The plasma region

In the plasma region, the 2-D Green’s theorem with the observation point on the plasma surface (i.e. the integration surface) can be obtained from the basic identity

(4.1) $$\begin{eqnarray}\displaystyle & & \displaystyle \boldsymbol{{\rm\nabla}}\times \left[G\boldsymbol{{\rm\nabla}}\times \left(\frac{{\it\psi}}{R}\boldsymbol{e}_{{\it\phi}}\right)-{\it\psi}\boldsymbol{{\rm\nabla}}\times \left(\frac{G}{R}\boldsymbol{e}_{{\it\phi}}\right)\right]\nonumber\\ \displaystyle & & \displaystyle \quad \qquad =G\boldsymbol{{\rm\nabla}}\times \boldsymbol{{\rm\nabla}}\times \left(\frac{{\it\psi}}{R}\boldsymbol{e}_{{\it\phi}}\right)-{\it\psi}\boldsymbol{{\rm\nabla}}\times \boldsymbol{{\rm\nabla}}\times \left(\frac{G}{R}\boldsymbol{e}_{{\it\phi}}\right).\end{eqnarray}$$

For vacuum fields the flux and 2-D Green’s function satisfy

(4.2) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \boldsymbol{{\rm\nabla}}\times \boldsymbol{{\rm\nabla}}\times \left(\frac{{\it\psi}}{R}\boldsymbol{e}_{{\it\phi}}\right)=\mathbf{0}\\ \displaystyle \boldsymbol{{\rm\nabla}}\times \boldsymbol{{\rm\nabla}}\times \left(\frac{G}{R}\boldsymbol{e}_{{\it\phi}}\right)={\it\delta}(R-R^{\prime }){\it\delta}(Z-Z^{\prime })\boldsymbol{e}_{{\it\phi}}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

In these expressions, unprimed and primed coordinates refer to the observation and integration points, respectively

The 2-D Green’s function is closely related to the flux function for a circular loop of wire. Specifically, the vector potential due to a wire loop, satisfies

(4.3) $$\begin{eqnarray}\boldsymbol{{\rm\nabla}}\times \boldsymbol{{\rm\nabla}}\times (A_{{\it\phi}}\boldsymbol{e}_{{\it\phi}})={\it\mu}_{0}J_{{\it\phi}}\boldsymbol{e}_{{\it\phi}}={\it\mu}_{0}I{\it\delta}(R-R^{\prime }){\it\delta}(Z-Z^{\prime })\boldsymbol{e}_{{\it\phi}}.\end{eqnarray}$$

The solution is

(4.4) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle RA_{{\it\phi}}=\frac{{\it\mu}_{0}I}{2{\rm\pi}}\left(\frac{R^{\prime }R}{k^{2}}\right)^{1/2}[(2-k^{2})K-2E]\\ \displaystyle k^{2}=\frac{4R^{\prime }R}{(R^{\prime }+R)^{2}+(Z^{\prime }-Z)^{2}}.\end{array}\right\}\end{eqnarray}$$

Here $K(k)=\int _{0}^{{\rm\pi}/2}(\text{d}{\it\theta}/\sqrt{1-k^{2}\sin ^{2}{\it\theta}})$ , $E(k)=\int _{0}^{{\rm\pi}/2}\sqrt{1-k^{2}\sin ^{2}{\it\theta}}\,\text{d}{\it\theta}$ are complete elliptic integrals. Thus, if we set ${\it\mu}_{0}I=1$ , we see that $RA_{{\it\phi}}=G$ ,

(4.5) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle G=\frac{1}{2{\rm\pi}}\left(\frac{R^{\prime }R}{k^{2}}\right)^{1/2}[(2-k^{2})K-2E]\\ \displaystyle k^{2}=\frac{4R^{\prime }R}{(R^{\prime }+R)^{2}+(Z^{\prime }-Z)^{2}}.\end{array}\right\}\end{eqnarray}$$

Also needed in the analysis is the normal derivative (in integration coordinates) of the Green’s function evaluated on the plasma surface. A short calculation yields

(4.6) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \frac{L_{P}}{2{\rm\pi}}(\boldsymbol{n}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}^{\prime }G)=\frac{1}{2}\frac{{\dot{Z}}^{\prime }}{R^{\prime }}(G-G^{\dagger })+\frac{{\dot{Z}}^{\prime }(R^{\prime }-R)-{\dot{R}}^{\prime }(Z^{\prime }-Z)}{(R^{\prime }-R)^{2}+(Z^{\prime }-Z)^{2}}G^{\dagger }\\ \displaystyle G^{\dagger }=\frac{1}{2{\rm\pi}}\left(\frac{R^{\prime }R}{k^{2}}\right)^{1/2}[2(1-k^{2})K-(2-k^{2})E].\end{array}\right\}\end{eqnarray}$$

Note that ${\dot{Z}}^{\prime },{\dot{R}}^{\prime }$ denote $\text{d}Z({\it\chi}^{\prime })/\text{d}{\it\chi}^{\prime }$ and $\text{d}R({\it\chi}^{\prime })/\text{d}{\it\chi}^{\prime }$ indicating that we have switched integration variables from $l^{\prime }$ to ${\it\chi}^{\prime }$ .

The next step is to apply Stokes theorem to (4.1) with the observation point on the plasma surface

(4.7) $$\begin{eqnarray}\frac{1}{2}{\it\psi}=\int _{L_{P}}\left[\frac{{\it\psi}^{\prime }}{R^{\prime }}\boldsymbol{{\rm\nabla}}^{\prime }G\times \boldsymbol{e}_{{\it\phi}}-\frac{G}{R^{\prime }}\boldsymbol{{\rm\nabla}}^{\prime }{\it\psi}^{\prime }\times \boldsymbol{e}_{{\it\phi}}\right]\boldsymbol{\cdot }\text{d}\boldsymbol{l}^{\prime }.\end{eqnarray}$$

