Another complicating factor is that loops and twists-writhes within them will surely become more intricate than posed in the simple (circuit) analogs shown in Part 1. One ought to expect, say in Delta sunspot groups, multiple loops of multiple twists and conceivably mutual inductances applicable to the system as well as self-inductances.

F1
= m_{0} N1 N2 l p r^{2} I1

If there occurs a change in the (interior or “twist”-generated) channel current I2 an EMF V1 will be induced in the writhe layer such that:

V1 = - m_{0} N1 N2 l p r^{2} dI2 /
dt

This EMF can be re-cast as:

V1 = - M12 dI2 / dt

Similarly, if a change occurs in the writhe channel current I1, an EMF V2 will be induced in the “twist” layer such that:

V2 = - m_{0} N1 N2 l p r^{2} dI1 /
dt =
- M21 dI1 / dt

where M12 and M21 are the mutual-inductances.

In general, provided the field lines’ writhe and twist are at the same distance r from the center of the loop:

M12 = M21 = M

In classical E-M theory, the total magnetic energy in a
circuit incorporating i loops may be expressed^{[1]}.

U(M) = ½ å
L_{i}I _{i} ^{2 } + ½ å
_{i} å _{j }_{ i} M_{ij} I_{i} I_{j}

where L is the
self-inductance for the ith element, and
M_{ij} = M_{ji}

for all j < i

The power P generated is (Jackson, *op. cit*.):

P = ¶U(M)/ ¶t = å ^{n} _{i
}I _{i }(dF_{i} )/dt

where F_{i} is the magnetic flux of the ith loop

and: F_{i} =
{ L_{i}I _{0i} + å
_{j .}_{ i}
M_{ij} I_{0j}} f _{t}

where f _{t } = (t/ T) denotes the fraction of the total
time T

to establish the currents I _{i}
which increase to final values I_{0j.}

In general, for the condition of linear force-free fields, we require currents:

I _{i}
= a _{i} (F_{i} ) / m_{0}

where the a _{i} are
the force free scale factors for the ith elements, whence the magnetic helicity: H = å i_{j}
L_{ij} F_{i} F_{j}

where L_{ij} = (a_{i}+_{ } a _{j}) (M_{ij} ) / m_{0}

In actual practice it is unrealistic to treat solar
circuits in this global fashion so that one wants to take into account the
volume of the current carrying tubes in calculating the total energy, as well
as the inductances. So for N-distinct current carrying circuits, we have (Wheatland and Farvis, 2004)^{[2]}:

E = ½ å^{N} _{ i=1}
L_{i}I _{i} ^{2 } + ½ å
_{i=1} å _{j > i}
M_{ij} I_{i} I_{j}

where now^{[3]} :

L_{i} = m_{0} /( 4p
Ii Ij ) òd ^{3}x ò_{Vi} d^{3} x [J(x_{i}) J(x_{j}’)]/ [x_{i} - x_{j}’]

M_{ij}
= m_{0} /( 4p
Ii Ij ) ò d ^{3}x_{i} ò_{Vi} d^{3} x_{j} [J(x_{i}) J(x_{j}’)]/ [x_{i} - x_{j}’]

where V_{i}
is the volume of the ith circuit.

In order for it to hold that M12 = M21 = M we must have
both currents in both coils for N1 and N2) at effectively the same distance r
from the center of the loop. However, it has been noted that more realistically
– and especially in a flare scenario – there will be different resonant
surfaces associated with the “coils” ^{[4]},
which means a radius differential, e.g. Dr = b – a. So, the outer surface may have radius b and the
inner surface a. However, this means: M12 ¹ M21

and the analogy breaks down.

What
one really needs, therefore, is a means to track the Gauss winding number
topologically, as opposed to the “coil” (inner or outer) for loop resonant
surfaces and the currents within them. Mutual helicity is first and foremost
about topology and not electro-dynamics. Strengthening this view, is the finding that – unlike the
magnetic helicity – only a part of the current helicity in the photosphere can
be inferred from photospheric vector magnetograms, by virtue of the limitations
imposed via ^{[5]} :

H_{c} = **B**· (**Ñ** **X B**) = B_{⊥}_{ }·
(**Ñ** **X B**) _{⊥} + B_{‖}_{ }· (**Ñ** **X B**) _{‖}

Where B_{⊥} and B_{‖}_{ } are the perpendicular and parallel magnetic field vectors, respectively. Analogous limitations apply to the linear force-free factor
^{[6]}

a = m
J _{z }/ B _{z}

for which: H_{c} = aB^{2}
= (B/ B_{‖})^{2} B_{‖}·(**Ñ** **X B**) _{‖}

The relevant estimates were given in Table I. In the equation, all the information is confined to the normal or vertical components, none to the horizontal.

Of course, these limits may account for why it is so
difficult to apply a circuit ansatz^{[8]} such
as used in this post, and also why the approach to flares, magnetic fields
and currents in solar active regions has bifurcated into two distinct
paradigms: the **B-v**, wherein emphasis is on magnetic fields, and the **E-J
**wherein the emphasis is on the behavior of currents.

One attempt to circumvent the reliability limits can be
treated by introduction of a “resonant cavity” model. E.g.

[1]
J.D. Jackson: 1998, *Classical
Electrodynamics*, John Wiley,

[2]
M.S. Wheatland and F.J. Farvis, Vol. 219, p. 109, *Solar Physics*, 2004.

[3]
See, e.g. M.S. Wheatland and F.J. Farvis, *op. cit*., 2004.

[4] See, e.g. P. Demoulin, E. Pariat and M.A.
Berger: ‘*Basic Properties of Mutual Magnetic Helicity’* in *Solar Physics*,
Vol. 233, p. 3, 2006. The authors obtain an accurate form for the mutual helicity
by using the Gauss linking number of the respective curves. This restores the
proper topological context and allows an explicit way to compute the mutual
helicity. The correct expression for two elementary flux tubes is given by
their equation (28). The central difference from the earlier referenced
formalism is the inclusion of a mulit-valued angular function, q ^{m} which keeps track of how fast one set of
footpoints is rotating about the other.

[5]
See, D.S. Spicer, *Solar Physics*, Vol. 53, p. 305, 1977. Spicer alludes
to families of resonant surfaces such that the total resistance will vary as: R
= (rmr(s)^{m-1 }) ^{m}
and r = ln r
- ¼, with r
the “mean island radius”. Spicer goes on to note the “novel” behavior in
plasmas vis-à-vis conventional electric circuits, especially that the (dL/dt)
term is no longer zero, unlike in prosaic circuits – where L = const.

[6]
S.D. Bao and H.Q. Zhang, Vol. 496, p. L43, *Astronomical Journal*, 1998.

[7]
In German, ‘*der Ansatz’* means’ basic approach,’ or ‘beginning’. In the case of the reliability circuit
approach, the definition fits perfectly. We are using a very primitive ‘basic
approach’ to nail down some particulars of the complex flare process in the
hope of making useable forecasts.

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