Consider an equilibrium magnetic flux-surface whose label is
.
Let
![$\displaystyle \frac{1}{\gamma(r)} =\frac{q}{g}\oint\frac{B\,R^{2}}{B_0\,R_0^{\,2}}\,\frac{d\theta}{2\pi},$](img4689.png) |
(A.8) |
where
, and
is the equilibrium magnetic field. Here,
,
,
,
,
, and
are specified in Sections 14.2 and 14.4.
It is helpful to define a new poloidal angle
such that
![$\displaystyle \frac{d{\mit\Theta}}{d\theta} = \frac{\gamma\,q}{g}\,\frac{B\,R^{2}}{B_0\,R_0^{\,2}}.$](img4692.png) |
(A.9) |
Let
![$\displaystyle I_1(r)$](img4693.png) |
![$\displaystyle =\oint \frac{B_0}{B}\,\frac{d{\mit\Theta}}{2\pi},$](img4694.png) |
(A.10) |
![$\displaystyle I_2(r)$](img4695.png) |
![$\displaystyle =\oint \frac{B}{B_0}\,\frac{d{\mit\Theta}}{2\pi},$](img4696.png) |
(A.11) |
![$\displaystyle I_3(r)$](img4697.png) |
![$\displaystyle =\oint\left(\frac{\partial B}{\partial{\mit\Theta}}\right)^2\frac{1}{B_0\,B}\,\frac{d{\mit\Theta}}{2\pi},$](img4698.png) |
(A.12) |
![$\displaystyle I_{4,j}(r)$](img4699.png) |
![$\displaystyle = \sqrt{2\,j}\oint\frac{\cos(j\,{\mit\Theta})}{B/B_0}\,\frac{d{\mit\Theta}}{2\pi},$](img4700.png) |
(A.13) |
![$\displaystyle I_{5,j}(r)$](img4701.png) |
![$\displaystyle = \sqrt{2\,j}\oint\frac{\cos(j\,{\mit\Theta})}{2\,(B/B_0)^{\,2}}\,\frac{d{\mit\Theta}}{2\pi},$](img4702.png) |
(A.14) |
![$\displaystyle I_6(r,\lambda)$](img4703.png) |
![$\displaystyle =\oint \frac{\sqrt{1-\lambda\,B/B_{\rm max}}}{B/B_0}\,\frac{d{\mit\Theta}}{2\pi},$](img4704.png) |
(A.15) |
where the integrals are taken at constant
,
is the maximum value of
on the magnetic
flux-surface, and
a positive integer.
The species-
transit frequency is written [7]
![$\displaystyle \omega_{t\,s}(r)= K_t\,\gamma\,v_{t\,s},$](img4706.png) |
(A.16) |
where
![$\displaystyle K_t(r) = \frac{I_1^{\,2}\,I_3}{I_2^{\,2}\,\sum_{j=1,\infty} I_{4,j}\,I_{5,j}},$](img4707.png) |
(A.17) |
and
![$\displaystyle v_{t\,s}(r) = \sqrt{\frac{2\,T_s}{m_s}}.$](img4708.png) |
(A.18) |
Here,
is the
species-
mass, and
the species-
temperature (in energy units). The fraction of passing particles
is [7]
![$\displaystyle f_p(r) = \frac{3\,I_2}{4}\,\frac{B_0^{\,2}}{B_{\rm max}^{\,2}}
\int_0^1 \frac{\lambda\,d\lambda}{I_6(r,\lambda)}$](img4711.png) |
(A.19) |
[See Equation (2.200).]
Finally, the dimensionless species-
collisionality parameter [see Equation (2.95)].
is written [7]
![$\displaystyle \nu_{\ast\,s} (r)= \frac{K_\ast\,g_t}{\omega_{t\,s}\,\tau_{ss}},$](img4712.png) |
(A.20) |
where
[See Equation (2.190).]
Here, the Coulomb logarithm,
[1], is assumed to take the same large constant value (i.e.,
),
independent of species.