Nonlinear Tearing Mode Theory

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

(7.217) |

Note that . It follows that magnetic field-lines run along contours of .

We can write

(7.218) |

where generates the equilibrium magnetic field, and generates the perturbed magnetic field associated with the tearing mode. Here, . In the vicinity of the interface, we have

(7.219) |

where is a constant. Here, we have made use of the fact that if the constant- approximation holds good (which is assumed to be the case).

Let and . It follows that the normalized perturbed magnetic flux function, , in the vicinity of the interface takes the form

(7.220) |

where , and

(7.221) |

Figure 7.8 shows the contours of plotted in - space. It can be seen that the tearing mode gives rise to the formation of a

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

(7.222) |

where

(7.223) |

is the jump in the logarithmic derivative of 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, , exceeds the normalized linear layer width, ), the growth-rate of the instability slows down considerably, until the mode eventually ends up growing on the extremely slow resistive timescale, . The tearing mode stops growing when it has attained a saturated island width , satisfying

(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 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 , centered on the interface, , the reconnection of magnetic field-lines that takes place in this region is capable of significantly modifying the whole magnetic field configuration.