11.2 Topological interpretation of magnetic helicity 339We next add another layer of ‘paint’ with more embeddedψflux, and also with embed-
dedΦflux so that bothψandΦincrease. The value ofΦcan be used to label the layers
of ‘paint’ so thatΦis the amount offlux influx tube #1 up to the layer of ‘paint’ labeled
byΦ. Furthermore, sinceψincreases with added layers of ‘paint’,ψmust be a function of
the layer of ‘paint’ and soψ=ψ(Φ).It is therefore possible to writedψ=ψ′dΦwhere
ψ′=dψ/dΦ.Thus, the amount of helicity added with each layer of ‘paint’ is
dK=2Φψ′dΦ (11.10)and so the sum of the helicity contributions from all the layers of ‘paint’ is
K=2
∫Φ
0Φψ′dΦ. (11.11)We now show thatψ′represents the twist of the embedded magnetic field. Letφbe the
angle the long way aroundflux tube #1 andθbe the angle the long way aroundflux tube
#2 as shown in Fig.11.2. Thus increasingφis in the direction of contourC 1 and increasing
θis in the direction of contourC 2 .The perimeter of a cross-section offlux tube #1 is in the
θdirection and the perimeter of a cross-section offlux tube #2 is in theφdirection. The
magnetic field influx tube #1 can be written as
B 1 =
1
2 π∇Φ×∇θ=1
2 π∇×Φ∇θ (11.12)which is in theφdirection since∇Φ is orthogonal to∇θ.HereΦis theflux linked by a
contour going in the direction of∇θ.To verify that this is the appropriate expression for
B 1 , theflux through the cross sectionS 1 offlux tube #1 is calculated as follows:
flux throughS 1 =∫
S 1ds·B 1=
1
2 π∫
S 1ds·∇×Φ∇θ=
1
2 π∫
C 2dl·Φ∇θ=
Φ
2 π∫
C 2dl·∇θ=
Φ
2 π∮
dθ= Φ. (11.13)ThefluxΦcan be factored from the integral in the third line above, becauseΦis theflux
linked by contourC 2 which goes in the direction of∇θ.Similarly, it is possible to write
B 2 =
1
2 π∇ψ×∇φ=1
2 π∇×ψ∇φ (11.14)which is in theθdirection since∇ψis orthogonal to∇φ.SinceB 1 is in theφdirection and
B 2 is in theθdirection, the total magnetic field can be written asB=B 1 ˆφ+B 2 ˆθwhich is
helical.