Thursday, May 6, 2010

Quantifying Reliability in Solar Compact Loops

One of the ongoing problems recurring in solar flare forecasting is how to narrow the windows for forecasting times, for specific flares and associated magnitudes. From about 1991, I have approached this from the perspective of employing an analog between the compact solar flare loop and a basic circuit in which one or more components fails. The failure then incepts a "short circuit" and the release of stored energy.

One such circuit analog is shown in the diagram for a line-tied arch(loop). By "line tied" I mean that the loop feet are firmly anchored in the photosphere at the level where the plasma beta approaches or exceeds 1. This can engender a "dynamo effect" whereby convective motions spawn electric currents which in turn create powerful magnetic fields in the loop. (The reader may refer here to Fig. 2 for a realistic portrayal of the loop.)

Assume the inductance (L) is distributed equally between two parallel branches of inductive magnitude (L/2) each, equivalent to lengths on either side of the loop apex. Then the dipole can be resolved into a 2-component (c1, c2) network. The associated monotonic structure function is:

(1) f (x1, x2) = 1 – (1 – x1)(1 – x2)

The implicit assumption in (1) is that the failure of two components is necessary to stop the current flow. That is, we require x1 = x2 = 0. The reliability function corresponding to (1) is written in terms of the separate component probabilities of failure(p1 for c1, and p2 for c2):

(2) H(p1, p2) = p1 + p2 – p1·p2

By way of example, for the line-tied loop, assume an effective (L=1) inductance for c1 and none for c2. Then:

f (x1, x2) = 1 – (1 – x1)(1 – x2) = 1 – (1 – 0)(1 – x2)
and f(x1,x2) = 1 – 1 – x2 = -x2

and: H(p1, p2) = p1 + p2 – p1·p2 = 0 + 1 – (0)(1) = 1

which leads to the interesting result that the monotonic structure function is (-x2) when the loop undergoes partial failure, and the probability is H = 1. In other words, only one polarity displays self-inductance.

A possible explanation for this may have to do with the sign of what is called the “magnetic helicity,” say incepted by a flow near one foot point that injects negative helicity near a neutral line[1].

The magnetic helicity of a field B within a volume V is defined:

H = INT_V A·B dV

where the integral (INT) is taken over the volume, V, B is the magnetic induction, and the vector potential A satisfies:

B = curl A

From Taylor’s hypothesis[2], the above integral is approximately invariant- so the minimum energy configuration is a “constant-alpha” (a) force free field, e.g. curl B = a B

In actual working solar conditions, one prefers a gauge-invariant form of H and this is provided by the “relative helicity” (H(R)) wherein one subtracts the helicity of some reference field (B (o)), e.g. associated with a = 0) and having the same distribution of the normal component of B on the surface (S).

It is hypothesized that shearing and twisting of the field “injects” helicity and that this may be useful in quantifying: a) how much magnetic free energy becomes available, and b) whether instability can be predicted based on observed indicators of helicity at the level of the photosphere-chromosphere. For example it may be possible to resolve H(R) observationally into two components based on twist and writhe for “relative helicity” such that:

(3) d H(R)/ dt = d H(R) [T] / dt + d H(R) [W] / dt

The sign of helicity will be positive or negative, depending on what is known as the “hemispheric helicity rule. That is, the force-free a characterizing each active region will have a tendency to be (+) in the southern solar hemisphere, and (-) in the northern solar hemisphere. Thus, in effect, in this case (-x2) -> a = (+ curl B_n / B_n), where B_n is the normal component of the field in the particular region.

Of key import(and an ongoing research issue) is how the actual curvilinear and cylindrical geometry affects all the properties posed above. Stay tuned!

[1] The neutral line or “magnetic inversion line” separate positive and negative polarities in a sunspot field or active region.

[2] J.B. Taylor: 1974, Phys. Rev. Letters, 33, 1139.

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