In this expression we need to be careful about the signs. The main point is that Stoke’s theorem requires $\text{d}\boldsymbol{l}^{\prime }$ to rotate in a right handed sense. Now, in the usual $R,{\it\phi},Z$ coordinate system as shown in figure 1, this implies that $\text{d}\boldsymbol{l}^{\prime }$ rotates in the clockwise direction. However, it is convenient and familiar to have ${\it\chi}^{\prime }$ rotate in the counter-clockwise direction. Thus, if we define a unit tangential vector $\boldsymbol{t}^{\prime }$ pointing in the counter-clockwise direction it then follows that

(4.8) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \text{d}\boldsymbol{l}^{\prime }=-\boldsymbol{t}^{\prime }\,\text{d}l^{\prime }=-\frac{L_{P}}{2{\rm\pi}}\boldsymbol{t}^{\prime }\,\text{d}{\it\chi}^{\prime }\\ \displaystyle \boldsymbol{t}^{\prime }=\frac{{\dot{R}}^{\prime }\boldsymbol{e}_{R}+{\dot{Z}}^{\prime }\boldsymbol{e}_{z}}{({\dot{R}}^{\prime 2}+{\dot{Z}}^{\prime 2})^{1/2}}\\ \displaystyle \boldsymbol{n}^{\prime }=\boldsymbol{e}_{{\it\phi}}\times \boldsymbol{t}^{\prime }=\frac{{\dot{Z}}^{\prime }\boldsymbol{e}_{R}-{\dot{R}}^{\prime }\boldsymbol{e}_{z}}{({\dot{R}}^{\prime 2}+{\dot{Z}}^{\prime 2})^{1/2}}.\end{array}\right\}\end{eqnarray}$$

Here, $\boldsymbol{n}^{\prime }$ is the outward pointing unit normal vector. With this sign convention (4.7) reduces to

(4.9) $$\begin{eqnarray}\frac{1}{2}{\it\psi}=\int _{L_{P}}\left[\frac{G}{R^{\prime }}\boldsymbol{n}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}^{\prime }{\it\psi}^{\prime }-\frac{{\it\psi}^{\prime }}{R^{\prime }}\boldsymbol{n}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}^{\prime }G\right]_{l,l^{\prime }}\,\text{d}l^{\prime }.\end{eqnarray}$$

A similar expression holds for the wall surface.

The calculation continues by substituting the Fourier expansions into (4.9) and then carrying out a Fourier analysis. A straightforward calculation leads to

(4.10) $$\begin{eqnarray}{\it\psi}_{m}+\mathop{\sum }_{n}\unicode[STIX]{x1D608}_{mn}^{11}{\it\psi}_{n}-\mathop{\sum }_{n}\unicode[STIX]{x1D609}_{mn}^{11}u_{n}=0\rightarrow (\unicode[STIX]{x1D644}+\unicode[STIX]{x1D63C}^{11})\boldsymbol{\cdot }{\bf\psi}-\unicode[STIX]{x1D63D}^{11}\boldsymbol{\cdot }\boldsymbol{u}=0,\end{eqnarray}$$

where the matrix elements $\unicode[STIX]{x1D608}_{mn}^{11}$ and $\unicode[STIX]{x1D609}_{mn}^{11}$ of $\unicode[STIX]{x1D63C}^{11}$ and $\unicode[STIX]{x1D63D}^{11}$ are given by

(4.11) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D608}_{mn}^{11}=\frac{2}{{\rm\pi}}\int \text{d}{\it\chi}^{\prime }\,\text{d}{\it\chi}\sin n{\it\chi}^{\prime }\sin m{\it\chi}\left[\frac{L_{P}}{2{\rm\pi}}\frac{\boldsymbol{n}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}^{\prime }G}{(R^{\prime }R)^{1/2}}\right]_{{\it\chi},{\it\chi}^{\prime }}\\ \displaystyle \unicode[STIX]{x1D609}_{mn}^{11}=\unicode[STIX]{x1D609}_{nm}^{11}=\frac{4}{{\rm\pi}}\int \text{d}{\it\chi}^{\prime }\,\text{d}{\it\chi}\sin n{\it\chi}^{\prime }\sin m{\it\chi}\left[\frac{G}{(R^{\prime }R)^{1/2}}\right]_{{\it\chi},{\it\chi}^{\prime }}.\end{array}\right\}\end{eqnarray}$$

For the matrix format the first superscript on $\unicode[STIX]{x1D63C}^{11}$ denotes the observation point while the second denotes integration point. The index 1 refers to the plasma surface, and the index 2 refers to the wall. This holds for all matrices that follow.

The outer vacuum region

The analysis of the outer vacuum region is very similar to that of the plasma. One simply has to switch surfaces and take into account the opposite sign of the outward surface normal. The basic equation for the outer vacuum region, assuming regularity at infinity, is given by

(4.12) $$\begin{eqnarray}\frac{1}{2}\hat{{\it\psi}}=-\int _{L_{W}}\left[\frac{G}{R^{\prime }}\boldsymbol{n}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}^{\prime }\hat{\hat{{\it\psi}}}^{\prime }-\frac{\hat{{\it\psi}}^{\prime }}{R^{\prime }}\boldsymbol{n}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}^{\prime }G\right]_{\hat{l},\hat{l}^{\prime }}\,\text{d}\hat{l}^{\prime }.\end{eqnarray}$$

On this surface, the Green’s function and its normal derivative are given by

(4.13) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle G=\frac{1}{2{\rm\pi}}\left(\frac{R^{\prime }R}{k^{2}}\right)^{1/2}[(2-k^{2})K-2E]\\ \displaystyle \frac{L_{W}}{2{\rm\pi}}(\boldsymbol{n}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}^{\prime }G)=\frac{1}{2}\frac{{\dot{Z}}^{\prime }}{R^{\prime }}(G-G^{\dagger })+\frac{{\dot{Z}}^{\prime }(R^{\prime }-R)-{\dot{R}}^{\prime }(Z^{\prime }-Z)}{(R^{\prime }-R)^{2}+(Z^{\prime }-Z)^{2}}G^{\dagger }\\ \displaystyle G^{\dagger }=\frac{1}{2{\rm\pi}}\left(\frac{R^{\prime }R}{k^{2}}\right)^{1/2}[2(1-k^{2})K-(2-k^{2})E].\end{array}\right\}\end{eqnarray}$$

