A Comparison Between Nonlinear Force-Free Field and Potential Field Models Using Full-Disk SDO/HMI Magnetogram

Measurements of magnetic fields and electric currents in the pre-eruptive corona are crucial to the study of solar eruptive phenomena, like flares and coronal mass ejections (CMEs). However, spectro-polarimetric measurements of certain photospheric lines permit a determination of the vector magnetic field only at the photosphere. Therefore, there is considerable interest in accurate modeling of the solar coronal magnetic field using photospheric vector magnetograms as boundary data. In this work, we model the coronal magnetic field above multiple active regions with the help of a potential field and a nonlinear force-free field (NLFFF) extrapolation code over the full solar disk using Helioseismic and Magnetic Imager (SDO/HMI) data as boundary conditions. We compare projections of the resulting magnetic field lines with full-disk coronal images from the Atmospheric Imaging Assembly (SDO/AIA) for both models. This study has found that the NLFFF model reconstructs the magnetic configuration closer to observation than the potential field model for full-disk magnetic field extrapolation. We conclude that many of the trans-equatorial loops connecting the two solar hemispheres are current-free.     

The global structure of the solar wind in June 1991

A numerical simulation of the global solar wind structure for Carrington rotation 1843 (31 May–28 June, 1991) is performed based on a fully three-dimensional, steady-state MHD model of the solar wind (Usmanov, 1993b). A self-consistent solution for 3-D MHD equations is constructed for the spherical shell extending from the solar photosphere up to 10 AU. Solar magnetic field observations are used to prescribe boundary conditions. The computed distribution of the magnetic field is compared with coronal hole observations and with the IMF measurements made by IMP-8 spacecraft at the Earth's orbit.     

Solar-interplanetary modeling: 3-D solar wind solutions in prescribed non-radial magnetic field geometries

A model is presented which describes the 3-dimensional non-radial solar wind expansion between the Sun and the Earth in a specified magnetic field configuration subject to synoptically observed plasma properties at the coronal base. In this paper, the field is taken to be potential in the inner corona based upon the Mt. Wilson magnetograph observations and radial beyond a certain chosen surface. For plasma boundary conditions at the Sun, we use deconvoluted density profiles obtained from synopticK-coronameter brightness observations. The temperature is taken to be 2 × 10⁶ K at the base of closed field lines and 1.6 x 10⁶K at the base of open field lines. For a sample calculation, we employ data taken during the period of the 12 November 1966 eclipse. Although qualitative agreement with observations at 1 AU is obtained, important discrepancies emerge which are not apparent from spherically symmetric models or those models which do not incorporate actual observations in the lower corona. These discrepancies appear to be due to two primary difficulties - the rapid geometric divergence of the open field lines in the inner corona as well as the breakdown in the validity of the Spitzer heat conduction formula even closer to the Sun than predicted by radial flow models. These two effects combine to produce conductively dominated solutions and lower velocities, densities, and field strengths at the Earth than those observed. The traditional difficulty in solar wind theory in that unrealistically small densities must be assumed at the coronal base in order to obtain observed densities at 1 AU is more than compensated for here by the rapid divergence of field lines in the inner corona. For these base conditions, the value ofβ(ratio of gas pressure to magnetic pressure) is shown to be significantly greater than one over most of the lower corona - suggesting that, for the coronal boundary conditions used here, the use of a potential or force-free magnetic field configuration may not be justified. The calculations of this paper point to the directions where future research on solar-interplanetary modelling should receive priority:

1. better models for the coronal magnetic field structure
2. improved understanding of the thermal conductivity relevant for the solar wind plasma.

     

Interplanetary magnetic field structure and solar wind parameters as inferred from solar magnetic field observations and by using a numerical 2-D MHD model

An attempt is made to infer parameters of the solar corona and the solar wind by means of a numerical, self-consistent MHD simulation. Boundary conditions for the magnetic field are given from the observations of the large-scale magnetic field at the Sun. A two-region, planar (the ecliptic plane is assumed) model for the solar wind flow is considered. Region I of transonic flow is assumed to cover the distances from the solar surface up to 10R _S (R _S is the radius of the Sun). Region II of supersonic, super-Alfvénic flow extends between 10R _S and the Earth's orbit. Treatment for region I is that for a mixed initial-boundary value problem. The solution procedure is similar to that discussed by Endler (1971) and Steinolfson, Suess, and Wu (1982): a steady-state solution is sought as a relaxation to the dynamic equilibrium of an initial state. To obtain a solution to the initial value problem in region II with the initial distribution of dependent variables at 10R _S (deduced from the solution for region I), a numerical scheme similar to that used by Pizzo (1978, 1982) is applied. Solar rotation is taken into account for region II; hence, the interaction between fast and slow solar wind streams is self-consistently treated. As a test example for the proposed formulation and numerical technique, a solution for the problem similar to that discussed by Steinolfson, Suess, and Wu (1982) is obtained. To demonstrate the applicability of our scheme to experimental data, solar magnetic field observations at Stanford University for Carrington rotation 1682 are used to prescribe boundary conditions for the magnetic field at the solar surface. The steady-state solution appropriate for the given boundary conditions was obtained for region I and then traced to the Earth's orbit through region II. We compare the calculated and spacecraft-observed solar wind velocity, radial magnetic field, and number density and find that general trends during the solar rotation are reproduced fairly well although the magnitudes of the density in comparison are vastly different.     

