Fundamentals of Plasma Physics

9.8 Static equilibria 281

which further implies that∇Imust be parallel to∇ψ.An arbitrary displacementdrresults
in respective changes in current and poloidalfluxdI=dr·∇Ianddψ=dr·∇ψso that

dI

dψ

=

dr·∇I

dr·∇ψ

;

since∇Iis parallel to∇ψ,the derivativedI/dψis always defined. ThusImust be a
function ofψand it is always possible to write

∇I(ψ)=I′(ψ)∇ψ (9.46)

where prime means derivative with respect to the argument. The poloidal current can there-
fore be expressed in terms of the poloidalflux function as

Jpol=

I′

2 π

∇ψ×∇φ. (9.47)

Substitution for the currents and magnetic fields in Eq.(9.44) gives the expression

∇P =

I′

2 π

(∇ψ×∇φ)×

μ 0 I

2 π

∇φ−∇φ

r^2

2 πμ 0

∇·

(

1

r^2

∇ψ

)

×

1

2 π

[∇ψ×∇φ]

= −

[

μ 0 II′

(2πr)^2

+

1

(2π)^2 μ 0

∇·

(

1

r^2

∇ψ

)]

∇ψ.
(9.48)
This gives the important result that∇Pmust also be parallel to∇ψwhich in turn implies
thatP =P(ψ)so that∇P =P′∇ψ.Equation (9.48) now has a common vector factor
∇ψ which may be divided out, so that the original vector equation reduces to thescalar
equation

∇·

(

1

r^2

∇ψ

)

+4π^2 μ 0 P′+

μ^20

r^2

II′=0. (9.49)

This equation, known as the Grad-Shafranov equation (Grad and Rubin 1958, Shafranov
1966), has the peculiarity thatψshows up as both an independent variable and as a de-
pendent variable, i.e., there are both derivatives ofψand derivatives with respect toψ.
Axisymmetry has made it possible to transform a three-dimensional vectorequation into a
one-dimensional scalar equation. It is not surprising that axisymmetry would transform a
three dimensional system into a two dimensional system, but the transformation of a three
dimensional system into a one dimensional system suggests more profound physics is in-
volved than just geometrical simplification.
The Grad-Shafranov equation can be substituted back into Eq.(9.42) to give

Jtor=

(

2 πr^2 P′+

μ 0

2 π

II′

)

∇φ (9.50)

so that the total current can be expressed as

J=Jpol+Jtor=

I′

2 π

∇ψ×∇φ+

(

2 πr^2 P′+

μ 0

2 π

II′

)

∇φ=2πr^2 P′∇φ+I′B.(9.51)

The last term is called the ‘force-free’ current because it is parallel to the magnetic field
and so provides no force. The first term on the right hand side is the diamagnetic current.