Potential Field Approximation Research Articles

Localization of geological objects according to magnetometric data with application of algorithms of automated selection

The method of carrying out quality and quantitative interpretation of basic magnetometric data on the basis of use of analytical approximation of potential fields and some set of derivative highest degrees is offered. In the course of the researches the following tasks are solved: creation of numerical model of the anomalous field is executed, transformant of magnetic field are calculated and quality interpretation of the received results is carried out, modeling local the anamalous of sources is executed. The developed software is tested on the model examples and magnetometric data obtained for the region of the Alushta and Batumi anomalous area in the water area of the Black Sea

Effective algorithms for sourcewise approximation of geopotential fields

A specialized class of fast algorithms for sourcewise approximation of potential fields, which is intended for mass use in the applications associated with handling large and extralarge measurement data, is developed. Pursuing the objective of the work, we included two informal requirements in the set of the criteria to be met by our algorithms: they should have a sufficiently simple mathematical structure which is transparent for the geophysicist and they should be quite easy to program. Both these requirements are dictated by the present-day practice existing in geology, when the absence of the centralized provision of program products by the geological organizations makes the interpreter create the necessary software themselves. Among the distinguishing features of the three suggested algorithms is the fact that the dimensionality of the approximating construction is not specified a priori but is determined a posteriori. Due to this, it turned out to be helpful to combine one of these algorithms with the fast wavelet transform, which enables the dimension of the future sourcewise approximation and the amount of computing resources required for the implementation of the main algorithm to be estimated a priori. The examples of the practical application of these technologies are presented. The promising paths for the future development of the algorithms for sourcewise approximation of the potential fields are outlined.

Quantifying the Difference Between the Flux-Tube Expansion Factor at the Source Surface and at the Alfvén Surface Using a Global MHD Model for the Solar Wind

The potential field approximation has been providing a fast, and computationally inexpensive estimation for the solar corona's global magnetic field geometry for several decades. In contrast, more physics-based global magnetohydrodynamic (MHD) models have been used for a similar purpose, while being much more computationally expensive. Here, we investigate the difference in the field geometry between a global MHD model and the potential field source surface model (PFSSM) by tracing individual magnetic field lines in the MHD model from the Alfven surface (AS), through the source surface (SS), all the way to the field line footpoint, and then back to the source surface in the PFSSM. We also compare the flux-tube expansion at two points at the SS and the AS along the same radial line. We study the effect of solar cycle variations, the order of the potential field harmonic expansion, and different magnetogram sources. We find that the flux-tube expansion factor is consistently smaller at the AS than at the SS for solar minimum and the fast solar wind, but it is consistently larger for solar maximum and the slow solar wind. We use the Wang--Sheeley--Arge (WSA) model to calculate the associated wind speed for each field line, and propagate these solar-wind speeds to 1AU. We find a more than five hours deviation in the arrival time between the two models for 20% of the field lines in the solar minimum case, and for 40% of the field lines in the solar maximum case.

A Green’s function method for calculating the potential magnetic field in solar active regions in a spherical geometry

A Green’s function solution of Laplace’s equation for the potential magnetic field in an external spherical region is found using the derivative of the potential along a selected direction as a boundary condition. A set of programs applying this solution to construct the potential magnetic-field lines in solar active regions based on the photospheric line-of-sight field component has been developed. The method is tested using some model fields, and the optimal step size is found for realistic conditions. The developed software is applied to four real solar active regions, adopting HMI/SDO magnetograms as the boundary conditions. The potential magnetic field in the chromosphere and corona have been reconstructed for the selected regions. The calculated field lines are compared with flux tubes observed by AIA/SDO in the EUV. This comparison is used as a basis to discuss the applicability of a potential field approximation to the magnetic fields in solar active regions.

Magnetic Energy Storage and Current Density Distributions for Different Force-Free Models

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.

Modelling stellar coronae from surface magnetograms: the role of missing magnetic flux

Recent advances in spectropolarimetry have allowed the reconstruction of stellar coronal magnetic fields. This uses Zeeman-Doppler magnetograms of the surface magnetic field as a lower boundary condition. The ZDI maps, however, suffer from the absence of information about the magnetic field over regions of the surface due to the presence of dark starspots and portions of the surface out of view due to a tilt in the rotation axis. They also suffer from finite resolution which leads to small scale field structures being neglected. This paper explores the effects of this loss of information on the extrapolated coronal fields. For this we use simulated stellar surface magnetic maps for two hypothetical stars. Using the potential field approximation, the coronal fields and emission measures are calculated. This is repeated for the cases of missing information due to, (i) starspots, (ii) a large area of the stellar surface out of view, (iii) a finite resolution. The largest effect on the magnetic field structure arises when a significant portion of the stellar surface remains out of view. This changes the nature of the field lines that connect to this obscured hemisphere. Nonetheless, the field structure in the visible hemisphere is reliably reproduced. Thus the calculation of the locations and surface filling factors of accretion funnels is reasonably well reproduced for the observed hemisphere. The decrease with height of the magnetic pressure, which is important in calculating disc truncation radii for accreting stars, is also largely unaffected in the equatorial plane. The fraction of surface flux that is open and therefore able to supply angular momentum loss in a wind, however, is often overestimated in the presence of missing flux.

