Numerical verification of the field-aligned coordinates

The generalized toroidal angle $\alpha$ is numerically calculated in my code. To verify $\mathbf{B} \cdot \nabla \alpha = 0$ along a magnetic field-line, figure 26 plots the values of $\alpha$ along a magnetic field line, which indicates that $\alpha$ is constant. This indicates the numerical implementation of the field-aligned coordinates is correct.

Figure 26: Left: Projection of a field line on the poloidal plane. Right: the value of $\Delta$ and $\alpha$ along the magnetic field line. Here $\alpha$ is defined by $\alpha = \phi - \Delta$, where $\phi$ is the usual cylindrical toroidal angle and $\Delta = \int_0^{\theta} \frac{\mathbf{B} \cdot \nabla \phi}{\mathbf{B} \cdot \nabla \theta} d \theta,$