Flares as Avalanches
Stir a magnetic field slowly and it does not give way slowly. It holds, holds, and then lets go, in flares of every size.
Motions at the Sun’s surface drag the feet of its magnetic field around and twist it, storing energy in the field. The field does not release that energy smoothly. It holds until somewhere the field is bent too sharply, that patch gives way, and the stress it sheds is handed to its neighbors, which may give way in turn. Most of the time a handful of patches go. Once in a while the whole region goes at once. The same stirring produces flares of every size, and their sizes follow a power law: small flares are common, large ones rare, and there is no typical size. The model below is the one that made that argument, a lattice of stressed patches with a threshold and a rule for passing stress on (Lu and Hamilton 1991).
A Stressed Field, Patch by Patch
Every Flare So Far
The Rule, in Three Lines
gives way when: |d| > 1
hands on: Bk −= ⅔ Z, each neighbor += ⅕ Z, Z = d/|d|
Stirring adds a small random amount to one patch at a time. Nothing else is put in by hand: no flare sizes, no timing, no threshold for what counts as a flare. A flare is just whatever cascade follows one patch giving way, counted until the lattice is quiet again.
The edge of the lattice is held at zero, so stress that reaches the edge leaves. That is what lets the field settle at a level it keeps: stirring puts stress in, flares carry it to the edge, and the average stress stops climbing.
Slopes, model against sky. Counting 60,000 flares here gives a slope of 1.55 for flare energy (above 100 units), 2.04 for the peak of a flare and 1.98 for duration (both above 10). Solar flares measured in hard X-rays give 1.62 ± 0.12, 1.73 ± 0.07 and 1.99 ± 0.35 (Aschwanden et al. 2016). Energy and duration land inside the observed range; the peak slope comes out steeper.
Why the Reading Is Open
Three things, classified separately. The program running the panel is an attractlet model under § 7.4, as on the Lorenz page. The critical state of the modeled lattice is a second thing. The Sun’s flaring corona is a third.
Name the boundary. Take X to be the stressed field, with the stirring supplied from outside X.
The case for attractlet. The critical state is held up by the stirring. Press Stop the stirring: the pattern freezes, and in 200,000 sweeps nothing fires again. That is the fifth test of § 7.4, a configuration that “relaxes to an inert or passive state when uncoupled.”
What is new here. This case splits the test used on the Weather page in two. A flare really does release energy the field itself stored, which the convection cells never do. But the release does not drive anything that feeds it: the next flare waits on the stirring, not on the last flare. Releasing stored energy and being driven by that release are two different claims, and only a system that does both is doing what the thunderstorm does. The site’s test asks for both in one sentence, and this page is the reason to say so.
The Shape of the Basin
One critical state, whatever the start. From a blank lattice, from a lattice jolted with strong random field, and from a different random seed, all three settle to the same place: average stress 0.596, 0.595 and 0.597 of the breaking point, mean flare 150, and size slopes of 1.55, 1.56 and 1.55. It takes about 250,000 flares to get there.
The climb is visible. Press Smooth the field. Early flares are tiny, and the largest flare of each batch of 20,000 grows as the stress builds: in one run, 391 units in the first batch, 3,030 by 100,000 flares, and about 35,000 once the state settled. The distribution is not there at the start; the lattice builds it. The slope in the readout counts every flare since the reset, so it stays steep until the early small ones are outnumbered.
A hard jolt is undone in one flare. Press Jolt it hard to scramble the lattice with strong random field: a single flare of about 190,000 units, 65 sweeps long, takes the jolt out and leaves the lattice at its usual stress. It is then below its usual tail, and has to build back: the biggest flare in the next 20,000 was 188 units.
Stop the stirring and nothing happens at all. The lattice holds its stress pattern exactly, with no flare in 200,000 sweeps. There is no decay, no relaxation, and no cycle: without the stirring the system is not quiet in the sense of resting, it is simply stopped.
Stir it during the flares and separate flares stop existing. The slider adds kicks between sweeps, which is what the Sun does not do: real stirring is far slower than a flare. At one kick per sweep the lattice went quiet 3 times in 200,000 sweeps, against 17,207 separate flares with the stirring held off during them. At five kicks per sweep it never went quiet at all, and the average stress rose from 0.60 to 1.06, above the breaking point. The counter stops counting flares, because there are no longer any.
What This Model Is For
The avalanche picture is a claim about why flares have no typical size: the corona is kept near its breaking point everywhere, so the same small disturbance can set off a small event or a large one, and which one it sets off depends on the state of its neighbors. That is the idea of self-organized criticality, brought to solar flares by Lu and Hamilton in 1991.
It is not a claim about the physics of any one flare. Reconnection, particle acceleration, the loops that brighten and the plasma that heats are all outside it. Nor is the argument settled: whether the corona is genuinely in a critical state, and whether these lattice rules stand in for magnetohydrodynamics in a defensible way, are live questions in the field.
Lu, E. T., & Hamilton, R. J. (1991). Avalanches and the distribution of solar flares.
The Astrophysical Journal Letters, 380, L89.
Aschwanden, M. J., et al. (2016). 25 years of
self-organized criticality: solar and astrophysics. Space Science Reviews, 198, 47.