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.*,
and
). The reconnecting magnetic fields are anti-parallel,
and of equal strength, . 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 suddenly changes.
This current sheet is assumed to be of thickness
and length .

Plasma is assumed to diffuse into the current layer, along its whole length,
at some relatively small inflow velocity, . The plasma is accelerated
along the layer, and eventually expelled from its two ends at some
relatively large exit velocity, . The inflow velocity
is simply an
velocity, so

(905) |

(906) |

(907) |

(908) |

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

(909) |

(910) |

(911) |

The above equations can be rearranged to give

(912) |

(913) |

(914) |

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 ,
, and
. 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 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

(915) |

(916) |

It follows that for reasonably large reconnection rates (*i.e.*,
) the length of the diffusion region becomes much smaller than the scale-size
of the system, , 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 that the shock waves make with
the interface is given approximately
by

(917) |

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

(918) |

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).