The expressions are the same as for the plasma region except that $L_{P}\rightarrow L_{W}$ in the second equation. Fourier analysis then leads to the following relation between Fourier coefficients

(4.14) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \hat{{\it\psi}}_{m}-\mathop{\sum }_{n}\unicode[STIX]{x1D608}_{mn}^{22}\hat{{\it\psi}}_{n}+\mathop{\sum }_{n}\unicode[STIX]{x1D609}_{mn}^{22}\hat{\hat{v}}_{n}=0\rightarrow (\unicode[STIX]{x1D644}-\unicode[STIX]{x1D63C}^{22})\boldsymbol{\cdot }\hat{{\bf\psi}}+\unicode[STIX]{x1D63D}^{22}\boldsymbol{\cdot }\hat{\hat{\boldsymbol{v}}}=0\\ \displaystyle \unicode[STIX]{x1D608}_{mn}^{22}=\frac{2}{{\rm\pi}}\int \text{d}\hat{{\it\chi}}^{\prime }\,\text{d}\hat{{\it\chi}}\sin n\hat{{\it\chi}}^{\prime }\sin m\hat{{\it\chi}}\left[\frac{L_{W}}{2{\rm\pi}}\frac{\boldsymbol{n}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}^{\prime }G}{(R^{\prime }R)^{1/2}}\right]_{\hat{{\it\chi}},\hat{{\it\chi}}^{\prime }}\\ \displaystyle \unicode[STIX]{x1D609}_{mn}^{22}=\unicode[STIX]{x1D609}_{mn}^{22}=\frac{4}{{\rm\pi}}\int \text{d}\hat{{\it\chi}}^{\prime }\,\text{d}\hat{{\it\chi}}\sin n\hat{{\it\chi}}^{\prime }\sin m\hat{{\it\chi}}\left[\frac{G}{(R^{\prime }R)^{1/2}}\right]_{\hat{{\it\chi}},\hat{{\it\chi}}^{\prime }}.\end{array}\right\}\end{eqnarray}$$

The inner vacuum region

The inner vacuum region is slightly more complicated to analyse because of the coupling of surface vectors between the plasma and wall surfaces. In this region, Green’s theorem must be used twice, once with the observation point on the plasma surface and once on the wall surface. The two basic equations are given by

Observation point on the plasma:

(4.15) $$\begin{eqnarray}\frac{1}{2}{\it\psi}(l)=-\int _{L_{P}}\left[\frac{G}{R^{\prime }}\boldsymbol{n}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}^{\prime }\hat{{\it\psi}}^{\prime }-\frac{{\it\psi}^{\prime }}{R^{\prime }}\boldsymbol{n}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}^{\prime }G\right]_{l,l^{\prime }}\text{d}l^{\prime }+\int _{L_{W}}\left[\frac{G}{R^{\prime }}\boldsymbol{n}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}^{\prime }\hat{{\it\psi}}^{\prime }-\frac{\hat{{\it\psi}}^{\prime }}{R^{\prime }}\boldsymbol{n}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}^{\prime }G\right]_{l,\hat{l}^{\prime }}\text{d}\hat{l}^{\prime }.\vphantom{\mathop{\sum }_{\mathop{\sum }_{\sum }}}\end{eqnarray}$$

Observation point on the wall:

(4.16) $$\begin{eqnarray}\frac{1}{2}\hat{{\it\psi}}(\hat{l})=-\int _{L_{P}}\left[\frac{G}{R^{\prime }}\boldsymbol{n}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}^{\prime }\hat{{\it\psi}}^{\prime }-\frac{\hat{{\it\psi}}^{\prime }}{R^{\prime }}\boldsymbol{n}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}^{\prime }G\right]_{\hat{l},l^{\prime }}\text{d}l^{\prime }+\int _{L_{W}}\left[\frac{G}{R^{\prime }}\boldsymbol{n}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}^{\prime }\hat{{\it\psi}}^{\prime }-\frac{\hat{{\it\psi}}^{\prime }}{R^{\prime }}\boldsymbol{n}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}^{\prime }G\right]_{\hat{l},\hat{l}^{\prime }}\text{d}\hat{l}^{\prime }.\vphantom{\mathop{\sum }_{\mathop{\sum }_{\sum }}}\end{eqnarray}$$

After carrying out the Fourier analysis, we arrive at two coupled equations for the Fourier amplitudes

(4.17) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle {\it\psi}_{m}-\mathop{\sum }_{n}\unicode[STIX]{x1D608}_{mn}^{11}{\it\psi}_{n}+\mathop{\sum }_{n}\unicode[STIX]{x1D609}_{mn}^{11}\hat{u} _{n}+\mathop{\sum }_{n}\unicode[STIX]{x1D608}_{mn}^{12}\hat{{\it\psi}}_{n}-\mathop{\sum }_{n}\unicode[STIX]{x1D609}_{mn}^{12}\hat{v}_{n}=0\\ \displaystyle \hat{{\it\psi}}_{m}+\mathop{\sum }_{n}\unicode[STIX]{x1D608}_{mn}^{22}{\it\psi}_{n}-\mathop{\sum }_{n}\unicode[STIX]{x1D609}_{mn}^{22}\hat{u} _{n}-\mathop{\sum }_{n}\unicode[STIX]{x1D608}_{mn}^{21}{\it\psi}_{n}+\mathop{\sum }_{n}\unicode[STIX]{x1D609}_{mn}^{21}\hat{v}_{n}=0\end{array}\right\}\end{eqnarray}$$

or in matrix form

(4.18) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle (\unicode[STIX]{x1D644}-\unicode[STIX]{x1D63C}^{11})\boldsymbol{\cdot }{\bf\psi}+\unicode[STIX]{x1D63D}^{11}\boldsymbol{\cdot }\hat{\boldsymbol{u}}+\unicode[STIX]{x1D63C}^{12}\boldsymbol{\cdot }\hat{{\bf\psi}}-\unicode[STIX]{x1D63D}^{12}\boldsymbol{\cdot }\hat{\boldsymbol{v}}=0\\ (\unicode[STIX]{x1D644}+\unicode[STIX]{x1D63C}^{22})\boldsymbol{\cdot }\hat{{\bf\psi}}-\unicode[STIX]{x1D63D}^{22}\boldsymbol{\cdot }\hat{\boldsymbol{v}}-\unicode[STIX]{x1D63C}^{21}\boldsymbol{\cdot }{\bf\psi}+\unicode[STIX]{x1D63D}^{21}\boldsymbol{\cdot }\hat{\boldsymbol{u}}=0.\end{array}\right\}\end{eqnarray}$$

