The (Equivalent) Electric Circuit Model For A Reliability Description of Compact Solar Flares

 Many, if not most problems in flare prediction arise from an inability to formulate a coherent, quantitative model or description of a flare. Part of this can be traced to a continued deficiency in instrumental resolution. Most flare researchers believe at least a minimal optical resolution of 0.1 arcsec is needed to distinguish the flare release site in the corona. No known instruments deliver this, although the Webb telescope might if adjustments are made.

   Apart from resolution issues, other serious problems have their genesis in the mode of description chosen, including the quantitative formulation. For example, treatments based on the analysis of the indigenous magnetic fields currently occupy a predominant role. The majority of these are based on MHD (magneto-hydrodynamic) theory wherein low frequency, long scale length and quasi-neutral (electric charge) assumptions are introduced into the plasma physics.

 In fact, MHD represents the end product of successive information loss in the transition from Vlasov theory to two-fluid theory to one-fluid theory and thence MHD, but remains the go-to approach e.g.

• Why The MHD Regime Is Preferred In Doing Solar Physics...

While it’s true that magnetic fields are more readily measured than currents (especially in the region of the solar atmosphere where solar flares occur) it is also true that currents are essential for a complete understanding of the flare process. More specifically, the circuit approach more readily ensures that boundary conditions will not be neglected. It also allows incorporation of plasma properties in a more general way.

The use of equivalent circuits for different magnetic field configurations is not new, as a survey of the literature can disclose (e.g. Alfven and Carlquist, 1967[1], Alfven, 1981[2], Spicer, 1982[3], and Leibacher and Stein, 1982[4]). An example of a closed magnetic configuration, incorporating a cavity resonator structure, is shown in Fig. 1.

                         Equivalent Circuits for a simple compact loop with L, C, R components

 The adoption of such equivalent circuits allows a novel flare forecast technique to be presented, an approach first noted in 1988[5]. This is in conjunction with the reliability concept used in electrical engineering systems. For years now, n-component systems have been shown to follow predictable survival laws based on how the components are connected in a network, and the probability of failure of the whole system when any given component (or combination) fails.[6]

In an analogous way, once a circuit approach is used, a solar flare can be depicted as a failure of an n-component system. These “components” are the current carrying arches-loops and finer structures therein linked to self-inductance, and resistance elements. Each loop or arch has associated with it a resistivity r in its contained plasma, and hence over a length L and constant cross-sectional area A, it will have a resistance:

R =  r L/ A

If the single loop or arch is symmetric and has two foot points, such that loop length can be equally apportioned, then it makes sense that one can have:

R1 =  r L1/ A  = r L2/ A 

where L1 = L2 =  L/2

Local space charge regions also occur within solar loops, over which large potential differences can be generated. Usually multiple double layers of a particular scale (e.g. 10 cm) are required, but the result is to act as an effective capacitor. Such a “capacitor” is depicted in the circuit as C.[7]

Transient Solution for Failure:

The simplest analog to a pre-flare circuit conforms to the configuration of Figure 1.Here we have a simple loop with opposite magnetic polarities at each end (“foot”) bounded by a much higher density photosphere. We say the footpoints of the loop are effectively “line tied.”  We assume also, without loss of generality, that the circuit is completed below the photosphere using conductive plasma.

The resulting L-C-R circuit has a current flowing parallel to the magnetic field provided the plasma Beta b < < 1. The loop itself is a plasma tube with semi-toroidal geometry (see the Frontispiece graphic – bottom of page 1, done using MathCad), bearing plasma of resistivity: h  = R L’ / A where R is the resistance, L’ is the total circuit length (in general L’ > L) and A is the uniform cross-sectional area of the tube. The capacitance C = e L’ where L’ is as before and e is the dielectric constant of the plasma.  This is given by[8]:

e  =   1  +  (i 4p s)/ w

 where  s is the conductivity and w the plasma frequency: 

(4p n _o e /m_e)^1/2.

The self-inductance is of the order (L/c), where c is the velocity of light.

Current in the system is generated from an emf E(t) which is motional so time-dependent. It arises from the relative displacement of the dipole feet at points P1 and P2 on the photosphere, such that:

1)                                                E(t) =   ò ^P2 _P1  (v X  B)  dS

where the integration is from P1 to P2 as the upper limit.  Identification of failure, or onset of the flare in the circuit is coincident with finding the solution to the differential equation:

2)                                      L (d² I/ dt² ) + R (dI/dt) + I/C =  dE/ dt

 

where all symbols are as previously defined. If (2) is divided through by L one obtains:

3)                                        (d² I/ dt² ) + R/ L (dI/dt) + I /LC =   1/L(dE/ dt)

The solution will be I(t),  a transient response function. This differential equation can be recast in terms of the associated Green’s function with I(t) º G = G(t,T) with the change in back emf dE/ dt replaced by the Dirac Delta function  d (t – T) and the problem treated from the viewpoint of distributions.

 

4)   d² G/ dt²   +  2 a (dG/dt) +   w_o² G =  d (t – T)/ L

where the response a = R/ 2L   and w_o =  1/ Ö(LC), or the natural frequency at which the dipole oscillates.  Note here that the impulse d (t – T) = 0  for t < T, hence I(t) = 0. This is the “steady state” solution: associated with the ongoing stability or “survival” of the dipole. By contrast, we are only interested in the case for t >  T, e.g. the transient or complementary solution. In particular, the solution we seek, on inverting the appropriate Laplace transforms is[9]:

5)   I(t) =  G(t, T_o) = G(t, 0) =    1/ w  [exp (- at) sin (w t)]                       t > T_o

Where w = [(w_o² -   a²)]^1/2

Not surprisingly, the current impulse becomes infinite when C = 0 and L = 0 at current interruption. In the case considered here, the failure bears a direct analogy to what happens in high power transmission lines when the current is suddenly switched off. That is, there is an explosive release of the inductive energy.  That this can be predicted makes the equivalent circuit approach a useful ansatz for compact solar flare behavior.

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[1] H. Alfven and P. Carlquist: 1967, Solar Phys., 1, 220.

[2] H. Alfven: 1981, Cosmic Plasma, D. Reidel, Dordrecht, Holland.

[3] D.S. Spicer: 1982, Space Science Rev., 31, 351.

[4] J. Leibacher and R.F. Stein: 1981, Smithsonian Astrophysical Observatory Special Report No. 392.

[5] P.A. Stahl: 1988, ‘A Reliability Description of Compact Solar Flares’ in The Bulletin of the American Astronomical Society, Vol. 20, No. 2, p.748.

[6] M.A. Kaufmann: 1969, La Confiance Technique, Dunod, Paris.

[7] A double layer is typically “the simplest space charge distribution that gives a potential drop in the layer and a vanishing electric field E on either side.” (Hannes Alfven, Cosmic Plasma, 1981, D. Reidel, Dordrecht, Holland.)

[8] In a more general sense, one can have: e = e(r) + ie (i) where e(r) is the real part and e (i) the imaginary part of the dielectric constant. In the case used in the text I have assumed: e (i) > > e(r).

[9] The problem posed will resemble that for the forced oscillations of a damped harmonic oscillator such that: x’’ + 2 ax’ + w_o² x = f(t). The Laplace transform of this is analogous to the one in the text for I(t).