With the neglect of plasma resistivity, the field configuration shown in Fig. 24
represents a *stable* equilibrium state, assuming, of course,
that we have normal pressure balance
across the interface. But, does the field configuration remain stable when we take
resistivity into account? If not, we expect an instability to develop which relaxes the
configuration to one possessing lower magnetic energy. As we shall see, this
type of relaxation process inevitably entails the breaking and reconnection of magnetic
field lines, and is, therefore, termed *magnetic reconnection*. The
magnetic energy released during the reconnection process eventually appears as
plasma thermal
energy. Thus, magnetic reconnection also involves plasma heating.

In the following, we shall outline the standard method for determining the
*linear* stability of the type of magnetic field configuration
shown in Fig. 26, taking into account the effect of plasma resistivity.
We are particularly
interested in plasma instabilities which are stable in the absence of resistivity,
and only grow when the resistivity is non-zero. Such instabilities are
conventionally termed *tearing modes*.
Since magnetic reconnection is, in fact, a *nonlinear* process, we shall
then proceed to investigate the nonlinear development of tearing modes.

The equilibrium magnetic field is written

(851) |

Here, is the equilibrium plasma density, the perturbed magnetic field, the perturbed plasma velocity, and the perturbed plasma pressure. The assumption of incompressible plasma flow is valid provided that the plasma velocity associated with the instability remains significantly smaller than both the Alfvén velocity and the sonic velocity.

Suppose that all perturbed quantities vary like

(856) |

respectively, where use has been made of Eqs. (854) and (855). Here, denotes .

It is convenient to normalize Eqs. (857)-(858) using a typical magnetic
field-strength, , and a typical scale-length, . Let us define the
*Alfvén time-scale*

(859) |

The ratio of these two time-scales is the Lundquist number:

(861) |

The term on the right-hand side of Eq. (862) represents plasma

It is assumed that the tearing instability grows on a *hybrid* time-scale
which is much less than but much greater than . It follows
that

(864) |

Equation (865) is simply the flux freezing constraint, which requires the plasma to move with the magnetic field. Equation (866) is the linearized, static force balance criterion: . Equations (865)-(866) are known collectively as the equations of

The ideal-MHD equations break down close to the interface because the neglect
of plasma resistivity and inertia becomes untenable as
.
Thus, there is a thin layer, in the immediate vicinity
of the interface, , where the behaviour of the plasma is governed
by the full MHD equations, (862)-(863). We can simplify these equations,
making use of the fact that and
in a
thin layer, to obtain the following layer equations:

Note that we have redefined the variables , , and , such that , , and . Here,

(869) |

The tearing mode stability problem reduces to solving the non-ideal-MHD layer equations,
(867)-(868), in the immediate vicinity of the interface, , solving
the ideal-MHD equations, (865)-(866), everywhere else in the plasma, matching
the two solutions at the edge of the layer, and applying physical
boundary conditions as
. This method
of solution was first described in a classic paper by Furth, Killeen, and
Rosenbluth.^{}

Let us consider the solution of the ideal-MHD equation (866) throughout
the bulk of the plasma. We could imagine launching a solution
at large positive
, which satisfies physical boundary conditions as
, and integrating this solution to the right-hand boundary of
the non-ideal-MHD layer at . Likewise, we could also launch
a solution at large negative , which satisfies physical boundary
conditions as
, and integrate this solution to
the left-hand boundary of the non-ideal-MHD layer at .
Maxwell's equations demand that must be continuous on either side
of the layer.
Hence, we can multiply our two solutions by appropriate factors, so as to ensure that
matches to the left and right of the layer. This leaves
the function undetermined
to an overall arbitrary multiplicative constant, just as we would expect in a
linear problem. In general,
is *not* continuous to the left and
right of the layer. Thus, the ideal solution can be characterized by the
real number

(870) |

The layer equations (867)-(868) possess a trivial solution (,
, where is independent of ), and
a nontrivial solution for which
and
.
The asymptotic behaviour of the nontrivial solution at the
edge of the layer is

where the parameter is determined by solving the layer equations, subject to the above boundary conditions. Finally, the growth-rate, , of the tearing instability is determined by the matching criterion

The layer equations (867)-(868) can be solved in a fairly straightforward manner
in Fourier transform space. Let

where . Equations (867)-(868) can be Fourier transformed, and the results combined, to give

where

(877) |

The most general small- asymptotic solution of Eq. (876) is written

(879) |

Thus, the matching parameter is determined from the small- asymptotic behaviour of the Fourier transformed layer solution.

Let us search for an unstable tearing mode, characterized by . It is
convenient to assume that

In the limit , Eq. (876)
reduces to

we can make use of the standard small argument asymptotic expansion of to write the most general solution to Eq. (876) in the form

Here, is an arbitrary constant.

In the limit

(885) |

(886) |

where and are arbitrary constants. Matching coefficients between Eqs. (884) and (887) in the range of satisfying the inequality (883) yields the following expression for the most general solution to Eq. (876) in the limit :

Finally, a comparison of Eqs. (878), (880), and (888) yields the result

The asymptotic matching condition (873) can be combined with the above
expression for to give the tearing mode dispersion relation

(891) |

From Eq. (882), the thickness of the non-ideal-MHD layer in -space
is

(892) |

When then , according to Eq. (890), giving . It is clear, therefore, that if the Lundquist number, , is very large then the non-ideal-MHD layer centred on the interface, , is

The time-scale for magnetic flux to diffuse across a layer of thickness
(in -space) is [*cf.*, Eq. (860)]

(895) |

(896) |