Fundamentals of Plasma Physics

11.4 Kinking and magnetic helicity 351

subribbon lying between

flux tube outer surface

and magnetic surface 



is flux through

this subribbon

location of magnetic surface 

Figure 11.7: ψisflux through cut subribbon located between magnetic surfaceΦand outer
surface offlux tube.

The vector potential for the entire field can be written as

A=Aaxis+

ψ(Φ)

2 π

∇φ. (11.49)

These results can be used in Eq.(11.43) to give the helicity content of theflux tube as

KVtube =

∫

Vtube

d^3 r

(

Aaxis+

ψ(Φ)

2 π

∇φ

)

·

(

Baxis+

1

2 π

∇ψ×∇φ

)

=

∫

Vtube

d^3 r

(

Aaxis·Baxis+

ψ(Φ)

2 π

∇φ·Baxis+Aaxis·

1

2 π

∇ψ×∇φ

)

(11.50)

and this may be decomposed into writhe and twist terms,

KVtube=Kwrithe+Ktwist

respectively defined as

Kwrithe=

∫

Vtube

d^3 rAaxis·Baxis (11.51)