Visualization of field-line-following coordinates

I am now developing a fully kinetic ions module in electromagnetic turbulence code GEM which uses field-line-following coordinates. Having an accurate understanding of the field-line-following coordinates is importan for writing the code. In this section, I try to visualize some aspects of the coordinates which are helpful for writing correct codes. The directions of the covariant basis vectors of $(\psi , \theta , \alpha )$ coordinates are as follows:

$$\frac{\partial \mathbf{r}}{\partial \alpha} \vert _{\psi, \theta}... ...\ensuremath{\operatorname{direction}} (\hat{\ensuremath{\boldsymbol{\phi}}}),$$

$$\frac{\partial \mathbf{r}}{\partial \theta} \vert _{\psi, \alpha}... ...ath{\operatorname{line}} \ensuremath{\operatorname{direction}}, \mathbf{B}/ B,$$

and the direction of $\partial \mathbf {r}/ \partial \psi \vert _{\theta , \alpha }$ is a combination of the radial and toroidal direction, as is shown in Fig. 15. A trivial but important thing to note is that the usual toroidal angle $\phi$ is changing when changing $\psi$ and/or $\theta$ while keeping the generalized toroidal angle $\alpha$ fixed. This is given by Eq. (282), i.e.,

$$\phi = \alpha + \int_0^{\theta} \frac{\mathbf{B} \cdot \nabla \phi}{\mathbf{B} \cdot \nabla \theta} d \theta \approx \alpha + q (\psi) \theta,$$ (284)

where the second equality becomes exact when $\theta$ is the straight-field-line poloidal angle defined in Sec. 5.3.4. Figure 15 shows how the usual toroidal angle $\phi$ changes when we change $\psi$ while keeping $\theta$ and $\alpha$ fixed.

Figure 15: (a) $\theta = 9 \times 2 \pi / 63$ contour on $\phi = 0$ plane. Middle: $\psi$ coordinate lines ( $\partial \mathbf {r}/ \partial \psi$ is tangent to these lines) on the isosurface of $\theta = 9 \times 2 \pi / 63$. Left: Grids on the isosurface of $\theta = 9 \times 2 \pi / 63$, where the red lines are $\alpha$ coordinate lines ( $\partial \mathbf {r}/ \partial \alpha$ is tangent to these lines), which are along the usual toroidal direction $\hat{\ensuremath{\boldsymbol{\phi}}}$; the blue lines are $\psi$ coordinate lines. Magnetic field from EAST discharge #59954@3.03s.

The relation $\phi \approx \alpha + q (\psi) \theta$ given by Eq. (284) indicates that the toroidal shift for a radial change form $\psi_1$ to $\psi_2$ is given by $(q (\psi_2) - q (\psi_2)) \theta$, which is larger on $\theta$ isosurface with larger value of $\theta$. An example for this is shown in Fig. 16, which has larger toroidal shift than that in Fig. 15.

Figure 16: (a) $\theta = 19 \times 2 \pi / 63$ contour on $\phi = 0$ plane, $\partial \mathbf {r}/ \partial \psi$ lines (upper left) and $\partial \mathbf {r}/ \partial \alpha$ lines (upper right) on isosurface of $\theta = 19 \times 2 \pi / 63$. $\partial \mathbf {r}/ \partial \alpha$ lines are identical with $\partial \mathbf {r}/ \partial \phi$ lines. Lower left: Grids on the isosurface of $\theta = 19 \times 2 \pi / 63$, which are the combinations of $\partial \mathbf {r}/ \partial \psi$ lines and $\partial \mathbf {r}/ \partial \alpha$ lines.

The $\partial \mathbf {r}/ \partial \psi \vert _{\theta , \alpha }$ lines can be understood in another way. Examine a family of magnetic field lines that start from the low-field-side midplane ( $\theta = 0$) with the same toroidal angle $(\phi = \phi_1)$ but different radial coordinates. When these field lines travel to an isosurfce of $\theta$ with $\theta \neq 0$, the intersecting points of these field lines with the $\theta$ isosurface will trace out a $\partial \mathbf {r}/ \partial \psi \vert _{\theta , \alpha }$ line. Examine another family of magnetic field lines similar to the above but with the starting toroidal angle $\phi = \phi_2$. They will trace out another $\partial \mathbf {r}/ \partial \psi \vert _{\theta , \alpha }$ line on the $\theta$ isosurface. Similarly choose another family of field lines with $\phi = \phi_3$, we get the third $\partial \mathbf {r}/ \partial \psi \vert _{\theta , \alpha }$ line. Continue the process, we finally get those lines in the upper-right panel of Fig. 15.

Figure 17 plots $\partial \mathbf {r}/ \partial \psi \vert _{\theta , \alpha }$ lines on the $\theta = 0, 2 \pi$ isosurfaces, which are chosen to be on the low-field-side midplane. On $\theta = 0$ surface, $\partial \mathbf {r}/ \partial \psi \vert _{\theta , \alpha }$ lines are idential to $\partial \mathbf{r}/ \partial \psi \vert _{\theta, \phi}$ lines. On $\theta = 2 \pi$ surface, $\partial \mathbf {r}/ \partial \psi \vert _{\theta , \alpha }$ lines have large $\phi$ shift. In my code, $\theta = 0, 2 \pi$ surface is chosen as the cut of $\theta$ and thus an inner boundary connection condition for the perturbations is needed on this surface. This connection condition is discussed in Sec. 8.2.1.

Figure 17: (a) $\theta = 0$ contour (blue line) in $\phi = 0$ plane; (b) a series of $\partial \mathbf {r}/ \partial \psi \vert _{\theta , \alpha }$ curves (with $\alpha _j = j 2 \pi / 20$, $j = 0, 1, 2, \ldots , 20$) on $\theta = 0$ isosurface, (c) a single $\partial \mathbf {r}/ \partial \psi \vert _{\theta , \alpha }$ curve (with $\alpha = 0$) on $\theta = 2 \pi$ isosurface. This curve finish about 4 torodial loops because $(q_{\max} - q_{\min}) 2 \pi = (5.56 - 1.79) \times 2 \pi \approx 4 \times 2 \pi$. The radial range is $\psi = 0.2 \rightarrow 0.9$, where $\psi$ is the normalized poloidal magnetic flux. Magnetic field from EAST discharge #59954@3.03s (gfile g059954.003030 provided by Hao BaoLong).