Fast Magnetic Reconnection

Up to now, we have only considered spontaneous magnetic reconnection, which develops from an instability of the plasma. As we have seen, such reconnection takes place at a fairly leisurely pace. Let us now consider forced magnetic reconnection in which the reconnection takes place as a consequence of an externally imposed flow or magnetic perturbation, rather than developing spontaneously. The principle difference between forced and spontaneous reconnection is the development of extremely large, positive ${\Delta}'$ values in the former case. Generally speaking, we expect ${\Delta}'$ to be $O(1)$ for spontaneous reconnection. By analogy with the previous analysis, we would expect forced reconnection to proceed faster than spontaneous reconnection (since the reconnection rate increases with increasing ${\Delta}'$). The question is, how much faster? To be more exact, if we take the limit ${\Delta}'\rightarrow \infty$, which corresponds to the limit of extreme forced reconnection, just how fast can we make the magnetic field reconnect? At present, this is a very controversial question, which is far from being completely resolved. In the following, we shall content ourselves with a discussion of the two ``classic'' fast reconnection models. These models form the starting point of virtually all recent research on this subject.

Let us first consider the Sweet-Parker model, which was first proposed by Sweet and Parker. The main features of the envisioned magnetic and plasma flow fields are illustrated in Fig. 26. The system is two dimensional and steady-state (i.e., $\partial/\partial z\equiv 0$ and $\partial/\partial t\equiv 0$). The reconnecting magnetic fields are anti-parallel, and of equal strength, $B_\ast$. We imagine that these fields are being forcibly pushed together via the action of some external agency. We expect a strong current sheet to form at the boundary between the two fields, where the direction of ${\bf B}$ suddenly changes. This current sheet is assumed to be of thickness $\delta$ and length $L$.

Figure 26: The Sweet-Parker magnetic reconnection scenario.

Plasma is assumed to diffuse into the current layer, along its whole length, at some relatively small inflow velocity, $v_0$. The plasma is accelerated along the layer, and eventually expelled from its two ends at some relatively large exit velocity, $v_\ast$. The inflow velocity is simply an ${\bf E}\times{\bf B}$ velocity, so

$$v_0 \sim \frac{E_z}{B_\ast}.$$ (905)

The $z$-component of Ohm's law yields

$$E_z \sim \frac{\eta\,B_\ast}{\mu_0\,\delta}.$$ (906)

Continuity of plasma flow inside the layer gives

$$L\,v_0 \sim \delta\,v_\ast,$$ (907)

assuming incompressible flow. Finally, pressure balance along the length of the layer yields

$$\frac{B_\ast^{~2}}{\mu_0} \sim \rho\,v_\ast^{~2}.$$ (908)

Here, we have balanced the magnetic pressure at the centre of the layer against the dynamic pressure of the outflowing plasma at the ends of the layer. Note that $\eta$ and $\rho$ are the plasma resistivity and density, respectively.

We can measure the rate of reconnection via the inflow velocity, $v_0$, since all of the magnetic field-lines which are convected into the layer, with the plasma, are eventually reconnected. The Alfvén velocity is written

$$V_A = \frac{B_\ast}{\sqrt{\mu_0\,\rho}}.$$ (909)

Likewise, we can write the Lundquist number of the plasma as

$$S = \frac{\mu_0\,L\,V_A}{\eta},$$ (910)

where we have assumed that the length of the reconnecting layer, $L$, is commensurate with the macroscopic length-scale of the system. The reconnection rate is parameterized via the Alfvénic Mach number of the inflowing plasma, which is defined

$$M_0 = \frac{v_0}{V_A}.$$ (911)

The above equations can be rearranged to give

$$v_\ast \sim V_A:$$ (912)

i.e., the plasma is squirted out of the ends of the reconnecting layer at the Alfvén velocity. Furthermore,

$$\delta \sim M_0\,L,$$ (913)

and

$$M_0 \sim S^{-1/2}.$$ (914)

We conclude that the reconnecting layer is extremely narrow, assuming that the Lundquist number of the plasma is very large. The magnetic reconnection takes place on the hybrid time-scale $\tau_A^{1/2}\,\tau_R^{1/2}$, where $\tau_A$ is the Alfvén transit time-scale across the plasma, and $\tau_R$ is the resistive diffusion time-scale across the plasma.

