Linear Tearing Mode Theory

With the neglect of plasma resistivity, the field configuration shown in Figure 7.7
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, then we expect an instability to develop that 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 Figure 7.7, taking into account the effect of plasma resistivity.
We are particularly
interested in plasma instabilities that are stable in the absence of resistivity,
and only grow when the resistivity is non-zero. Such instabilities are
conventionally termed *tearing modes*.
Because 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

where . There is assumed to be no equilibrium plasma flow. The linearized equations of resistive-MHD [i.e., Equations (7.1)-(7.4), with Equation (7.3) replaced by Equation (7.102)], assuming incompressible flow, take the form

Here, is the equilibrium plasma density, the perturbed magnetic field, the perturbed plasma velocity, the perturbed plasma pressure, and use has been made of Maxwell's equation. 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

where is the instability growth-rate. The -component of Equation (7.172), and the -component of the curl of Equation (7.173), reduce to

respectively, where use has been made of Equations (7.174) and (7.175). Here, denotes .

It is convenient to normalize Equations (7.177) and (7.178) using a typical magnetic
field-strength,
, and a typical lengthscale,
. Let us define the
*Alfvén timescale*

(7.179) |

where is the Alfvén velocity, and the

The ratio of these two timescales is the Lundquist number:

(7.181) |

Let , , , , , , and . It follows that

The term on the right-hand side of Equation (7.182) represents plasma resistivity, whereas the term on the left-hand side of Equation (7.183) represents plasma inertia.

It is assumed that the tearing instability grows on a hybrid timescale that is much less than , but much greater than . It follows that

(7.184) |

Thus, throughout most of the plasma, we can neglect the right-hand side of Equation (7.182), and the left-hand side of Equation (7.183), which is equivalent to the neglect of plasma resistivity and inertia. In this case, Equations (7.182) and (7.183) reduce to

Equation (7.185) is simply the flux-freezing constraint, which requires the plasma to move with the magnetic field. Equation (7.186) is the linearized, static force balance criterion: . Equations (7.185) and (7.186) 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 behavior of the plasma is governed by the resistive-MHD equations, (7.182) and (7.183). 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,

(7.189) |

is the so-called

The tearing mode stability problem reduces to solving the resistive-MHD layer equations, (7.187) and (7.188), in the immediate vicinity of the interface, , solving the ideal-MHD equations, (7.185) and (7.186), 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 (Furth, Killeen, and Rosenbluth 1963).

Let us consider the solution of the ideal-MHD equation (7.186) 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 resistive-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 resistive-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

(7.190) |

that is, by the jump in the logarithmic derivative of to the left and right of the layer. This parameter is known as the

The layer equations, (7.187) and (7.188), possess a trivial solution ( , , where is independent of ), and a nontrivial solution for which and . The asymptotic behavior of the nontrivial solution at the edge of the layer is

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

The layer equations, (7.187) and (7.188), can be solved in a fairly straightforward manner in Fourier transform space. Let

where . Equations (7.187) and (7.188) can be Fourier transformed, and the results combined, to give

where

(7.197) |

The most general small- asymptotic solution of Equation (7.196) is written

where and are independent of , and it is assumed that . When inverse Fourier transformed, the previous expression leads to the following expression for the asymptotic behavior of at the edge of the resistive-MHD layer (Erdéyli 1954):

(7.199) |

It follows from a comparison with Equations (7.191) and (7.192) that

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

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

This ordering, which is known as the

In the limit , Equation (7.196) reduces to

The solution to this equation that is well behaved in the limit is written , where is a standard parabolic cylinder function (Abramowitz and Stegun 1965e). In the limit

we can make use of the standard small argument asymptotic expansion of to write the most general solution to Equation (7.196) in the form (Abramowitz and Stegun 1965e)

Here, is an arbitrary constant.

In the limit

(7.205) |

Equation (7.196) reduces to

(7.206) |

The most general solution to this equation is written

where and are arbitrary constants. Matching coefficients between Equations (7.204) and (7.207) in the range of satisfying the inequality (7.203) yields the following expression for the most general solution to Equation (7.196) in the limit :

Finally, a comparison of Equations (7.198), (7.200), and (7.208) gives the result

The asymptotic matching condition (7.193) can be combined with the previous expression for to give the tearing mode dispersion relation

Here, use has been made of the definitions of and . According to the dispersion relation, the tearing mode is unstable whenever , and grows on the hybrid timescale . It is easily demonstrated that the tearing mode is stable whenever . According to Equations (7.193), (7.201), and (7.209), the constant- approximation holds provided that

(7.211) |

that is, provided that the tearing mode does not become too unstable.

According to Equation (7.202), the thickness of the resistive-MHD layer in -space is

(7.212) |

It follows from Equations (7.194) and (7.195) that the thickness of the layer in -space is

When then , according to Equation (7.210), giving . It is clear, therefore, that if the Lundquist number, , is very large then the resistive-MHD layer centered on the interface, , is extremely narrow.

The timescale for magnetic flux to diffuse across a layer of thickness (in -space) is [see Equation (7.180)]

If

(7.215) |

then the tearing mode grows on a timescale that is far longer than the timescale on which magnetic flux diffuses across the resistive layer. In this case, we would expect the normalized ``radial'' magnetic field, , to be approximately constant across the layer, because any non-uniformities in would be smoothed out via resistive diffusion. It follows from Equations (7.213) and (7.214) that the constant- approximation holds provided that

(7.216) |

(i.e., ), which is in agreement with Equation (7.201).