Flux-Surface Average Operator
The flux-surface average operator,
, is defined [8]
![\begin{displaymath}\langle A(\varsigma,{\mit\Omega},\zeta,T)\rangle
\equiv \left...
.../2}}\,\frac{d\zeta}{2\pi}
&&{\mit\Omega}> 1
\end{array}\right.,\end{displaymath}](img2805.png) |
(8.67) |
where
and
.
It follows that
![$\displaystyle \langle\{A,{\mit\Omega}\}\rangle = 0$](img2808.png) |
(8.68) |
for any
. It is helpful to define
![$\displaystyle \tilde{A} \equiv A - \frac{\langle A\rangle}{\langle 1\rangle}.$](img2810.png) |
(8.69) |
It follows that
![$\displaystyle \langle \tilde{A}\rangle =0$](img2811.png) |
(8.70) |
for any
.
Equation (8.66) yields
![$\displaystyle {\cal J}_0({\mit\Omega},\zeta,T)= \overline{\cal J}_0({\mit\Omega...
...al_{\mit\Omega}\!\left[M\left(M+\frac{L}{1+\tau}\right)\right]\widetilde{X^2} ,$](img2812.png) |
(8.71) |
where
is an undetermined flux-surface function.