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Timedependent Stochastic Modeling of Solar Active Region Energy 

Mike Wheatland Submitted: 20100803 12:09
A timedependent model for the energy of a flaring solar
active region is presented based on an existing
stochastic jumptransition 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 frequencysize and waitingtime distributions, but modeling
presented to date has considered only the timeindependent choices of
constant energy input and constant flare transition rates with a
powerlaw 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 OctoberNovember 2009 (Wheatland 2010).
Time variation is incorporated into the jumptransition 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
flarelike event statistics. In each case the frequencyenergy
distribution is a power law for flare energies less than a
timedependent 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 waitingtime 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 waitingtime 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 frequencyenergy and waitingtime 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: 20100804 06:37



MonteCarlo Simulation of Solar ActiveRegion Energy 

Mike Wheatland Submitted: 20090206 18:49
A MonteCarlo approach to solving a stochastic jump
transition model for activeregion 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
timedependent situations, and investigation of details of
event statistics. The MonteCarlo approach is illustrated by
application to a Gaussian test case, and to the class of
flarelike models presented in
Wheatland (2008), which are steadystate
models with constant rates of energy supply, and powerlaw
distributed jump transition rates. These models have two
free parameters: an index (delta ), which defines the
dependence of the jump transition rates on activeregion
energy, and a nondimensional ratio (overline{r}) of
total flaring rate to rate of energy supply. For
overline{r}ll 1 the nondimensional mean energy langle
overline{E}
angle of the activeregion satisfies langle
overline{E}
angle gg 1, resulting in a powerlaw
distribution of flare events over many decades in energy.
The MonteCarlo method is used to explore the behavior of
the waitingtime distributions for the flarelike 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
activeregion 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: 20090207 09:40




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