Evaluation of Integrals

Figure: 11.1 The integrals $I_4$ and $I_5$ evaluated as functions of $\lambda _{\theta \varphi }$.

It is helpful to define the new magnetic flux-surface label $k=[(1+{\mit\Omega})/2]^{1/2}$. (See Section 8.11.) It follows from Equation (11.101) that $k=0$ at the O-points of the magnetic island chain, and $k=1$ on the magnetic separatrix. Hence, we can write

$$I_1 = \int_0^\infty \frac{4\,[(2\,k^2-1)\,{\cal A}-2\,k^2\,{\cal C}]^{2}}{{\cal A}}\,dk,$$ (11.155)

$$I_2 = 16\int_1^\infty \left(\frac{\cal D}{\cal C} - \frac{1}{{\cal A}}\right)k^2\,dk,$$ (11.156)

$$I_3 = \frac{2\pi}{3}-\int_1^\infty \frac{4}{\cal C}\left(\frac{{\cal E}\,{\cal A}}{{\cal C}^{2}}-1\right)dk,$$ (11.157)

$$I_4 = \int_1^\infty \left(k\,{\cal C}\,d_k{\cal F} - {\cal A}\,{\cal F}\right)\left(\frac{{\cal E}}{{\cal C}^{2}}-\frac{1}{{\cal A}}\right)k\,dk,$$ (11.158)

$$I_5 = \int_1^\infty {\cal F}\,d_k{\cal F}\left({\cal E}- \frac{{\cal C}^{2}}{\cal A}\right)k^3\,dk.$$ (11.159)

where

$$d_k\!\left[{\cal C}\,d_k(k\,{\cal C}\,{\cal F})\right] = \lambda_{\theta\varphi}\,({\cal A}\,{\cal C}-1)\,k\,{\cal F},$$ (11.160)

$d_k\equiv d/dk$, and

$$\lambda_{\theta\varphi} = \frac{4\,\epsilon_\theta}{\epsilon_\varphi},$$ (11.161)

$${\cal F}(1) =0,$$ (11.162)

$${\cal F}(\infty) = 1.$$ (11.163)

Here, the functions ${\cal A}(k)$, ${\cal C}(k)$, ${\cal D}(k)$, and ${\cal E}(k)$ are defined in Section 8.11. Note that the factor $2\pi/3$ in Equation (11.157) is generated by the discontinuity in the function $L({\mit\Omega})$ (i.e., the discontinuity in the pressure gradient) across the separatrix of the magnetic island chain [23]. If this contribution is omitted then the sign of the integral $I_3$ is reversed.

The values of the first three integrals are

$$I_1 = 0.8227,$$ (11.164)

$$I_2 = 1.5835,$$ (11.165)

$$I_3 = 1.3814.$$ (11.166)

Figure 11.1 shows the values of the integrals $I_4$ and $I_5$ as functions of $\lambda _{\theta \varphi }$. Note that $I_4$ and $I_5$ are both only very weak functions of $\lambda _{\theta \varphi }$.