Thursday, November 4, 2021

Looking Again At Magnetic Helicity


Solar prominence - lower right- illustrating magnetic helicity

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

H =  ò V  A·B dV

where the vector potential A satisfies:

B = curl A   


From Taylor’s hypothesis[1], the above integral is approximately invariant- so the minimum energy configuration is a “constant-alpha” force free field, e.g. curl B = a B.  

A force -free field is one for which the conditions apply:

1) curl B = a B. 

2) (B ·  Ñ )  a   =   0

Where a is a scale factor, and B the magnetic induction. More fundamentally, the force-free assumption requires that we have for the Lorentz force:

J X B = 0

where J is the current density. Thus, the current density is essentially parallel to the magnetic induction.

In actual working solar conditions, one prefers a gauge-invariant form of H and this is provided by the “relative helicity” – 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).  Thence:


H r  = òV     A  B dV  -   ò Vo A o Bo dVo


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. 

 In this case it is also possible to write about relative helicity.  We may then resolve H r   observationally into two components based on twist and writhe for “relative helicity” such that[1b]:

H / dt =  H  [T] / dt    +  H r [W] / dt   

Where T on the RHS refers to "twist" and W refers to "writhe".  Both T and W are needed to fully describe relative magnetic helicity.   The sign of helicity will be positive or negative, depending on what is known as the “hemispheric helicity rule".[2] That is, the force-free parameter a characterizing each active region will have a tendency to be (+) in the southern solar hemisphere, and (-) in the northern solar hemisphere. 

When you think about it this makes eminent sense. (Think of the Coriolis force causing a preferred sign or handedness, relative to convective flows in the northern and southern hemispheres of a planet like Earth) If there is a preferred “handedness” (or chirality) associated with magnetic flux, it would be expected to exhibit a different sign in each hemisphere.

Observations confirm that this sign asymmetry exists throughout the solar atmosphere: in the corona, the solar wind and the photosphere. (For the latter evidence, see, e.g. A.A. Pevtsov et al, The Astrophysical Journal, Vol. 473, p. 533, 1996)

We see evidence of the Sun’s magnetic helicity in the solar corona as well as the solar wind that streams past Earth. In eruptive prominences, for example see the one in the attached image, we actually have the graphical detection of the twist and writhe (or helical structure) associated with topological helicity – and which is magnetic helicity in the Sun’s magnetic environment.

Also, if you carefully inspect the prominence in the lower right of the top image, you can discern both twist and writhe in the plasma filaments. Evidently then, prominences are capable of transporting magnetic helicity in the solar corona.

[1] M.A. Berger: 1984, Geophys. Astrophys. Fluid Dyn., 30, 79.

[1b] K. Kusano et al., Ap. J., 577, p. 501, 2002.

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

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