Radial coordinate to be deleted

We know that the toroidal flux $\psi_t$, safety factor $q$, and the $\Psi$ in the GS equation are related by the following equations:

$$d \psi_t = 2 \pi q d \Psi$$ (532)

$$\Longrightarrow \psi_t = 2 \pi \int_0^{\Psi} q d \Psi$$ (533)

Define:

$$\rho \equiv \sqrt{\frac{\psi_t}{\pi}}$$ (534)

(In the Toray_ga code, the radial coordinate $\rho$ is defined as

$$\rho \equiv \sqrt{\frac{\psi_t}{\pi B_{t 0}}},$$ (535)

where $B_{t 0}$ is a constant factor.$\rho$ defined this way is of length dimension, which is an effective geometry radius obtained by approximating the flux surface as circular.)

I use Eq. (534) to define $\rho$. Then we have

$$\psi_t = \pi \rho^2$$ (536)

$$\Longrightarrow \frac{d \psi_t}{d \rho} = 2 \pi \rho$$ (537)

$$\Longrightarrow \frac{d \psi_t}{d \psi} \frac{d \psi}{d \rho} = 2 \pi \rho$$ (538)

$$\Longrightarrow 2 \pi q \frac{d \psi}{d \rho} = 2 \pi \rho$$ (539)

$$\Longrightarrow \frac{d \psi}{d \rho} = \frac{\rho}{q}$$ (540)

Eq. (540) is used to transform between $\psi$ and $\rho$.

$$d \rho = \frac{1}{\sqrt{\Phi / \pi R_0^2 B_0}} \frac{1}{2} \frac{... ...}{\pi B_0 R_0^2} 2 \pi q d \psi = \frac{1}{\rho} \frac{1}{B_0 R_0^2} q d \psi$$

$$\Rightarrow d \psi = \frac{\rho B_0 R_0^2}{q} d \rho$$

Figure 36: to be delted, Isosurface of $\alpha = 2 \pi / 8$. The surface is made of a family of contours of $\alpha = 2 \pi / 8$, which are all magnetic field lines. These field lines are traced by starting from a series of points on the low-field-side midplane $(\theta = 0)$ at different radial locations and the field lines are followed by a complete poloidal loop. Magnetic field from EAST discharge #59954@3.03s.