Modeling coronal magnetic field using spherical geometry: cases with several active regions

The magnetic fields in the solar atmosphere structure the plasma, store free magnetic energy and produce a wide variety of active solar phenomena, like flare and coronal mass ejections (CMEs). The distribution and strength of magnetic fields are routinely measured in the solar surface (photosphere). Therefore, there is considerable interest in accurately modeling the 3D structure of the coronal magnetic field using photospheric vector magnetograms. Knowledge of the 3D structure of magnetic field lines also help us to interpret other coronal observations, e.g., EUV images of the radiating coronal plasma. Nonlinear force-free field (NLFFF) models are thought to be viable tools for those task. Usually those models use Cartesian geometry. However, the spherical nature of the solar surface cannot be neglected when the field of view is large. In this work, we model the coronal magnetic field above multiple active regions using NLFFF extrapolation code using vector magnetograph data from the Synoptic Optical Long-term Investigations of the Sun survey (SOLIS)/Vector Spectromagnetograph (VSM) as a boundary conditions. We compare projections of the resulting magnetic field lines solutions with their respective coronal EUV-images from the Atmospheric Imaging Assembly (SDO/AIA) observed on October 15, 2011 and November 13, 2012. This study has found that the NLFFF model in spherical geometry reconstructs the magnetic configurations for several active regions which agrees to some extent with observations. During October 15, 2011 observation, there are substantial number of trans-equatorial loops carrying electric current.     

A numerical study of the pre-ejection, magnetically-sheared corona as a free boundary problem

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.     

Catastrophe of coronal magnetic rope in partly open multipolar magnetic field

Catastrophe of coronal magnetic rope embedded in a partly open multipolar background magnetic field is studied by using a 2-dimensional, 3-component ideal MHD model in spherical coordinates. The background field is composed of three closed bipolar fields of a coronal streamer and an open field with an equatorial current sheet. The magnetic rope lies below the central bipolar field, and it is characterized by its annular and axial magnetic fluxes. For a given annual flux, there is a critical value of the axial flux, and for a given axial flux, there is a critical value of annual flux such that, below the critical value, the magnetic rope is attached to the solar surface and the system stays in equilibrium, but when the critical value is exceeded, the magnetic rope breaks free and erupts upward. This implies that catastrophe can occur in a coronal magnetic rope embedded in a partly open multipolar background magnetic field. Our computation gives a threshold value of magnetic energy that is about 15% greater than the energy of the partly open magnetic field (the central bipolar field open and the fields on either side closed). The excess energy may serve as source for solar explosions such as coronal mass ejections.     

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.     

MHD waves at a spherical interface modelling EIT waves

Eruptive events such as flares and coronal mass ejections (CMEs) are known to generate global waves propagating over distances comparable to the solar radius in different layers of the solar atmosphere. Here we investigate the propagation of coronal EIT waves, modelled as fast magnetoacoustic modes propagating at a spherical interface in the presence of a purely radial magnetic field. Based on a simplified equilibrium we derive the dispersion relation of the waves. The generation and propagation of EIT waves at the spherical interface is studied numerically for different values of spherical degree and preliminary conclusions are reached regarding the properties of EIT waves. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Advanced Technology Solar Telescope

High-resolution studies of the Sun’s magnetic fields are needed for a better understanding of the fundamental processes responsible for solar variability. The generation of magnetic fields through dynamo processes, the amplification of fields through the interaction with plasma flows, and the destruction of fields are poorly understood. There is incomplete insight into physical mechanisms responsible for chromospheric and coronal structure and heating, causes of variations in the radiative output of the Sun, and mechanisms that trigger flares and coronal mass ejections. Progress in answering these critical questions requires study of the interaction of the magnetic field and convection with a resolution sufficient to observe scale fundamental to these processes. The planned 4 m aperture ATST will be a unique scientific tool, with excellent angular resolution, a large wavelength range, and low scattered light. With its integrated adaptive optics, the ATST will achieve a spatial resolution nearly 10 times better than any existing solar telescope. The ATST design and development phase began in 2001 and it is now ready to begin construction in 2009.     

