Collisionality Parameters

Consider an equilibrium magnetic flux-surface whose label is $r$. Let

$$\frac{1}{\gamma(r)} =\frac{q}{g}\oint\frac{B\,R^{2}}{B_0\,R_0^{\,2}}\,\frac{d\theta}{2\pi},$$ (A.8)

where $B=\vert{\bf B}\vert$, and ${\bf B}$ is the equilibrium magnetic field. Here, $q(r)$, $g(r)$, $R$, $\theta$, $B_0$, and $R_0$ are specified in Sections 14.2 and 14.4. It is helpful to define a new poloidal angle ${\mit\Theta}$ such that

$$\frac{d{\mit\Theta}}{d\theta} = \frac{\gamma\,q}{g}\,\frac{B\,R^{2}}{B_0\,R_0^{\,2}}.$$ (A.9)

Let

$$I_1(r) =\oint \frac{B_0}{B}\,\frac{d{\mit\Theta}}{2\pi},$$ (A.10)

$$I_2(r) =\oint \frac{B}{B_0}\,\frac{d{\mit\Theta}}{2\pi},$$ (A.11)

$$I_3(r) =\oint\left(\frac{\partial B}{\partial{\mit\Theta}}\right)^2\frac{1}{B_0\,B}\,\frac{d{\mit\Theta}}{2\pi},$$ (A.12)

$$I_{4,j}(r) = \sqrt{2\,j}\oint\frac{\cos(j\,{\mit\Theta})}{B/B_0}\,\frac{d{\mit\Theta}}{2\pi},$$ (A.13)

$$I_{5,j}(r) = \sqrt{2\,j}\oint\frac{\cos(j\,{\mit\Theta})}{2\,(B/B_0)^{\,2}}\,\frac{d{\mit\Theta}}{2\pi},$$ (A.14)

$$I_6(r,\lambda) =\oint \frac{\sqrt{1-\lambda\,B/B_{\rm max}}}{B/B_0}\,\frac{d{\mit\Theta}}{2\pi},$$ (A.15)

where the integrals are taken at constant $r$, $B_{\rm max}(r)$ is the maximum value of $B$ on the magnetic flux-surface, and $j$ a positive integer. The species-$s$ transit frequency is written [7]

$$\omega_{t\,s}(r)= K_t\,\gamma\,v_{t\,s},$$ (A.16)

where

$$K_t(r) = \frac{I_1^{\,2}\,I_3}{I_2^{\,2}\,\sum_{j=1,\infty} I_{4,j}\,I_{5,j}},$$ (A.17)

and

$$v_{t\,s}(r) = \sqrt{\frac{2\,T_s}{m_s}}.$$ (A.18)

Here, $m_s$ is the species-$s$ mass, and $T_s(r)$ the species-$s$ temperature (in energy units). The fraction of passing particles is [7]

$$f_p(r) = \frac{3\,I_2}{4}\,\frac{B_0^{\,2}}{B_{\rm max}^{\,2}} \int_0^1 \frac{\lambda\,d\lambda}{I_6(r,\lambda)}$$ (A.19)

[See Equation (2.200).] Finally, the dimensionless species-$s$ collisionality parameter [see Equation (2.95)]. is written [7]

$$\nu_{\ast\,s} (r)= \frac{K_\ast\,g_t}{\omega_{t\,s}\,\tau_{ss}},$$ (A.20)

where

$$g_t(r) =\frac{f_p}{1-f_p},$$ (A.21)

$$K_\ast(r) = \frac{3}{8\pi}\,\frac{I_2}{I_3}\,K_t^{\,2},$$ (A.22)

$$\frac{1}{\tau_{ss}(r)} = \frac{4}{3\!\sqrt{\pi}}\,\frac{4\pi\,n_s\,e_s^{\,4}\,\ln{\mit\Lambda}}{(4\pi\,\epsilon_0)^{\,2}\,m_s^{\,2}\,v_{t\,s}^{\,3}}.$$ (A.23)

[See Equation (2.190).] Here, the Coulomb logarithm, $\ln{\mit\Lambda}$ [1], is assumed to take the same large constant value (i.e., $\ln{\mit\Lambda}\simeq 16$), independent of species.