Width of a Magnetic Island

At each mode rational surface, $q(x)=m/n$, the expression

$$nB^{0\zeta}(x)-mB^{0\theta}(x)=[nq(x)-m]B^{0\theta}(x)$$

goes to zero, thus giving the appearance that equations (8) and (10) are singular at mode rational surfaces. However, the existence of a magnetic island or stochastic region, which is implied by a nonzero $B^{1x}_{mn}$ at the mode rational surfaces, locally flattens the $p^{0}(x)$ and $J^{0\zeta}(x)/B^{0\zeta}(x)$ profiles. This removes the singularity in Eqs (8) and (10) [3,8] as is shown in the subsection below. The perturbation affects the background profiles and permits a saturated island solution, in contrast to a linear tearing mode solution.

Consider a stream function $\psi$ that is uniform along magnetic field lines:

$${\bf B}\cdot\nabla\psi = 0$$ (26)

If $\psi$ is a smooth function with a local minimum at the mode rational surface, the Taylor series for $\psi$ that satisfies Eq. (26) yields

$$\psi = \psi^{0}_{mn}(x_{mn}) - \left( \frac{\psi^{1}_{mn}nB^... ... (x-x_{mn})^{2} + \cdots +\psi^{1}_{mn}\cos (m\theta - n\zeta)$$ (27)

where $x_{mn}$ is the radius of the mode rational surface and where $\psi^{1}_{mn}\cos (m\theta -n\zeta)$ is the lowest order harmonic perturbation. With the value of $\psi$ at the edge of the widest part of the island ( $m\theta - n\zeta = 0$, $x-x_{mn}=W/2$) set equal to the value of $\psi$ at the x-point ( $m\theta - n\zeta = \pi$, $x-x_{mn}=0$), the following equation for the width of the island is obtained

$$W_{mn}=4\sqrt{\left[\frac{-iB^{1x}_{mn}}{nB^{0\theta}(dq/dx)} \right]_{x=x_{mn}}}$$ (28)

Higher order terms in the Taylor series (27) may be used to determine the asymmetry in the island width and other details of the island's shape. Note that the island width is measured in terms of the flux surface label $x$. The geometrical width of the island may vary poloidally around a flux surface, being wider where the background flux surfaces are more spread apart and narrower where the background flux surfaces are squeezed together. This effect is automatically taken account when using flux coordinates.