The newly introduced matrix elements are defined by

(4.19) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D608}_{mn}^{12}=\frac{2}{{\rm\pi}}\int \text{d}\hat{{\it\chi}}^{\prime }\,\text{d}{\it\chi}\sin n\hat{{\it\chi}}^{\prime }\sin m{\it\chi}\left[\frac{L_{W}}{2{\rm\pi}}\frac{\boldsymbol{n}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}^{\prime }G}{(R^{\prime }R)^{1/2}}\right]_{{\it\chi},\hat{{\it\chi}}^{\prime }}\\ \displaystyle \unicode[STIX]{x1D608}_{mn}^{21}=\frac{2}{{\rm\pi}}\int \text{d}{\it\chi}^{\prime }\,\text{d}\hat{{\it\chi}}\sin n{\it\chi}^{\prime }\sin m\hat{{\it\chi}}\left[\frac{L_{P}}{2{\rm\pi}}\frac{\boldsymbol{n}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}^{\prime }G}{(R^{\prime }R)^{1/2}}\right]_{\hat{{\it\chi}},{\it\chi}^{\prime }}\\ \displaystyle \unicode[STIX]{x1D609}_{mn}^{12}=\unicode[STIX]{x1D609}_{nm}^{12}=\frac{4}{{\rm\pi}}\int \text{d}\hat{{\it\chi}}^{\prime }\,\text{d}{\it\chi}\sin n\hat{{\it\chi}}^{\prime }\sin m{\it\chi}\left[\frac{G}{(R^{\prime }R)^{1/2}}\right]_{{\it\chi},\hat{{\it\chi}}^{\prime }}\\ \displaystyle \unicode[STIX]{x1D609}_{mn}^{21}=\unicode[STIX]{x1D609}_{nm}^{21}=\frac{4}{{\rm\pi}}\int \text{d}{\it\chi}^{\prime }\,\text{d}\hat{{\it\chi}}\sin n{\it\chi}^{\prime }\sin m\hat{{\it\chi}}\left[\frac{G}{(R^{\prime }R)^{1/2}}\right]_{\hat{{\it\chi}},{\it\chi}^{\prime }}.\end{array}\right\}\end{eqnarray}$$

Note that because of the symmetry $G(R,Z,R^{\prime },Z^{\prime })=G(R^{\prime },Z^{\prime },R,Z)$ it follows that $\unicode[STIX]{x1D609}_{mn}^{12}=\unicode[STIX]{x1D609}_{nm}^{21}$ implying that $\unicode[STIX]{x1D63D}^{21}=(\unicode[STIX]{x1D63D}^{12})^{\text{T}}$ .

The four constraint relations given by (4.10), (4.14), and (4.18) can now be written in a compact form as

(4.20) $$\begin{eqnarray}\unicode[STIX]{x1D63E}^{\text{T}}\boldsymbol{\cdot }\boldsymbol{x}=\mathbf{0},\end{eqnarray}$$

where $\unicode[STIX]{x1D63E}^{\text{T}}$ is a $4M\times 6M$ matrix given by

(4.21) $$\begin{eqnarray}\unicode[STIX]{x1D63E}^{\text{T}}=\left|\begin{array}{@{}cccccc@{}}\unicode[STIX]{x1D644}+\unicode[STIX]{x1D63C}^{11} & \mathbf{0} & -\unicode[STIX]{x1D63D}^{11} & \mathbf{0} & \mathbf{0} & \mathbf{0}\\ \mathbf{0} & \unicode[STIX]{x1D644}-\unicode[STIX]{x1D63C}^{22} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \unicode[STIX]{x1D63D}^{22}\\ \unicode[STIX]{x1D644}-\unicode[STIX]{x1D63C}^{11} & \unicode[STIX]{x1D63C}^{12} & \mathbf{0} & \unicode[STIX]{x1D63D}^{11} & -\unicode[STIX]{x1D63D}^{12} & \mathbf{0}\\ -\unicode[STIX]{x1D63C}^{21} & \unicode[STIX]{x1D644}+\unicode[STIX]{x1D63C}^{22} & \mathbf{0} & \unicode[STIX]{x1D63D}^{21} & -\unicode[STIX]{x1D63D}^{22} & \mathbf{0}\end{array}\right|.\end{eqnarray}$$

Definition of the normalized wall radius $b/a$

As explained in Paper I, the plasma surface is determined by inverting the implicit equation ${\it\Psi}=0$ given by the Solov’ev equilibrium. On this surface, the outer equatorial point has coordinates $(R,Z)=(R_{0}+a,0)$ , the top point has coordinates $(R,Z)=(R_{0}-{\it\delta}a,{\it\kappa}a)$ and the inner equatorial point has coordinates $(R,Z)=(R_{0}-a,0)$ . Now, our wall model has a shape similar to the plasma. It is characterized by three free input parameters: the normalized inner midplane gap ${\it\Delta}_{i}/a$ , the normalized outer midplane gap ${\it\Delta}_{o}/a$ and the normalized vertical gap ${\it\Delta}_{v}/a$ . These in turn are easily related to the more familiar normalized wall radius $b/a$ and wall elongation ${\it\kappa}_{w}$ . The geometry is illustrated in figure 1. In the numerical studies, two of the three gap parameters, ${\it\Delta}_{i}/a$ and ${\it\Delta}_{v}/a$ , are held fixed. Changing the wall radius corresponds to varying only the outer gap; that is, the single parameter ${\it\Delta}_{o}/a$ or equivalently $b/a$ . This choice of variation is motivated by experimental observations (McCracken et al. Reference McCracken, Lipschultz, LaBombard, Goetz, Granetz, Jablonski, Lisgo, Ohkawa, Stangeby and Terry1997), which show that the impurity influx in divertor tokamaks from the outboard midplane area is substantially greater than from the inboard side. Consequently, in order to achieve better impurity isolation in future experiments, it may be necessary to increase the outboard midplane gap ${\it\Delta}_{o}/a$ .