Coronal Mass Ejections as Expanding Force-Free Structures

We model solar coronal mass ejections (CMEs) as expanding force-free magnetic structures and find the self-similar dynamics of configurations with spatially constant ??, where J=?? B, in spherical and cylindrical geometries, expanding spheromaks and Lundquist fields, respectively. The field structures remain force-free, under the conventional non-relativistic assumption that the dynamical effects of the inductive electric fields can be neglected. While keeping the internal magnetic field structure of the stationary solutions, expansion leads to complicated internal velocities and rotation, caused by inductive electric fields. The structure depends only on overall radius R(t) and rate of expansion $\dot{R}(t)$ measured at a given moment, and thus is applicable to arbitrary expansion laws. In case of cylindrical Lundquist fields, magnetic flux conservation requires that both axial and radial expansion proceed with equal rates. In accordance with observations, the model predicts that the maximum magnetic field is reached before the spacecraft reaches the geometric center of a CME.     

ICME Identification from Solar Wind Ion Measurements

Interplanetary coronal mass ejections (ICMEs), the interplanetary counterpart of coronal mass ejections (CMEs), are most commonly identified by their enhanced magnetic field strengths and rotating magnetic field orientation. However, there are other frequent signatures in the plasma. We use a pair of these signatures, a linearly decreasing plasma bulk velocity and a cool (< 20 km s⁻¹) ion thermal speed, to identify candidate ICMEs. Many ICMEs, identified through their magnetic signatures, are also found by their ion signatures alone. However, many are not. These missed ICMEs appear not to be expanding, even when they are accompanied by leading shocks. The ICMEs with both the magnetic and ion signatures appear to be expanding as judged from either set of observations. The most clearly defined ICMEs have transit times from the Sun and growth times to the observed size that are equal. These ropes fit the paradigm of compact magnetic structures arising low in the corona and expanding uniformly in time, as they travel at constant center of mass speed toward 1 AU.     

Photospheric and heliospheric magnetic fields

The magnetic field in the heliosphere evolves in response to the photospheric field at its base. This evolution, together with the rotation of the Sun, drives space weather through the continually changing conditions of the solar wind and the magnetic field embedded within it. We combine observations and simulations to investigate the sources of the heliospheric field from 1996 to 2001. Our algorithms assimilate SOHO/MDI magnetograms into a flux-dispersal model, showing the evolving field on the full sphere with an unprecedented duration of 5.5 yr and temporal resolution of 6 hr. We demonstrate that acoustic far-side imaging can be successfully used to estimate the location and magnitude of large active regions well before they become visible on the solar disk. The results from our assimilation model, complemented with a potential-field source-surface model for the coronal and inner-heliospheric magnetic fields, match Yohkoh/SXT and KPNO/He?10830 Å coronal hole boundaries quite well. Even subject to the simplification of a uniform, steady solar wind from the source surface outward, our model matches the polarity of the interplanetary magnetic field (IMF) at Earth ～3% of the time during the period 1997–2001 (independent of whether far-side acoustic data are incorporated into the simulation). We find that around cycle maximum, the IMF originates typically in a dozen disjoint regions. Whereas active regions are often ignored as a source for the IMF, the fraction of the IMF that connects to magnetic plage with absolute flux densities exceeding 50 Mx cm^?2 increases from ?10% at cycle minimum up to 30–50% at cycle maximum, with even direct connections between sunspots and the heliosphere. For the overall heliospheric field, these fractions are ?1% to 20–30%, respectively. Two case studies based on high-resolution TRACE observations support the direct connection of the IMF to magnetic plage, and even to sunspots. Parallel to the data assimilation, we run a pure simulation in which active regions are injected based on random selection from parent distribution functions derived from solar data. The global properties inferred for the photospheric and heliospheric fields for these two models are in remarkable agreement, confirming earlier studies that no subtle flux-emergence patterns or field-dispersal properties are required of the solar dynamo beyond those that are included in the model in order to understand the large-scale solar and heliospheric fields.

