Shape parameters of a closed magnetic surface

This section introduces parameters that characteristic the shape of the projection of a magnetic surface on the poloidal plane. The ``midplane'' is defined as the plane that passes through the magnetic axis and is perpendicular to the symmetric axis ($Z$ axis). For a up-down symmetric (about the midplane) magnetic surface, its shape can be roughly characterized by four parameters, namely, the $R$ coordinate of the innermost and outermost points on the midplane, $R_{\ensuremath {\operatorname {in}}}$ and $R_{\ensuremath {\operatorname {out}}}$; the $(R, Z)$ coordinators of the highest point of the magnetic surface, $(R_{\ensuremath {\operatorname {top}}}, Z_{\ensuremath {\operatorname {top}}})$. These four parameters are indicated in Fig. 4.

Figure 4: Four parameters characterizing the shape of a flux surface: the $R$ coordinate of the innermost and outermost points in the middle-plane, $R_{\ensuremath {\operatorname {in}}}$ and $R_{\ensuremath {\operatorname {out}}}$; the $(R, Z)$ coordinators of the highest point of the flux surface, $(R_{\ensuremath {\operatorname {top}}}, Z_{\ensuremath {\operatorname {top}}})$.

In terms of these four parameters, we can define the major radius of a magnetic surface

$$R_0 = \frac{R_{\ensuremath{\operatorname{in}}} + R_{\ensuremath{\operatorname{out}}}}{2},$$ (21)

(which is the $R$ coordinate of the geometric center of the magnetic surface), the minor radius of a magnetic surface

$$a = \frac{R_{\ensuremath{\operatorname{out}}} - R_{\ensuremath{\operatorname{in}}}}{2},$$ (22)

the triangularity of a magnetic surface

$$\delta = \frac{R_0 - R_{\ensuremath{\operatorname{top}}}}{a},$$ (23)

and, the ellipticity of a magnetic surface

$$\kappa = \frac{Z_{\ensuremath{\operatorname{top}}}}{a} .$$ (24)

Strictly speaking, the ellipticity $\kappa$ defined above is different from the elongation, which is define by $(\kappa - 1)$ and thus is zero for a circular magnetic surface. However, many authors simply consider the ellipticity to be the elongation. Usually, we specify the value of $R_0$, $a$, $\delta$, and $\kappa$, instead of $(R_{\ensuremath{\operatorname{in}}}, R_{\ensuremath{\operatorname{out}}}, R_{\ensuremath{\operatorname{top}}}, Z_{\ensuremath{\operatorname{top}}})$, to characterize the shape of a magnetic surface. Besides, using $a$ and $R_0$, we can define another useful parameter $\varepsilon \equiv a / R_0$, which is called the inverse aspect ratio.

The four shape parameters for the typical Last-Closed-Flux-Surface (LCFS) of EAST tokamak are: major radius $R_0 = 1.85 m$ (can reach $1.9 m$), minor radius $a = 0.45 m$, ellipticity $\kappa = 1.8$ (can be in the range from 1.7 to 1.9), triangularity $\delta = 0.6$ (can be in the range from 0.5 to 0.7). Note that the major radius $R_0$ of the LCFS is usually different from $R_{\ensuremath{\operatorname{axis}}}$ (the $R$ coordinate of the magnetic axis). Usually we have $R_{\ensuremath{\operatorname{axis}}} > R_0$ due to the Shafranov shift.