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Achieving Self-Consistent Nonlinear Force-free Modeling of Solar Active Regions  

Mike Wheatland   Submitted: 2010-12-16 16:07

A nonlinear force-free solution is constructed for the coronal magnetic field in NOAA solar active region AR 10953 based on a photospheric vector magnetogram derived from Hinode satellite observations on 30 April 2007, taking into account uncertainties in the boundary data and using improved methods for merging multiple-instrument data. The solution demonstrates the ''self-consistency'' procedure of Wheatland & Regnier (2009), for the first time including uncertainties. The self-consistency procedure addresses the problem that photospheric vector magnetogram data are inconsistent with the force-free model, and in particular that the boundary conditions on vertical electric current density are over-specified and permit the construction of two different nonlinear force-free solutions. The procedure modifies the boundary conditions on current density during a sequence of cycles until the two nonlinear force-free solutions agree. It hence constructs an accurate single solution to the force-free model, with boundary values close, but not matched exactly, to the vector magnetogram data. The inclusion of uncertainties preserves the boundary conditions more closely at points with smaller uncertainties. The self-consistent solution obtained for active region AR 10953 is significantly non-potential, with magnetic energy E/E_0 = 1.08, where E_0 is the energy of the reference potential (current-free) magnetic field. The self-consistent solution is shown to be robust against changes in the details of the construction of the two force-free models at each cycle. This suggests that reliable nonlinear force-free modeling of active regions is possible if uncertainties in vector magnetogram boundary data are included.

Authors: M.S. Wheatland & K.D. Leka
Projects: Hinode/SOT,SoHO-MDI

Publication Status: To appear in the Astrophysical Journal (accepted 15 Dec 2010)
Last Modified: 2010-12-17 09:40
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Time-dependent Stochastic Modeling of Solar Active Region Energy  

Mike Wheatland   Submitted: 2010-08-03 12:09

A time-dependent model for the energy of a flaring solar active region is presented based on an existing stochastic jump-transition model (Wheatland and Glukhov 1998; Wheatland 2008; Wheatland 2009). The magnetic free energy of an active region is assumed to vary in time due to a prescribed (deterministic) rate of energy input and prescribed (random) jumps downwards in energy due to flares. The existing model reproduces observed flare statistics, in particular flare frequency-size and waiting-time distributions, but modeling presented to date has considered only the time-independent choices of constant energy input and constant flare transition rates with a power-law distribution in energy. These choices may be appropriate for a solar active region producing a constant mean rate of flares. However, many solar active regions exhibit time variation in their flare productivity, as exemplified by NOAA active region AR 11029, observed during October-November 2009 (Wheatland 2010). Time variation is incorporated into the jump-transition model for two cases: 1. a step change in the rates of flare transitions; and 2. a step change in the rate of energy supply to the system. Analytic arguments are presented describing the qualitative behavior of the system in the two cases. In each case the system adjusts by shifting to a new stationary state over a relaxation time which is estimated analytically. The model exhibits flare-like event statistics. In each case the frequency-energy distribution is a power law for flare energies less than a time-dependent rollover set by the largest energy the system is likely to attain at a given time. The rollover is not observed if the mean free energy of the system is sufficiently large. For Case 1, the model exhibits a double exponential waiting-time distribution, corresponding to flaring at a constant mean rate during two intervals (before and after the step change), if the average energy of the system is large. For Case 2 the waiting-time distribution is a simple exponential, again provided the average energy of the system is large. Monte Carlo simulations of Case 1 are presented which confirm the estimate for the relaxation time, and confirm the expected forms of the frequency-energy and waiting-time distributions. The simulation results provide a qualitative model for observed flare statistics in active region AR 11029.

Authors: M. Kanazir and M.S. Wheatland
Projects: None

Publication Status: To appear in Solar Physics (accepted 31 July 2010)
Last Modified: 2010-08-04 06:37
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Evidence for departure from a power-law flare size distribution for a small solar active region  

Mike Wheatland   Submitted: 2010-01-09 17:46

Active region 11029 was a small, highly flare-productive solar active region observed at a time of extremely low solar activity. The region produced only small flares: the largest of the >70 Geostationary Observational Environmental Satellite (GOES) events for the region has a peak 1-8mbox{AA} flux of 2.2 imes 10-6,{ m W},{ m m}-2 (GOES C2.2). The background-subtracted GOES peak-flux distribution suggests departure from power-law behavior above 10-6,{ m W},{ m m}-2, and a Bayesian model comparison strongly favors a power-law plus rollover model for the distribution over a simple power-law model. The departure from the power law is attributed to this small active region having a finite amount of energy. The rate of flaring in the region varies with time, becoming very high for two days coinciding with the onset of an increase in complexity of the photospheric magnetic field. The observed waiting-time distribution for events is consistent with a piecewise-constant Poisson model. These results present challenges for models of flare statistics and of energy balance in solar active regions.

