Determination of Electron Flows

Making use of Equations (2.169), (2.177), (2.196), and (2.203), we obtain

$$f_t\,[\mu_e]\,(u_{\theta\,e}) = -[F_{ee}]\,(u_{\parallel\,e}) +[F... ...\,(u_{\parallel\,i}) +\langle{\mit\Omega}_e\rangle\,\tau_{ee}\,(E_{\parallel}).$$ (2.233)

It follows from Equations (2.183) and (2.233) that

$$([F_{ee}]+f_t\,[\mu_e])\,(u_{\theta\,e}) = (-[F_{ee}]+[F_{ei}])\,(V_E) - [F_{ee}]\,(V_{\ast\,e}) + [F_{ei}]\,\{(u_{\theta\,i})+(V_{\ast\,i})\}$$

$$\phantom{=} + \langle{\mit\Omega}_{e}\rangle\,\tau_{ee}\,(E_\parallel).$$ (2.234)

However, $(-[F_{ee}]+[F_{ei}])\,(V_E)=(0)$ because $F_{ee\,j1}=F_{ei\,j1}$, for $j=1,2$, and $V_{E\,2}=0$. Hence, we get

$$([F_{ee}]+f_t\,[\mu_e])\,(u_{\theta\,e}) =- [F_{ee}]\,(V_{\ast\,e}) + [F_{ei}]\,([1]-[L_{ii}])\,(V_{\ast\,i}) + \langle{\mit\Omega}_{e}\rangle\,\tau_{ee}\,(E_\parallel),$$ (2.235)

where use has been made of Equation (2.214). It follows that

$$(u_{\theta\,e}) = -[L_{ee}]\,(V_{\ast\,e}) +[L_{ei}]\,(V_{\ast\,i}) + \langle{\mit\Omega}_e\rangle\,\tau_{ee}\,[Q_{ee}]\,(E_{\parallel}),$$ (2.236)

where

$$[L_{ee}] = ([F_{ee}] + f_t\,[\mu_e])^{-1}\,[F_{ee}],$$ (2.237)

$$[L_{ei}] = ([F_{ee} ]+ f_t\,[\mu_e])^{-1}\,[F_{ei}]\,([1] - [L_{ii}]),$$ (2.238)

$$[Q_{ee}] = ([F_{ee}] + f_t\,[\mu_e])^{-1}.$$ (2.239)

Making use of Equations (2.197), (2.198), and (2.216), and only retaining terms up to first order in the small parameter $f_t$, we get

$$[L_{ee}] \begin{align*}= \left[ \begin{array}{cc} 1- \beta_{11}\,f_t, & -\beta_{12}\,f_t\\ [0.5ex] -\beta_{21}\,f_t,& 1-\beta_{22}\,f_t \end{array}\right],\end{align*}$$ (2.240)

$$[L_{ei}] \begin{align*}= \left[ \begin{array}{cc} 1- \beta_{11}\,f_t, & \alpha_1\,(1-[\al... ...x] -\beta_{21}\,f_t,& -\alpha_1\,\beta_{21}\,f_t \end{array}\right],\end{align*}$$ (2.241)

$$[Q_{ee}] \begin{align*}= \left[ \begin{array}{cc} \gamma_{11}-\delta_{11}\,f_t, & \gamma_... ...}-\delta_{21}\,f_t,&\gamma_{22}-\delta_{22}\,f_t \end{array}\right],\end{align*}$$ (2.242)

where

$$\beta_{11} = \frac{(\!\sqrt{2}+13/4)\,\mu_{e\,11} - (3/2)\,\mu_{e\,12}}{\sqrt{2}+1}=1.64,$$ (2.243)

$$\beta_{12} = \frac{(\!\sqrt{2}+13/4)\,\mu_{e\,12} - (3/2)\,\mu_{e\,22}}{\sqrt{2}+1}=1.23,$$ (2.244)