The specific shape of our wall is denoted by the coordinates $\hat{R}$ , $\hat{Z}$ and is given by

(6.1) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\hat{R}=\hat{R}_{0}+b\cos ({\it\tau}+\hat{{\it\delta}}_{0}\sin {\it\tau})\\ \hat{Z}={\it\kappa}_{w}b\sin {\it\tau}.\end{array}\right\}\end{eqnarray}$$

Note that the average horizontal wall radius $b/a$ and wall elongation ${\it\kappa}_{w}$ are related to the gap widths and plasma elongation by

(6.2) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \frac{b}{a}=1+\frac{1}{2}\left(\frac{{\it\Delta}_{i}}{a}+\frac{{\it\Delta}_{o}}{a}\right)\\ \displaystyle {\it\kappa}_{w}=\frac{a}{b}\left({\it\kappa}+\frac{{\it\Delta}_{v}}{a}\right).\end{array}\right\}\end{eqnarray}$$

The parameters $\hat{R}_{0}$ and $\hat{{\it\delta}}_{0}$ can also be expressed in terms of the gap widths by utilizing the assumption that the maximum heights of both the wall and plasma occur at the same  $R$ .

(6.3) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \frac{\hat{R}_{0}}{R_{0}}=1+\frac{1}{2}\left(\frac{{\it\Delta}_{o}}{a}-\frac{{\it\Delta}_{i}}{a}\right){\it\varepsilon}\\ \displaystyle \hat{{\it\delta}}_{0}=\frac{a}{b}\left[{\it\delta}+\frac{1}{2}\left(\frac{{\it\Delta}_{o}}{a}-\frac{{\it\Delta}_{i}}{a}\right)\right].\end{array}\right\}\end{eqnarray}$$

Also, for the numerics, it is convenient to normalize and parameterize the wall coordinates as follows: $\hat{R}=R_{0}\hat{X}$ , $\hat{Z}=R_{0}{\hat{Y}}$ with

(6.4) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \hat{X}=1+\left(\frac{b}{a}-1-\frac{{\it\Delta}_{i}}{a}\right){\it\varepsilon}+\left(\frac{b}{a}\right){\it\varepsilon}\cos ({\it\tau}+\hat{{\it\delta}}_{0}\sin {\it\tau})\\ \displaystyle {\hat{Y}}=\left(\frac{b}{a}\right){\it\kappa}_{w}{\it\varepsilon}\sin {\it\tau}.\end{array}\right\}\end{eqnarray}$$

Once the gap widths and plasma geometry are specified, the wall coordinates given by (6.4) are completely determined. From this, it is then a straightforward task to calculate the angular arc length coordinate $\hat{{\it\chi}}$ on the wall surface.

The reference case

The next step is to define a reference case. The goal is to determine a typical set of values for the parameters of interest: ${\it\beta}_{p}$ , $b/a$ and ${\it\gamma}{\it\tau}_{w}$ . To accomplish this task, we examine the data for several major large tokamak experiments as shown in table 1: ASDEX Upgrade (AUG) (Ryter et al. Reference Ryter, Alexander, Gruber, Vollmer, Becker, Gehre, Gentle, Lackner, Leuterer and Peeters1996), Alcator C-Mod (C-Mod) (Greenwald et al. Reference Greenwald, Boivin, Bombarda, Bonoli, Fiore, Garnier, Goetz, Golovato, Graf and Granetz1997), DIII-D (Petty et al. Reference Petty, Luce, Burrell, Chiu, de Grassie, Forest, Gohil, Greenfield, Groebner and Harvey1995), JET (Balet, Campbell & Christiansen Reference Balet, Campbell and Christiansen1995), NSTX (Sabbagh et al. Reference Sabbagh, Kaye, Menard, Paolettia, Bellb, Bellb, Bialeka, Bitterb, Fredricksonb and Gatesb2001), and ITER (Aymar et al. Reference Aymar, Barabaschi and Shimomura2002).

Table 1.

Parameters for high performance elongated tokamaks. Here $\bar{p}$ is the volume averaged pressure and $l_{i}$ is the internal inductance per unit length of the Solov’ev profile calculated by $l_{i}=2\int _{V_{p}}B_{p}^{2}\,\text{d}\boldsymbol{r}/({\it\mu}_{0}^{2}I_{{\it\phi}}^{2}R_{0})$ where $I_{{\it\phi}}$ is the total toroidal current.

Each set of data corresponds to a high performance (i.e. high pressure) discharge. Observe first that a reasonable value of poloidal beta for the reference case can be chosen as

(6.5) $$\begin{eqnarray}{\it\beta}_{p}=1.\end{eqnarray}$$

Next, by combining the experimental data with machine drawings, we conclude that the measured inner and outer gap widths are approximately equal (i.e.  ${\it\Delta}_{o}={\it\Delta}_{i}$ ) and are set to the values listed in the tables. Also listed is the corresponding value of $b/a$ as calculated from (6.2). From the table we then assume that the reference values for the gaps and wall radius are given by

(6.6) $$\begin{eqnarray}\frac{{\it\Delta}_{i}}{a}=\frac{{\it\Delta}_{o}}{a}=0.1\rightarrow \frac{b}{a}=1.1.\end{eqnarray}$$

The reference wall elongation is an additional, but not independent, geometric parameter which enters the analysis but is more difficult to estimate. The walls have different shapes and the spacing between plasma and wall is different on the top and bottom because of the divertor. Even for a single experiment, it is not clear how to relate the wall shapes in the drawings to the simplified up–down symmetric wall parameter  ${\it\kappa}_{w}$ .

