11.4 Kinking and magnetic helicity 34711.4.1KVtube,helicity content of theflux tube
We now consider the helicity content of the possibly helicalflux tube,
KVtube=∫
VtubeA·Bd^3 r. (11.40)The magnetic field in theflux tube is decomposed into a componentBaxisparallel to the
flux tube axis and an orthogonal componentBazimuthalwhich goes the short way around
the axis;thus the magnetic field inside theflux tube is
B=Baxis+Bazimuthal. (11.41)The field lines in theflux tube are assumed to lie in successive layers (magnetic surfaces)
wrapped around theflux tube axis. The axial and azimuthal magnetic fields are derived
from respective vector potentialsAaxisandAazimuthalsuch that
Baxis = ∇×Aaxis
Bazimuthal = ∇×Aazimuthal. (11.42)It should be emphasized that these definitions say nothing about the direction ofAaxisor
Aazimuthal and so, unlike axisymmetric situations, here neitherAaxisnorAazimuthal
should be construed to be in a particular direction because of the possibility that theflux
tube is helical. All that can be said is that the curl of Aaxisgives theflux tube axial
magnetic field and the curl ofAazimuthalgives the azimuthal field.
The helicity content of theflux tube is thus
KVtube=∫
Vtube(Aaxis+Aazimuthal)·(Baxis+Bazimuthal)d^3 r (11.43)and this is true even though the axis of theflux tube could be helical.
By definition, each layer of field lines constituting a magnetic surface encloses aflux
Φ.An equivalent definition is to state thatΦis the magneticflux linked by a contour in the
magnetic surface going the short way around theflux tube axis. The magnetic surface is la-
beled byΦand soΦcan be considered to be a coordinate having its gradient always normal
to theflux surface. In effectΦis a re-scaled minor radius, sinceΦincreases monotonically
with minor radius.
These definitions are sufficiently general to allow for the axis of theflux tube to be a
helix, a knot, or a simple closed curve lying in a plane. If the axis is just a simple closed
curve lying in a plane, then∇×φˆis normal to the plane and therefore normal toφ.ˆ In the
slightly more general case where the axis is not in a plane butˆφ·∇׈φ=0,the path traced
out byφˆcan be considered to be the perimeter of a some bumpy surface. However, in the
most general case whereˆφ·∇×φˆis finite, the path traced out byˆφis not the perimeter of
even a bumpy surface (Barnes 1977). There might also be situations whereφˆ·∇׈φ=0
but the sign ofˆφ·∇׈φalternates. If the average ofφˆ·∇׈φover the length of the axis
is zero, then the situation would be similar to the case whereφˆ·∇׈φ=0everywhere,
because twists in the axis could be squeezed together axially until mutually cancelling out.