Solovév equilibrium

For most choices of $P (\Psi)$ and $g (\Psi)$, the GS equation has to be solved numerically. For the particular choice of $P$ and $g$ profiles,

$$\frac{d P}{d \Psi} = - \frac{c_1}{\mu_0},$$ (69)

$$g \frac{d g}{d \Psi} = - c_2 R_0^2,$$ (70)

analytical solution to the GS equation can be found, which is given by[9]

$$\Psi = \frac{1}{2} (c_2 R_0^2 + c_0 R^2) Z^2 + \frac{1}{8} (c_1 - c_0) (R^2 - R_0^2)^2,$$ (71)

where $c_0$, $c_1$, $c_2$, and $R_0$ are arbitrary constants. [Proof: By direct substitution, we can verify $\Psi$ of this form is indeed a solution to the GS equation (53).] A useful choice for tokamak application is to set $c_0 = B_0 / (R_0^2 \kappa_0 q_0)$, $c_1 = B_0 (\kappa_0^2 + 1) / (R_0^2 \kappa_0 q_0)$, and $c_2 = 0$. Then Eq. (71) is written

$$\Psi = \frac{B_0}{2 R_0^2 \kappa_0 q_0} \left[ R^2 Z^2 + \frac{\kappa_0^2}{4} (R^2 - R_0^2)^2 \right],$$ (72)

which can be solved analytically to give the explicit form of the contour of $\Psi$ on $(R, Z)$ plane:

$$Z = \pm \frac{1}{R} \sqrt{\frac{2 R_0^2 \kappa_0 q_0}{B_0} \Psi - \frac{\kappa_0^2}{4} (R^2 - R_0^2)^2},$$ (73)

which indicates the magnetic surfaces are up-down symmetrical. Using Eq. (69), i.e.,

$$\frac{d P}{d \Psi} = - \frac{c_1}{\mu_0} = - \frac{B_0 (\kappa_0^2 + 1)}{\mu_0 R_0^2 \kappa_0 q_0},$$ (74)

the pressure is written

$$P = P_0 - \frac{B_0 (\kappa_0^2 + 1)}{\mu_0 R_0^2 \kappa_0 q_0} \Psi,$$ (75)

where $P_0$ is a constant of integration. Note Eq. (72) indicates that that $\Psi = 0$ at the magnetic axis ( $R = R_0, Z = 0$). Therefore, Eq. (75) indicates that $P_0$ is the pressure at the magnetic axis. The toroidal field function $g$ is a constant in this case, which implies there is no poloidal current in this equilibrium. (For the Solovev equilibrium (72), I found numerically that the value of the safety factor at the magnetic axis ( $R = R_0, Z = 0$) is equal to $q_0 g / (R_0 B_0)$. This result should be able to be proved analytically. I will do this later. In calculating the safety factor, we also need the expression of $\vert \nabla \Psi \vert$, which is given analytically by

$$\vert \nabla \Psi \vert = \sqrt{\left( \frac{\partial \Psi}{\partial R} \right)^2 + \left( \frac{\partial \Psi}{\partial Z} \right)^2}$$

$$= \frac{B_0}{2 R_0^2 \kappa_0 q_0} \sqrt{[2 R Z^2 + \kappa_0^2 (R^2 - R_0^2) R]^2 + (2 R^2 Z)^2} .$$ (76)

)

Define $\Psi_0 = B_0 R_0^2$, and $\overline{\Psi} = \Psi / \Psi_0$, then Eq. (72) is written as

$$\overline{\Psi} = \frac{1}{2 \kappa_0 q_0} \left[ \overline{R}^2 \overline{Z}^2 + \frac{\kappa_0^2}{4} (\overline{R}^2 - 1)^2 \right],$$ (77)

where $\overline{R} = R / R_0$, $\overline{Z} = Z / R_0$. From Eq. (77), we obtain

$$\overline{Z} = \pm \frac{1}{\overline{R}} \sqrt{2 \kappa_0 q_0 \overline{\Psi} - \frac{\kappa_0^2}{4} (\overline{R}^2 - 1)^2} .$$ (78)

Given the value of $\kappa_0$, $q_0$, for each value of $\overline {\Psi }$, we can plot a magnetic surface on $(\overline{R}, \overline{Z})$ plane. An example of the nested magnetic surfaces is shown in Fig. 6.

Figure 6: Flux surfaces of Solovév equilibrium for $\kappa _0 = 1.5$ and $q_0 = 1.5$, with $\overline {\Psi }$ varying from zero (center) to 0.123 (edge). The value of $\overline {\Psi }$ on the edge is determined by the requirement that the minimum of $\overline {R}$ is equal to zero. (To prevent ``divided by zero'' that appears in Eq. (78) when $R = 0$, the value of $\overline {\Psi }$ on the edge is shifted to $0.123 - \varepsilon$ when plotting the above figure, where $\varepsilon$ is a small number, $\varepsilon = 10^{- 3}$ in this case.)

The minor radius of a magnetic surface of the Solovev equilibrium can be calculated by using Eq. (73), which gives

$$R_{\ensuremath{\operatorname{in}}} = \sqrt{R_0^2 - \sqrt{A \Psi}},$$ (79)

$$R_{\ensuremath{\operatorname{out}}} = \sqrt{R_0^2 + \sqrt{A \Psi}},$$ (80)

and thus

$$a = \frac{R_{\ensuremath{\operatorname{out}}} - R_{\ensuremath{\o... ...}}{2} = \frac{\sqrt{R_0^2 + \sqrt{A \Psi}} - \sqrt{R_0^2 - \sqrt{A \Psi}}}{2} .$$ (81)

where $A = 8 R_0^2 q_0 / (B_0 \kappa_0)$. In my code of constructing Solovev magnetic surface, the value of $a$ is specified by users, and then Eq. (81) is solved numerically to obtain the value of $\Psi$ of the flux surface. Note that the case $\Psi = 0$ corresponds to $R_{\ensuremath{\operatorname{in}}} = R_{\ensuremath{\operatorname{out}}} = R_0$, i.e., the magnetic axis, while the case $\Psi = R_0^2 B_0 \kappa_0 / (8 q_0)$ corresponds to $R_{\ensuremath{\operatorname{in}}} = 0$. Therefore, the reasonable value of $\Psi$ of a magnetic surface should be in the range $0 \leqslant \Psi < R_0^2 B_0 \kappa_0 / (8 q_0)$. This range is used as the interval bracketing a root in the bisection root finder.

Using Eq. (81), the inverse aspect ratio of a magnetic surface labeled by $\Psi$ can be approximated as[9]

$$\varepsilon \approx \sqrt{\frac{2 q_0 \Psi}{\kappa_0 R_0^2 B_0}} .$$ (82)

Therefore, the value of $\Psi$ of a magnetic surface with the inverse aspect ratio $\varepsilon$ is approximately given by

$$\Psi = \frac{\varepsilon^2 \kappa_0 R_0^2 B_0}{2 q_0} .$$ (83)