Fundamentals of Plasma Physics

9.8 Static equilibria 279

then using Eq.(9.22) to determine a correspondingB(r)with associated currentJ(r);alter-
natively one could imagine prescribingB(r)and then using Eq.(9.22) to determineP(r).
Unfortunately, neither of these approaches work in general because solutions to Eq.(9.22)
only exist for a very limited set of functions.
The reason why an arbitrary magnetic field cannot be specified is that∇×∇P =0
is always true by virtue of a mathematical identity whereas∇×(J×B)=0is true
only for certain types ofB(r).It is also true that an arbitrary equilibrium pressure profile
P(r)cannot be prescribed (see assignment #2 where it is demonstrated that noJ×B
exists which can confine a spherically symmetric pressure profile). Equilibria thus exist
only for certain specific situations, and these situations typically require symmetry in some
direction. We shall now examine a very important example, namely static equilibria which
are azimuthally symmetric about an axis (typically defined as thezaxis). This symmetry
applies to a wide variety of magnetic confinement devices used in magnetic fusion research,
for example tokamaks, reversed field pinches, spheromaks, and field reversed theta pinches.
We start the analysis by assuming azimuthal symmetry about thezaxis of a cylindri-
cal coordinate systemr,φ,zso that any physical quantityfhas the property∂f/∂φ=0.
Because of the identity∇×∇φ=0,algebraic manipulations become considerably sim-
plified if vectors in theφdirection are expressed in terms of∇φ=φ/rˆ rather than in terms
ofˆφ.As sketched in Fig.9.10 the term toroidal denotes vectors in theφdirection (long way
around a torus) and the term poloidal denotes vectors in ther−zplane.

z

toroidal

vector

poloidal

vector

Figure 9.10: Toroidal vectors and poloidal vectors.

The most general form for an axisymmetric magnetic field is

B=

1

2 π

(∇ψ×∇φ+μ 0 I∇φ). (9.34)

ψ(r,z)is called the poloidalflux andI(r,z)is the current linked by a circle of radiusr
with center on the axis at axial locationz.The toroidal magnetic field then is

Btor=Bφφˆ=

μ 0 I

2 π

∇φ=

μ 0 I

2 πr

φˆ (9.35)