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

$\displaystyle I_1$ $\displaystyle = \int_0^\infty \frac{4\,[(2\,k^2-1)\,{\cal A}-2\,k^2\,{\cal C}]^{2}}{{\cal A}}\,dk,$ (11.155)
$\displaystyle I_2$ $\displaystyle = 16\int_1^\infty \left(\frac{\cal D}{\cal C} - \frac{1}{{\cal A}}\right)k^2\,dk,$ (11.156)
$\displaystyle I_3$ $\displaystyle = \frac{2\pi}{3}-\int_1^\infty \frac{4}{\cal C}\left(\frac{{\cal E}\,{\cal A}}{{\cal C}^{2}}-1\right)dk,$ (11.157)
$\displaystyle I_4$ $\displaystyle = \int_1^\infty \left(k\,{\cal C}\,d_k{\cal F} - {\cal A}\,{\cal F}\right)\left(\frac{{\cal E}}{{\cal C}^{2}}-\frac{1}{{\cal A}}\right)k\,dk,$ (11.158)
$\displaystyle I_5$ $\displaystyle = \int_1^\infty {\cal F}\,d_k{\cal F}\left({\cal E}- \frac{{\cal C}^{2}}{\cal A}\right)k^3\,dk.$ (11.159)


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

$d_k\equiv d/dk$, and

$\displaystyle \lambda_{\theta\varphi}$ $\displaystyle = \frac{4\,\epsilon_\theta}{\epsilon_\varphi},$ (11.161)
$\displaystyle {\cal F}(1)$ $\displaystyle =0,$ (11.162)
$\displaystyle {\cal F}(\infty)$ $\displaystyle = 1.$ (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

$\displaystyle I_1$ $\displaystyle = 0.8227,$ (11.164)
$\displaystyle I_2$ $\displaystyle = 1.5835,$ (11.165)
$\displaystyle I_3$ $\displaystyle = 1.3814.$ (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 $\lambda _{\theta \varphi }$.