284 Chapter 9. MHD equilibria
separatrixand is given by the ellipse
r^2 +4α^2 z^2 =2r^20. (9.56)
On the magnetic axisψis a local maximum ifψ 0 is positive or a local minimum ifψ 0
is negative (the sign depends on the sense of the toroidal current). Thus, one can imagine
that a hill (or valley) of poloidalflux exists with apex (or bottom) at the magnetic axis
and in the vicinity of the apex (bottom) the contours of constantψ are circles or ellipses
enclosing the magnetic axis. The degree of ellipticity of these surfaces is determined by
the value ofα.These closedflux surfaces are a set of nested toroidal surfaces sharing the
same magnetic axis. The projection of the total magnetic field lies in aflux surface since
Eq.(9.34) shows
B·∇ψ=0,
i.e., the magnetic field does not have a component in the direction normal to theflux surface.
Direct substitution of Eq.(9.55) into Eq.(9.54) gives
λ=2 ψ 0
π^2 r^40 μ 0(
1+α^2)
(9.57)
so that the pressure is
P(ψ)=P 0 +
2 ψ 0
π^2 r 04 μ 0(
1+α^2)
ψ. (9.58)Pressure vanishes at the plasma edge so ifψedgeis the poloidalflux at the edge, then the
constantP 0 is determined and Eq.(9.58) becomes
P(ψ)=2 ψ 0
π^2 r^40 μ 0(
1+α^2)[
ψ−ψedge]
. (9.59)
In the earlier discussion of single particle motion it was shown that particles were at-
tached toflux surfaces. Particles that start on a particularflux surface always stay on that
flux surface, although they may travel to any part of thatflux surface. The equilibrium pre-
scribed by Eq.(9.55) is essentially a three-dimensional vortex with pressure peaking on the
magnetic axis and then falling monotonically to zero at the edge.
This is a particularly simple solution to the Grad-Shafranov equation, butnevertheless
demonstrates several important features:
- Three dimensional equilibria can incorporate closed nested poloidalflux surfaces
which are concentric about a magnetic axis and correspond to local maxima orminima
of thefluxψ. - There can also be openflux surfaces;i.e.,flux surfaces which go to infinity.
- Theflux surface separating closed and openflux surfaces is called a separatrix.
- The total magnetic field projects into theflux surfaces sinceB·∇ψ=0.
- Finite pressure corresponds to a depression (or elevation) of the poloidalflux. The
pressure maximum andflux extrema are located at the magnetic axis. - The poloidal magnetic field and associated poloidalfluxψare responsible for plasma
confinement and result from the toroidal current. Thustoroidalcurrent is essential for
confinement in an axisymmetric geometry. For a givenP(ψ)the toroidal magnetic