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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:

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

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

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

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

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

$\displaystyle \overline{W} = 4\sqrt{\frac{{\mit\Psi}}{F'(0)}}.$ (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.
\epsfysize =3in

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

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


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

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

next up previous
Next: Fast Magnetic Reconnection Up: Magnetohydrodynamic Fluids Previous: Linear Tearing Mode Theory
Richard Fitzpatrick 2016-01-23