M. S. Wheatland

An Optimization Approach to Reconstructing Force-free Fields

A new method for reconstructing force-free magnetic fields from their boundary values, based on minimizing the global departure of an initial field from a force-free and solenoidal state, is presented. The method is tested by application to a known nonlinear solution. We discuss the obstacles to be overcome in the application of this method to the solar case: the reconstruction of force-free fields in the corona from measurements of the vector magnetic field in the low atmosphere.

A Critical Assessment of Nonlinear Force-Free Field Modeling of the Solar Corona for Active Region 10953

Nonlinear force-free field (NLFFF) models are thought to be viable tools for investigating the structure, dynamics, and evolution of the coronae of solar active regions. In a series of NLFFF modeling studies, we have found that NLFFF models are successful in application to analytic test cases, and relatively successful when applied to numerically constructed Sun-like test cases, but they are less successful in application to real solar data. Different NLFFF models have been found to have markedly different field line configurations and to provide widely varying estimates of the magnetic free energy in the coronal volume, when applied to solar data. NLFFF models require consistent, force-free vector magnetic boundary data. However, vector magnetogram observations sampling the photosphere, which is dynamic and contains significant Lorentz and buoyancy forces, do not satisfy this requirement, thus creating several major problems for force-free coronal modeling efforts. In this paper, we discuss NLFFF modeling of NOAA Active Region 10953 using Hinode/SOT-SP, Hinode/XRT, STEREO/SECCHI-EUVI, and SOHO/MDI observations, and in the process illustrate three such issues we judge to be critical to the success of NLFFF modeling: (1) vector magnetic field data covering larger areas are needed so that more electric currents associated with the full active regions of interest are measured, (2) the modeling algorithms need a way to accommodate the various uncertainties in the boundary data, and (3) a more realistic physical model is needed to approximate the photosphere-to-corona interface in order to better transform the forced photospheric magnetograms into adequate approximations of nearly force-free fields at the base of the corona. We make recommendations for future modeling efforts to overcome these as yet unsolved problems.

Nonlinear Force‐free Field Modeling of a Solar Active Region around the Time of a Major Flare and Coronal Mass Ejection

Solar flares and coronal mass ejections are associated with rapid changes in field connectivity and are powered by the partial dissipation of electrical currents in the solar atmosphere. A critical unanswered question is whether the currents involved are induced by the motion of preexisting atmospheric magnetic flux subject to surface plasma flows or whether these currents are associated with the emergence of flux from within the solar convective zone. We address this problem by applying state-of-the-art nonlinear force-free field (NLFFF) modeling to the highest resolution and quality vector-magnetographic data observed by the recently launched Hinode satellite on NOAA AR 10930 around the time of a powerful X3.4 flare. We compute 14 NLFFF models with four different codes and a variety of boundary conditions. We find that the model fields differ markedly in geometry, energy content, and force-freeness. We discuss the relative merits of these models in a general critique of present abilities to model the coronal magnetic field based on surface vector field measurements. For our application in particular, we find a fair agreement of the best-fit model field with the observed coronal configuration, and argue (1) that strong electrical currents emerge together with magnetic flux preceding the flare, (2) that these currents are carried in an ensemble of thin strands, (3) that the global pattern of these currents and of field lines are compatible with a large-scale twisted flux rope topology, and (4) that the ~1032 erg change in energy associated with the coronal electrical currents suffices to power the flare and its associated coronal mass ejection.

The origin of the solar flare waiting-time distribution

It was recently pointed out that the distribution of times between solar flares (the flare waiting-time distribution) follows a power law for long waiting times. Based on 25 years of soft X-ray flares observed by Geostationary Operational Environmental Satellite instruments, it is shown that (1) the waiting-time distribution of flares is consistent with a time-dependent Poisson process and (2) the fraction of time the Sun spends with different flaring rates approximately follows an exponential distribution. The second result is a new phenomenological law for flares. It is shown analytically how the observed power-law behavior of the waiting times originates in the exponential distribution of flaring rates. These results are argued to be consistent with a nonstationary avalanche model for flares.

The Waiting-Time Distribution of Solar Flare Hard X-Ray Bursts

A waiting-time distribution is constructed for 8 yr of solar flare hard X-ray bursts observed by the ICE/ISEE 3 spacecraft. The observed distribution is compared with a simulated waiting-time distribution produced by a time-dependent Poisson process constructed using rates estimated from the observations. The observed distribution shows an overabundance of short waiting times (10 s-10 minutes) in comparison with the simulation. This result implies that the hard X-ray bursts are not independent events. The implications of this result for the existence of sympathetic flaring and to models of flare statistics are discussed, and the result is compared with previous determinations of waiting-time distributions for solar hard X-ray events.