To circumvent this difficulty, we assume that for each experiment the vertical gap is three times the measured inner horizontal gap to allow for a larger vertical space to accommodate the divertor: ${\it\Delta}_{v}/a=3{\it\Delta}_{i}/a$ . Thus, for the table, the reference case and all future numerical studies, the wall elongation is given by

(6.7) $$\begin{eqnarray}{\it\kappa}_{w}=\frac{a}{b}\left({\it\kappa}+3\frac{{\it\Delta}_{i}}{a}\right).\end{eqnarray}$$

Here ${\it\kappa}$ is arbitrary, ${\it\Delta}_{i}/a$ is specified either experimentally or at its reference value and $b/a$ is obtained from (6.2).

Using the data in table 1 we have carried out numerical calculations to determine the value of ${\it\gamma}{\it\tau}_{w}$ that leads to a numerical eigenvalue ${\it\lambda}_{\mathit{min}}=0$ for each experiment. By construction, this defines the elongation at high performance that the feedback system, characterized by ${\it\gamma}{\it\tau}_{w}$ , can safely stabilize. By comparing the ${\it\gamma}{\it\tau}_{w}$ data from the different experiments, but omitting DIII-D, we deduce that a typical value of the feedback parameter is

(6.8) $$\begin{eqnarray}{\it\gamma}{\it\tau}_{w}=1.5.\end{eqnarray}$$

Interestingly, the value of ${\it\gamma}{\it\tau}_{w}$ for DIII-D is substantially higher than for the other experiments and the question is ‘Why?’ We suggest that a much stronger feedback system (i.e. a much larger ${\it\gamma}{\it\tau}_{w}$ ) is needed to achieve the high triangularity ${\it\delta}=0.85$ for the listed shot. Furthermore, this stronger feedback is possible in DIII-D since the feedback coils are located inside the TF coils, much closer to the plasma. In future fusion grade experiments, this will probably not be possible because of neutron radiation. This is the reason why a low weight is given to DIII-D when estimating a ‘reference’ value for ${\it\gamma}{\it\tau}_{w}$ . The DIII-D data is discussed in more detail shortly.

Lastly, we note that we could also compute a reference internal inductance $l_{i}$ from the data, with a value of the order of 0.4, as can be seen in table 1. This value is below the values usually obtained in experiments, and cannot be varied much in our model because of the fixed Solov’ev profiles. For the modes under consideration, we do not expect this limitation to have qualitative consequences on the scaling relations we calculate (Bernard et al. Reference Bernard, Berger, Gruber and Troyon1978), but it has quantitative implications, as we discuss shortly.

Having defined the reference case, we now proceed with a series of numerical calculations to shed insight onto the scaling of maximum achievable ${\it\kappa}$ versus ${\it\varepsilon}$ as a function of the experimental parameters, including the feedback system.

The reference case

To establish a baseline we calculate curves of maximum ${\it\kappa}={\it\kappa}({\it\varepsilon})$ and the corresponding ${\it\delta}={\it\delta}({\it\varepsilon})$ for the reference case. To do this, we set ${\it\beta}_{p}=1$ , ${\it\Delta}_{i}/a=0.1$ , ${\it\Delta}_{v}/a=0.3$ , ${\it\Delta}_{o}/a=0.1$ (corresponding to $b/a=1.1$ ) and ${\it\gamma}{\it\tau}_{w}=1.5$ . The value of ${\it\kappa}_{w}$ is set in accordance with (6.7). The desired scaling curves are obtained by choosing a value for ${\it\varepsilon}$ and then iterating on ${\it\kappa}$ and ${\it\delta}$ such that the eigenvalue ${\it\lambda}_{\mathit{min}}=0$ for each pair of values. The result can then be plotted as a curve of ${\it\kappa}$ versus ${\it\delta}$ for the given ${\it\varepsilon}$ , as shown in figure 2. We see that there is an optimized value of ${\it\delta}$ for which ${\it\kappa}$ is a maximum. Even so, the expanded scale indicates that the maximum is relatively flat in the vicinity of the optimum.

Figure 2.

Plot of ${\it\kappa}$ versus ${\it\delta}$ for ${\it\varepsilon}=0.3$ and the reference values ${\it\beta}_{p}=1$ , ${\it\gamma}{\it\tau}_{w}=1.5$ , ${\it\Delta}_{i}/a={\it\Delta}_{o}/a={\it\Delta}_{v}/3a=0.1$ . Observe that there is an optimum ${\it\delta}$ at which ${\it\kappa}$ is a maximum.

The procedure is repeated for a range of ${\it\varepsilon}$ , thereby generating a curve of maximum ${\it\kappa}={\it\kappa}({\it\varepsilon})$ and corresponding ${\it\delta}={\it\delta}({\it\varepsilon})$ which is illustrated in figure 3. Observe that, in agreement with previous work (Wesson Reference Wesson1978) and experimental data, the maximum achievable elongation increases as the aspect ratio becomes tighter. As $a/R_{0}$ increases from 0.1 to 0.8, the maximum ${\it\kappa}$ increases from 1.89 to 2.88. The optimum triangularity also increases as the aspect ratio gets tighter, but in a stronger way. Over the same range of $a/R_{0}$ , the triangularity increases from 0.05 to 0.65. At $a/R_{0}=0.3$ , the maximum elongation and optimum triangularity have the values ${\it\kappa}=2.06$ and ${\it\delta}=0.18$ .

Figure 3.

Curves of maximum ${\it\kappa}$ (a) and corresponding optimum ${\it\delta}$ (b) versus ${\it\varepsilon}$ for the reference case ${\it\beta}_{p}=1$ , ${\it\gamma}{\it\tau}_{w}=1.5$ , ${\it\Delta}_{i}/a={\it\Delta}_{o}/a={\it\Delta}_{v}/3a=0.1$ .

The values of ${\it\kappa}$ in figure 3 are in general slightly higher than those listed in table 1. The experimental values of ${\it\delta}$ are substantially higher. This may be explained by the fact that the values in table 1 correspond to high performance as measured by high average pressure. However, high performance is not determined solely by $n=0$ MHD considerations. Kink stability and transport play a comparably important role, and experimental data indicates that the highest experimental pressure may be achieved by operating at a larger value of ${\it\delta}$ than the $n=0$ MHD optimum because of more than compensatory gains in ${\it\beta}_{N}$ and ${\it\tau}_{E}$ (Lomas & JET Team Reference Lomas2000). We will return to this point in § 8 of the article.