The height stratification of solar magnetic fields in cycles 21–23

The radial component of the solar magnetic field, Br, was calculated in the potential approximation in the height range from 1 to 2.5 solar radii, Ro. According to these data, synoptic maps of the magnetic field for solar cycles 21–23 were constructed. For each 10-degree latitudinal zone, the proportion of its area, S _+field, that was occupied by the “+” field in each rotation was found. In the entire latitudinal zone, the radial component of the field is assumed to be positive if S_+field ≥ 80% and negative if S _+field ≤ 20%. The field proved to be virtually unipolar at the level of the photosphere (R = Ro) during most of the cycle, from the poles to the north and south latitude ≈60°. In the vicinity of minimums between cycles 21 and 22, as well as cycles 22 and 23, for a few rotations of the Sun, the field was almost unipolar within the range of latitudes (?40°)-90°. At R = 2.5 Ro, for most of each cycle, the field was unipolar in the range of latitudes (?20°-(-90°)) and (20°–90°). According to our interpretation, the shift of the polar-field boundary to the equator with height reflects superradial expansion of open magnetic flux tubes from the polar coronal holes. It was found that the reversal of the polar fields began with 1–2 rotations and ended from 2 to 14 solar rotations earlier at great heights than at the surface of the Sun. This indicates that the reversal of the large-scale field occurs first and then that of the small-scale one. In the study of the sectoral structure of the magnetic field at different heights it was found that the boundaries that rotate with a period of less than the Carrington revolution extend to greater heights than the boundaries with a Carrington or longer period. We assume that the boundaries of the first type are formed by the large-scale structures of the magnetic field and the boundaries of the second type are determined by the active regions.     

Meridional flow in the solar wind in the presence of latitudinally dependent boundary conditions

The influence of latitudinally dependent boundary conditions on the large radius values of meridional flow in the distant solar wind is examined through a double perturbation expansion of the magnetohydrodynamic equations. A general result is derived for the meridional velocity which allows arbitrary specification of radial velocity, radial magnetic field, and mass flux, as a function of colatitude at some coronal reference surface. Three specific examples are treated, including the model of Pneuman and Kopp (1971). The latter example indicates that there may be flow toward the equator at large radii, as opposed to the pure equatorial divergence of internally generated motion due to a flow which is latitudinally uniform at the reference radius. A solar cycle effect most probably averages the boundary conditions so that only the equatorial divergence from an average spherically symmetric corona is seen in comet-tail observations. This may also explain the off-and-on-again nature of the meridional gradient in the radial velocity of the solar wind as seen in radio scintillation observations.     

Modeling of the magnetosphere of Mercury at the time of the first MESSENGER flyby

The MESSENGER spacecraft flyby of Mercury on 14 January 2008 provided a new opportunity to study the intrinsic magnetic field of the innermost planet and its interaction with the solar wind. The model presented in this paper is based on the solution of the three-dimensional, bi-fluid equations for solar wind protons and electrons in the absence of mass loading. In this study we provide new estimates of Mercury’s intrinsic magnetic field and the solar wind conditions that prevailed at the time of the flyby. We show that the location of the boundary layers and the strength of the magnetic field along the spacecraft trajectory can be reproduced with a solar wind ram pressure P_sw = 6.8 nPa and a planetary magnetic dipole having a magnitude of 210 R_M³ − nT and an offset of 0.18 R_M to the north of the equator, where R_M is Mercury’s radius. Analysis of the plasma flow reveals the existence of a stable drift belt around the planet; such a belt can account for the locations of diamagnetic decreases observed by the MESSENGER Magnetometer. Moreover, we determine that the ion impact rate at the northern cusp was four times higher than at the southern cusp, a result that provides a possible explanation for the observed north-south asymmetry in exospheric sodium in the neutral tail.     

