Fundamentals of Plasma Physics

280 Chapter 9. MHD equilibria

showing that the functional form of Eq.(9.34) is consistent with Ampere’s law

∮

B·dl=μ 0 I.
The poloidal magnetic field is

Bpol=

1

2 π

(∇ψ×∇φ). (9.36)

Integration of the poloidal magnetic field over the area of a circle of radiusrwith center at
axial locationzgives
∫r

0

Bpol·ds=

∫r

0

1

2 π

∇ψ×∇φ·zˆ 2 πr′dr′=ψ(r,z); (9.37)

thusψ(r,z)is the poloidalflux at locationr,z.The concept of poloidalflux depends on
the existence of axisymmetry so that a circle of radiusrcan always be associated with a
locationr,z.
Axisymmetry also provides a useful relationship between toroidal and poloidal vectors.
In particular, the curl of a toroidal vector is poloidal since

∇×Btor=

μ 0

2 π

∇I×∇φ (9.38)

and similarly the curl of a poloidal vector is toroidal since

∇×Bpol=∇×(Brrˆ+Bzzˆ)=ˆφ

(

∂Br

∂z

−

∂Bz

∂r

)

. (9.39)

The curl of the poloidal magnetic field is a Laplacian-like operator onψsince

∇φ·∇×Bpol=∇·(Bpol×∇φ)=∇·

(

1

2 π

[∇ψ×∇φ]×∇φ

)

=−

1

2 π

∇·

(

1

r^2

∇ψ

)

,

(9.40)

a relationship established using the vector identity∇·(F×G)=G·∇×F−F·∇×G.
Because∇×Bpolis purely toroidal andφˆ=r∇φ,one can write

∇×Bpol=−

r^2

2 π

∇·

(

1

r^2

∇ψ

)

∇φ. (9.41)

Ampere’s law states that∇×B=μ 0 J. Thus, from Eqs. (9.38) and (9.41) the respective
toroidal and poloidal currents are

Jtor=−

r^2

2 πμ 0

∇·

(

1

r^2

∇ψ

)

∇φ (9.42)

and

Jpol=

1

2 π

∇I×∇φ. (9.43)

We are now in a position to evaluate the magnetic force in Eq.(9.22). After decomposing
the magnet field and current into toroidal and poloidal components, Eq.(9.22)becomes

∇P=Jpol×Btor+Jtor×Bpol+Jpol×Bpol. (9.44)

TheJpol×Bpolterm is in the toroidal direction and is the only toroidally directed term on
the right hand side of the equation. However∂P/∂φ=0 because all physical quantities
are independent ofφand soJpol×Bpolmust vanish. This implies

(∇I×∇φ)×(∇ψ×∇φ)=0 (9.45)