Magnetic surface average

Define the magnetic surface average of a physical quantiy $G$ by

$$\langle G \rangle \equiv \lim_{\Delta \Psi \rightarrow 0} \left( ... ...t \int \int_{\Delta \Psi} G d^3 V}{\int \int \int_{\Delta \Psi} d^3 V} \right),$$ (207)

where the volume integration is over the small volume between two adjacent flux surfaces with $\Psi$ difference being $\triangle \Psi$. The differential volume element is given by $d^3 V = \vert \mathcal{J} \vert d \psi d \theta d \phi$, where $\mathcal{J}$ is the Jacobian of $(\psi , \theta , \phi )$ coordinates. Using this, equation (207) is written as

$$\langle G \rangle = \lim_{\Delta \Psi \rightarrow 0} \left( \frac{\int \int \int_{\De... ...\int \int_{\Delta \Psi} \vert \mathcal{J} \vert d \psi d \theta d \phi} \right)$$

$$= \frac{\int \int G \vert \mathcal{J} \vert d \theta d \phi}{\int \int \vert \mathcal{J} \vert d \theta d \phi},$$ (208)

which is an averaging over a magnetic surface and thus is called magnetic surface averaging. Sometimes, we do not want the Jacobian explicitly to appear in the formula. This can be achived by writing the differential volume element as

$$d^3 V = R d \phi \frac{d \Psi}{\vert \nabla \Psi \vert} d l_p .$$ (209)

Using $B_p = \vert \nabla \Psi \vert / R$, the volume element is further written as

$$d^3 V = d \phi \frac{d \Psi}{B_p} d l_p$$ (210)

Using this, the averaging defined in Eq. (207) is written as

$$\langle G \rangle = \lim_{\Delta \Psi \rightarrow 0} \frac{\int \int \int_{\Delta \Ps... ...\Psi}{B_p} d l_p}{\int \int \int_{\Delta \Psi} d \phi \frac{d \Psi}{B_p} d l_p}$$

$$= \frac{\int \int G \frac{1}{B_p} d \phi d l_p}{\int \int \frac{1}{B_p} d \phi d l_p} .$$ (211)

If $G$ is axisymmetric, then the above equation is written as

$$\langle G \rangle = \frac{\oint G \frac{1}{B_p} d l_p}{\oint \frac{1}{B_p} d l_p} .$$ (212)

(Equation (212) is used in the GTAW code to calculate the magnetic surface averaging.) Using Eq. (188) and $B_p = \vert \nabla \Psi \vert / R$, equation (212) can also be written as

$$\langle G \rangle = \frac{\int_0^{2 \pi} G \vert\mathcal{J}\vert d \theta}{\int_0^{2 \pi} \vert\mathcal{J}\vert d \theta} .$$ (213)

Noting that the Jacobian does not change sign, the above equation is written as

$$\langle G \rangle = \frac{\int_0^{2 \pi} G\mathcal{J}d \theta}{\int_0^{2 \pi} \mathcal{J}d \theta} .$$ (214)

Using the expression of the volume element $d \tau = \vert\mathcal{J}\vert d \theta d \phi d \psi$, the volume within a magnetic surface is written

$$V (\psi) = \int d \tau = \int \vert\mathcal{J}\vert d \theta d \p... ...\pi \int_{\psi_0}^{\psi} \int_0^{2 \pi} \vert\mathcal{J}\vert d \theta d \psi .$$ (215)

Using this, the differential of $V$ with respect to $\psi$ is written as

$$V' \equiv \frac{d V}{d \psi} = 2 \pi \int_0^{2 \pi} \vert\mathcal{J}\vert d \theta .$$ (216)

Using this, Eq. (213) is written as

$$\langle G \rangle = \frac{2 \pi}{V'} \int_0^{2 \pi} G \vert\mathcal{J}\vert d \theta$$ (217)