On the stationary configuration of the heliospheric sheet

The expansion of solar coronal plasma is considered for the model described in Koutchmy et al. (1999). In addition to a spherical solar surface, the initial configuration represents a heliospheric sheet of dense plasma in the dipole equatorial plane. The heliospheric-sheet current decreases with distance as 1/r ², with its sign being opposite to the sign of the initial-dipole current. The latter follows from the fact that the plasma sheet is denser than the surrounding corona and that the equilibrium condition for the sheet in the gravitational and magnetic fields is satisfied. The field lines of this configuration are nearly straight. We have obtained a general solution of the steady-state MHD equations, which depends not only on distance r but also on latitude θ. Applicability of the solution to interpreting observational data, in particular, those obtained from the Ulysses spacecraft, is discussed.     

Global Solar Free Magnetic Energy and Electric Current Density Distribution of Carrington Rotation 2124

Solar eruptive phenomena, like flares and coronal mass ejections (CMEs), are governed by magnetic fields. To describe the structure of these phenomena one needs information on the magnetic flux density and the electric current density vector components in three dimensions throughout the atmosphere. However, current spectro-polarimetric measurements typically limit the determination of the vector magnetic field to only the photosphere. Therefore, there is considerable interest in accurate modeling of the solar coronal magnetic field using photospheric vector magnetograms as boundary data. In this work, we model the coronal magnetic field for global solar atmosphere using nonlinear force-free field (NLFFF) extrapolation codes implemented to a synoptic maps of photospheric vector magnetic field synthesized from the Vector Spectromagnetograph (VSM) on Synoptic Optical Long-term Investigations of the Sun (SOLIS) as boundary condition. Using the resulting three-dimensional magnetic field, we calculate the three-dimensional electric current density and magnetic energy throughout the solar atmosphere for Carrington rotation 2124 using our global extrapolation code. We found that spatially, the low-lying, current-carrying core field demonstrates a strong concentration of free energy in the active-region core, from the photosphere to the lower corona (about 70 Mm). The free energy density appears largely co-spatial with the electric current distribution.     

The Influence of Coronal Background Structure with Nested and Closed Magnetic Fields on the Triggering of CME

The influence of coronal streamer background with nested and closed magnetic fields on the of the triggering of coronal mass ejections (CMEs) is investigated in the meridian plane. In the coronal streamers’ background magnetic structure there are three small-scale closed magnetic fields, of which the middle one has a direction opposite to that of the global dipolar field of coronal streamers. The trigger model of CMEs emerges from beneath this small-scale closed magnetic field and possesses a concentric circular structure with radius of a = 0.1R_s (R_s being the solar radius). The direction of the magnetic field in the front half of the CME trigger model is opposite to that of the small-scale closed field and is the same as that of the streamers’ global dipolar field. As revealed by numerical simulation, when the ratio of the plasma pressure at the center of the CME trigger model to the boundary pressure is m ≤ 2, then the emerging model can trigger CMEs. When m < 2, then it cannot. The error in this critical value of 2 is less than 1%.     

Differential Rotation of the Solar Corona from Magnetic Field Data

A method for investigating the differential rotation of the solar corona using the coronal magnetic field as a tracer is proposed. The magnetic field is calculated in the potential approximation from observational data at the photospheric level. The time interval from June 24, 1976, to December 31, 2004, is considered. The magnetic field has been calculated for all latitudes from the equator to ±75? with a 5? step at distances from the base of the corona 1.0 R_⊙ to 2.45 R_⊙ near the source surface. The coronal rotation periods at 14 distances from the solar center have been determined by the method of periodogram analysis. The coronal rotation is shown to become progressively less differential with increasing heliocentric distance; it does not become rigid even near the source surface. The change in the coronal rotation periods with time is considered. At the cycleminimumthe rotation has been found to bemost differential, especially at small distances from the solar center. The change in coronal rotation with time is consistent with the tilt of the solar magnetic equator. The results from the magnetic field are compared with those obtained from the brightness of the green coronal Fe XIV 530.3 nm line. The consistency between these results confirms the reliability of the proposed method for studying the coronal rotation. Studying the rotation of the coronal magnetic field gives hope for the possibility of using this method to diagnose the differential rotation in subphotospheric layers.     

The solar corona as a minimum energy system?

