Evaluation of Integrals
Figure: 11.1
The integrals
and
evaluated as functions of
.
|
It is helpful to define the new magnetic flux-surface label
. (See Section 8.11.)
It follows from Equation (11.101) that
at the O-points of the magnetic island chain, and
on the magnetic separatrix.
Hence, we can write
 |
![$\displaystyle = \int_0^\infty \frac{4\,[(2\,k^2-1)\,{\cal A}-2\,k^2\,{\cal C}]^{2}}{{\cal A}}\,dk,$](img3625.png) |
(11.155) |
 |
 |
(11.156) |
 |
 |
(11.157) |
 |
 |
(11.158) |
 |
 |
(11.159) |
where
![$\displaystyle d_k\!\left[{\cal C}\,d_k(k\,{\cal C}\,{\cal F})\right] = \lambda_{\theta\varphi}\,({\cal A}\,{\cal C}-1)\,k\,{\cal F},$](img3630.png) |
(11.160) |
, and
 |
 |
(11.161) |
 |
 |
(11.162) |
 |
 |
(11.163) |
Here, the functions
,
,
, and
are defined in Section 8.11.
Note that the factor
in Equation (11.157) is generated by the discontinuity in the function
(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
is reversed.
The values of the first three integrals are
 |
 |
(11.164) |
 |
 |
(11.165) |
 |
 |
(11.166) |
Figure 11.1 shows the values of the integrals
and
as functions of
.
Note that
and
are both only very weak functions of
.