A second important effect is associated with the fact that any given experiment has a fixed wall shape. Thus, the typical way to increase elongation is by shrinking the minor radius of the plasma. The effective increase in wall radius leads to a higher resistive wall growth rate requiring more feedback and the smaller plasma volume leads to reduced performance because of smaller  ${\it\tau}_{E}$ . Both lead to a reduced  ${\it\kappa}$ .

A final contributing factor is associated with the fact that the equilibrium Solov’ev current profile used in our analysis is somewhat broader than typical experimental profiles. Specifically, whereas the Solov’ev internal inductance is always approximately $l_{i}\approx 0.4$ , the more peaked experimental profiles have internal inductances that typically lie in the range $l_{i}\sim 0.5{-}1.0$ . This implies that the Solov’ev profile has a higher current density close to the wall than the experimental profiles, and therefore is more strongly affected by wall stabilization. The result is a slightly higher ${\it\kappa}$ for the Solov’ev profile.

Based on this discussion, we see that the numerical results presented here and below should be viewed in the context of future experimental designs where the wall to plasma radius can remain fixed as the plasma geometry is varied. Even so, if the designs are based primarily on empirical ${\it\tau}_{E}$ scaling, the impact of triangularity will not be accurately taken into account.

Having established and discussed the reference case, we now focus on the scaling of maximum elongation with various physical parameters.

Scaling with ${\it\beta}_{p}$

In the first set of studies, as well as all that follow, we fix ${\it\Delta}_{i}/a=0.1$ , ${\it\Delta}_{v}/a=0.3$ . The initial studies focus on scaling with ${\it\beta}_{p}$ . As such we fix ${\it\Delta}_{o}/a=0.1$ (which is equivalent to $b/a=1.1$ ) and ${\it\gamma}{\it\tau}_{w}=1.5$ . The value of ${\it\kappa}_{w}$ is again set in accordance with (6.7).

The desired scaling curves are calculated by repeating the procedure described for the reference case but for various values for ${\it\beta}_{p}$ . In figure 4, a set of curves is shown for four values of ${\it\beta}_{p}=0$ , 0.5, 1, 1.5. An examination of these curves indicates only a weak scaling of ${\it\kappa}$ with ${\it\beta}_{p}$ . Noticeable differences occur only for tight aspect ratios, $a/R_{0}>0.5$ . With regard to triangularity, observe that the optimum ${\it\delta}$ increases with increasing ${\it\beta}_{p}$ although the values, even at ${\it\beta}_{p}=1.5$ , are still below the peak performance values given in table 1, presumably because of the reasons discussed with the reference case.

Figure 4.

Curves of maximum ${\it\kappa}$ (a) and corresponding optimum ${\it\delta}$ (b) versus ${\it\varepsilon}$ for various values of ${\it\beta}_{p}$ at fixed ${\it\gamma}{\it\tau}_{w}=1.5$ , ${\it\Delta}_{i}/a={\it\Delta}_{o}/a={\it\Delta}_{v}/3a=0.1$ .

A possible reason for the larger triangularity as ${\it\beta}_{p}$ increases is as follows. As ${\it\beta}_{p}$ increases, the contribution to the toroidal current density at the outer-midplane $R>R_{0}$ becomes larger than the current density at the inner-midplane $R<R_{0}$ . Since the outer midplane toroidal curvature is unfavourable, its effect is minimized by reducing the area on the outside of the plasma. This is accomplished by increasing the triangularity. Hence ${\it\delta}$ increases with increasing ${\it\beta}_{p}$ . However, ${\it\delta}$ cannot become too large because of the corresponding increase in unfavourable poloidal curvature at the vertical tips of the plasma.

Scaling with $b/a$

In the second set of studies we fix ${\it\beta}_{p}=1$ , ${\it\gamma}{\it\tau}_{w}=1.5$ and vary the wall radius $b/a$ . As previously stated we do this by setting ${\it\Delta}_{i}/a=0.1$ , ${\it\Delta}_{v}/a=0.3$ and varying the outer gap parameter ${\it\Delta}_{o}/a$ . The values of $b/a$ and ${\it\kappa}_{w}$ are then determined from (6.2).

Following the procedure described above, we compute curves of ${\it\kappa}={\it\kappa}({\it\varepsilon})$ and the corresponding ${\it\delta}={\it\delta}({\it\varepsilon})$ for various ${\it\Delta}_{o}/a$ . These curves are illustrated in figure 5 for the values ${\it\Delta}_{o}/a=0.1$ , 0.3, 0.5 or equivalently $b/a=1.1$ , 1.2, 1.3. A comparative plot of the geometries for each elongation is shown in figure 6.

Figure 5.

Curves of maximum ${\it\kappa}$ (a) and corresponding optimum ${\it\delta}$ (b) versus ${\it\varepsilon}$ for various values of ${\it\Delta}_{o}/a$ at fixed ${\it\beta}_{p}=1$ , ${\it\gamma}{\it\tau}_{w}=1.5$ , ${\it\Delta}_{i}/a={\it\Delta}_{v}/3a=0.1$ .

Figure 6.

Comparative wall geometries for ${\it\Delta}_{o}/a=0.1$ , 0.3, 0.5 corresponding to $b/a=1.1$ , 1.2, 1.3 at fixed ${\it\Delta}_{i}/a={\it\Delta}_{v}/3a=0.1$ .

The numerical results show that, as expected, moving the wall further out leads to a lower maximum elongation. However, the decrease in maximum elongation is smaller than the increase in wall radius. Specifically, for any ${\it\varepsilon}$ , a change in $b/a=0.2$ leads to an approximate change in ${\it\kappa}\approx 0.1$ . Also, the change in triangularity is small, at approximately 0.05 over the whole range of ${\it\varepsilon}$ for the same change in $b/a=0.2$ .