     

Are interplanetary magnetic clouds manifestations of coronal transients at 1 AU?

Using proxy data for the occurrence of those mass ejections from the solar corona which are directed earthward, we investigate the association between the post-1970 interplanetary magnetic clouds of Klein and Burlaga (1982) and coronal mass ejections. The evidence linking magnetic clouds following shocks with coronal mass ejections is striking; six of nine clouds observed at Earth were preceded an appropriate time earlier by meter-wave type II radio bursts indicative of coronal shock waves and coronal mass ejections occurring near central meridian. During the selected control periods when no clouds were detected near Earth, the only type II bursts reported were associated with solar activity near the limbs. Where the proxy solar data to be sought are not so clearly suggested, that is, for clouds preceding interaction regions and clouds within cold magnetic enhancements, the evidence linking the clouds and coronal mass ejections is not as clear; proxy data usually suggest many candidate mass-ejection events for each cloud. Overall, the data are consistent with and support the hypothesis suggested by Klein and Burlaga that magnetic clouds observed with spacecraft at 1 AU are manifestations of solar coronal mass ejection transients.     

X-ray Ejecta,White-Light CMEs and a Coronal Shock Wave

We report on a coronal shock wave inferred from the metric type II burst of 13 January 1996. To identify the shock driver, we examined mass motions in the form of X-ray ejecta and white-light coronal mass ejections (CMEs). None of the ejections could be considered fast (> 400 km s^–1) events. In white light, two CMEs occurred in quick succession, with the first one associated with X-ray ejecta near the solar surface. The second CME started at an unusually large height in the corona and carried a dark void in it. The first CME decelerated and stalled while the second one accelerated, both in the coronagraph field of view. We identify the X-ray ejecta to be the driver of the coronal shock inferred from metric type II burst. The shock speed reported in the Solar Geophysical Data (1000–2000 km s^–1) seems to be extremely large compared to the speeds inferred from X-ray and white-light observations. We suggest that the MHD fast-mode speed in the inner corona could be low enough that the X-ray ejecta is supermagnetosonic and hence can drive a shock to produce the type II burst.     

The Magnetic Field at the Inner Boundary of the Heliosphere Around Solar Minimum

STEREO A and B observations of the radial magnetic field between 1 January 2007 and 31 October 2008 show significant evidence that in the heliosphere, the ambient radial magnetic field component with any dynamic effects removed is uniformly distributed. Based on this monopolar nature of the ambient heliospheric field we find that the surface beyond which the magnetic fields are in the monopolar configuration must be spherical, and this spherical surface can be defined as the inner boundary of the heliosphere that separates the monopole-dominated heliospheric magnetic field from the multipole-dominated coronal magnetic field. By using the radial variation of the coronal helmet streamers belts and the horizontal current – current sheet – source surface model we find that the spherical inner boundary of the heliosphere should be located around 14 solar radii near solar minimum phase.     

Magnetic Energy of Force-Free Fields with Detached Field Lines

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

Coronal Magnetic Field Structure and Evolution for Flaring AR 11117 and Its Surroundings

In this study, photospheric vector magnetograms obtained with the Synoptic Optical Long-term Investigations of the Sun (SOLIS) survey are used as boundary conditions to model three-dimensional nonlinear force-free (NLFF) coronal magnetic fields as a sequence of NLFF equilibria in spherical geometry. We study the coronal magnetic field structure inside an active region and its temporal evolution. We compare the magnetic field configuration obtained from NLFF extrapolation before and after the flaring event in active region (AR) 11117 and its surroundings observed on 27 October 2010, and we also compare the magnetic field topologies and the magnetic energy densities and study the connectivities between AR 11117 and its surroundings. During the investigated time period, we estimate the change in free magnetic energy from before to after the flare to be 1.74×10³²?erg, which represents about 13.5?% of the NLFF magnetic energy before the flare. In this study, we find that electric currents from AR 11117 to its surroundings were disrupted after the flare.     