This paper is an exploration of the possibility that the large-scale equilibrium of plasma and magnetic fields in the solar corona is a minimum energy state. Support for this conjecture is sought by considering the simplest form of that equilibrium in a dipole solar field, as suggested by the observed structure of the corona at times of minimum solar activity. Approximate, axisymmetric solutions to the MHD equations are constructed to include both a magnetically closed, hydrostatic region and a magnetically open region where plasma flows along field lines in the form of a transonic, thermally-driven wind. Sequences of such solutions are obtained for various degrees of magnetic field opening, and the total energy of each solution is computed, including contributions from both the plasma and magnetic field. It is shown that along a sequence of increasingly closed coronal magnetic field, the total energy curve is a non-monotonic function of the parameter measuring the degree of magnetic field opening, with a minimum occurring at moderate field opening.For reasonable choices of model parameters (coronal temperature, base density, base magnetic field strength, etc.), the morphology of the minimum energy solution resembles the observed quiet, solar minimum corona. The exact location energy minimum along a given sequence depends rather sensitively on some of the adopted parameter values. It is nevertheless argued that the existence of an energy minimum along the sequences of solutions should remain a robust property of more realistic coronal wind models that incorporate the basic characteristics of the equilibrium corona- the presence of both open and closed magnetic regions.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

A joined model for solar dynamo and differential rotation

A model for the solar dynamo, consistent in global flow and numerical method employed with the differential rotation model, is developed. The magnetic turbulent diffusivity is expressed in terms of the entropy gradient, which is controlled by the model equations. The magnetic Prandtl number and latitudinal profile of the alpha-effect are specified by fitting the computed period of the activity cycle and the equatorial symmetry of magnetic fields to observations. Then, the instants of polar field reversals and time-latitude diagrams of the fields also come into agreement with observations. The poloidal field has a maximum amplitude of about 10 Gs in the polar regions. The toroidal field of several thousand Gauss concentrates near the base of the convection zone and is transported towards the equator by the meridional flow. The model predicts a value of about 10³⁷ erg for the total magnetic energy of large-scale fields in the solar convection zone.     

Reflection Properties of Gravito-MHD Waves in an Inhomogeneous Horizontal Magnetic Field

We derive the dispersion equation for gravito-magnetohydrodynamical (MHD) waves in an isothermal, gravitationally stratified plasma with a horizontal inhomogeneous magnetic field. Sound and Alfvén speeds are constant. Under these conditions, it is possible to derive analytically the equations for gravito-MHD waves. The high values of the viscous and magnetic Reynolds numbers in the solar atmosphere imply that the dissipative terms in the MHD equations are negligible, except in layers around the positions where the frequency of the MHD wave equals the local Alfvén or slow wave frequency. Outside these layers the MHD waves are accurately described by the equations of ideal MHD. We consider waves that propagate energy upward in the atmosphere. For the plane boundary, z=0, between two isothermal plasma regions with horizontal but different magnetic fields, we discuss the boundary conditions and derive the equations for the reflection and transmission coefficients. In the simpler case of a gravitationally stratified plasma without magnetic field, these coefficients describe the reflection and transmission properties of gravito-acoustic waves.     

Kink mode <Emphasis Type="Italic">m</Emphasis> = 1 in a thin plasma filament with discontinuous vertical magnetic field

In this paper, we study the conditions of realization and stability of kink modes with azimuthal wave numbers m = ±1 in a cylindrical plasma filament with a twisted magnetic field and a homogeneous current along its axis. We assume that there are vertical constant magnetic fields inside and outside of the filament; the filament is surrounded by current-free plasma; and outside of its boundary, the azimuthal magnetic field decreases inversely in proportion to the distance from the filament’s border. The dispersion equations for stable and unstable modes are obtained in the approximation of “thin” plasma filament. The analysis of the equations for the case of discontinuous vertical magnetic field at the filament’s boundary is provided. The conditions of propagation of the wave modes have been defined. We have obtained that the unstable modes with m = ±1 cannot be realized. The results of this work can be applied to the interpretation of the solar magnetic flux tubes’ behavior using measurements provided by the spacecrafts.     

Different Types of Coronal Mass Ejections At Minimum and Maximum of Solar Activity and Their Relation to Magnetic Field Evolution

Coronal mass ejections (CMEs) are considered to be associated with large-scale, closed magnetic field structures in the corona. These structures change throughout the solar activity cycle following the evolution of the general solar magnetic field. To study the variation of CME characteristics with the evolution of coronal magnetic structures, we compute the 3-D coronal magnetic field at minimum and maximum of activity with a source-surface potential field model. In particular, we study the central latitude distribution of CMEs and the frequency of occurrence of the different CME types in these two periods. We find that most CMEs are indeed associated with large-scale, magnetically closed structures, and their latitudinal distribution follows the solar cycle latitudinal changes of the location of these structures. We also find that different CME types, which constitute different fractions of the total during the maximum and the minimum, are associated with different shapes and orientations of the closed structures at different times of the solar cycle.     

