Toroidal Plasma Equilibrium
As before, let , , be a set of righthanded cylindrical coordinates whose symmetry axis corresponds to that of the
plasma equilibrium.
On the other hand, let , , be a set of righthanded flux coordinates such that labels the equilibrium magnetic fluxsurfaces, and increases by
for every poloidal circuit of a given fluxsurface. We can assume that
without loss of generality. (Note that is a generalization of the poloidal angle introduced in Section 2.7 that does not assume that the fluxsurfaces have circular crosssections.) As before, we shall set on the outboard midplane. Note that
. The Jacobean of our fluxcoordinate
system is defined

(2.121) 
Now, a general vector field, , can be written

(2.122) 
Moreover [29],

(2.123) 
The axisymmetric equilibrium magnetic field of a tokamak can be expressed in the following manifestly
divergencefree manner:

(2.124) 
where
. It follows that
where

(2.128) 
is the safetyfactor profile [29]. Note that the previous expression reduces to expression (1.76) in the large aspectratio,
circular magnetic fluxsurface limit.
It is convenient to specialize to a coordinate system in which

(2.129) 
It follows that

(2.130) 
The equilibrium electric field is written
Here,
, and
. Note that the previous equation automatically satisfies
.
Finally, we expect the plasma equilibrium to be characterized by number density, temperature, and pressure profiles that
are fluxsurface functions [18]. In other words,
,
, and
. (See
Section 2.25.)