Time-dependent Stochastic Modeling of Solar Active Region Energy |
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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
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Publication Status: To appear in Solar Physics (accepted 31 July 2010)
Last Modified: 2010-08-04 06:37
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Monte-Carlo Simulation of Solar Active-Region Energy |
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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
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Publication Status: Accepted for publication in Solar Physics
Last Modified: 2009-02-07 09:40
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