Flux-Surface Average Operator

It is helpful to define the flux-surface average operator:

$$\langle \cdots\rangle = \left.\oint R^2\,(\cdots)\,\frac{d\theta}{2\pi}\right/\oint R^2\,\frac{d\theta}{2\pi}.$$ (2.148)

According to Equations (2.129) and (2.130),

$$\langle {\bf B}\cdot\nabla A\rangle = 0,$$ (2.149)

where $A(R,Z)$ is a general axisymmetric scalar field. Moreover, $\langle 1\rangle = 1$.

Making use of Equations (2.121)–(2.123) and (2.129), it is easily demonstrated that

$$\langle\nabla\cdot{\bf A}\rangle = \frac{d\langle {\bf A}\cdot\nabla{\cal V}\rangle}{d{\cal V}},$$ (2.150)

where ${\bf A}$ is a general axisymmetric (i.e., $\partial/\partial\varphi = 0$) vector field, and

$${\cal V}(\psi) = \int_0^\psi\oint\oint {\cal J}(\psi',\theta)\,d\... ...\pi)^2\int_0^\psi \frac{q}{I}\left(\oint R^2\,\frac{d\theta}{2\pi}\right)d\psi'$$ (2.151)

is the volume contained within the magnetic flux-surface whose label is $\psi$.