Faraday's Law in Solar Flares: A Cautionary Message

Nugget
Number:	480
1st Author:	Michael FARADAY
2nd Author:
Published:
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Introduction

Faraday's law, in its original formulation, dealt with the electromagnetic force (EMF) observed when a conductor moves through a magnetic field. Essentially an electric field E = - V x B appears in the conductor, where V is its velocity and B is the ambient magnetic field. In the Maxwell-Heaviside equations, this becomes ∂B/∂t = -∇ x E.

In the plasma constituting a solar flare, we have rapid motions and high conductivity in a highly ionized medium, so it is natural to ask about the nature of the motional EMFs that must develop. The flare plasma contains embedded currents, including "unbalanced" ones (e.g., Ref. [1]). We could assume a spatial scale of 10 Mm and a magnetic field of 0.1 T as ambient conditions in the core of an active region prior to a flare. Then, in the flare, we could consider a velocity of 10 km/s. We thus find the quite remarkable electrical potential drop of magnitude L v B = 10 Mm · 10 km/s · 0.1 T = 10 GV in a dimensional approximation. One might think that this field would not accelerate particles because v x B is perpendicular to the imposed field B, but of course the plasma motions themselves dynamically alter B in some complicated way that we have no theory for.

There would be no reason not to conclude that powerful particle acceleration must occur via the direct application of Faraday's law. Such phenomena cannot be described within MHD theory (Ref. [3]) but the full set of Maxwell's equations are clear and correct about the presence of the EMF.

Plasma Motions in Flares

These are observationally quite obvious: coronal mass ejections, shrinkage, implosion, dipolarization, and implosion all have considerable literature. These all imply plasma motions with components perpendicular to the ambient (imposed) field, and of course there are sprays, surges, and jets of all sorts. The idea of implosion (Ref. [3]) is quite general: to extract magnetic free energy one requires magnetic restructuring.

Relationship to particle acceleration

Faraday's Law describes the development of a stressed electromagnetic field, which is not directly relevant to particles. Most theories of particle acceleration in the solar/heliospheric context describe the behavior of test particles in prescribed environments, but we think that the way the environment may evolve can differ substantially from the description "prescribed." Ref. [4] describes a toy model that links the large-scale field development with particle acceleration, but there is otherwise very little literature on this topic.

Conclusions

"Collapsing traps" (Ref. [5]) and many other examples of test particles exploring MHD structures abound, with descriptions of Fermi and betatron acceleration (and other mechanisms), but these theories are usually not self-consistent electrodynamically with the global configuration (e.g., they may use periodic boundary conditions). They describe steady-state fluids and do not include electrostatic potentials nor particle distribution functions. This Nugget just reminds us that there is likely to be rich physics to explore in the application of Faraday's Law in solar-flare plasma theory, and we speculate that the lack of much literature on this subject probably means that it involves difficult theoretical and modeling work.