Fundamentals of Plasma Physics

9.9 Dynamic equilibria:flows 287

definition differs slightly from the conventional definition of vorticity (curl of velocity)
because it includes an extra factor ofr. This slight change of definition is essentially a
matter of semantics, but makes the algebra more transparent.
Since∇φ=φ/rˆ it is seen that

∇×U=

φˆ

r

rφˆ·∇×U=χ∇φ (9.66)

and so

∇×

(

1

2 π

∇ψ×∇φ

)

=χ∇φ. (9.67)

Dotting Eq.(9.67) with∇φand using the vector identity∇φ·∇×Q=∇·(Q×∇φ)
gives

∇·

(

1

2 π

(∇ψ×∇φ)×∇φ

)

=

χ

r^2

(9.68)

or

r^2 ∇·

(

1

r^2

∇ψ

)

=− 2 πχ (9.69)

which is a Poisson-like relation between the vorticity and the stream function. The vorticity
plays the role of the source (charge-density) and the stream function plays the role of the
potential function (electrostatic potential). However, the ellipticoperator is not exactly a
Laplacian and when expanded has the form

r^2 ∇·

(

1

r^2

∇ψ

)

= r

∂

∂r

(

1

r

∂ψ

∂r

)

+

∂^2 ψ

∂z^2

(9.70)

which is just the operator in the Grad-Shafranov equation.
Since the current was assumed to be purely poloidal, the magnetic field must be
purely toroidal and so can be written as

B=

μ 0

2 π

I∇φ (9.71)

whereIis the current through a circle of radiusrat locationz. This is consistent with the
integral form of Ampere’s law,

∮

B·dl=μ 0 I.The curl of Eq.(9.71) gives

∇×B=

μ 0

2 π

∇I×∇φ (9.72)

showing thatIacts like a stream-function for the magnetic field.