Glasser-Greene-Johnson Parameters
It is convenient to add to this appendix a generalized calculation of the magnetic curvature length,
, that appears in the Rutherford island width evolution
equation, (12.15).
Let
,
, and
.
Furthermore, let
![$\displaystyle J_1(r)$](img4855.png) |
![$\displaystyle = \oint\frac{1}{\hat{B}}\,\frac{d{\mit\Theta}}{2\pi},$](img4856.png) |
(A.89) |
![$\displaystyle J_2(r)$](img4857.png) |
![$\displaystyle = \oint \hat{B}\,\frac{d{\mit\Theta}}{2\pi},$](img4858.png) |
(A.90) |
![$\displaystyle J_3(r)$](img4859.png) |
![$\displaystyle = \oint\frac{1}{\hat{B}^{\,3}}\,\frac{d{\mit\Theta}}{2\pi},$](img4860.png) |
(A.91) |
![$\displaystyle J_4(r)$](img4861.png) |
![$\displaystyle = \oint\frac{1}{\hat{B}\,\vert\hat{\nabla}\psi_p\vert^{\,2}}\,\frac{d{\mit\Theta}}{2\pi},$](img4862.png) |
(A.92) |
![$\displaystyle J_5(r)$](img4863.png) |
![$\displaystyle = \oint\frac{\hat{B}}{\vert\hat{\nabla}\psi_p\vert^{\,2}}\,\frac{d{\mit\Theta}}{2\pi},$](img4864.png) |
(A.93) |
![$\displaystyle J_6(r)$](img4865.png) |
![$\displaystyle = \oint\frac{1}{\hat{B}^{\,3}\,\vert\hat{\nabla}\psi_p\vert^{\,2}}\,\frac{d{\mit\Theta}}{2\pi}.$](img4866.png) |
(A.94) |
It follows that [6]
where
, and
is the equilibrium plasma pressure.
Finally,
![$\displaystyle D_R(r)= E + F + H^{2}.$](img4873.png) |
(A.98) |
The value of the dimensionless parameter
at a given rational magnetic flux-surface is related to the magnetic curvature length,
, introduced in Section 11.4, according to
![$\displaystyle D_R(r)= - \frac{2\,c_\beta^{\,2}\,L_s^{\,2}}{L_c\,L_p},$](img4875.png) |
(A.99) |
Here,
is a dimensionless measure of the plasma pressure at the rational surface [see Equations (4.65) and (4.66)],
the magnetic shear-length at the rational surface [see Equation (5.27)], and
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-
, tokamak plasma with magnetic flux-surfaces of circular cross-section.