The Sweet-Parker reconnection ansatz is undoubtedly correct. It has been simulated numerically innumerable times, and was recently confirmed experimentally in the Magnetic Reconnection Experiment (MRX) operated by Princeton Plasma Physics Laboratory. The problem is that Sweet-Parker reconnection takes place far too slowly to account for many reconnection processes which are thought to take place in the solar system. For instance, in solar flares $S\sim 10^8$, $V_A\sim 100\,{\rm km}\,{\rm s}^{-1}$, and $L\sim 10^4\,{\rm km}$. According to the Sweet-Parker model, magnetic energy is released to the plasma via reconnection on a typical time-scale of a few tens of days. In reality, the energy is released in a few minutes to an hour. Clearly, we can only hope to account for solar flares using a reconnection mechanism which operates far faster than the Sweet-Parker mechanism.

One, admittedly rather controversial, resolution of this problem was suggested by Petschek. He pointed out that magnetic energy can be converted into plasma thermal energy as a result of shock waves being set up in the plasma, in addition to the conversion due to the action of resistive diffusion. The configuration envisaged by Petschek is sketched in Fig. 27. Two waves (slow mode shocks) stand in the flow on either side of the interface, where the direction of ${\bf B}$ reverses, marking the boundaries of the plasma outflow regions. A small diffusion region still exists on the interface, but now constitutes a miniature (in length) Sweet-Parker system. The width of the reconnecting layer is given by

$$\delta = \frac{L}{M_0\,S},$$ (915)

just as in the Sweet-Parker model. However, we do not now assume that the length, $L_\ast$, of the layer is comparable to the scale-size, $L$, of the system. Rather, the length may be considerably smaller than $L$, and is determined self-consistently from the continuity condition

$$L_\ast = \frac{\delta}{M_0},$$ (916)

where we have assumed incompressible flow, and an outflow speed of order the Alfvén speed, as before. Thus, if the inflow speed, $v_0$, is much less than $V_A$ then the length of the reconnecting layer is much larger than its width, as assumed by Sweet and Parker. On the other hand, if we allow the inflow velocity to approach the Alfvén velocity then the layer shrinks in length, so that $L_\ast$ becomes comparable with $\delta$.

Figure 27: The Petschek magnetic reconnection scenario.

It follows that for reasonably large reconnection rates (i.e., $M_0\rightarrow 1$) the length of the diffusion region becomes much smaller than the scale-size of the system, $L$, so that most of the plasma flowing into the boundary region does so across the standing waves, rather than through the central diffusion region. The angle $\theta$ that the shock waves make with the interface is given approximately by

$$\tan\theta \sim M_0.$$ (917)

Thus, for small inflow speeds the outflow is confined to a narrow wedge along the interface, but as the inflow speed increases the angle of the outflow wedges increases to accommodate the increased flow.

It turns out that there is a maximum inflow speed beyond which Petschek-type solutions cease to exist. The corresponding maximum Alfvénic Mach number,

$$(M_0)_{\rm max} = \frac{\pi}{8 \ln S},$$ (918)

can be regarded as specifying the maximum allowable rate of magnetic reconnection according to the Petschek model. Clearly, since the maximum reconnection rate depends inversely on the logarithm of the Lundquist number, rather than its square root, it is much larger than that predicted by the Sweet-Parker model.

It must be pointed out that the Petschek model is very controversial. Many physicists think that it is completely wrong, and that the maximum rate of magnetic reconnection allowed by MHD is that predicted by the Sweet-Parker model. In particular, Biskamp wrote an influential and widely quoted paper reporting the results of a numerical experiment which appeared to disprove the Petschek model. When the plasma inflow exceeded that allowed by the Sweet-Parker model, there was no acceleration of the reconnection rate. Instead, magnetic flux ``piled up'' in front of the reconnecting layer, and the rate of reconnection never deviated significantly from that predicted by the Sweet-Parker model. Priest and Forbes later argued that Biskamp imposed boundary conditions in his numerical experiment which precluded Petschek reconnection. Probably the most powerful argument against the validity of the Petschek model is the fact that, more than 30 years after it was first proposed, nobody has ever managed to simulate Petschek reconnection numerically (except by artificially increasing the resistivity in the reconnecting region--which is not a legitimate approach).