Determination of Ion Flows

Making use of Equations (2.169), (2.177), (2.191), and (2.203), we obtain

$\displaystyle f_t\,[\mu_i]\,(u_{\theta\,i}) = -[F_{ii}]\,(u_{\parallel\,i}).$ (2.212)

Note that we have neglected terms that are ${\cal O}(\!\sqrt{m_e/m_i})$ smaller than the leading-order terms in the previous equation. This simplifying assumption allows the ion parallel flow to be decoupled from the electron parallel flow [38]. It follows from Equations (2.183) and (2.212) that

$\displaystyle ([F_{ii}] + f_t\,[\mu_i])\,(u_{\theta\,i}) = - [F_{ii}]\,(V_E) - [F_{ii}]\,(V_{\ast\,i}).$ (2.213)

However, $[F_{ii}]\,(V_E)= (0)$ because $F_{ii\,j1}=0$, for $j=1,2$, and $V_{E\,2}=0$. Thus, we get

$\displaystyle (u_{\theta\,i}) = -[L_{ii}]\,(V_{\ast\,i}),$ (2.214)


$\displaystyle [L_{ii}]= ([F_{ii}+f_t\,[\mu_i])^{-1}\,[F_{ii}].$ (2.215)

Employing Equation (2.193), and only retaining terms up to first order in the small parameter $f_t$, we obtain

$\displaystyle [L_{ii}] =\left[\begin{array}{cc} 0,&-\alpha_1\,(1-\alpha_2\,f_t)\\ [0.5ex]
0,&1-\alpha_2\,f_t\end{array}\right],$ (2.216)


$\displaystyle \alpha_1$ $\displaystyle = \frac{\mu_{i\,12}}{\mu_{i\,11}}= 1.17,$ (2.217)
$\displaystyle \alpha_2$ $\displaystyle = \frac{\mu_{i\,11}\,\mu_{i\,22}- \mu_{i\,12}\,\mu_{i\,21}}{\sqrt{2}\,\mu_{i\,11}}=0.461.$ (2.218)

Here, use has been made of Equations (2.204)–(2.206). Incidentally, $f_t\sim \epsilon^{1/2}$ is only a small parameter in the large aspect-ratio limit $\epsilon \ll 1$.

It follows from Equations (2.165), (2.166), (2.170), (2.181)–(2.185), (2.214), and (2.216) that

$\displaystyle \frac{{\bf V}_i\cdot\nabla\theta}{{\bf B}\cdot\nabla\theta}$ $\displaystyle = \alpha_1\,(1-\alpha_2\,f_t)\,\frac{V_{\ast\,i\,2}}{\langle B^2\rangle},$ (2.219)
$\displaystyle \frac{{\bf q}_i\cdot\nabla\theta}{p_i\,{\bf B}\cdot\nabla\theta}$ $\displaystyle = \frac{5}{2} \,(1-\alpha_2\,f_t)\,\frac{V_{\ast\,i\,2}}{\langle B^2\rangle},$ (2.220)


$\displaystyle \langle {\bf V}_i\cdot{\bf B}\rangle$ $\displaystyle = V_{E\,1} + V_{\ast\,i\,1} + \alpha_1\,(1-\alpha_2\,f_t)\,V_{\ast\,i\,2},$ (2.221)
$\displaystyle \frac{\langle {\bf q}_i\cdot{\bf B}\rangle}{p_i}$ $\displaystyle = -\frac{5}{2}\,\alpha_2\,f_t\,V_{\ast\,i\,2}.$ (2.222)

Consider the circular magnetic flux-surface limit. In this limit, $I\simeq R_0\,B$ and $d\psi/dr\simeq R_0\,B_\theta$, so

$\displaystyle B_\theta\,V_{\ast\,s\,1}$ $\displaystyle \simeq -\frac{B}{e_s\,n_s}\,\frac{dp_s}{dr},$ (2.223)
$\displaystyle B_\theta\,V_{\ast\,s\,2}$ $\displaystyle \simeq \frac{B}{e_s}\,\frac{dT_s}{dr},$ (2.224)
$\displaystyle B_\theta\,V_{E\,1}$ $\displaystyle \simeq - B\,\frac{d{\mit\Phi}}{dr},$ (2.225)

where use has been made of Equations (2.171)–(2.174). Thus, Equations (2.219) and (2.220) reduce to [30,52]