Authors: M.S. Wheatland
Projects: None

Publication Status: Accepted by ApJ (9 Jan 2010)
Last Modified: 2010-01-11 13:37
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A self-consistent nonlinear force-free solution for a solar active region magnetic field  

Mike Wheatland   Submitted: 2009-06-24 01:13

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/SOT observations of NOAA active region 10953.

Authors: M.S. Wheatland and S. Regnier
Projects: Hinode/SOT

Publication Status: Accepted for publication in ApJ Letters
Last Modified: 2009-06-24 07:27
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Monte-Carlo Simulation of Solar Active-Region Energy  

Mike Wheatland   Submitted: 2009-02-06 18:49

A Monte-Carlo approach to solving a stochastic jump transition model for active-region energy (Wheatland and Glukhov, Astrophys. J. 494, 1998; Wheatland, Astrophys. J. 679, 2008) is described. The new method numerically solves the stochastic differential equation describing the model, rather than the equivalent master equation. This has the advantages of allowing more efficient numerical solution, the modelling of time-dependent situations, and investigation of details of event statistics. The Monte-Carlo approach is illustrated by application to a Gaussian test case, and to the class of flare-like models presented in Wheatland (2008), which are steady-state models with constant rates of energy supply, and power-law distributed jump transition rates. These models have two free parameters: an index (delta ), which defines the dependence of the jump transition rates on active-region energy, and a non-dimensional ratio (overline{r}) of total flaring rate to rate of energy supply. For overline{r}ll 1 the non-dimensional mean energy langle overline{E} angle of the active-region satisfies langle overline{E} angle gg 1, resulting in a power-law distribution of flare events over many decades in energy. The Monte-Carlo method is used to explore the behavior of the waiting-time distributions for the flare-like models. The models with deltaeq 0 are found to have waiting times which depart significantly from simple Poisson behavior when langle overline{E} angle gg 1. The original model from Wheatland and Glukhov (1998), with delta=0 (no dependence of transition rates on active-region energy), is identified as being most consistent with observed flare statistics.

Authors: M.S. Wheatland
Projects: None

Publication Status: Accepted for publication in Solar Physics
Last Modified: 2009-02-07 09:40
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Subject will be restored when possible  

Mike Wheatland   Submitted: 2008-02-26 16:45

A stochastic model for the energy of a flaring solar active region is presented, generalising and extending the approach of Wheatland & Glukhov (1998). The probability distribution for the free energy of an active region is described by the solution to a master equation involving deterministic energy input and random jump transitions downwards in energy (solar flares). It is shown how two observable distributions, the flare frequency-energy distribution and the flare waiting-time distribution, may be derived from the steady-state solution to the master equation, for given choices for the energy input and for the rates of flare transitions. An efficient method of numerical solution of the steady-state master equation is presented. Solutions appropriate for flaring, involving a constant rate of energy input and power-law distributed jump transition rates, are numerically investigated. The flare-like solutions exhibit power-law flare frequency-energy distributions below a high energy rollover, set by the largest energy the active region is likely to have. The solutions also exhibit approximately exponential (i.e. Poisson) waiting-time distributions, despite the rate of flaring depending on the free energy of the system.

Authors: M.S. Wheatland
Projects: None

Publication Status: To appear in the Astrophysical Journal
Last Modified: 2008-09-23 21:22
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Subject will be restored when possible  

Mike Wheatland   Submitted: 2007-09-27 20:20

Improvements to an existing method for calculating nonlinear force-free magnetic fields (Wheatland 2006, Solar Phys. 238, 29) are described. In particular a solution of the 3-D Poisson equation using 2-D Fourier transforms is presented. The improved nonlinear force-free method is demonstrated in application to linear force-free test cases with localized non-zero values of the normal component of the field in the boundary. These fields provide suitable test cases for nonlinear force-free calculations because the boundary conditions involve localized non-zero values of the normal components of the field and of the current density, and because (being linear force-free fields) they have more direct numerical solutions. Despite their simplicity, fields of this kind have not been recognised as test cases for nonlinear methods before. The examples illustrate the treatment of the boundary conditions on current in the nonlinear force-free method, and in particular the limitations imposed by field lines which connect outside of the boundary region.

Authors: M.S. Wheatland
Projects: None

Publication Status: Solar Physics (in press)
Last Modified: 2007-09-28 05:01
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The Origin of the Solar Flare Waiting-time Distribution  

Mike Wheatland   Submitted: 2000-06-05 04:05

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 (GOES) 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 distribut- distribution of flaring rates. These results are argued to be consistent with a non-stationary avalanche model for flares.

