Miller's formula for shaped flux surfaces

According to Refs. [5,10], Miller's formula for a series of shaped flux surfaces is given by

$$R = R_0 (r) + r \cos \{ \theta + \arcsin [\delta (r) \sin \theta] \},$$ (518)

$$Z = \kappa (r) r \sin \theta,$$ (519)

where $\kappa (r)$ and $\delta (r)$ are elongation and triangularity profile, $R_0 (r)$ is the Shafranov shift profile, which is given by

$$R_0 (r) = R_0 (a) - \frac{a R_0'}{2} \left[ 1 - \left( \frac{r}{a} \right)^2 \right],$$ (520)

where $R_0'$ is a constant, $R_0 (a)$ is the major radius of the center of the boundary flux surface. The triangularity profile is

$$\delta (r) = \delta_0 \left( \frac{r}{a} \right)^2,$$ (521)

and the elongation profile is

$$\kappa (r) = \kappa_0 - 0.3 + 0.3 \left( \frac{r}{a} \right)^4 .$$ (522)

The nominal ITER parameters are $\kappa _0 = 1.8$, $\delta _0 = 0.5$ and $R_0' = - 0.16$. I wrote a code to plot the shapes of the flux surface (/home/yj/project/miller_flux_surface). An example of the results is given in Fig. 33.

Figure 33: Flux-surfaces given by Eqs. (518) and (519) with $r / a$ varying from 0.1 to 1.0 (corresponding boundary surface). Other parameters are $R_0 (a) / a = 3$, $\kappa _0 = 1.8$, $\delta _0 = 0.5$, $R_0' = - 0.16$.