As is well known, the smallness of the toroidal magnetization parameter,
, ensures that the dominant parallel
viscosity tensors,
![$\pi$](img478.png)
and
![$\Theta$](img893.png)
, take the so-called Chew-Goldberger-Low forms [11,29]:
where
. Incidentally, it is clear from Equation (2.60) that the parallel viscosity tensor in the classical closure scheme does indeed take this form.
Now, if
![$\pi$](img478.png)
takes the Chew-Goldberger-Low form then
These results follow because
.
Furthermore,
where, in Cartesian coordinates,
![$\displaystyle \left(\widetilde{\bf A}\right)_{jk} = \frac{1}{2}\left(\frac{\partial A_j}{\partial r_k} + \frac{\partial A_k}{\partial r_j}\right),$](img903.png) |
(2.157) |
and use has been made of the fact that
![$\pi$](img478.png)
is a symmetric tensor. In fact, it can be shown that [19]
![$\displaystyle \widetilde{\nabla}\left(R^2\,\nabla\varphi\right) = {\bf0}.$](img904.png) |
(2.158) |
Hence, we deduce that
Making use of Equation (2.150), we obtain
![$\displaystyle \left\langle R^2\,\nabla\varphi \cdot\nabla\cdot\mbox{\boldmath$\...
...rphi\cdot\mbox{\boldmath$\pi$}_{\parallel\,s}\cdot\nabla {\cal V}\right\rangle.$](img906.png) |
(2.160) |
However, if
![$\pi$](img478.png)
takes the Chew-Goldberger-Low form then
![$\nabla\varphi\cdot$](img907.png)
![$\pi$](img478.png)
,
because
. It follows that [32,34]
![$\displaystyle \left\langle R^2\,\nabla\varphi \cdot\nabla\cdot\mbox{\boldmath$\pi$}_{\parallel\,s} \right\rangle =0.$](img910.png) |
(2.161) |
Likewise, if
![$\Theta$](img893.png)
takes the Chew-Goldberger-Low form then
![$\displaystyle \left\langle R^2\,\nabla\varphi \cdot\nabla\cdot\mbox{\boldmath$\Theta$}_{\parallel\,s} \right\rangle =0.$](img911.png) |
(2.162) |