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1.
A new fast mathematical method is described for computing potential magnetic field solutions in the solar atmosphere from the observed line-of-sight component of the photospheric magnetic field. As in a standard Neumann boundary problem the orthogonality relation of the spherical harmonics is used to determine the coefficients of the harmonic expansion. This leads to a very simple set of recursion formulae that determines the harmonic coefficients successively.  相似文献   

2.
Green's function methods for potential magnetic fields   总被引：1，自引：0，他引：1  
The Green's function method to calculate potential magnetic field on the Sun, which was first established by Schmidt (1964) in the case that the field component normal to a flat boundary plane is specified, is extended to the following three cases: (a) The field component along the line of sight, which is not generally normal to the flat boundary plane, is specified; (b) the line of sight component on a spherical boundary surface is specified; (c) the normal component on a spherical surface is specified, together with the condition that the field becomes approximately radial on an outer spherical surface (the so-called source surface). Properties of these Green's functions are examined, and the applicability of these methods to solar magnetic data is discussed.On leave of absence from Department of Astronomy, University of Tokyo.  相似文献   

3.
V. S. Titov 《Solar physics》1992,139(2):401-404
The method of calculating two-dimensional potential magnetic configurations with current sheets (CS) is proposed, the number of CS in corona and the degree of asymmetry being both arbitrary. As a given boundary value the component of magnetic field normal to the photosphere is considered.  相似文献   

4.
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   

5.
This paper is concerned with the Laplace boundary-value problem with the directional derivative, corresponding to the specific nature of measurements of the longitudinal component of the photospheric magnetic field. The boundary conditions are specified by a distribution on the sphere of the projection of the magnetic field vector into a given direction, i.e., they exactly correspond to the data of daily magnetograms distributed across the full solar disk. It is shown that the solution of this problem exists in the form of a spherical harmonic expansion, and uniqueness of this solution is proved. A conceptual sketch of numerical determination of the harmonic series coefficients is given. The field of application of the method is analyzed with regard to the peculiarities of actual data. Results derived from calculating magnetic fields from real magnetograms are presented. Finally, we present differences in results derived from extrapolating the magnetic field from a synoptic map and a full-disk magnetogram.  相似文献   

6.
The potential magnetic field from a finite planar boundary is extrapolated into the upper hemisphere using information from all three magnetic field components. The method determines, first, the transverse field associated with the observed normal magnetic intensity. Then by subtraction, the method determines the associated transverse magnetic field observed in the interior (i.e., in the field of view) of the magnetogram which is due to the normal flux exterior to the field of view of the magnetogram. Inverting this information gives an approximation to the exterior normal flux. The combination of the observed normal flux of the interior and the approximation of the exterior normal flux is employed to calculate the potential field. The formulation of the problem results in an ill-posed integral inversion problem in which a regularized solution is obtained using the singular value decomposition (SVD) technique in conjunction with an appropriate Tikhonov-Phillips filter. The technique can be applied to correcting potential field calculations which are influenced by out-of-view fluxes, e.g., for a high spatial resolution vector magnetogram with a small field of view in which there is no supporting exterior data. The problem studied is also important in providing a regularized solution of the Cauchy potential problem. The method provides a much larger range of convergence than the method of Gary and Musielak (1992), and, in fact, is stable in the total upper hemisphere.The U.S. Government right to retain a non-exclusive, royalty-free licence in and to any copyright is acknowledged.  相似文献   

7.
The Weber-Davis model of the solar wind is generalized to include the effects of latitude. The principal assumptions of perfect electrical conductivity, rotational symmetry, a polytropic relation between pressure and density, and a flow aligned magnetic field in a system rotating with the Sun, are retained. A flow aligned magnetic field in the rotating system may be expressed in terms of the flow velocity and density. Rotational symmetry fixes the longitudinal flow velocity Vφ in terms of the flow in the r?θ plane. Thus, the original three dimensional magnetohydrodynamic flow problem is reduced to a two dimensional hydrodynamic flow problem in the r?θ plane.There are three critical surfaces associated with the equations which supply conditions to determine three of six required boundary conditions. The specified boundary conditions at the base of the corona are the temperature, density, and magnitude of the magnetic field. The equations are then expanded about the radial, nonrotating Parker solution and an analytic solution is obtained for the resulting first order equations. The results show that for constant coronal boundary conditions there is a latitudinal flow toward the solar poles, as a result of magnetic stresses, which persists out to large distances for the Sun. Associated with this flow is a latitudinal component of the magnetic field. The radial flow parameters are, to within small first order differences, in agreement with those of the Parker and the Weber-Davis models of the solar wind.The equations are further generalized to permit first order latitudinal variations in the specified coronal boundary conditions. Results at 1 a.u. are presented for 5 per cent latitudinal differences between the equatorial and polar values. These results show that the solution at 1 a.u. is most sensitive to a latitudinal dependence in the boundary temperature and least sensitive to a latitudinal dependence in the magnetic field magnitude.A solution is then obtained for an approximate dipolar variation in the coronal magnetic field magnitude. This solution predicts that the latitudinal flow is initially toward the Equator due to magnetic channeling; however, this effect is rapidly overcome and the latitudinal flow at 1 a.u. is toward the pole and not significantly different from the solution for constant boundary conditions.  相似文献   

