Cylindrical Tokamak Equilibrium
Consider a low-
, 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,
,
,
. The
equilibrium magnetic flux-surfaces lie on surfaces of constant
. The system is
assumed to be periodic in the
(“toroidal”) direction, with periodicity length
,
where
is the simulated major radius of the plasma. Let
be the minor radius of the plasma. The equilibrium magnetic
field is written
![$\displaystyle {\bf B} = B_\theta(r)\,{\bf e}_\theta + B_z(r)\,{\bf e}_z,$](img1416.png) |
(3.1) |
where
is the poloidal magnetic field-strength, and
the toroidal magnetic field-strength.
Here,
and
. The safety-factor profile takes the form
![$\displaystyle q(r)= \frac{r\,B_z(r)}{R_0\,B_\theta(r)}.$](img1420.png) |
(3.2) |
[See Equation (1.76).] It is assumed that
.
The equilibrium current density is written
![$\displaystyle {\bf j} = j_\theta(r) \,{\bf e}_\theta+j_z(r)\,{\bf e}_z,$](img1422.png) |
(3.3) |
where
the poloidal and toroidal current densities
take the respective forms
and
denotes
. The plasma equilibrium satisfies the force balance criterion [see Equation (2.377)],
![$\displaystyle {\bf j}\times {\bf B} - \nabla p = 0,$](img1429.png) |
(3.6) |
where
is the total plasma pressure. It follows from Equations (3.1) and (3.3)–(3.5) that
![$\displaystyle \frac{d}{dr}\!\left(p + \frac{B_\theta^{\,2} + B_z^{\,2}}{2\,\mu_0}\right) + \frac{B_\theta^{\,2}}{\mu_0\,r} = 0.$](img1430.png) |
(3.7) |