Helium abundance variations in the solar wind

Helium abundance variations in the solar wind have been studied using data obtained with Los Alamos plasma instrumentation on IMP 6, 7, and 8 from 1971 through 1978. For the first time, average flow characteristics have been determined as a function of helium abundance, A(He). Low and average values of A(He) are each preferentially identified with a different characteristic plasma ‘state’ these correspond to what have previously been recognized as the signatures of interplanetary magnetic field polarity reversals and high speed streams, respectively. Helium enhancements at 1 AU also can be identified with a characteristic plasma state, which includes high magnetic field intensity and low proton temperature. This is further evidence that such enhancements are a signal of coronal transient mass ejections. Long-term averages of A(He) at least partially reflect the relative frequency with which coronal streamers, holes, and transients extend their influence into the ecliptic plane at 1 AU. As a result, there is a real and pronounced solar cycle variation of solar wind H(He).     

The mean coronal magnetic field determined from HELIOS Faraday rotation measurements

Coronal Faraday rotation of the linearly polarized carrier signals of the HELIOS spacecraft was recorded during the regularly occurring solar occultations over almost a complete solar cycle from 1975 to 1984. These measurements are used to determine the average strength and radial variation of the coronal magnetic field at solar minimum at solar distances from 3–10 solar radii, i.e., the range over which the complex fields at the coronal base are transformed into the interplanetary spiral. The mean coronal magnetic field in 1975–1976 was found to decrease with radial distance according to r ^?α, where α = 2.7 ± 0.2. The mean field magnitude was 1.0 ± 0.5 × 10 ^?5 tesla at a nominal solar distance of 5 solar radii. Possibly higher magnetic field strengths were indicated at solar maximum, but a lack of data prevented a statistical determination of the mean coronal field during this epoch.     

On Thermal-Pulse-Driven Plasma Flows in Coronal Funnels as Observed by the Hinode/EUV Imaging Spectrometer (EIS)

Using one-arcsecond-slit-scan observations from the Hinode/EUV Imaging Spectrometer (EIS) on 5 February 2007, we find the plasma outflows in the open and expanding coronal funnels at the eastern boundary of AR 10940. The Doppler-velocity map of Fe?xii 195.120 Å shows the diffuse closed-loop system to be mostly red-shifted. The open arches (funnels) at the eastern boundary of AR exhibit blue-shifts with a maximum speed of about 10?–?15 km?s^?1. This implies outflowing plasma through these magnetic structures. In support of these observations, we perform a 2D numerical simulation of the expanding coronal funnels by solving the set of ideal MHD equations in appropriate VAL-III C initial temperature conditions using the FLASH code. We implement a rarefied and hotter region at the footpoint of the model funnel, which results in the evolution of slow plasma perturbations propagating outward in the form of plasma flows. We conclude that the heating, which may result from magnetic reconnection, can trigger the observed plasma outflows in such coronal funnels. This can transport mass into the higher corona, giving rise to the formation of the nascent solar wind.     

Simulation of the Unusual Solar Minimum with 3D SIP-CESE MHD Model by Comparison with Multi-Satellite Observations

The observations both near the Sun and in the heliosphere during the activity minimum between solar cycles 23 and 24 exhibit different phenomena from those typical of the previous solar minima. In this paper, we have chosen Carrington rotation 2070 in 2008 to investigate the properties of the background solar wind by using the three-dimensional (3D) Solar?CInterPlanetary Conservation Element/Solution Element Magnetohydrodynamic (MHD) model. We also study the effects of polar magnetic fields on the characteristics of the solar corona and the solar wind by conducting simulations with an axisymmetric polar flux added to the observed magnetic field. The numerical results are compared with the observations from multiple satellites, such as the Solar and Heliospheric Observatory (SOHO), Ulysses, Solar Terrestrial Relations Observatory (STEREO), Wind and the Advanced Composition Explorer (ACE). The comparison demonstrates that the first simulation with the observed magnetic fields reproduces some observed peculiarities near the Sun, such as relatively small polar coronal holes, the presence of mid- and low-latitude holes, a tilted and warped current sheet, and the broad multiple streamers. The numerical results also capture the inconsistency between the locus of the minimum wind speed and the location of the heliospheric current sheet, and predict slightly slower and cooler polar streams with a relatively smaller latitudinal width, broad low-latitude intermediate-speed streams, and globally weak magnetic field and low density in the heliosphere. The second simulation with strengthened polar fields indicates that the weak polar fields in the current minimum play a crucial role in determining the states of the corona and the solar wind.