Fundamentals of Plasma Physics

10.6 Magnetic helicity 321

The question immediately arises whether this definition makes sense, i.e., is it reasonable
to defineA·Bas an intensive property andKas an extensive property. An obvious
problem is thatAis undefined with respect to a gauge, sinceAcan be redefined to be
A′=A+∇fwherefis an arbitrary scalar function without affecting the magnetic field,
because∇×A=∇×(A+∇f).The above definition for magnetic helicity would be of
little use ifKdepended on choice of gauge. However, if no magnetic field penetrates the
surfaceSenclosing the volumeV,the proposed definition ofKis gauge-independent. This
is becauseB·ds=0everywhere on the surface if no magnetic field penetrates the surface.
If this is so, then
∫

V

d^3 r(A+∇f)·B =

∫

V

d^3 rA·B+

∫

V

d^3 r∇·(fB)

=

∫

V

d^3 rA·B+

∫

S

ds·(fB)

=

∫

V

d^3 rA·B (10.129)

and so the helicity is gauge-independent even though the helicity density is gauge-dependent.
Let us consider the situation where there is no vacuum region between the plasma and
an impermeable wall. Thus the normalfluid velocityu 1 ⊥must vanish at the wall. From
Ohm’s law the component of the perturbed electric field tangential to the wallE 1 tis just

E 1 t=−u 1 ⊥×B 0 =0 (10.130)

sinceB 0 lies in the plane of the wall. Thus, an impermeable wall is equivalentto a con-
ducting wall. If the magnetic field initially does not penetrate the wall,i.e. B·ds=0
initially, then the field will always remain tangential to the wall and the helicityKin the
volume enclosed by the wall will always be a well-defined quantity (i.e., will always be
gauge-invariant).
The time derivative of the magnetic helicity density together with Faraday’s law gives
a conservation equation for helicity density. This is seen by direct calculation:

−

∂

∂t

(A·B)= −

∂A

∂t

·B−A·

∂B

∂t

=(E+∇φ)·B+A·∇×E

=2E·B+∇·(φB+E×A)

(10.131)

whereφis the electrostatic potential. Since the ideal MHD Ohm’s law givesE·B=0,
Eq. (10.131) can be rearranged in the form of a helicity conservation equation

∂

∂t

(A·B)+∇·(φB+E×A)=0. (10.132)

Integration of Eq.(10.132) over the entire magnetofluid volume gives a global helicity con-
servation relation

∂

∂t

[∫

d^3 rA·B

]

+

∫

dsˆn·(φB+E×A)=0. (10.133)