Toroidal Plasma Equilibrium
As before, let
,
,
be a set of right-handed cylindrical coordinates whose symmetry axis corresponds to that of the
plasma equilibrium.
On the other hand, let
,
,
be a set of right-handed flux coordinates such that
labels the equilibrium magnetic flux-surfaces, and
increases by
for every poloidal circuit of a given flux-surface. 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 flux-surfaces have circular cross-sections.) As before, we shall set
on the outboard midplane. Note that
. The Jacobean of our flux-coordinate
system is defined
![$\displaystyle {\cal J}(R,Z) = (\nabla\psi\times \nabla\theta\cdot\nabla\varphi)^{-1}.$](img834.png) |
(2.121) |
Now, a general vector field,
, can be written
![$\displaystyle {\bf A} = A^\psi\,{\cal J}\,\nabla\theta\times\nabla\varphi+ A^\t...
...nabla\varphi\times\nabla\psi+A^\varphi\,{\cal J}\,\nabla\psi\times\nabla\theta.$](img835.png) |
(2.122) |
Moreover [29],
![$\displaystyle {\cal J}\,\nabla\cdot{\bf A} = \frac{\partial({\cal J}\,A^\psi)}{...
...\theta)}{\partial\theta}+\frac{\partial({\cal J}\,A^\varphi)}{\partial\varphi}.$](img836.png) |
(2.123) |
The axisymmetric equilibrium magnetic field of a tokamak can be expressed in the following manifestly
divergence-free manner:
![$\displaystyle {\bf B} = \nabla\varphi\times \nabla\psi + \nabla{\mit\Psi}\times \nabla\theta,$](img837.png) |
(2.124) |
where
. It follows that
where
![$\displaystyle q(\psi)=\frac{d{\mit\Psi}}{d\psi}$](img844.png) |
(2.128) |
is the safety-factor profile [29]. Note that the previous expression reduces to expression (1.76) in the large aspect-ratio,
circular magnetic flux-surface limit.
It is convenient to specialize to a coordinate system in which
![$\displaystyle {\cal J} = \frac{q(\psi)\,R^2}{I(\psi)}.$](img845.png) |
(2.129) |
It follows that
![$\displaystyle {\bf B} = I\,\nabla\varphi + \nabla\varphi\times \nabla\psi.$](img846.png) |
(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 flux-surface functions [18]. In other words,
,
, and
. (See
Section 2.25.)