Flux-Surface Average Operator

The flux-surface average operator, $\langle\cdots\rangle$, is defined [8]

$$\langle A(\varsigma,{\mit\Omega},\zeta,T)\rangle \equiv \left... .../2}}\,\frac{d\zeta}{2\pi} &&{\mit\Omega}> 1 \end{array}\right.,$$ (8.67)

where $\zeta_0=\cos^{-1}({\mit\Omega})$ and $0\leq\zeta_0\leq\pi$. It follows that

$$\langle\{A,{\mit\Omega}\}\rangle = 0$$ (8.68)

for any $A(\varsigma,{\mit\Omega},\zeta,T)$. It is helpful to define

$$\tilde{A} \equiv A - \frac{\langle A\rangle}{\langle 1\rangle}.$$ (8.69)

It follows that

$$\langle \tilde{A}\rangle =0$$ (8.70)

for any $A(\varsigma,{\mit\Omega},\zeta,T)$.

Equation (8.66) yields

$${\cal J}_0({\mit\Omega},\zeta,T)= \overline{\cal J}_0({\mit\Omega... ...al_{\mit\Omega}\!\left[M\left(M+\frac{L}{1+\tau}\right)\right]\widetilde{X^2} ,$$ (8.71)

where $\overline{{\cal J}}_0({\mit\Omega},T)$ is an undetermined flux-surface function.