Metastable Magnetic Configurations and Their Significance for Solar Eruptive Events

Solar flares and coronal mass ejections (CMEs) involve the sudden release of magnetic energy that can lead to the ejection from the Sun of large masses of gas with entrained magnetic field. In dynamical systems, such sudden events are characteristic of metastable configurations that are stable against small perturbations but unstable to sufficiently large perturbations. Linear stability analysis indicates whether or not the first requirement is met, and energetic analysis can indicate whether or not the second requirement is met: if a magnetic configuration that is stable against small perturbations can make a transition to a lower energy state, then it is metastable. In this paper, we consider a long twisted flux tube, anchored at both ends in the photosphere and restrained by an overlying magnetic arcade. We argue from a simple order-of-magnitude calculation that, for appropriate parameter values, it is energetically favorable for part of the flux tube to erupt into interplanetary space, even when the configuration is stable according to linear MHD stability theory. The properties of metastable magnetic configurations may be relevant to CMEs and to other explosive astrophysical events such as solar flares.

A Bayesian Approach to Solar Flare Prediction

A number of methods of flare prediction rely on classification of physical characteristics of an active region, in particular optical classification of sunspots, and historical rates of flaring for a given classification. However, these methods largely ignore the number of flares the active region has already produced, in particular the number of small events. The past history of occurrence of flares (of all sizes) is an important indicator of future flare production. We present a Bayesian approach to flare prediction, which uses the flaring record of an active region together with phenomenological rules of flare statistics to refine an initial prediction for the occurrence of a big flare during a subsequent period of time. The initial prediction is assumed to come from one of the extant methods of flare prediction. The theory of the method is outlined, and simulations are presented to show how the refinement step of the method works in practice.

Rotational Signature and Possible r-Mode Signature in the GALLEX Solar Neutrino Data

Recent analysis of the Homestake data has yielded evidence that the solar neutrino flux varies in time—more specifically, that it exhibits a periodic variation that may be attributed to rotational modulation occurring deep in the solar interior, either in the tachocline or in the radiative zone. Here we present a spectral analysis of the GALLEX data that yields supporting evidence for this rotational modulation. The most prominent peak in the power spectrum occurs at the synodic frequency of 13.08 yr (cycles per year) and is estimated to be significant 21

USING CORONAL LOOPS TO RECONSTRUCT THE MAGNETIC FIELD OF AN ACTIVE REGION BEFORE AND AFTER A MAJOR FLARE

The shapes of solar coronal loops are sensitive to the presence of electrical currents that are the carriers of the non-potential energy available for impulsive activity. We use this information in a new method for modeling the coronal magnetic field of active region (AR) 11158 as a nonlinear force-free field (NLFFF). The observations used are coronal images around the time of major flare activity on 2011 February 15, together with the surface line-of-sight magnetic field measurements. The data are from the Helioseismic and Magnetic Imager and Atmospheric Imaging Assembly on board the Solar Dynamics Observatory. The model fields are constrained to approximate the coronal loop configurations as closely as possible, while also being subject to the force-free constraints. The method does not use transverse photospheric magnetic field components as input and is thereby distinct from methods for modeling NLFFFs based on photospheric vector magnetograms. We validate the method using observations of AR 11158 at a time well before major flaring and subsequently review the field evolution just prior to and following an X2.2 flare and associated eruption. The models indicate that the energy released during the instability is about 1 × 1032 erg, consistent with what is needed to power such a large eruptive flare. Immediately prior to the eruption, the model field contains a compact sigmoid bundle of twisted flux that is not present in the post-eruption models, which is consistent with the observations. The core of that model structure is twisted by ≈0.9 full turns about its axis.

A self-consistent nonlinear force-free solution for a solar active region magnetic field

Nonlinear force-free solutions for the magnetic field in the solar corona constructed using photospheric vector magnetic field boundary data suffer from a basic problem: the observed boundary data are inconsistent with the nonlinear force-free model. Specifically, there are two possible choices of boundary conditions on vertical current provided by the data, and the two choices lead to different force-free solutions. A novel solution to this problem is described. Bayesian probability is used to modify the boundary values on current density, using field-line connectivity information from the two force-free solutions and taking into account uncertainties, so that the boundary data are more consistent with the two nonlinear force-free solutions. This procedure may be iterated until a set of self-consistent boundary data (the solutions for the two choices of boundary conditions are the same) is achieved. The approach is demonstrated to work in application to Hinode/Solar Optical Telescope observations of NOAA active region 10953.