Constant-$\psi$ Magnetic Islands

Figure: 9.5 Equally-spaced contours of the magnetic flux-function, $\hat{\psi}(\hat{x},\xi)$, in the vicinity of a constant-$\psi$ magnetic island chain. The magnetic separatrix is shown as a dashed line.

Suppose that the constant-$\psi$ approximation is valid. According to Section 9.5, this implies that the perturbed magnetic flux, $\psi_1(x)$, is approximately constant in the vicinity of the resonant layer. Let $\psi(x,y,t)=-B_0\,a\,\hat{\psi}(x,y,t)$ and $\psi_1(x)\,{\rm e}^{\,\gamma\,t}= -B_0\,a\,\skew{3}\hat{\mit\Psi}(t)\,{\rm e}^{-{\rm i}\,\varphi}$, where $\skew{3}\hat{\mit\Psi}$ is real and positive, and $\varphi$ is real. The physical magnetic flux, which is the real part of Equation (9.18), reduces to

$$\hat{\psi}(\hat x,\xi,\hat{t})=\frac{\hat{x}^2}{2}+\skew{3}\hat{\mit\Psi}(\hat{t})\,\cos\xi$$ (9.83)

in the limit $\vert\hat{x}\vert\ll 1$, where $\xi= k\,y-\varphi$. Let

$$\hat{W} = 4\,\skew{3}\hat{\mit\Psi}^{1/2}.$$ (9.84)

Equation (9.83) yields

$$\frac{\hat{\psi}(\hat x,\xi)}{\skew{3}\hat{\mit\Psi}}=8\left(\frac{\hat{x}}{\hat{W}}\right)^2+\cos\xi.$$ (9.85)

Figure 9.5 shows the contours of $\hat{\psi}(\hat{x},\xi)$ specified in the previous equation. Recall that the contours of $\hat{\psi}$ correspond to magnetic field-lines. It can be seen that the tearing mode has changed the topology of the magnetic field in the immediate vicinity of the resonant surface, $\hat{x}=0$. In fact, as the tearing mode grows in amplitude (i.e., as $\skew{3}\hat{\mit\Psi}$ increases), magnetic field-lines pass through the magnetic “X-points” (which are located at $\hat{x}=0$, $\xi=n\,2\pi$, where $n$ is an integer), at which time they break (or “tear”) and then reconnect to form new field-lines that do not extend over all values of $\xi$. The magnetic field-line that forms the boundary between the unreconnected and reconnected regions is known as the magnetic separatrix, and corresponds to the contour $\hat{\psi}(\hat{x},\xi) = \skew{3}\hat{\mit\Psi}$. The reconnected regions within the magnetic separatrix are termed magnetic islands. The full width (in $\hat{x}$) of the magnetic separatrix, which is known as the magnetic island width, is $\hat{W}$. It can be seen from Equation (9.84) that the magnetic island width is proportional to the square-root of the quantity $\skew{3}\hat{\mit\Psi}$, which is termed the (normalized) reconnected magnetic flux. (In fact, the magnetic flux, per unit length in the $z$-direction, that passes through a surface (whose normal lies in the $x$-$y$ plane) linking the center of a magnetic island to the separatrix is $2\,a\,B_0\,\skew{3}\hat{\mit\Psi}$.)

Consider the term $[\phi,\psi]$, appearing in the reduced-MHD Ohm's law, (9.11). With $\hat{\psi}$ specified by Equation (9.83), the term in question reduces to

$$[\phi,\psi] = B_0\,k\,\hat{x}\,\frac{\partial\phi}{\partial\xi} + B_0\,k\,\frac{\partial\phi}{\partial \hat{x}}\,\skew{3}\hat{\mit\Psi}\,\sin\xi.$$ (9.86)

The first term on the right-hand side of the previous equation is linear (i.e., it is first order in the perturbed quantities $\phi$ and $\skew{3}\hat{\mit\Psi}$), whereas the second is nonlinear (i.e., it is second order in perturbed quantities). Thus, linear layer theory is only valid when the second term is negligible with respect to the first. Estimating both $(\partial \phi/\partial \xi)/(\partial\phi/\partial \hat{x})$ and $\hat{x}$ as $\hat{\delta}$, where $\hat{\delta}$ is the normalized constant-$\psi$ linear layer width, and recalling that $\skew{3}\hat{\mit\Psi}\propto \hat{W}^{\,2}$, the criterion for the validity of linear layer theory becomes

$$\hat{\delta}\gg \hat{W}.$$ (9.87)

In other words, linear layer theory is only valid when the magnetic island width is much less than the linear layer width. Given that linear layers in a high Lundquist number plasmas are very narrow (i.e, $\hat{\delta}\sim S^{-2/5}$), this implies that linear layer theory breaks down before the tearing mode has significantly modified the topology of the magnetic field.