8.
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   

9.
In this paper we develop a statistical approach to resolve the transport problem for the tangential fluctuations of the geomagnetic field in the mantle. For the sake of simplicity we treat the mantle as a thick layer of vacuum and assume in addition that only a radial component of the magnetic field of the core penetrates through the core-mantle boundary. These assumptions allow us to find exact expressions for the tangential field components throughout the mantle. By using such expressions we construct a correlation tensor of tangential components and then, since the mantle is thick enough, study its asymptotic properties on the Earth surface. Incidentally, the correlation tensor trace happens to be equal to the correlation function of the radial component that was obtained by Pilipenko and Sokoloff (1992). Indeed, we provide a simple boundary problem which initially describes the diffusion functions. We also pay a special attention to transformation properties of the correlation tensor and find here some interesting analogies with secular variation data of the geomagnetic field  相似文献   

10.
Although for many solar physics problems the desirable or meaningful boundary is the radial component of the magnetic field \(B_{\mathrm {r}}\), the most readily available measurement is the component of the magnetic field along the line of sight to the observer, \(B_{\mathrm {los}}\). As this component is only equal to the radial component where the viewing angle is exactly zero, some approximation is required to estimate \(B_{\mathrm {r}}\) at all other observed locations. In this study, a common approximation known as the “\(\mu\)-correction”, which assumes all photospheric field to be radial, is compared to a method that invokes computing a potential field that matches the observed \(B_{\mathrm {los}}\), from which the potential field radial component, \(B_{\mathrm {r}}^{\mathrm {pot}}\) is recovered. We demonstrate that in regions that are truly dominated by a radially oriented field at the resolution of the data employed, the \(\mu\)-correction performs acceptably if not better than the potential-field approach. However, it is also shown that for any solar structure that includes horizontal fields, i.e. active regions, the potential-field method better recovers both the strength of the radial field and the location of magnetic neutral line.  相似文献   

11.
A class of magnetostatic equilibria with axial symmetry outside a unit sphere in the presence of plasma pressure and an r –2 gravitational field is constructed. The structure contains a localized current-carrying region confined by a background bipolar potential field, and the shape of the region changes subject to the variation of the electric current. The continuity requirement for the magnetic field and plasma pressures at the outer boundary of the cavity defines a free boundary problem, which is solved numerically using a spectral boundary scheme. The model is then used to study the expansion of the current-carrying region, caused by the buildup of magnetic shear, against the background confining field. The magnetic shear in our model is induced by the loading of an azimuthal field, accompanied by a depletion of plasma density.We show that due to the additional effect of confinement by the dense surrounding plasma, the energy of the magnetic field can exceed the energy of its associated open field, presumably a necessary condition for the onset of coronal mass ejections. (However, the plasma beta of the confining fluid is higher than that in the outer boundary of a realistic helmet-streamer structure.) Furthermore, under the assumption that coronal mass ejections are driven by magnetic buoyancy, the result from our model study lends further support to the notion of a suspended magnetic flux rope in the low-density cavity of a helmet-streamer as a promising pre-ejection configuration.The National Center for Atmospheric Research is sponsored by the National Science Foundation.  相似文献   