Authors: Wheatland, M.S.
Projects:

Publication Status: ApJ Letters (in press)
Last Modified: 2000-06-05 04:05
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Do solar flares exhibit an interval-size relationship?  

Mike Wheatland   Submitted: 1999-11-24 05:03

Some models for flare statistics predict or assume that there is a relationship between the times between flares and the energy of flares. This question is examined observationally using the WATCH solar X-ray burst catalogue. A rank correlation test applied to the data finds strong evidence for a correlation between the time since the last event, t_b, and the size (peak count rate) of an event, and for a correlation between the time to the next event, t_a, and the size of an event. A more sophisticated statistical test, taking into account a probable bias in event selection, does not support the hypothesis that event size depends on t_b or t_a.

Authors: Wheatland, M.S.
Projects:

Publication Status: Solar Physics (in press)
Last Modified: 1999-11-24 05:03
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Frequency-energy distributions of flares and active region transient brightenings  

Mike Wheatland   Submitted: 1999-11-17 04:04

Some time ago Uchida and Shibata proposed a model for compact loop flares as due to the collision of two large amplitude torsional Alfvén wave packets coming up along a coronal magnetic loop, leaking out from the subphotospheric convective layers of the solar atmosphere. We investigate the possibility that active region transient brightenings occur when a single torsional Alfvén wave packet transits a coronal loop. Assuming this related origin for flares and transient brightenings, the statistics of the two phenomena must also be closely related. It is shown that the observed power-law frequency-energy distributions of flares and transient brightenings may be accounted for in a natural way if the energy distribution of the underlying torsional Alfvén wave packets is itself a power law.

Authors: Wheatland, M.S. and Uchida, Y.
Projects:

Publication Status: Solar Physics (in press)
Last Modified: 1999-11-17 04:04
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Are electric currents in solar active regions neutralized?  

Mike Wheatland   Submitted: 1999-11-17 03:03

Models for energy storage in the solar corona due to the twisting of coronal magnetic fields by subphotospheric flows imply that coronal currents should be neutralized (i.e. observed currents over a given polarity of the magnetic field should sum to zero). The neutralized current hypothesis is quantitatively tested by examining vector magnetic field data from 21 active regions observed by the Solar Magnetic Field Telescope of the Huariou Solar Observing Station of Beijing Astronomical Observatory. For each active region the current over the positive polarity of the field, I_+, is estimated, as well as the current over the negative polarity, I_-, and the total current over both polarities, I_tot. In no case is the total current I_tot significantly different from zero. The currents I_+ and I_- are found to be significantly different from zero (at the 3 sigma level) in more than half of the active regions studied, implying that large scale currents in active regions are typically unneutralized. The implications of this result and the relationship of this study to related studies (e.g. of current helicity) are discussed.

Authors: Wheatland, M.S.
Projects:

Publication Status: ApJ (in press)
Last Modified: 1999-11-17 03:03
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Flare frequency-size distributions for individual active regions  

Mike Wheatland   Submitted: 1999-11-17 03:02

It is well known that the size (peak count rate of emission, or inferred energy) distribution of flares observed on the Sun follows a power law. However, the distributions that are constructed include flares from different active regions that have a variety of physical properties. In this paper cumulative peak count rate distributions are constructed for individual active regions using events listed in the WATCH solar X-ray burst catalog. The individual distributions are found to be consistent with the distribution constructed from events in all active regions named in the catalog. This result suggests that the flare power-law size distribution reflects an intrinsic property of the flare mechanism, and is not a product of the distributions of physical parameters between active regions. The result is consistent with the avalanche model for flares, and more generally argues against macroscopic flare models.

Authors: Wheatland, M.S.
Projects:

Publication Status: ApJ (in press)
Last Modified: 1999-11-17 03:02
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Abstracts by Author
Achieving Self-Consistent Nonlinear Force-free Modeling of Solar Active Regions
Time-dependent Stochastic Modeling of Solar Active Region Energy
Evidence for departure from a power-law flare size distribution for a small solar active region
A self-consistent nonlinear force-free solution for a solar active region magnetic field
Monte-Carlo Simulation of Solar Active-Region Energy
Subject will be restored when possible
Subject will be restored when possible
The Origin of the Solar Flare Waiting-time Distribution
Do solar flares exhibit an interval-size relationship?
Frequency-energy distributions of flares and active region transient brightenings
Are electric currents in solar active regions neutralized?
Flare frequency-size distributions for individual active regions

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