9.8 Static equilibria 283Figure 9.11: Contours of constantflux of Solov’ev solution to Grad-Shafranov equation.
Even though it is an idealization, the Solov’ev solution illustrates manyimportant fea-
tures of three-dimensional, static MHD equilibria. Let us now examine some of these
features. The constants in Eq.(9.55) have been chosen so thatψ=ψ 0 atr=r 0 ,z=0
(in the figure this point corresponds tor=1,z=0since lengths in the figure have been
normalized tor 0 ).The locationr=r 0 ,z=0is called themagnetic axisbecause it is the
axis linked by the closed contours of poloidalflux.
Let us temporarily assume thatψ 0 is positive. Examination of Eq.(9.55) shows that
ifψis positive, then the quantity 2 r 02 −r^2 − 4 α^2 z^2 is positive and vice versa. For any
negative value ofψ, Eq.(9.55) can be satisfied by makingrvery small andzvery large. In
particular, ifris infinitesimal thenzmust become infinite. Thus, all contours for whichψ
is negative go toz=±∞;these contours are the open or type (i) contours.
On the other hand, ifψis positive then it has a maximum value ofψ 0 which occurs on
the magnetic axis and if 0 <ψ <ψ 0 thenψmust be located at some point outside the
magnetic axis, but inside the curve 2 r^20 −r^2 − 4 α^2 z^2 =0.Hence all contours of positive
ψmust lie inside the curve 2 r 02 −r^2 − 4 α^2 z^2 =0and correspond to field lines that do not
go to infinity. These contours are the closed or type (ii) contours, since they form closed
curves in ther,zplane.
The contour separating the closed contours from the contours going to infinity isthe