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

$$J_1(r) = \oint\frac{1}{\hat{B}}\,\frac{d{\mit\Theta}}{2\pi},$$ (A.89)

$$J_2(r) = \oint \hat{B}\,\frac{d{\mit\Theta}}{2\pi},$$ (A.90)

$$J_3(r) = \oint\frac{1}{\hat{B}^{\,3}}\,\frac{d{\mit\Theta}}{2\pi},$$ (A.91)

$$J_4(r) = \oint\frac{1}{\hat{B}\,\vert\hat{\nabla}\psi_p\vert^{\,2}}\,\frac{d{\mit\Theta}}{2\pi},$$ (A.92)

$$J_5(r) = \oint\frac{\hat{B}}{\vert\hat{\nabla}\psi_p\vert^{\,2}}\,\frac{d{\mit\Theta}}{2\pi},$$ (A.93)

$$J_6(r) = \oint\frac{1}{\hat{B}^{\,3}\,\vert\hat{\nabla}\psi_p\vert^{\,2}}\,\frac{d{\mit\Theta}}{2\pi}.$$ (A.94)

It follows that [6]

$$E(r) = - \frac{dP/d\psi_p}{(dq/d\psi_p)^{2}}\,\frac{1}{\gamma}\left[\f... ...t(\frac{J_1}{\gamma}\right) - g\,\frac{dq}{d\psi_p}\,\frac{J_1}{J_2}\right]J_5,$$ (A.95)

$$F(r) = \frac{(dP/d\psi_p)^{\,2}}{(dq/d\psi_p)^{2}}\,\frac{1}{\gamma^{2}}\left[g^{2}\left(J_5\,J_6- J_4^{\,2}\right)+ J_5\,J_3\right],$$ (A.96)

$$H(r) = \frac{dP/d\psi_p}{dq/d\psi_p}\,\frac{g}{\gamma}\left(J_4 - \frac{J_1\,J_5}{J_2}\right),$$ (A.97)

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

$$D_R(r)= E + F + H^{2}.$$ (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

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