Magnetic Island Chains

The analysis of Section 5.16 suggests that a non-zero value of the reconnected magnetic flux at the $k$th rational surface, ${\mit\Psi}_k$, will cause a helical magnetic island chain, with $m_k$ periods in the poloidal direction, and $n$ periods in the toroidal direction, to open in the immediate vicinity of the surface. Let us investigate the properties of such a chain.

We can write

$$\delta {\bf B} = \nabla\times \delta {\bf A},$$ (14.71)

where

$$\nabla \cdot\delta {\bf A} = 0.$$ (14.72)

Suppose that all terms in the previous equation are of equal importance. It follows from Equation (14.13) that

$$r\,\delta A^{r}\sim r^2\,\delta A^{\theta} \sim \frac{n}{m}\,r^2\,\delta A^{\varphi},$$ (14.73)

and, hence, that

$$r\,\delta A_r\sim \delta A_\theta\sim \frac{n}{m}\,\frac{r^2}{R^2}\,\delta A_\varphi.$$ (14.74)

Thus, Equations (14.14)–(14.16) and (14.71) yield

$${\cal J}\,\delta B^{\,r} \simeq \frac{\partial \,\delta A_\varphi}{\partial\theta},$$ (14.75)

$${\cal J}\,\delta B^{\,\theta} \simeq - \frac{\partial \,\delta A_\varphi}{\partial r},$$ (14.76)

$${\cal J} \,\delta B^{\,\varphi} \simeq 0,$$ (14.77)

where the neglected terms are, at least, of order $(n/m)\,(r/R)$ smaller than the retained terms. The previous three equations are consistent with Equations (14.44), (14.51), and (14.52) provided that

$$\delta A_\varphi(r,\theta,\varphi,t) \simeq R_0\sum_j\frac{\psi_j(r,t)}{m_j} \,{\rm e}^{\,{\rm i}\,(m_j\,\theta-n\,\varphi)}.$$ (14.78)

Let us search for a function, $F(r,\theta,\varphi,t)$, which is such that

$$({\bf B} + \delta{\bf B})\cdot\nabla F = 0.$$ (14.79)

It follows from Equations (14.2), (14.4), and (14.6) that

$$(B^{\,r}+\delta B^{\,r})\,\frac{\partial F}{\partial r} + (B^{\,\... ...+ (B^{\,\varphi}+\delta B^{\,\varphi})\,\frac{\partial F}{\partial \varphi} =0.$$ (14.80)

Equations (14.3), (14.19)–(14.21), and (14.75)–(14.77) yield

$$\frac{\partial F}{\partial r}\,\frac{\partial\delta A_\varphi}{\p... ...ac{\partial F}{\partial\theta} +q\,\frac{\partial F}{\partial\varphi}\right)=0.$$ (14.81)

Suppose that

$$F(r,\theta,\varphi,t) = F_0(r) + \delta A_\varphi(r,\theta,\varphi,t).$$ (14.82)

The previous two equations give

$$\frac{dF_0}{dr}\,\frac{\partial \delta A_\varphi}{\partial\theta}... ...\partial\theta} +q\,\frac{\partial \delta A_\varphi}{\partial\varphi}\right)=0.$$ (14.83)

According to Equation (14.78), we can write

$$\delta A_\varphi(r,\theta,\varphi,t) \simeq R_0\,\frac{\psi_k(r,t)}{m_k} \,{\rm e}^{\,{\rm i}\,(m_k\,\theta-n\,\varphi)}$$ (14.84)

in the vicinity of the $k$th rational surface, where we have neglected the non-resonant components of $\delta A_\varphi$ (because we do not expect them to open up an island chain at this surface) [26]. The previous two equations yield

$$\frac{dF_0}{dr} = - \frac{B_0\,R_0\,f}{m_k}\,(m_k-n\,q)\simeq B_0\left(\frac{s\,g}{q}\right)_{r_k}(r-r_k),$$ (14.85)

where $s=d\ln q/d\ln r$ is the magnetic shear, and use has been made of Equation (14.18), as well as the fact that $q(r_k)=m_k/n$. Finally, Equations (14.65), (14.82), (14.84), and (14.85) can be combined to give

$$F(r,\theta,\varphi,t) = \frac{B_0}{2}\left(\frac{s\,g}{q}\right)_{r_k}(r-r_k)^2 + R_0\,\vert{\mit\Psi}_k\vert\,\cos\xi,$$ (14.86)

where $\xi = m_k\,\theta-n\,\varphi+{\rm arg}({\mit\Psi}_k)$. Here, we have taken the (physical) real part of $F$, and use has been made of the constant-$\psi$ approximation that ${\mit\psi}_k(r,t)\simeq m_k\,{\mit\Psi}_k(t)$ in the immediate vicinity of the $k$th rational surface. (See Chapter 5.)

The previous equation can be written

$$\frac{F(r,\xi,t)}{R_0\,\vert{\mit\Psi}_k\vert} = 8\left(\frac{r-r_k}{W_k}\right)^2 + \cos\xi,$$ (14.87)

where

$$W_k = 4\,R_0\left[\left(\frac{q}{g\,s}\right)_{r_k}\,\frac{\vert{\mit\Psi}_k\vert}{B_0\,R_0}\right]^{1/2}.$$ (14.88)

Now, according to Equation (14.79), the contours of the function $F(r,\xi,t)$ map out the perturbed magnetic flux-surfaces in the immediate vicinity of the $k$th rational surface. These contours are shown in Figure 5.7 [with $(r-r_k)/W_k$ playing the role of $\hat{x}/\hat{W}$]. It can be seen that the reconnected magnetic flux at the $k$th rational surface has indeed opened up a helical magnetic island chain, with $m_k$ periods in the poloidal direction, and $n$ periods in the toroidal direction, at the surface. Moreover, the full radial width of the island chain (in $r$) is $W_k$. Incidentally, the previous equation is the toroidal generalization of the cylindrical result, (5.129).