Time-Dependent Stochastic Modeling of Solar Active Region Energy |
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| Authors: | M Kanazir M S Wheatland |
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| Institution: | (1) School of Physics, University of Sydney, NSW, 2006, Australia |
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| Abstract: | 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 in Astrophys. J.
494, 858, 1998; Wheatland in Astrophys. J.
679, 1621, 2008 and Solar Phys.
255, 211, 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 in Astrophys. J.
710, 1324, 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 the expected forms of the frequency – energy
and waiting-time distributions. The simulation results provide a qualitative model for observed flare statistics in AR 11029. |
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