Validation of a synoptic solar wind model

We present a validation of a three‐dimensional magnetohydrodynamic model for the solar corona and the inner heliosphere. We compare the results of the model with long‐term satellite data at 1 AU for a 1 year period during solar minimum and another year period of solar maximum. Overall, the model predicts rather well the magnitude of the magnetohydrodynamical variables for solar minimum conditions. For solar maximum conditions, the magnitude of the magnetic field predicted by the model is too low. This result is consistent with the assumption that during solar maximum, a significant portion of the heliospheric open magnetic flux is not captured by the magnetogram input and the potential field approximation.

Some properties of solar magnetic fields as deduced from NSO/KP data and from calculations

In this paper we carry out a correlative analysis of the measurements of the photospheric magnetic field Bd (the magnetic component along the line of sight) from NSO/KP, and of the calculations of the field components on the visible solar disk from these measurements using Rudenko's (2001) potential field approximation method. The analysis revealed the following features: (1) near the equator the character of variation of langle|Bd| rangle(ρ) varies with the range of the values of |Bd| used in the analysis (here the brackets langle rangle denote an averaging over time and over the area of the portion of the size 10°×10°, and ρ is the angle between this portion and the central meridian (CM)); (2) generally near the equator there is a relatively small field `radiality': the value of langle|BR| rangle/ langle|BF| rangle near the CM varies over the range 1.25–0.9 and increases from the CM to the limb (here BR and BF are the calculated radial and toroidal field components); (3) the magnetic field measurements near the equator are different for different longitudes; (4) in the region of coronal holes (He i λ10 830 A) and at the calculated bases of open magnetic tubes (OMT) there is an increased `radiality' of the magnetic field in comparison with solar disk areas near the equator. The largest recorded values of the ratios of area-averaged coronal holes (CH) and OMT of the moduli of the radial and transverse field components were 9.2 in CH, and 4 in OMT.

Stability of accretion discs threaded by a strong magnetic field

We study the stability of poloidal magnetic fields anchored in a thin accretion disc. The two-dimensional hydrodynamics in the disc plane is followed by a grid-based numerical simulation including the vertically integrated magnetic forces. The 3--dimensional magnetic field outside the disc is calculated in a potential field approximation from the magnetic flux density distribution in the disc. For uniformly rotating discs we confirm numerically the existence of the interchange instability as predicted by Spruit, Stehle & Papaloizou (1995). In agreement with predictions from the shearing sheet model, discs with Keplerian rotation are found to be stabilized by the shear, as long as the contribution of magnetic forces to support against gravity is small. When this support becomes significant, we find a global instability which transports angular momentum outward and allows mass to accrete inward. The instability takes the form of a $m=1$ rotating `crescent', reminiscent of the purely hydrodynamic nonlinear instability previously found in pressure-supported discs. A model where the initial surface mass density $\Sigma(r)$ and $B_{\mathrm{z}}(r)$ decrease with radius as power laws shows transient mass accretion during about 6 orbital periods, and settles into a state with surface density and field strength decreasing approximately exponentially with radius. We argue that this instability is likely to be the main angular momentum transport mechanism in discs with a poloidal magnetic field sufficiently strong to suppress magnetic turbulence. It may be especially relevant in jet-producing discs.

Statistical Mechanical Equilibrium Theory of Selective Ion Channels

A rigorous statistical mechanical formulation of the equilibrium properties of selective ion channels is developed, incorporating the influence of the membrane potential, multiple occupancy, and saturation effects. The theory provides a framework for discussing familiar quantities and concepts in the context of detailed microscopic models. Statistical mechanical expressions for the free energy profile along the channel axis, the cross-sectional area of the pore, and probability of occupancy are given and discussed. In particular, the influence of the membrane voltage, the significance of the electric distance, and traditional assumptions concerning the linearity of the membrane electric field along the channel axis are examined. Important findings are: 1) the equilibrium probabilities of occupancy of multiply occupied channels have the familiar algebraic form of saturation properties which is obtained from kinetic models with discrete states of denumerable ion occupancy (although this does not prove the existence of specific binding sites; 2) the total free energy profile of an ion along the channel axis can be separated into an intrinsic ion-pore free energy potential of mean force, independent of the transmembrane potential, and other contributions that arise from the interfacial polarization; 3) the transmembrane potential calculated numerically for a detailed atomic configuration of the gramicidin A channel embedded in a bilayer membrane with explicit lipid molecules is shown to be closely linear over a distance of 25Å along the channel axis. Therefore, the present analysis provides some support for the constant membrane potential field approximation, a concept that has played a central role in the interpretation of flux data based on traditional models of ion permeation. It is hoped that this formulation will provide a sound physical basis for developing nonequilibrium theories of ion transport in selective biological channels.

