## Thursday, October 28, 2010

### A Look at Magnetic Helicity

As the new solar cycle ramps up, attention turns again to the forecasting of large solar flares and their possible terrestrial effects. These can include everything from interference in radio and TV communications (since satellites can be saturated with x-rays), to disturbance of navigational controls aboard aircraft, and the melting of large conducting transmission lines in power grids (as occurred in Ottawa, due to a large flare in 1989).

Factoring in new flare indicators is an ongoing task, and as each critical morphological or physical feature is included, it must be examined carefully in the whole soalr flare dynamic process. One of the more recent flare indicators to have made the cut - certainly in the last decade or so - is the magnetic helicity.

In this blog I want to explore a bit what it is about and why we want to integrate it into our forecast schemes.

Before going on to magnetic helicity, it is useful to get a general idea of helicity – for example in the topological sense.

To do this very easily, you can simply cut out a rectangular strip of paper with the rough dimensions shown below:

!----------------------L----------------------------------------------]
W
!---------------------------------------------------------------------]

Now take the strip and put a kink or half-twist into it about two –thirds from one end, then tape the free ends. What you will have is called a Moebius strip.

The Moebius strip has one part twist and one part writhe and this is the fundamental basis of helicity. You can get a pictorial idea by going to:

http://en.wikipedia.org/wiki/M%C3%B6bius_strip

“Magnetic helicity” was probably first introduced by K. Moffat in the late 1950s as a topological invariant that describes the complexity of a magnetic field. Like the pure tolpological helicity, this magnetic helicity also has “twist” and “writhe” components. It is written as a function of the vector potential (A) and the magnetic field (B), and measures the topological linkage of magnetic fluxes (F)

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

H = INT V A*B dV

where INT denotes integral over the volume V and A is the vector potential, B is 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 the force-free parameter alpha = 0) and having the same distribution of the normal component of B on the surface (S). Thus,

H(R) = [INT V A B dV - INT V_o 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.

H(R) can then be resolved into two components such that:

d H(R)/ dt = d H(R) [T] / dt + d H(R) [W] / dt

where term 1 on the RHS refers to the “twist” and term 2 to the “writhe”

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 and study the prominence in the upper right of the image below, you can discern both twist and writhe in the plasma filaments. Evidently then, prominences are capable of transporting magnetic helicity in the solar corona.

http://antwrp.gsfc.nasa.gov/apod/ap060416.html

The magnetic helicity is also visible in the gas filaments of the prominence depicted in the lower left of the image below:

http://antwrp.gsfc.nasa.gov/apod/ap051109.html

Solar eruption, especially coronal mass ejections (CMEs) carry magnetic flux as well as helicity from the Sun. When the erupted magnetic field reaches the Earth it interacts with the magnetosphere, causing magnetic substorms and auroras.

Some recent research also reveals remarkable aspects of magnetic helicity in the solar environment. For example, it seems that magnetic helicity of different signs or polarities (+ or -) can occur, depending on which hemisphere of the Sun it’s measured.

Specifically, the sign of helicity will be positive or negative, depending on what is known as the “hemispheric helicity rule.[1] That is, the force-free a characterizing each active region will have a tendency to be (+) in the southern solar hemisphere, (-) in the northern solar hemisphere. Thus, in effect, in this case (-x2) -> alpha = (+ curl B_n / B_n), where B_n is the normal component of the field in the particular region. and 'alpha' is the force free constant (the (assumed) force-free parameter associated with the vertical current density, J_z, and vertical magnetic induction, B_z, is such that: alpha = u J_z/ B_z, where u denotes the magnetic permeability of free space, or u = 4 pi x 10^-7 H/m)

When you think about it, though, 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)

Next: Specific applications in flares

[1] K. Kusano et al., Astrophysical Journal, 577, p. 501, 2002.