Fundamentals of Plasma Physics

11.2 Topological interpretation of magnetic helicity 337

BecauseBvanishes outside of the twoflux tubes, the helicity integral is finite only in the

volumesV 1 andV 2 of the two tubes and so the helicity can be expressed as (Moffatt 1978)

K=

∫

V 1

A·Bd^3 r+

∫

V 2

A·Bd^3 r. (11.2)

The contribution to the helicity from integrating over the volume offlux tube #1 is

K 1 =

∫

V 1

A·Bd^3 r. (11.3)

In order to evaluate this integral it is recalled that the magneticflux through a surfaceS

with perimeterCcan be expressed as

Φ=

∫

S

B·ds=

∮

C

A·dl. (11.4)

Influx tube #1,d^3 r=dl·Bˆ∆S,where∆Sis the cross-sectional area offlux tube #1 anddl

is an element of length alongC 1 .It is therefore possible to recast the integrand in Eq.(11.3)

as

A·Bd^3 r = A·Bdl·Bˆ∆S

= A·dlΦ 1 (11.5)

sincedlis parallel toBandB∆S=Φ 1 .BecauseΦ 1 is by definition constant along the

length offlux tube #1, it may be factored from theK 1 integral, giving

K 1 =Φ 1

∫

C 1

A·dl. (11.6)

However, theflux linked by contourC 1 is precisely theflux in tube #2, i.e.,

∫

C 1 A·dl=Φ^2

and so

K 1 =Φ 1 Φ 2. (11.7)

The same analysis applied toflux tube#1leads toK 2 =Φ 1 Φ 2 and so the helicity content

of the linkedflux tubes is therefore

K=K 1 +K 2 =2Φ 1 Φ 2. (11.8)

flux 
 1

contourC 1

volume V 1

flux 
 2

contourC 2

volume V 2

B B

Figure 11.1: Two linked thin untwistedflux tubes. Magnetic field is zero outside theflux
tubes.