Study of the structure of streamer belts and chains in the solar corona

A comparison is made of polarization brightness distributions of the white-light corona based on the data from Mark III (MLSO) with calculated magnetic configurations in the corona (a potential-field approximation) between adjacent coronal holes (or associated open magnetic tubes) with magnetic fields of single polarity. It is shown that in these coronal regions, which were referred to as ‘chains of streamers’ in earlier papers of these authors, magnetic structures in meridional and azimuthal cross-sections typically have the aspect of helmet streamers. The calculated magnetic field configurations under the helmets of chains of streamers have an even number of systems of arched features closely adjacent to each other (unlike the structure of the streamer belt with a neutral line where their number is odd). The height of the helmet top of chains of streamers remains smaller than that of the source surface but changes along the length of the chain and reaches maximum values at the ends of the chain.

Smooth approximation of potential fields from noisy scattered data

Geophysical potential‐field data generally are observed at scattered sampling points and are contaminated by measurement and preprocessing errors. Analysis and further treatment require a representation of the underlying potential fields on a regular grid. In the presence of noise, an approximation of the data is more appropriate than an exact interpolation. Analysis of the smoothness of potential fields indicates an approximation by band‐limited functions. The resulting least‐squares problem can be formulated as a linear system of equations in which the system matrix is of block‐Toeplitz type. This special structure allows a computationally attractive and robust solution via the fast Fourier transform and the conjugate gradient algorithm. On this basis, we present a gridding method that incorporates additional knowledge about the physical properties of potential fields and the statistics of the data. Experimental results for synthetic and real‐world gravity data demonstrate good performance for highly scattered noisy data.

Comparison of heliospheric current sheet structure obtained from potential magnetic field computations and from observed polarization coronal brightness

We compare the structure of the heliospheric current sheet early in sunspot cycle 21 as computed from the observed photospheric magnetic field with a potential field approximation and as inferred from synoptic maps of the observed coronal polarization brightness. On most of the solar rotations compared, the two methods give essentially the same results; the basic shape of the warped current sheet and the amplitude (in solar latitude) of the displacements of the sheet from the solar equator are similar. On one rotation the current sheet computed with the potential field approximation appears to be distorted by a large photospheric region of unbalanced magnetic flux.

Coronal general magnetic field evolution as a new parameter of the solar cycle

The magnetic field lines of the corona associated with the solar-cycle surface general magnetic field are calculated by a potential-field approximation to study the solar-cycle evolution of the geometry of the coronal field. The surface field evolution used here is the radial field evolution, predicted by a model of the solar cycle driven by the dynamo action of the global convection, and justified observationally using Mount Wilson magnetic synoptic chart data. The evolution of the calculated coronal general field is now good for comparison with observations and shows the following. (i) The field of the polar and high-altitude corona has dipolar structure in almost all phases of the solar cycle except in a short time interval around maximum phase despite the quadrupolar structure of the general magnetic field at the surface; quadrupolar field forms loop-like structure in the lower corona. The almost-dipolar structure of the polar and high-latitude corona and the loop formation of the equatorial lower corona explain the appearance of the undisturbed minimum corona observed at eclipses. (ii) The polar field lines are directed almost radially at the minimum phase, which should be responsible for polar plumes. The field lines slowly open up to participate in the loop-like structure of the equatorial lower corona, and rapidly change their structure and polarity at the maximum phase, to resume the almost radial configuration slowly, (iii) During the rapidly changing maximum phase, the field lines do not penetrate deep into the interplanetary space resulting in the absence of polar plumes and the appearance of the circular corona- the maximum corona. In this phase, the coronal field should not be approximated by a dipole field. The surface field evolution which can explain such behaviors of the corona is characteristic of the solar-cycle process dominated by the latitudinal gradient of the differential rotation. If the radial gradient dominated in the subsurface process, the coronal evolution would look quite different and would show latitudinal propagation of enhancement of activity. Although nonaxisymmetric features should be superposed on the axisymmetric general field to express the real corona, the general field can be a basic coronal field in studying long-term interaction between the convection zone and the interstellar space especially in studying the magnetic braking of the solar rotation.

Time Variations in Coronal Active Regions

SummaryThe AS&amp;E X-ray telescope experiment on Skylab has obtained images of the solar X-ray corona with a variety of time resolutions ranging from 21/2 s to the regular 12 ± 2 h synoptic observation rate. The form and brightness of coronal active region structures are seen to vary on time scales ranging from seconds, for flare associated changes, to several solar rotations for long term evolution of the regions. The extrapolation of photospheric magnetic fields into the corona, using the potential field approximation, results in a good morphological agreement between the form of the computed coronal field lines and the structure of many of the active regions observed. Thus, in general, the coronal active region structures follow potential field lines and the long term evolutionary changes can be explained on the basis of the spreading of the fields. Short term changes in active region structure frequently take the form of selective brightening or dimming of pre-existing loops due to changes in the pressure of the emitting coronal plasma. In these cases, variations in the non-potential component of the coronal fields supporting and containing the plasma are implied.