Let
be the equilibrium plasma pressure profile.
The equilibrium force balance relation (see Section 2.25)
![$\displaystyle {\bf j}\times{\bf B} = \nabla p$](img4028.png) |
(14.30) |
gives
![$\displaystyle {\cal J}\,(j^{\,\theta}\,B^{\,\varphi}- j^{\,\varphi}\,B^{\,\theta})= \frac{dp}{dr},$](img4029.png) |
(14.31) |
where use has been made of Equations (14.5) and (14.7).
The previous equation reduces to the Grad-Shafranov equation [19,26,32],
![$\displaystyle \frac{f}{r}\,\frac{\partial}{\partial r}\!\left(r\,f\,\vert\nabla...
...dg}{dr} + \frac{\mu_0}{B_0^{\,2}}\left(\frac{R}{R_0}\right)^2\frac{dp}{dr} = 0,$](img4030.png) |
(14.32) |
where use has been made of Equations (14.20), (14.21), (14.28), and (14.29). The Grad-Shafranov
equation can be written in the alternative form
![$\displaystyle \frac{\partial^2 \hat{\mit\Psi}_p}{\partial \hat{R}^2} -\frac{1}{...
...g\,\frac{dg}{d\hat{\mit\Psi}_p}+ \hat{R}^2\frac{d\hat{p}}{d\hat{\mit\Psi}_p}=0,$](img4031.png) |
(14.33) |
where
,
,
, and
.