Glasser-Greene-Johnson Parameters

It is convenient to add to this appendix a generalized calculation of the magnetic curvature length, $L_c$, that appears in the Rutherford island width evolution equation, (12.15). Let $\hat{B} = \vert{\bf B}\vert/B_0$, $\hat{\nabla} = R_0\,\nabla$, and $d\psi_p/dr=f(r)/R_0$. Furthermore, let

$\displaystyle J_1(r)$ $\displaystyle = \oint\frac{1}{\hat{B}}\,\frac{d{\mit\Theta}}{2\pi},$ (A.89)
$\displaystyle J_2(r)$ $\displaystyle = \oint \hat{B}\,\frac{d{\mit\Theta}}{2\pi},$ (A.90)
$\displaystyle J_3(r)$ $\displaystyle = \oint\frac{1}{\hat{B}^{\,3}}\,\frac{d{\mit\Theta}}{2\pi},$ (A.91)
$\displaystyle J_4(r)$ $\displaystyle = \oint\frac{1}{\hat{B}\,\vert\hat{\nabla}\psi_p\vert^{\,2}}\,\frac{d{\mit\Theta}}{2\pi},$ (A.92)
$\displaystyle J_5(r)$ $\displaystyle = \oint\frac{\hat{B}}{\vert\hat{\nabla}\psi_p\vert^{\,2}}\,\frac{d{\mit\Theta}}{2\pi},$ (A.93)
$\displaystyle J_6(r)$ $\displaystyle = \oint\frac{1}{\hat{B}^{\,3}\,\vert\hat{\nabla}\psi_p\vert^{\,2}}\,\frac{d{\mit\Theta}}{2\pi}.$ (A.94)

It follows that [6]

$\displaystyle E(r)$ $\displaystyle = - \frac{dP/d\psi_p}{(dq/d\psi_p)^{2}}\,\frac{1}{\gamma}\left[\f...
...t(\frac{J_1}{\gamma}\right) - g\,\frac{dq}{d\psi_p}\,\frac{J_1}{J_2}\right]J_5,$ (A.95)
$\displaystyle F(r)$ $\displaystyle = \frac{(dP/d\psi_p)^{\,2}}{(dq/d\psi_p)^{2}}\,\frac{1}{\gamma^{2}}\left[g^{2}\left(J_5\,J_6- J_4^{\,2}\right)+ J_5\,J_3\right],$ (A.96)
$\displaystyle H(r)$ $\displaystyle = \frac{dP/d\psi_p}{dq/d\psi_p}\,\frac{g}{\gamma}\left(J_4 - \frac{J_1\,J_5}{J_2}\right),$ (A.97)

where $P(r) = \mu_0\,p(r)/B_0^{\,2}$, and $p(r)$ is the equilibrium plasma pressure. Finally,

$\displaystyle D_R(r)= E + F + H^{2}.$ (A.98)

The value of the dimensionless parameter $D_R(r_s)$ at a given rational magnetic flux-surface is related to the magnetic curvature length, $L_c$, introduced in Section 11.4, according to

$\displaystyle D_R(r)= - \frac{2\,c_\beta^{\,2}\,L_s^{\,2}}{L_c\,L_p},$ (A.99)

Here, $c_\beta^{\,2}$ is a dimensionless measure of the plasma pressure at the rational surface [see Equations (4.65) and (4.66)], $L_s$ the magnetic shear-length at the rational surface [see Equation (5.27)], and $L_p$ the effective pressure gradient scale-length at the rational surface [see Equation (8.35)]. The previous equation is a generalization of Equation (11.57). The latter equation only holds in a large-aspect ratio, low-$\beta $, tokamak plasma with magnetic flux-surfaces of circular cross-section.