Nonlinear Tearing Mode Theory

We have seen that if ${\Delta}'>0$ then a magnetic field configuration of the type shown in Fig. 24 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:

$${\bf B} = B_0\,a\,\nabla\psi\times\hat{\bf z}.$$ (897)

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

We can write

$$\psi(\bar{x},\bar{y}) \simeq \psi_0(\bar{x}) + \psi_1(\bar{x},\bar{y}),$$ (898)

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

$$\psi \simeq - \frac{F'(0)}{2}\,\bar{x}^{~2} + {\Psi}\,\cos \bar{k}\, \bar{y},$$ (899)

where ${\Psi}$ 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 $\chi = -\psi/{\Psi}$ 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

$$\chi = 8\,X^2 - \cos\theta,$$ (900)

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

$$\bar{W} = 4\sqrt{\frac{{\Psi}}{F'(0)}}.$$ (901)

Figure 25 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 centred 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 $\bar{x}$-space) is given by the island width, $a\,\bar{W}$. Note that the island width is proportional to the square root of the perturbed ``radial'' magnetic field at the interface (i.e., $\bar{W}\propto \sqrt{\Psi}$).

Figure 25: Magnetic field-lines in the vicinity of a magnetic island.

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

$$0.823\,\tau_R\,\frac{d\bar{W}}{dt} = {\Delta}'(\bar{W}),$$ (902)

where

$${\Delta}'(\bar{W}) = \left[\frac{1}{\psi}\frac{d\psi}{d\bar{x}}\right]_{-\bar{W}/2}^{+\bar{W}/2}$$ (903)

is the jump in the logarithmic derivative of $\psi$ taken across the island. It is clear that once the tearing mode enters the nonlinear regime (i.e., once the normalized island width, $\bar{W}$, 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 time-scale, $\tau_R$. The tearing mode stops growing when it has attained a saturated island width $\bar{W}_0$, satisfying

$${\Delta}'(\bar{W}_0) = 0.$$ (904)

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