$$\beta_{21} = \frac{-(3/2)\,\mu_{e\,11} + \mu_{e\,12}}{\sqrt{2}+1}=-0.0722,$$ (2.245)

$$\beta_{22} = \frac{-(3/2)\,\mu_{e\,12} + \mu_{e\,22}}{\sqrt{2}+1}=0.600,$$ (2.246)

$$\gamma_{11} = \frac{\sqrt{2}+13/4}{\sqrt{2}+1}=1.93,$$ (2.247)

$$\gamma_{12} = - \frac{3/2}{\sqrt{2}+1}=-0.621,$$ (2.248)

$$\gamma_{21} = - \frac{3/2}{\sqrt{2}+1}=-0.621,$$ (2.249)

$$\gamma_{22} = \frac{1}{\sqrt{2}+1}=0.414,$$ (2.250)

$$\delta_{11} = \beta_{11}\,\gamma_{11} + \beta_{12}\,\gamma_{21}=2.41,$$ (2.251)

$$\delta_{12} = \beta_{11}\,\gamma_{12} + \beta_{12}\,\gamma_{22}=-0.512,$$ (2.252)

$$\delta_{21} = \beta_{21}\,\gamma_{11} + \beta_{22}\,\gamma_{21}=-0.512,$$ (2.253)

$$\delta_{22} = \beta_{21}\,\gamma_{12} + \beta_{22}\,\gamma_{22}=0.293.$$ (2.254)

Here use has been made of Equations (2.209)–(2.211), (2.217), and (2.218).

It follows from Equations (2.165), (2.166), (2.170), (2.181)–(2.185), (2.236), and (2.240)–(2.242) that

$$\frac{{\bf V}_e\cdot\nabla\theta}{{\bf B}\cdot\nabla\theta} =-(1-\beta_{11}\,f_t)\,\frac{V_{\ast\,e\,1}}{\langle B^2\rangle} ... ...,V_{\ast\,e\,2} +(1-\beta_{11}\,f_t)\,\frac{V_{\ast\,i\,1}}{\langle B^2\rangle}$$

$$\phantom{=} +\alpha_1\, [1-(\alpha_2+\beta_{11})\,f_t]\,\frac{V_{... ...elta_{11}\,f_t)\,\frac{\langle {\bf E}\cdot{\bf B}\rangle}{\langle B^2\rangle},$$ (2.255)

$$\frac{{\bf q}_e\cdot\nabla\theta}{p_e\,{\bf B}\cdot\nabla\theta} = -\frac{5}{2}\,\beta_{21}\,f_t\,V_{\ast\,e\,1}+\frac{5}{2}\,(1-\... ...rac{5}{2}\,\alpha_1\,\beta_{21}\,f_t\,\frac{V_{\ast\,i\,2}}{\langle B^2\rangle}$$

$$\phantom{=} +\frac{5}{2}\,\frac{e\,\tau_{e}}{m_e}\,(\gamma_{21}-\delta_{21}\,f_t)\,\frac{\langle {\bf E}\cdot{\bf B}\rangle}{\langle B^2\rangle},$$ (2.256)

and

$$\langle{\bf V}_e\cdot{\bf B}\rangle =V_{E\,1} +\beta_{11}\,f_t\,V_{\ast\,e\,1} +\beta_{12}\,f_t\,V_{\ast\,e\,2} +(1-\beta_{11}\,f_t)\,V_{\ast\,i\,1}$$

$$\phantom{=} + \alpha_1\,[1-(\alpha_2+\beta_{11})\,f_t]\,V_{\ast\,... ..._{e}}{m_e}\,(\gamma_{11}-\delta_{11}\,f_t)\,\langle {\bf E}\cdot{\bf B}\rangle,$$ (2.257)

$$\frac{\langle {\bf q}_e\cdot{\bf B}\rangle}{p_e} = -\frac{5}{2}\,\beta_{21}\,f_t\,V_{\ast\,e\,1} -\frac{5}{2}\,\be... ...21}\,f_t\,V_{\ast\,i\,1}+\frac{5}{2}\,\alpha_1\,\beta_{21}\,f_t\,V_{\ast\,i\,2}$$

$$\phantom{=} +\frac{5}{2}\,\frac{e\,\tau_{e}}{m_e}\,(\gamma_{21}-\delta_{21}\,f_t)\,\langle {\bf E}\cdot{\bf B}\rangle.$$ (2.258)

