**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 **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:

**= 0**

*J X B*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} = ò_{V
} _{ }**A** **B **dV
- ò_{
Vo} **A** _{o }**B**_{o} dV_{o}

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]:

d H _{r }/ dt = d H _{r } [T] / dt + d 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)

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.

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