$\displaystyle V_{\theta\,i}$ $\displaystyle \simeq \alpha_1\,(1-\alpha_2\,f_t)\,\frac{1}{e\,B}\,\frac{dT_i}{dr} = 1.17\,(1-0.67\,\epsilon^{1/2})\,\frac{1}{e\,B}\,\frac{dT_i}{dr},$ (2.226)
$\displaystyle \frac{q_{\theta\,i}}{p_i}$ $\displaystyle \simeq \frac{5}{2}\,(1-\alpha_2\,f_t)\,\frac{1}{e\,B}\,\frac{dT_i}{dr}= \frac{5}{2}\,(1-0.67\,\epsilon^{1/2})\,\frac{1}{e\,B}\,\frac{dT_i}{dr},$ (2.227)

whereas Equations (2.221) and (2.222) yield

$\displaystyle V_{\parallel\,i}$ $\displaystyle \simeq -\frac{1}{B_\theta}\,\frac{d{\mit\Phi}}{dr} - \frac{1}{e\,...
...p_i}{dr} + \alpha_1\,(1-\alpha_2\,f_t)\,\frac{1}{e\,B_\theta}\,\frac{dT_i}{dr},$ (2.228)
$\displaystyle \frac{q_{\parallel\,i}}{p_i}$ $\displaystyle = - \frac{5}{2}\,\alpha_2\,f_t\,\frac{1}{e\,B_\theta}\,\frac{dT_i}{dr},$ (2.229)

where use has been made of Equations (2.202), (2.217), and (2.218).

According to Equation (2.227), the poloidal component of the ion heat flux is equivalent to the poloidal component of the diamagnetic heat flux specified in Equation (2.137), except that it is reduced by a factor $1-0.67\,\epsilon^{1/2}$. The explanation for this reduction is that trapped ions cannot carry a net poloidal heat flux around a magnetic flux-surface, because they are forced to periodically reverse their poloidal drift direction. Hence, the diamagnetic ion heat flux is reduced by a factor that is roughly equal to the fraction of passing particles. [See Equation (2.201).] This reduction is accomplished by the parallel heat flux specified in Equation (2.229).

Equation (2.226) shows that the poloidal component of the ion fluid velocity is directly proportional to the poloidal component of the ion heat flux. The most surprising feature of Equation (2.226) is the absence of a contribution from the E-cross-B velocity, despite the fact that this velocity possesses a non-zero poloidal component. [See Equation (2.138).] The explanation for this absence is that parallel friction forces drive an ion flow parallel to the magnetic field that cancels out the poloidal component of the E-cross-B velocity, as well as the poloidal component of the ion diamagnetic velocity. The requisite flow, which is specified in Equation (2.228), is quite large, being a factor $q/\epsilon \sim 9$ larger than a typical diamagnetic velocity. Equation (2.226) implies that a tokamak plasma is not free to rotate in the poloidal direction. In fact, any deviation of the ion poloidal flow velocity from the so-called neoclassical velocity specified by Equation (2.226) is opposed by friction between trapped and passing ions, and consequently damps away on a timescale that is proportional to $\tau _i$. This effect is known as poloidal flow damping [48]. (See Section 2.24.)

Incidentally, in the large aspect-ratio limit, Equation (2.189) reduces to

$\displaystyle (u_{\varphi\,s}) \simeq (u_{\parallel\,s}),$ (2.230)

because the omitted term is ${\cal O}[(\epsilon/q)^2]$ smaller than the other terms. Thus, it follows from Equations (2.130), (2.165), (2.166), (2.181), and (2.186) that

$\displaystyle V_{\varphi\,s}$ $\displaystyle \simeq V_{\parallel\,s},$ (2.231)
$\displaystyle q_{\varphi\,s}$ $\displaystyle \simeq q_{\parallel\,s}.$ (2.232)

In other words, the toroidal flow velocities and heat fluxes are approximately equal to the parallel flow velocities and heat fluxes, respectively. This is true for both ions and electrons.