Grad-Shafranov Equation

Let $p(r)$ be the equilibrium plasma pressure profile. The equilibrium force balance relation (see Section 2.25)

$\displaystyle {\bf j}\times{\bf B} = \nabla p$ (14.30)


$\displaystyle {\cal J}\,(j^{\,\theta}\,B^{\,\varphi}- j^{\,\varphi}\,B^{\,\theta})= \frac{dp}{dr},$ (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],

$\displaystyle \frac{f}{r}\,\frac{\partial}{\partial r}\!\left(r\,f\,\vert\nabla...
...dg}{dr} + \frac{\mu_0}{B_0^{\,2}}\left(\frac{R}{R_0}\right)^2\frac{dp}{dr} = 0,$ (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

$\displaystyle \frac{\partial^2 \hat{\mit\Psi}_p}{\partial \hat{R}^2} -\frac{1}{...
...g\,\frac{dg}{d\hat{\mit\Psi}_p}+ \hat{R}^2\frac{d\hat{p}}{d\hat{\mit\Psi}_p}=0,$ (14.33)

where $\hat{\mit\Psi}_p= {\mit\Psi}_p/(B_0\,R_0^{\,2})$, $\hat{R} = R/R_0$, $\hat{Z}=Z/R_0$, and $\hat{p}= \mu_0\,p/B_0^{\,2}$.