The presumable explanation is that, even though the outer part of the wall is being moved further away from the plasma, the strong resistive wall image currents stay approximately the same on the inner, top and bottom of the first wall since these gaps have been held fixed. In other words, the effectiveness of the feedback system is not primarily driven by the proximity of the outer wall to the plasma. One might wonder whether larger decreases in maximum ${\it\kappa}$ would occur by instead increasing the inner or upper/lower gaps. This turns out to not be the case based on separate numerical studies that we have carried out (but which for brevity are not reported here). The conclusion is that the maximum ${\it\kappa}$ depends significantly on the size of the gap but not its location.

Scaling with ${\it\gamma}{\it\tau}_{w}$

The final set of numerical studies examines the scaling with the feedback parameter ${\it\gamma}{\it\tau}_{w}$ . For these studies we fix the wall gaps to ${\it\Delta}_{i}/a=0.1$ , ${\it\Delta}_{v}/a=0.3$ , ${\it\Delta}_{o}/a=0.1$ and beta poloidal to ${\it\beta}_{p}=1$ . These are the reference values. The values of $b/a$ and ${\it\kappa}_{w}$ are again determined from (6.7).

Curves are generated of ${\it\kappa}={\it\kappa}({\it\varepsilon})$ and the corresponding ${\it\delta}={\it\delta}({\it\varepsilon})$ for ${\it\gamma}{\it\tau}_{w}=0$ , 1, 2, 3 as shown in figure 7. Observe that the curve for ${\it\gamma}{\it\tau}_{w}=0$ represents an experiment without a vertical stability feedback system. It therefore approximates the results for earlier natural elongation studies (see for example Hakkarainen et al. (Reference Hakkarainen, Betti, Freidberg and Gormely1990)). The achievable elongations are indeed quite modest, for example ${\it\kappa}=1.17$ , ${\it\delta}=0.17$ for $a/R_{0}=0.3$ .

Figure 7.

Curves of maximum ${\it\kappa}$ (a) and corresponding optimum ${\it\delta}$ (b) versus ${\it\varepsilon}$ for various values of ${\it\gamma}{\it\tau}_{w}$ at fixed, ${\it\beta}_{p}=1$ , ${\it\Delta}_{i}/a={\it\Delta}_{o}/a={\it\Delta}_{v}/3a=0.1$ .

Figure 8.

Plot of the required ${\it\gamma}{\it\tau}_{w}$ versus ${\it\delta}$ at fixed ${\it\varepsilon}=0.35$ , ${\it\beta}_{p}=1$ , ${\it\Delta}_{i}/a={\it\Delta}_{o}/a={\it\Delta}_{v}/3a=0.1$ . The minimum in the curve corresponds to ${\it\gamma}{\it\tau}_{w}=2$ , ${\it\kappa}=2.37$ , and ${\it\delta}=0.20$ .

For higher values of ${\it\gamma}{\it\tau}_{w}$ , we see that increases in the feedback system capabilities lead to substantial increases in the maximum achievable elongation. Again, for $a/R_{0}=0.3$ , the maximum ${\it\kappa}$ increases from 1.17 to 2.77 as ${\it\gamma}{\it\tau}_{w}$ increases from 0 to 3. The optimized triangularity is insensitive to ${\it\gamma}{\it\tau}_{w}$ for small to moderate ${\it\varepsilon}$ , but decreases appreciably for tight aspect ratios.

A final quite interesting point concerns a different aspect of triangularity as evidenced in the data from DIII-D in table 1. To illustrate the point, we have carried out a series of calculations assuming a starting point with values of ${\it\kappa}=2.37$ and ${\it\delta}=0.20$ at ${\it\varepsilon}=0.35$ from the ${\it\gamma}{\it\tau}_{w}=2$ curve. We then vary ${\it\delta}$ holding ${\it\kappa}$ and ${\it\varepsilon}$ fixed. At each new ${\it\delta}$ , we recompute the value of ${\it\gamma}{\it\tau}_{w}$ required to make the eigenvalue ${\it\lambda}_{\mathit{min}}=0$ . This results in a curve of ${\it\gamma}{\it\tau}_{w}$ versus ${\it\delta}$ , as shown in figure 8. In other words, how much must the feedback capability be increased to stabilize a triangularity that is away from its optimum value? We see that the minimum in ${\it\gamma}{\it\tau}_{w}$ is relatively flat in the vicinity of ${\it\gamma}{\it\tau}_{w}=2$ but that a large increase is needed for high triangularities. For example, to achieve a triangularity of 0.71 requires a doubling of the feedback capacity to ${\it\gamma}{\it\tau}_{w}=4$ even though the elongation has remained unchanged. Some insight into this strong behaviour can be obtained by noting that the ratio of the pressure driven term to the line bending term in the ideal MHD ${\it\delta}W_{F}$ scales as

(7.1) $$\begin{eqnarray}\frac{2{\it\mu}_{0}({\bf\xi}_{\bot }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}p)({\bf\xi}_{\bot }^{\ast }\boldsymbol{\cdot }{\bf\kappa})}{|\boldsymbol{Q}_{\bot }|^{2}}\sim \frac{{\it\beta}_{p}}{1-{\it\delta}^{2}}.\end{eqnarray}$$

The $1-{\it\delta}^{2}$ factor arises from increasing unfavourable poloidal curvature at the top and bottom of the plasma as ${\it\delta}$ becomes larger. This leads to increased instability requiring a larger feedback capability, which is consistent with the DIII-D data.

Figure 9 shows the values of elongation and triangularity observed in a large sample of discharges in the DIII-D experiment. We can see that the maximum elongation ( ${\it\kappa}=2.3$ ) occurs for moderate triangularity, at approximately ${\it\delta}=0.5$ instead of the highest achievable triangularity, which is around ${\it\delta}=0.9$ . The existence of an optimal triangularity for the maximum elongation obtained in experiments agrees with the theoretical prediction in this paper.

Figure 9.

Plot of the total stored energy (Wtot) in joule, elongation ( ${\it\kappa}$ ) and triangularity ( ${\it\delta}$ ) for a large sample of discharges in the DIII-D experiment. The samples are taken from the ITER H-mode database db3.v11. The discharge in the red square (shot number 73334) has the maximum store energy at ${\it\kappa}=2.05$ and ${\it\delta}=0.85$ .