Coronal Mass Ejections and the Index of Effective Solar Multipole

The paper considers the relationship between the cyclic variations in the velocity of coronal mass ejections (CME) and the large-scale magnetic field structure (LSMF) in cycles 21??C?23. To characterize a typical size of the LSMF structure, we have used the index of the effective solar multipole (ESMI). The cyclic behavior of the CME occurrence rate and velocity proved to be similar to that of ESMI. The hysteresis observed in variations of the CME maximum velocity is interpreted as a manifestation of different contributions from the two field structures (local and global magnetic fields) in different phases of the 11-year activity cycle. It is suggested that cyclic variations in the maximum velocity of coronal mass ejections are due to different conditions for the formation of the complexes of active regions connected by coronal arch systems, which are the main source of high-velocity CMEs.     

Microwave Mode Coupling Above Active Regions as a Coronal Density Diagnostic

It is well recognized that the phenomenon of depolarization (the conversion of polarized radio emission into unpolarized emission) of microwaves over solar active regions can be used to infer the coronal electron density once the coronal magnetic field is known. In this paper we explore this technique using an active region for which we have excellent radio data showing depolarization at two frequencies, and for which we have an excellent magnetic field model which has been tested against observations. We show that this technique for obtaining coronal densities is very sensitive to a number of factors. When Cohen's (1960) theory where depolarization is due to magnetic field rotation alone is used, the result is particularly sensitive to the location of the surface on which the magnetic field is orthogonal to the line of sight. Depending on whether we take into account the presence of electric currents in the photosphere or not, their extrapolation into the corona can result in very different heights being deduced for the location of the depolarization strip, and this changes the density which is then deduced from the depolarization condition. Such extreme sensitivity to the magnetic field model requires that field extrapolations be able to accurately predict the polarity of magnetic fields up to coronal heights as high as 10⁵ km in order to exploit depolarization as a density diagnostic.     

3-D non-linear evolution of a magnetic flux tube in a spherical shell: The isentropic case

We present recent 3-D MHD numerical simulations of the non-linear dynamical evolution of magnetic flux tubes in an adiabatically stratified convection zone in spherical geometry, using the anelastic spherical harmonic (ASH) code.We seek to understand the mechanism of emergence of strong toroidal fields from the base of the solar convection zone to the solar surface as active regions. We confirm the results obtained in cartesian geometry that flux tubes that are not twisted split into two counter vortices before reaching the top of the convection zone. Moreover, we find that twisted tubes undergo the poleward-slip instability due to an unbalanced magnetic curvature force which gives the tube a poleward motion both in the non-rotating and in the rotating case. This poleward drift is found to be more pronounced on tubes originally located at high latitudes. Finally, rotation is found to decrease the rise velocity of the flux tubes through the convection zone, especially when the tube is introduced at low latitudes. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Solar filaments as tracers of subsurface processes

Solar filaments are discussed in terms of two contrasting paradigms. The standard paradigm is that filaments are formed by condensation of coronal plasma into magnetic fields that are twisted or dimpled as a consequence of motions of the fields&amp;#x2019; sources in the photosphere. According to a new paradigm, filaments form in rising, twisted flux ropes and are a necessary intermediate stage in the transfer to interplanetary space of dynamo-generated magnetic flux. It is argued that the accumulation of magnetic helicity in filaments and their coronal surroundings leads to filament eruptions and coronal mass ejections. These ejections relieve the Sun of the flux generated by the dynamo and make way for the flux of the next cycle.     

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%.