Plasma current density

Using the equilibrium constraint in the $\mathbf{R}$ direction, $J_{\phi}$ given by Eq. (41) is written

$$J_{\phi} = - \frac{1}{\mu_0 R} \triangle^{\ast} \Psi$$ (61)

$$= R \frac{d P}{d \Psi} + \frac{1}{\mu_0 R} \frac{d g}{d \Psi} g.$$ (62)

The poloidal current density $\mathbf{J}_p$ is written

$$\mathbf{J}_p \equiv J_R \hat{\mathbf{R}} + J_Z \hat{\mathbf{Z}}$$

$$= \frac{1}{\mu_0} \left( - \frac{1}{R} \frac{\partial g}{\partial Z... ...thbf{R}} + \frac{1}{R} \frac{\partial g}{\partial R} \hat{\mathbf{Z}} \right) .$$

$$= \frac{1}{\mu_0} \nabla g \times \nabla \phi$$ (63)

Using $g = g (\Psi)$, Eq. (63) is written

$$\mathbf{J}_p = \frac{1}{\mu_0} \frac{d g}{d \Psi} \nabla \Psi \times \nabla \phi$$

$$= \frac{1}{\mu_0} \frac{d g}{d \Psi} \mathbf{B}_p .$$

The parallel (to the magnetic field) current density is written as

$$J_{\parallel} \equiv \frac{\mathbf{J} \cdot \mathbf{B}}{B}$$

$$= \frac{J_{\phi} B_{\phi} +\mathbf{J}_p \cdot \mathbf{B}_p}{B}$$

$$= \frac{\left( R \frac{d P}{d \Psi} + \frac{1}{\mu_0 R} \frac{d g}{... ...+ \frac{1}{\mu_0} \frac{d g}{d \Psi} \left( \frac{\nabla \Psi}{R} \right)^2}{B}$$

$$= \frac{g \frac{d P}{d \Psi} + \frac{1}{\mu_0} \frac{d g}{d \Psi} \... ...eft( \frac{g}{R} \right)^2 + \left( \frac{\nabla \Psi}{R} \right)^2 \right]}{B}$$

$$= \frac{g \frac{d P}{d \Psi} + \frac{1}{\mu_0} \frac{d g}{d \Psi} B^2}{B} .$$ (64)

For later use, define

$$\sigma \equiv \frac{J_{\parallel}}{B}$$

$$= g \frac{d P}{d \Psi} \frac{1}{B^2} + \frac{1}{\mu_0} \frac{d g}{d \Psi} .$$ (65)

Equation (65) is used in GTAW code to calculate $J_{\parallel} / B$ (actually calculated is $\mu_0 J_{\parallel} / B$)[8]. Note that the expression for $J_{\parallel} / B$ in Eq. (65) is not a magnetic surface function. Define $\sigma_{\ensuremath{\operatorname{ps}}}$ as

$$\sigma_{\ensuremath{\operatorname{ps}}} \equiv \sigma - \langle \sigma \rangle$$

$$= g \frac{d P}{d \Psi} \left[ \frac{1}{B^2} - \left\langle \frac{1}{B^2} \right\rangle \right]$$ (66)

$$\equiv \frac{J_{\parallel}^{\ensuremath{\operatorname{ps}}}}{B},$$ (67)

where $J_{\parallel}^{\ensuremath{\operatorname{ps}}}$ is called Pfirsch-Schluter (PS) current. In cylindrical geometry, due to the poloidal symmetry, the Pfiersch-Schluter current is obviously zero. In toroidal geometry, due to the poloidal asymmetry, the PS current is generally nonzero. Thus, this quantity characterizes a toroidal effect.

Another useful quantity is $\mu_0 \langle \mathbf{J} \cdot \mathbf{B} \rangle$, which is written as

$$\mu_0 \langle \mathbf{J} \cdot \mathbf{B} \rangle = \mu_0 g \frac{d P}{d \Psi} + \frac{d g}{d \Psi} \langle B^2 \rangle,$$ (68)

where $\langle \ldots \rangle$ is flux surface averaging operator, which will be defined later in this note.