Nonlinear Tearing Mode Theory

We have seen that if ${\mit\Delta}'>0$ then a magnetic field configuration of the type shown in Figure 7.7 is unstable to a tearing mode. Let us now investigate how a tearing instability affects the field configuration as it develops.

It is convenient to write the magnetic field in terms of a flux-function:

$${\bf B} = B_0\,a\,\nabla\psi\times{\bf e}_z.$$ (7.217)

Note that ${\bf B}\cdot\nabla\psi=0$ . It follows that magnetic field-lines run along contours of $\psi(x,y)$ .

We can write

$$\psi(\bar{x},\bar{y}) \simeq \psi_0(\bar{x}) + \psi_1(\bar{x},\bar{y}),$$ (7.218)

where $\psi_0$ generates the equilibrium magnetic field, and $\psi_1$ generates the perturbed magnetic field associated with the tearing mode. Here, $\bar{y}= y/a$ . In the vicinity of the interface, we have

$$\psi(\bar{x},\bar{y}) \simeq - \frac{F'(0)}{2}\,\bar{x}^{\,2} + {\mit\Psi}\,\cos( \bar{k}\, \bar{y}),$$ (7.219)

where ${\mit\Psi}$ is a constant. Here, we have made use of the fact that $\psi_1(\bar{x},\bar{y})\simeq \psi_1(\bar{y})$ if the constant-$\psi$ approximation holds good (which is assumed to be the case).

Let $\chi = -\psi/{\mit\Psi}$ and $\theta=\bar{k}\, \bar{y}$ . It follows that the normalized perturbed magnetic flux function, $\chi$ , in the vicinity of the interface takes the form

$$\chi = 8\,X^{\,2} - \cos\theta,$$ (7.220)

where $X = \bar{x}/\overline{W}$ , and

$$\overline{W} = 4\sqrt{\frac{{\mit\Psi}}{F'(0)}}.$$ (7.221)

Figure 7.8 shows the contours of $\chi$ plotted in $X$ -$\theta$ space. It can be seen that the tearing mode gives rise to the formation of a magnetic island centered on the interface, $X=0$ . Magnetic field-lines situated outside the ``separatrix'' are displaced by the tearing mode, but still retain their original topology. By contrast, field-lines inside the separatrix have been broken and reconnected, and now possess quite different topology. The reconnection obviously takes place at the ``X-points,'' which are located at $X=0$ and $\theta = j\,2\pi$ , where $j$ is an integer. The maximum width of the reconnected region (in $x$ -space) is given by the island width, $a\,\overline{W}$ . Note that the island width is proportional to the square root of the perturbed ``radial'' magnetic field at the interface (i.e., $\overline{W}\propto \sqrt{\mit\Psi}$ ).

Figure 7.8: Magnetic field-lines in the vicinity of a magnetic island.

According to a result first established in a very elegant paper by Rutherford (Rutherford 1973), the nonlinear evolution of the island width is governed by

$$0.823\,\tau_R\,\frac{d\overline{W}}{dt} = {\mit\Delta}'(\overline{W}),$$ (7.222)

where

$${\mit\Delta}'(\overline{W}) = \left[\frac{1}{\psi}\frac{d\psi}{d\bar{x}}\right]_{-\overline{W}/2}^{+\overline{W}/2}$$ (7.223)

is the jump in the logarithmic derivative of $\psi$ taken across the island (White, Monticello, Rosenbluth, and Waddell 1977). It is clear that once the tearing mode enters the nonlinear regime (i.e., once the normalized island width, $\overline{W}$ , exceeds the normalized linear layer width, $S^{-2/5}$ ), the growth-rate of the instability slows down considerably, until the mode eventually ends up growing on the extremely slow resistive timescale, $\tau_R$ . The tearing mode stops growing when it has attained a saturated island width $\overline{W}_0$ , satisfying

$${\mit\Delta}'(\overline{W}_0) = 0.$$ (7.224)

The saturated width is a function of the original plasma equilibrium, but is independent of the resistivity. There is no particular reason why $\overline{W}_0$ should be small. In general, the saturated island width is comparable with the characteristic lengthscale of the magnetic field configuration. We conclude that, although ideal-MHD only breaks down in a narrow region of relative width $S^{-2/5}$ , centered on the interface, $x=0$ , the reconnection of magnetic field-lines that takes place in this region is capable of significantly modifying the whole magnetic field configuration.