Some discussions

The fact $\mathbf{B} \cdot \nabla \alpha = 0$ implies that $\alpha$ is constant along a magnetic field line. At first glance, a magnetic line on an irrational surface seems to sample all the points on the surface. This seems to indicate that $\alpha$ is a flux surface label for irrational surface. However, $\alpha$ must be a non-flux-surface-function so that it can provide a suitable toroidal coordinate. I had once been confused by this conflict for a long time. The key point to resolve this confusion is to realize that it is wrong to say there is only one magnetic line on an irrational surface, i.e. it is wrong to say a magnetic line on an irrational surface samples all the points on the surface. There are still infinite number of magnetic field lines that can not be connected with each other on an irrational surface. Then the fact $\mathbf{B} \cdot \nabla \alpha = 0$ does not imply that $\alpha$ must be the same on these different magnetic field lines. In fact, although $\mathbf{B} \cdot \nabla \alpha = 0$, the gradient of $\alpha$ on a flux-surface along the perpendicular (to $\mathbf {B}$) direction is nonzero, i.e., $\mathbf{B} \times \nabla \Psi \cdot \nabla \alpha \neq 0$. [Proof:

$$\mathbf{B} \times \nabla \Psi \cdot \nabla \alpha = \nabla \Psi \times \nabla \alpha \cdot \mathbf{B}$$

$$= B^2$$ (303)

which is obviously nonzero.] This indicates that $\alpha$ is not constant on a flux-surface.

In $(\psi , \theta , \phi )$ coordinates, $\nabla \phi$ is perpendicular to $\nabla \psi$. However, in field-line-following coordinates $(\psi , \theta , \alpha )$, $\nabla \alpha$ is not perpendicular to $\nabla \psi$. [Proof:

$$\mathbf{B} \times \nabla \psi \cdot \nabla \alpha = \nabla \psi \times \nabla \alpha \cdot \mathbf{B}$$

$$= \Psi' (\nabla \psi \times \nabla \alpha) \times \nabla \psi \cdot \nabla \alpha$$

$$= \Psi' [\vert \nabla \psi \vert^2 \nabla \alpha - (\nabla \alpha \cdot \nabla \psi) \nabla \psi] \cdot \nabla \alpha$$

$$= \Psi' [\vert \nabla \psi \vert^2 \vert \nabla \alpha \vert^2 - (\nabla \alpha \cdot \nabla \psi)^2] .$$ (304)

on the other hand

$$\mathbf{B} \times \nabla \psi \cdot \nabla \alpha = \frac{B^2}{\Psi'} \neq 0$$

therefore

$$\Psi' [\vert \nabla \psi \vert^2 \vert \nabla \alpha \vert^2 - (\nabla \alpha \cdot \nabla \psi)^2] \neq 0,$$

i.e., $\nabla \alpha$ is not perpendicular to $\nabla \psi$.]

(In GEM code, the radial coordinate $\psi$ is chosen to be the minor radius $r$ of magnetic surfaces on the low-field-side of the midplane, and the field-aligned coordinates $(x, y, z)$ used in the code are defined by $x = r - r_0, y = \alpha r_0 / q_0$, and $z = \theta q_0 R_0$, where $r_0$ and $R_0$ are constant quantities of length dimension, $q_0$ is a dimensionless constant.)

(check** may be wrong, In the practical use of the field-aligned coordinates, the $\theta$ coordinate, which is along the magnetic field line, can not be infinite, i.e., we can not follow a magnetic field for infinite distance. It must be truncated into a finite interval.

[check** may be wrong. **Next, I explain the practical use of the field-aligned coordinates in the flux-tube turbulence simulations. The flux-tube means a region that is in the vicinity of a magnetic line and follows the magnetic field line. On every magnetic surface, we follow the magnetic field line for a distance longer than the $2 \pi / k_{\parallel \min}$, where $k_{\parallel \min}$ is the smallest parallel wave number included in the simulation.

**may be wrong** check**The perpendicular width of the flux-tube is determined by the width of $\alpha$ coordinate, which is chosen so that the perpendicular width of the flux tube on a magnetic surface is much larger than the perpendicular wavelength of the turbulence. Then we can use the periodic condition at $\alpha = \alpha_{\min}$ and $\alpha = \alpha_{\max}$ where $[\alpha_{\min}, \alpha_{\max}]$ is the width of $\alpha$ coordinate chosen by us. The radial range of the flux tube is chosen to be much larger than the largest radial wave-length of the turbulence. The radial profiles of all the equilibrium quantities are assumed to be constant and the effects of the radial gradient of the equilibrium quantities enter the model through the drive terms in the gyro-kinetic equation.**]

By the way, note that $(\mathbf{B} \cdot \nabla \alpha) / (\mathbf{B} \cdot \nabla \theta) = 0$, i.e., the magnetic field lines are straight with zero slope on $(\theta, \alpha)$ plane.

Figure 23: The value of $\theta$, $\alpha$ and their gradients on an annulus on the poloidal plane $(R, Z)$. Note that $\nabla \theta$ is single-valued while $\partial \alpha / \partial R$ and $\nabla \alpha$ are multi-valued and thus there is a jump near the branch cut when a single branch is chosen.

Figure: The same as Fig. 23, but gradients are computed in cylindrical coordinates. $\overline{\delta} = \int_0^{\theta} \hat{q} d \theta$, $\alpha = \phi - \overline {\delta }$. The results agrees with those of Fig. 23, which provides the confidence in the correctness of the numerical implemenation.

Figure 25: The value of $\theta$, $\overline {\delta }$ on an annulus on the poloidal plane $(R, Z)$.