Toroidal Plasma Equilibrium

As before, let $R$, $\varphi $, $Z$ be a set of right-handed cylindrical coordinates whose symmetry axis corresponds to that of the plasma equilibrium. On the other hand, let $\psi $, $\theta$, $\varphi $ be a set of right-handed flux coordinates such that $\psi(R,Z)$ labels the equilibrium magnetic flux-surfaces, and $\theta$ increases by $2\pi$ for every poloidal circuit of a given flux-surface. We can assume that $\theta=\theta(R,Z)$ without loss of generality. (Note that $\theta$ 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 $\theta=0$ on the outboard midplane. Note that $\vert\nabla\varphi\vert= 1/R$. The Jacobean of our flux-coordinate system is defined

$\displaystyle {\cal J}(R,Z) = (\nabla\psi\times \nabla\theta\cdot\nabla\varphi)^{-1}.$ (2.121)

Now, a general vector field, ${\bf A}$, can be written

$\displaystyle {\bf A} = A^\psi\,{\cal J}\,\nabla\theta\times\nabla\varphi+ A^\t...
...nabla\varphi\times\nabla\psi+A^\varphi\,{\cal J}\,\nabla\psi\times\nabla\theta.$ (2.122)

Moreover [29],

$\displaystyle {\cal J}\,\nabla\cdot{\bf A} = \frac{\partial({\cal J}\,A^\psi)}{...
...\theta)}{\partial\theta}+\frac{\partial({\cal J}\,A^\varphi)}{\partial\varphi}.$ (2.123)

The axisymmetric equilibrium magnetic field of a tokamak can be expressed in the following manifestly divergence-free manner:

$\displaystyle {\bf B} = \nabla\varphi\times \nabla\psi + \nabla{\mit\Psi}\times \nabla\theta,$ (2.124)

where ${\mit\Psi}= {\mit\Psi}(\psi)$. It follows that

$\displaystyle {\cal J}\,B^\psi$ $\displaystyle =0,$ (2.125)
$\displaystyle {\cal J}\,B^\theta$ $\displaystyle =1,$ (2.126)
$\displaystyle {\cal J}\,B^\varphi$ $\displaystyle = q,$ (2.127)


$\displaystyle q(\psi)=\frac{d{\mit\Psi}}{d\psi}$ (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

$\displaystyle {\cal J} = \frac{q(\psi)\,R^2}{I(\psi)}.$ (2.129)

It follows that

$\displaystyle {\bf B} = I\,\nabla\varphi + \nabla\varphi\times \nabla\psi.$ (2.130)

The equilibrium electric field is written

$\displaystyle {\bf E}$ $\displaystyle =\skew{3}\dot{\psi}\,\nabla\varphi-\nabla{\mit\Phi}.$ (2.131)

Here, $\dot{\psi}\equiv \partial \psi/\partial t$, and ${\mit\Phi}= {\mit\Phi}(\psi)$. Note that the previous equation automatically satisfies $\nabla\times {\bf E}=-\partial{\bf B}/\partial t$.

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, $n_s=n_s(\psi)$, $T_s=T_s(\psi)$, and $p_s=p_s(\psi)$. (See Section 2.25.)