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We have seen that if 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 fluxfunction:

(897) 
Note that
. It follows that magnetic fieldlines
run along contours of .
We can write

(898) 
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

(899) 
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

(900) 
where
, and

(901) 
Figure 25 shows the contours of plotted in  space. It can
be seen that the tearing mode gives rise to
the formation of a magnetic island centred on the interface, .
Magnetic fieldlines situated outside the separatrix are displaced by the
tearing mode, but still retain their original topology. By contrast, fieldlines
inside the separatrix have been broken and reconnected, and now possess
quite different topology. The reconnection obviously takes place at the ``Xpoints,''
which are located at and
, where is an integer.
The maximum width of the reconnected region (in space) is given by
the island width, . Note that the island width is proportional
to the square root of the perturbed ``radial'' magnetic field at the interface
(i.e.,
).
Figure 25:
Magnetic fieldlines 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

(902) 
where

(903) 
is the jump in the logarithmic derivative of taken across the island.
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 growthrate 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

(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 should
be small: i.e., in general, the saturated island width is comparable
with the scalelength of the magnetic field configuration.
We conclude that, although idealMHD only breaks down in a narrow region of
width , centered on the interface, , the reconnection of
magnetic fieldlines which takes place
in this region is capable of significantly
modifying the whole magnetic field configuration.
Next: Fast Magnetic Reconnection
Up: Magnetohydrodynamic Fluids
Previous: Linear Tearing Mode Theory
Richard Fitzpatrick
20110331