To zeroth order in
, Equation (11.72) yields
![$\displaystyle \partial_X^{\,2}{\mit\Psi}_0 = 1.$](img2779.png) |
(11.100) |
Solving this equation subject to the boundary condition (11.83), we obtain
![$\displaystyle {\mit\Psi}_0 = {\mit\Omega}(X,\zeta)\equiv \frac{X^2}{2} + \cos\zeta.$](img2780.png) |
(11.101) |
Thus, we again conclude that, to lowest order in our expansion, the magnetic flux-surfaces in the island region have the constant-
structure pictured in Figure 5.7. The island O-points correspond to
and
(where
is an integer), the
X-points correspond to
and
, and the magnetic separatrix corresponds to
.
To zeroth order in
, Equation (11.71) yields
![$\displaystyle \{{\cal N}_0, {\mit\Omega}\}= 0,$](img2785.png) |
(11.102) |
where use has been made of Equation (11.101). Given that
is an odd function of
, it
follows that
![$\displaystyle {\cal N}_0(X,\zeta,T) = \varsigma\, {\cal N}_{(0)}({\mit\Omega},T).$](img2786.png) |
(11.103) |
By symmetry,
inside the magnetic separatrix of the island chain.
Let
![$\displaystyle L({\mit\Omega},T) = \frac{\partial{\cal N}_{(0)}}{\partial{\mit\Omega}}.$](img2789.png) |
(11.104) |
Note that
. Equations (11.84) and (11.101) imply that
![$\displaystyle L({\mit\Omega}\rightarrow\infty,T) = \frac{1}{\sqrt{2\,{\mit\Omega}}}.$](img2791.png) |
(11.105) |
To zeroth order in
, Equation (11.68) yields
![$\displaystyle \{{\mit\Phi}_0,{\mit\Omega}\}=0,$](img2792.png) |
(11.106) |
where use has been made of Equations (11.101) and (11.103). Given that
is an odd function of
, it follows that
![$\displaystyle {\mit\Phi}_0(X,\zeta,T) = \varsigma\, {\mit\Phi}_{(0)}({\mit\Omega},T).$](img2793.png) |
(11.107) |
By symmetry,
inside the magnetic separatrix of the island chain. Let
![$\displaystyle M({\mit\Omega},T) = \frac{\partial{\mit\Phi}_{(0)}}{\partial{\mit\Omega}}.$](img2796.png) |
(11.108) |
Note that
. Equations (11.85) and (11.101) imply that
![$\displaystyle M({\mit\Omega}\rightarrow\infty,T) = \frac{v(T)}{\sqrt{2\,{\mit\Omega}}}+ v'(T).$](img2798.png) |
(11.109) |
To zeroth order in
, Equation (11.69) yields
![$\displaystyle \{{\cal V}_0,{\mit\Omega}\} = 0,$](img2799.png) |
(11.110) |
where use has been made of Equations (11.101), (11.103), and (11.107). Given that
is an even function of
, we can write
![$\displaystyle {\cal V}_0(X,\zeta,T) = {\cal V}_{(0)}({\mit\Omega},T).$](img2800.png) |
(11.111) |
Finally, to zeroth order in
, Equation (11.70) gives
where
![$\displaystyle F({\mit\Omega}, T) = M({\mit\Omega}, T) + \frac{1}{1+\tau}\left(1-\alpha_\theta\,\frac{\eta_i}{1+\eta_i}\right)L({\mit\Omega},T),$](img3550.png) |
(11.113) |
and use has been made of Equations (11.101), (11.103), (11.104), (11.107), and (11.108). Moreover,
. Note that
.
Let us write
![$\displaystyle {\cal J}_0({\mit\Omega},\zeta,T) = {\cal J}_0^{(c)}({\mit\Omega},\zeta,T) + {\cal J}_0^{(s)}({\mit\Omega},\zeta,T) ,$](img3552.png) |
(11.114) |
where
has the symmetry of
, whereas
has the symmetry of
.
It follows from Equations (11.112) and (11.114) that
![$\displaystyle \{{\cal J}_0^{(c)},{\mit\Omega}\} =-\zeta_g\,\{L\,\vert X\vert,{\...
...Omega}\!\left[M\left(M+\frac{L}{1+\tau}\right)\right]X^2, {\mit\Omega}\right\},$](img3555.png) |
(11.115) |
which implies that
![$\displaystyle {\cal J}_0^{(c)}({\mit\Omega},\zeta,T)= \overline{\cal J}_0({\mit...
...ial_{\mit\Omega}\!\left[M\left(M+\frac{L}{1+\tau}\right)\right]\widetilde{X^2},$](img3556.png) |
(11.116) |
where the
operator is defined in Section 8.6, and
is an undetermined flux-surface function.
Equations (11.112) and (11.114) also yield
![$\displaystyle \{{\cal J}^{(s)}_0,{\mit\Omega}\} =-\frac{\epsilon_\theta}{\epsil...
...\partial_{\mit\Omega}\!\left(\xi^{-1}\,{\cal V}_{(0)} - \vert X\vert\,F\right).$](img3558.png) |
(11.117) |
The flux-surface average (see Section 8.6) of the previous equation gives
![$\displaystyle \langle X\,\partial_{\mit\Omega}(\xi^{-1}\,{\cal V}_{(0)} - \vert X\vert\,F)\rangle =0.$](img3559.png) |
(11.118) |
Inside the separatrix of the magnetic island chain, recalling that
, the previous equation reduces to
![$\displaystyle \langle X\rangle\, \partial_{\mit\Omega} {\cal V}_{(0)} = 0,$](img3560.png) |
(11.119) |
whereas outside the separatrix, it gives
![$\displaystyle \partial_{\mit\Omega}\!\left(\xi^{-1}\,\langle\vert X\vert\rangle \,{\cal V}_{(0)} - \langle X^{\,2}\rangle\, F\right)= 0,$](img3561.png) |
(11.120) |
where use has been made of the easily proved identity
.
Thus, we conclude that
![$\displaystyle {\cal V}_{(0)} ({\mit\Omega}, T)= c_0(T)$](img3563.png) |
(11.121) |
inside the separatrix, and
![$\displaystyle {\cal V}_{(0)}({\mit\Omega}, T) = \xi\,\langle X^{\,2}\rangle \,F({\mit\Omega},T) + c_1(T)$](img3564.png) |
(11.122) |
outside the separatrix. Here, use has been made of
outside the separatrix.
However, the boundary condition (11.86), combined with Equations (11.101), (11.105), (11.109), and (11.113), implies that
. Finally, Equations (11.117), (11.121), and (11.122) yield
![$\displaystyle \{{\cal J}^{(s)},{\mit\Omega}\} =- \frac{\epsilon_\theta}{\epsilo...
...l_{\mit\Omega}\!\left[\left(\langle X^{\,2}\rangle-\vert X\vert\right)F\right].$](img3567.png) |
(11.123) |