Useful Results

In the following, $(u)$ represents a $1\times 2$ vector, whereas $[A]$ represents a $2\times 2$ matrix. The elements of $(u)$ and $[A]$ are denoted $u_j$ and $A_{kj}$, respectively, where $k=1,2$ indexes the rows, and $j=1,2$ indexes the columns. Furthermore, $(0)$ denotes a null vector, and $[1]$ denotes an identity matrix.

Equations (2.175) and (2.176) can be written in the compact form [38]

$$\langle {\bf B}\cdot\nabla\cdot(\pi_{\parallel\,s})\rangle = \langle {\bf B}\cdot (F_s)\rangle + e_s\,n_s\,\langle B\rangle\,(E_\parallel),$$ (2.177)

where

$$({\mit\pi}_{\parallel\,s}) = \left(\begin{array}{c}\mbox{\boldmath$\pi$}_{\parallel\,s}\\ [0.5ex]\mbox{\boldmath$\Theta$}_{\parallel\,s}\end{array}\right),$$ (2.178)

$$(F_s) = \left(\begin{array}{c}{\bf F}_{s\,1}\\ [0.5ex]{\bf F}_{s\,2}\end{array}\right),$$ (2.179)

$$(E_\parallel) = \left(\begin{array}{c}\langle {\bf E}\cdot{\bf B}\rangle/\langle B\rangle\\ [0.5ex] 0 \end{array}\right).$$ (2.180)

It is helpful to define

$$(u_{\parallel\,s}) = \left(\begin{array}{c} \langle u_{\parallel\,s\,1}\,B\rangle\\ [0.5ex] \langle u_{\parallel\,s\,2}\,B\rangle \end{array}\right),$$ (2.181)

$$(u_{\theta\,s}) = \left(\begin{array}{c}u_{\theta\,s\,1}\,\langle B^2\rangle\\ [0.5ex]u_{\theta\,s\,2}\,\langle B^2\rangle\end{array}\right).$$ (2.182)

Equations (2.170)–(2.174), (2.181), and (2.182) can be combined to give

$$(u_{\parallel\,s}) = (u_{\theta\,s}) + (V_E) + (V_{\ast\,s}),$$ (2.183)

where

$$(V_E) = \left(\begin{array}{c} V_{E\,1}\\ [0.5ex] 0 \end{array}\right),$$ (2.184)

$$(V_{\ast\,s}) = \left(\begin{array}{c} V_{\ast\,s\,1}\\ [0.5ex] V_{\ast\,s\,2} \end{array}\right).$$ (2.185)

It is also helpful to define

$$(u_{\varphi\,s}) = \frac{I}{\langle R^2\rangle}\left(\begin{array... ...[0.5ex] \langle R^2\,\nabla\varphi\cdot{\bf u}_{s\,2}\rangle\end{array}\right).$$ (2.186)

It follows from Equations (2.130), (2.134)–(2.137), (2.165), (2.166), and (2.171)–(2.174) that

$$(u_{\varphi\,s}) = \frac{I^{\,2}}{\langle R^2\rangle}\left(\begin... ...{\vert\nabla\psi\vert^{\,2}}{B^2}\right\rangle\left[(V_E)+(V_{\ast\,s})\right].$$ (2.187)

However, Equations (2.170) and (2.182) yield

$$\left(\begin{array}{c}\langle u_{\parallel\,s\,1}/B\rangle\\ [0.5... ...theta) + \left\langle\frac{1}{B^2}\right\rangle\left[(V_E)+(V_{\ast\,s})\right]$$ (2.188)

Finally, the previous two equations can be combined to give

$$(u_{\varphi\,s}) = \left(\frac{I^{\,2}}{\langle R^2\rangle\,\langle B^2\rangle}-1\right)(u_{\theta\,s}) + (u_{\parallel\,s}),$$ (2.189)

where use has been made of Equations (2.130) and (2.183).