Here, we have made use of the fact that $\tau_{ee}=\tau_e$.

In the circular magnetic flux-surface limit, Equations (2.255) and (2.256) reduce to

$$V_{\theta\,e} \simeq-(1-\beta_{11}\,f_t)\,\frac{1}{e\,n_e\,B}\,\frac{dp_e}{dr} ... ...\,B}\,\frac{dT_e}{dr}-(1-\beta_{11}\,f_t)\,\frac{1}{e\,n_e\,B}\,\frac{dp_i}{dr}$$

$$\phantom{=} +\alpha_1\, [1-(\alpha_2+\beta_{11})\,f_t]\,\frac{1}{... ...u_{e}}{m_e}\,\frac{B_\theta}{B}\,(\gamma_{11}-\delta_{11}\,f_t)\,E_{\parallel},$$ (2.259)

$$\frac{q_{\theta\,e}}{p_e} \simeq -\frac{5}{2}\,\beta_{21}\,f_t\,\frac{1}{e\,n_e\,B}\,\frac{... ...c{dT_e}{dr} -\frac{5}{2}\,\beta_{21}\,f_t\,\frac{1}{e\,n_e\,B}\,\frac{dp_i}{dr}$$

$$\phantom{=} +\frac{5}{2}\,\alpha_1\,\beta_{21}\,f_t\,\frac{1}{e\,... ...tau_{e}}{m_e}\,\frac{B_\theta}{B}\,(\gamma_{21}-\delta_{21}\,f_t)\,E_\parallel,$$ (2.260)

whereas Equations (2.257)–(2.258) yield

$$V_{\parallel\,e} \simeq-\frac{1}{B_\theta}\,\frac{d{\mit\Phi}}{dr}+\beta_{11}\,f_t... ...frac{dT_e}{dr}-(1-\beta_{11}\,f_t)\,\frac{1}{e\,n_e\,B_\theta}\,\frac{dp_i}{dr}$$

$$\phantom{=} +\alpha_1\, [1-(\alpha_2+\beta_{11})\,f_t]\,\frac{1}{... ..._i}{dr}-\frac{e\,\tau_{e}}{m_e}\,(\gamma_{11}-\delta_{11}\,f_t)\,E_{\parallel},$$ (2.261)

$$\frac{q_{\parallel\,e} }{p_e} \simeq -\frac{5}{2}\,\beta_{21}\,f_t\,\frac{1}{e\,n_e\,B_\theta}\... ...{dr} -\frac{5}{2}\,\beta_{21}\,f_t\,\frac{1}{e\,n_e\,B_\theta}\,\frac{dp_i}{dr}$$

$$\phantom{=} +\frac{5}{2}\,\alpha_1\,\beta_{21}\,f_t\,\frac{1}{e\,... ...ac{5}{2}\,\frac{e\,\tau_{e}}{m_e}\,(\gamma_{21}-\delta_{21}\,f_t)\,E_\parallel.$$ (2.262)

Expressions (2.259)–(2.262) have many features in common with the corresponding expressions, (2.226)–(2.229), for the ions, despite being much more complicated in nature. In particular, the E-cross-B velocity makes no contribution to the poloidal component of the electron fluid velocity. Moreover, the poloidal component of the electron fluid velocity is similar in magnitude to a diamagnetic velocity, whereas the parallel component is larger by a factor $q/\epsilon$.