically: i) the original null point stays nearly at the same location, ii) the field-line geometry and connectivity are slightly modified, iii) even if the free magnetic energy is significantly increased, the energy storage happens at the same location. This extensive
study of different force-free models for a simple magnetic configuration shows that some topological elements of an observed
active region, such as null points, can be reproduced with confidence only by considering the potential-field approximation.
This study is a preliminary work aiming at understanding the effects of electric currents generated by characteristic photospheric
motions on the structure and evolution of the coronal magnetic field. 相似文献
17.
Carolus J. Schrijver Marc L. Derosa Thomas R. Metcalf Yang Liu Jim Mctiernan Stéphane Régnier Gherardo Valori Michael S. Wheatland Thomas Wiegelmann 《Solar physics》2006,235(1-2):161-190
We compare six algorithms for the computation of nonlinear force-free (NLFF) magnetic fields (including optimization, magnetofrictional,
Grad–Rubin based, and Green's function-based methods) by evaluating their performance in blind tests on analytical force-free-field
models for which boundary conditions are specified either for the entire surface area of a cubic volume or for an extended
lower boundary only. Figures of merit are used to compare the input vector field to the resulting model fields. Based on these
merit functions, we argue that all algorithms yield NLFF fields that agree best with the input field in the lower central
region of the volume, where the field and electrical currents are strongest and the effects of boundary conditions weakest.
The NLFF vector fields in the outer domains of the volume depend sensitively on the details of the specified boundary conditions;
best agreement is found if the field outside of the model volume is incorporated as part of the model boundary, either as
potential field boundaries on the side and top surfaces, or as a potential field in a skirt around the main volume of interest.
For input field (B) and modeled field (b), the best method included in our study yields an average relative vector error En = 〈 |B−b|〉/〈 |B|〉 of only 0.02 when all sides are specified and 0.14 for the case where only the lower boundary is specified, while the total
energy in the magnetic field is approximated to within 2%. The models converge towards the central, strong input field at
speeds that differ by a factor of one million per iteration step. The fastest-converging, best-performing model for these
analytical test cases is the Wheatland, Sturrock, and Roumeliotis (2000) optimization algorithm as implemented by Wiegelmann
(2004). 相似文献
18.
We present a method to include stereoscopic information about the three-dimensional structure of flux tubes into the reconstruction of the coronal magnetic field. Due to the low plasma beta in the corona we can assume a force-free magnetic field, with the current density parallel to the magnetic field lines. Here we use linear force-free fields for simplicity. The method uses the line-of-sight magnetic field on the photosphere as observational input. The value of is determined iteratively by comparing the reconstructed magnetic field with the observed structures. The final configuration is the optimal linear force-free solution constrained by both the photospheric magnetogram and the observed plasma structures. As an example we apply our method to SOHO MDI/EIT data of an active region. In the future it is planned to apply the method to analyse data from the SECCHI instrument aboard the STEREO mission. 相似文献
19.
Using the boundary element method (BEM) for constant-, force-free fields, the vector magnetic field distributions in the chromosphere of a flare-productive active region. AR 6659 in June 1991, are obtained by extrapolating from the observed vector magnetograms at the photosphere. The calculated transverse magnetic fields skew highly from the photosphere to the chromosphere in the following positive polarity sunspot whereas they skew only slightly in the main preceding sunspot. This suggests that more abundant energy was stored in the former area causing flares. Those results demostrate the validity of the BEM solution and the associations between the force-free magnetic field and the structure of the AR 6659 region. It shows that the features of the active region can be revealed by the constant- force-free magnetic field approximation. 相似文献
20.
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. 相似文献