Discussion about the poloidal current function, check!

Note that, on both an irrational surface and a rational surface, there are infinite number of magnetic field lines that are not connected with each other (it is wrong to say there is only one magnetic field line on a irrational surface). Sine $\mathbf{B} \cdot \nabla \Psi = 0$, the value $\Psi$ is a constant along any one of the magnetic field lines. Now comes the question: whether the values of $\Psi$ on different field lines are equal to each other? To answer this question, we can choose a direction different from $\mathbf {B}$ on the magnetic surface and examine whether $\Psi$ is constant or not along this direction, i.e, whether $\mathbf{k} \cdot \nabla \Psi$ equals zero or not, where $\mathbf{k}$ is the chosen direction. For axsiymmetric magnetic surfaces, it is ready to see that $\mathbf{k}= \hat{\ensuremath{\boldsymbol{\phi}}}$ is a direction on the magnetic surface and it is usually not identical with $\mathbf{B}/ B$. Then we obtain

$$\mathbf{k} \cdot \nabla \Psi = \hat{\ensuremath{\boldsymbol{\phi}}} \cdot \nabla \Psi = \frac{1}{R} \frac{\partial \Psi}{\partial \phi} = 0.$$ (527)

Then, since $\mathbf{B} \cdot \nabla \Psi = 0$ and $\mathbf{k} \cdot \nabla \Psi = 0$, where $\mathbf {B}$ and $\mathbf{k}$ are two different directions on the magnetic surfaces, we know that $\Psi$ is constant on the surface, i.e., the values of $\Psi$ on different field lines are equal to each other. This reasoning is for the case of axsiymmetric magnetic surfaces. It is ready to do the same reasoning for non-axisymmetrica magnetic surface after we find a convient direction $\mathbf{k}$ on the magnetic surface.

[check***As discussed in Sec. 2.1, the force balance equation of axisymmetric plasma requires that $\mathbf{B} \cdot \nabla g = 0$. From this and the fact $\mathbf{B} \cdot \nabla \Psi = 0$, we conclude that $g$ is a function of $\Psi$, i.e., $g = g (\Psi)$. However, this reasoning is not rigorous. Note the concept of a function requires that a function can not be a one-to-more map. This means that $g = g (\Psi)$ indicates that the values of $g$ must be equal on two different magnetic field lines that have the same value of $\Psi$. However, the two equations $\mathbf{B} \cdot \nabla g = 0$ and $\mathbf{B} \cdot \nabla \Psi = 0$ do not require this constraint. To examine whether this constraint removes some equilibria from all the possible ones, we consider a system with an $X$ point. Inside one of the magnetic islands, we use

$$g = \Psi^2 / (B_0 R_0^3),$$ (528)

and inside the another, we use

$$g = \Psi^3 / (B_0^2 R_0^5),$$ (529)

Then solve the two GS equations respectively within the boundary of the two islands. It is easy to obtain two magnetic surfaces that have the same value of $\Psi$ respectively inside the two islands. Equations (528) and (529) indicate that the values of $g$ on the two magnetic surfaces are different from each other. It is obvious the resulting equilibrium that contain the two islands can not be recovered by directly solving a single GS equation with a given function $g (\Psi)$.**check]