Grad-Shafranov equation in $(r, \theta )$ coordinates

Consider $(r, \theta )$ coordinates, which are related to the cylindrical coordinates $(R, Z)$ by $Z = r \sin \theta$ and $R = R_0 + r \cos \theta$, as shown in Fig. 28.

Figure 28: The relation between $(r, \theta )$ and $(R, Z)$ coordinates.

Next, we transform the GS equation from $(R, Z)$ coordinates to $(r, \theta )$ coordinates. Using the relations $R = R_0 + r \cos \theta$ and $Z = r \sin \theta$, we have

$$r = \sqrt{(R - R_0)^2 + Z^2} \Rightarrow \frac{\partial r}{\partial Z} = \frac{Z}{r} = \sin \theta$$ (369)

$$\sin \theta = \frac{Z}{\sqrt{(R - R_0)^2 + Z^2}} \Rightarrow \cos \theta \frac{\partial \theta}{\partial Z} = \frac{r - Z \frac{Z}{r}}{r^2} .$$ (370)

$$\Rightarrow \frac{\partial \theta}{\partial Z} = \frac{r - Z \fra... ... (R - R_0)} = \frac{1 - \sin^2 \theta}{R - R_0} = \frac{\cos^2 \theta}{R - R_0}$$ (371)

$$\frac{\partial \sin \theta}{\partial Z} = \frac{r - Z \frac{Z}{r}}{r^2} = \frac{1 - \sin^2 \theta}{r} = \frac{\cos^2 \theta}{r}$$ (372)

The GS equation in $(R, Z)$ coordinates is given by

$$\frac{\partial^2 \Psi}{\partial Z^2} + R \frac{\partial}{\partial... ...ial R} \right) = - \mu_0 R^2 \frac{d P}{d \Psi} - \frac{d g}{d \Psi} g (\Psi) .$$ (373)

The term $\partial \Psi / \partial Z$ is written as

$$\frac{\partial \Psi}{\partial Z} = \frac{\partial \Psi}{\partial r} \frac{\partial r}{\partial Z} + \frac{\partial \Psi}{\partial \theta} \frac{\partial \theta}{\partial Z}$$

$$= \frac{\partial \Psi}{\partial r} \sin \theta + \frac{\partial \Psi}{\partial \theta} \frac{\cos^2 \theta}{R - R_0} .$$ (374)

Using Eq. (374), $\partial^2 \Psi / \partial Z^2$ is written as

$$\frac{\partial^2 \Psi}{\partial Z^2} = \frac{\partial}{\partial Z} \left( \frac{\partial \Psi}{\partial ... ...ft( \frac{\partial \Psi}{\partial \theta} \frac{\cos^2 \theta}{R - R_0} \right)$$

$$= \sin \theta \frac{\partial}{\partial Z} \left( \frac{\partial \Ps... ...theta} \frac{\partial}{\partial Z} \left( \frac{\cos^2 \theta}{R - R_0} \right)$$

$$= \sin \theta \left( \frac{\partial^2 \Psi}{\partial r^2} \sin \the... ...\frac{\partial^2 \Psi}{\partial \theta^2} \frac{\cos^2 \theta}{R - R_0} \right)$$

$$- \frac{\partial \Psi}{\partial \theta} \frac{1}{R - R_0} 2 \cos \theta \sin \theta \frac{\cos^2 \theta}{R - R_0}$$ (375)

$$\frac{\partial r}{\partial R} = \frac{\partial}{\partial R} \sqrt{(R - R_0)^2 + Z^2} = \frac{R - R_0}{r} = \cos \theta$$ (376)

$$\sin \theta = \frac{Z}{\sqrt{(R - R_0)^2 + Z^2}} .$$

$$\cos \theta \frac{\partial \theta}{\partial R} = - Z \frac{R - R_0}{r^3}$$

$$\frac{\partial \theta}{\partial R} = - \frac{Z}{r^2} .$$ (377)

$$\cos \theta = \frac{R - R_0}{\sqrt{(R - R_0)^2 + Z^2}}$$

$$\frac{\partial}{\partial R} \cos \theta = \frac{r - \frac{R - R_0}{r} (R - R_0)}{r^2} = \frac{1 - \cos^2 \theta}{r} = \frac{\sin^2 \theta}{r}$$ (378)

$$R \frac{\partial}{\partial R} \left( \frac{1}{R} \frac{\partial \Psi}{\partial R} \right) = R \frac{\partial}{\partial R} \left[ \frac{1}{R} \left( \frac{\pa... ...rtial \Psi}{\partial \theta} \frac{\partial \theta}{\partial R} \right) \right]$$

$$= R \frac{\partial}{\partial R} \left[ \frac{1}{R} \left( \frac{\pa... ...os \theta - \frac{\partial \Psi}{\partial \theta} \frac{Z}{r^2} \right) \right]$$

$$= \frac{\partial}{\partial R} \left( \frac{\partial \Psi}{\partial ... ...ial \Psi}{\partial \theta} \frac{Z}{r^2} \right) \left( - \frac{1}{R^2} \right)$$

$$= \left( \frac{\partial^2 \Psi}{\partial r^2} \frac{\partial r}{\pa... ...al r} \cos \theta - \frac{\partial \Psi}{\partial \theta} \frac{Z}{r^2} \right)$$

$$= \left( \frac{\partial^2 \Psi}{\partial r^2} \cos \theta - \frac{\... ...al r} \cos \theta - \frac{\partial \Psi}{\partial \theta} \frac{Z}{r^2} \right)$$

$$= \frac{\partial^2 \Psi}{\partial r^2} \cos^2 \theta + \frac{\parti... ...theta}{r} + \frac{\partial \Psi}{\partial \theta} Z \frac{1}{r^3} 2 \cos \theta$$ (379)

$$- \frac{1}{R} \left( \frac{\partial \Psi}{\partial r} \cos \theta - \frac{\partial \Psi}{\partial \theta} \frac{1}{r} \sin \theta \right)$$ (380)

Sum of the expression on r.h.s of Eq. (375) and the expression on line (379) can be reduced to

$$\frac{\partial^2 \Psi}{\partial r^2} + \frac{\partial \Psi}{\partial r} \frac{1}{r} + \frac{1}{r^2} \frac{\partial^2 \Psi}{\partial \theta^2}$$ (381)

Using these, the GS equation is written as

$$\frac{\partial^2 \Psi}{\partial r^2} + \frac{\partial \Psi}{\part... ...u_0 (R_0 + r \cos \theta)^2 \frac{d P}{d \Psi} - \frac{d g}{d \Psi} g (\Psi),$$

which can be arranged in the form

$$\left( \frac{1}{r} \frac{\partial}{\partial r} r \frac{\partial}{... ...\mu_0 (R_0 + r \cos \theta)^2 \frac{d P}{d \Psi} - \frac{d g}{d \Psi} g (\Psi),$$ (382)

which agrees with Eq. (3.6.2) in Wessson's book[12], where $f$ is defined by $f = R B_{\phi} / \mu_0$, which is different from $g \equiv R B_{\phi}$ by a $1 / \mu_0$ factor.