Cylindrical Tokamak Equilibrium

Consider a low-$\beta$, large aspect-ratio, tokamak plasma equilibrium whose magnetic flux-surfaces map out (almost) concentric circles in the poloidal plane. Such an equilibrium can be approximated as a periodic cylinder [4,14]. Let us employ a conventional set of right-handed cylindrical coordinates, $r$, $\theta$, $z$. The equilibrium magnetic flux-surfaces lie on surfaces of constant $r$. The system is assumed to be periodic in the $z$ (“toroidal”) direction, with periodicity length $2\pi\,R_0$, where $R_0$ is the simulated major radius of the plasma. Let $a$ be the minor radius of the plasma. The equilibrium magnetic field is written

$${\bf B} = B_\theta(r)\,{\bf e}_\theta + B_z(r)\,{\bf e}_z,$$ (3.1)

where $B_\theta(r)$ is the poloidal magnetic field-strength, and $B_z(r)$ the toroidal magnetic field-strength. Here, ${\bf e}_\theta\equiv \nabla\theta/\vert\nabla\theta\vert$ and ${\bf e}_z\equiv \nabla z/\vert\nabla z\vert$. The safety-factor profile takes the form

$$q(r)= \frac{r\,B_z(r)}{R_0\,B_\theta(r)}.$$ (3.2)

[See Equation (1.76).] It is assumed that $q\sim{\cal O}(1)$. The equilibrium current density is written

$${\bf j} = j_\theta(r) \,{\bf e}_\theta+j_z(r)\,{\bf e}_z,$$ (3.3)

where the poloidal and toroidal current densities take the respective forms

$$\mu_0\,j_\theta = -B_z',$$ (3.4)

$$\mu_0\,j_z = \frac{(r\,B_\theta)'}{r},$$ (3.5)

and $'$ denotes $d/dr$. The plasma equilibrium satisfies the force balance criterion [see Equation (2.377)],

$${\bf j}\times {\bf B} - \nabla p = 0,$$ (3.6)

where $p(r)$ is the total plasma pressure. It follows from Equations (3.1) and (3.3)–(3.5) that

$$\frac{d}{dr}\!\left(p + \frac{B_\theta^{\,2} + B_z^{\,2}}{2\,\mu_0}\right) + \frac{B_\theta^{\,2}}{\mu_0\,r} = 0.$$ (3.7)