12.
M. J. Hagyard 《Solar physics》1987,107(2):239-246
In this paper we investigate the changes that occur in measured magnetic fields when they are transformed into a heliographic coordinate system. To carry out this investigation we took measurements of the vector magnetic field of an active region that was observed at 1/3 the solar radius from disk center and transformed the observed field into heliographic coordinates. We also examined differences in the calculated potential field that occur when the heliographic normal component of the field is used as the boundary condition rather than the observed line-of-sight component. The results of this analysis show (1) that the observed fields of sunspots more closely resemble the generally accepted picture of the distribution of umbral fields if they are displayed in heliographic coordinates, (2) that the differences in the potential calculations are less than 200 G in field strength and 20° in field azimuth outside sunspots, and (3) that differences in the two potential calculations in the sunspot areas are no more than 400 G in field strength but range from 60 to 80° in field azimuth in localized umbral areas.  相似文献   

13.
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   

14.
The inverse problem in a spherical shell to find the two-dimensional spatial distributions of the α-effect and differential rotation in a mean-field dynamo model has been solved. The derived distributions lead to the generation of a magnetic field concentrated inside the convection zone. The magnetic field is shown to have no time to rise from the region of maximum generation located in the lower layers to the surface in the polarity reversal time due to magnetic diffusion. The ratio of the maximum magnetic energy in the convection zone to its value at the outer boundary reaches two orders of magnitude or more. This result is important in interpreting the observed stellar and planetary magnetic fields. The proposed method of solving the inverse nonlinear dynamo problem is easily adapted for a wide class of mathematical-physics problems.  相似文献   

15.
Yan  Yihua  Sakurai  Takashi 《Solar physics》2000,195(1):89-109
A boundary integral equation to describe a force-free magnetic field with finite energy content in the open space above the solar surface is found. This is a new representation for a 3-D nonlinear force-free field in terms of the boundary field and its normal gradient at the boundary. Therefore the magnetic field observed on the solar surface can be incorporated into the formulation directly and a standard numerical technique, the boundary element method, can be applied to solve the field. A numerical test case demonstrates the power of the method by recovering the analytical solution to the desired accuracy and its application to practical solar magnetic field problems is straightforward and promising.  相似文献   

16.
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17.
Rudenko  G.V. 《Solar physics》2001,198(2):279-287
A technique is proposed for constructing self-consistent 3-D solutions satisfying the magnetohydrostatic (MHS) equations, and fitting observations along the line of sight of the magnetic field at the photosphere. The technique is a generalization of a potential-field extrapolation method (Rudenko, 2001) using full-disk magnetogram data. The solution of the problem under consideration is based on representing the magnetic field in terms of a scalar function, with its subsequent harmonic expansion in terms of the functional basic set of spherical functions that satisfies the specified boundary conditions. It is expected that a numerical realization of the proposed method will make possible a real-time modeling of the three-dimensional magnetic field, temperature, pressure and density distributions.  相似文献   

18.
The influence on the rate of angular momentum loss from the Sun of magnetic geometries which are not spherically symmetric is estimated. Departures from spherical symmetry are expected to influence significantly the loss rate by two effects - the presence of closed magnetic field regions with no loss and also the variability in the radial distance to the Alfvénic point, as stressed by Mestel (1968).The loss rate is calculated for an MHD solar wind model with a solar magnetic field whose normal component at the surface is that of a north-south dipole. In contrast to Mestel's work, where the field was assumed dipolar within a certain surface and radial outside, the coupling between the solar wind and magnetic field is here taken into account exactly. For equivalent boundary conditions at the surface, the resulting field configuration yields an angular momentum loss rate which is only 15% of that for the monopole field normally used in angular momentum loss estimates. If, instead of equating boundary conditions at the Sun, one equates the two losses at the equator to that observed at 1 AU by spacecraft, then the ratio of the total loss for the distended dipole to that for the monopole is about 40%.On Leave from the Department of Applied Mathematics, The University, St. Andrews, Scotland.The National Center for Atmospheric Research is sponsored by the National Science Foundation.  相似文献   

19.
A numerical method to determine the electromagnetic field of a steadily rotating magnetosphere with an inclined magnetic moment under a given boundary condition on an arbitrary shaped boundary surface is presented. The region may include the light cylinder. The present method, together with a companion method giving particle motion and creation, makes an iterative scheme to obtain a global model of the pulsar magnetosphere. A key problem for explaining the particle acceleration in pulsars is to solve field-aligned electric field in an accelerating region bounded by an ideal-MHD region. The present method is fit to connect a solution for the non-ideal-MHD region with another solution for the ideal-MHD region on a boundary surface whose location should also be solved (i.e., a floating boundary). The integration scheme is based on the boundary element method and it has great advantage as compared with other methods like the finite difference method and the Fourier transformation method.  相似文献   

20.
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.  相似文献   

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