282 Chapter 9. MHD equilibria
The Grad-Shafranov equation is a non-linear equation inψ and, in general, cannot
be solved analytically. It does not in itself determine the equilibrium because it involves
three independent quantities,ψ,P(ψ), andI(ψ).Thus, use of the Grad-Shafranov equation
involves specifying two of these functions and then solving for the third. TypicallyP(ψ)
andI(ψ)are specified (either determined from other equations or from experimental data)
and then the Grad-Shafranov equation is used to determineψ.
Although the Grad-Shafranov equation must in general be solved numerically, there
exist a limited number of analytic solutions. These can be used as idealized examples that
demonstrate typical properties of axisymmetric equilibria. We shall now examine one such
analytic solution, the Solov’ev solution (Solovev 1976).
The Solov’ev solution is obtained by invoking two assumptions: first, the pressure is
assumed to be a linear function ofψ,
P=P 0 +λψ (9.52)and second,I is assumed to be constant within the plasma so
I′=0. (9.53)
The second assumption corresponds to having all thez-directed currentflowing on the
z-axis, so that away from thez-axis, the toroidal field is a vacuum field (cf. Eq.(9.35)).
This arrangement is equivalent to having a zero-radius current-carrying wire going up the
zaxis acting as the sole source for the toroidal magnetic field. The region over which the
Solev’ev solution applies excludes thez−axis and so the problem is analogous to a central
force problem where the source of the central force is a singularity at theorigin. The
excluded region of thez−axis corresponds to the ‘hole in the doughnut’ of a tokamak. In
the more general case whereI′is finite, the plasma is either diamagnetic (toroidal field is
weaker than the vacuum field) or paramagnetic (toroidal field is stronger than the vacuum
field).
Using these two simplifying assumptions the Grad-Shafranov equation reduces to
r∂
∂r(
1
r∂ψ
∂r)
+
∂^2 ψ
∂z^2
+4π^2 r^2 μ 0 λ=0 (9.54)which has the exact solution (Solov’ev solution)
ψ(r,z)=ψ 0r^2
r 04(
2 r^20 −r^2 − 4 α^2 z^2)
(9.55)
whereψ 0 ,r 0 , andαare constants. Figure 9.11 shows a contour plot ofψas a function
ofr/r 0 andz/z 0 for the caseα=1. Note that there are three distinct types of curves in
Fig.9.11, namely (i) open curves going toz=±∞,(ii) concentric closed curves, and (iii)
a single curve called the separatrix which separates the first two types of curves.