Boundary magnetic surface and initial coordinates

In the fixed boundary equilibrium problem, the shape of the boundary magnetic surface (it is also the boundary of the computational region, which is usually the last-closed-flux-surface) is given while the shape of the inner flux surface is to be solved. A simple analytical expression for a D-shaped magnetic surface takes the form

$$R = R_0 + a \cos (\theta + \Delta \sin \theta),$$ (454)

$$Z = \kappa a \sin \theta,$$ (455)

with $\theta$ changing from 0 to $2 \pi$. According to the definition in Eqs. (21), (22), and (24) we can readily verify that the parameters $a$, $R_0$, $\kappa$ appearing in Eqs. (454) and (455) are indeed the minor radius, major radius, and ellipticity, respectively. According to the definition of triangularity Eq. (23), the triangularity $\delta$ for the magnetic surface defined by Eqs. (454) and (455) is written as

$$\delta = \sin \Delta .$$ (456)

Another common expression for the shape of a magnetic surface was given by Miller[5,10], which is written as

$$R = R_0 + a \cos [\theta + \arcsin (\Delta \sin \theta)],$$ (457)

$$Z = \kappa a \sin \theta .$$ (458)

Note that Miller's formula is only slightly different from the formula given in Eqs. (454) and (455). For Miller's formula, it is easy to prove that the triangularity $\delta$ is equal to $\Delta$ (instead of $\delta = \sin \Delta$ as given in Eq. (456)).

In the iterative metric method[7] for solving the fixed boundary equilibrium problem, we need to provide an initial guess of the shape of the inner flux surface (this initial guess is used to construct a initial generalized coordinates system). A common guess of the inner flux surfaces is given by

$$R = \psi^{\alpha} (R_{\ensuremath{\operatorname{LCFS}}} - R_0) + R_0,$$ (459)

$$Z = \psi^{\alpha} Z_{\ensuremath{\operatorname{LCFS}}},$$ (460)

where $\alpha$ is a parameter, $\psi$ is a label parameter of flux surface. If the shape of the LCFS is given by Eqs. (454) and (455), then Eqs. (459) and (460) are written as

$$R = R_0 + \psi^{\alpha} a \cos (\theta + \Delta \sin \theta),$$ (461)

$$Z = \psi^a \kappa a \sin \theta .$$ (462)

Fig. 29 plots the shape given by Eqs. (461) and (462) for $a =$0.4, $R_0 = 1.7$, $\kappa = 1.7$, $\Delta = \arcsin (0.6)$, and $\alpha = 1$ with $\psi$ varying from zero to one.

Figure 29: Shape of flux surface given by Eqs. (461) and (462). Left figure: points with the same value of $\psi$ are connected to show $\psi$ coordinate surface; Right figure: dot plot. Parameters are $a =$0.4m, $R_0 = 1.7 m$, $\kappa = 1.7$, $\Delta = \arcsin (0.6)$, $\alpha = 1$ with $\psi$ varying from zero (center) to one (boundary). The shape parameters of $\ensuremath {\operatorname {LCFS}}$ are chosen according to the parameters of EAST tokamak.