Magnetic Reconnection

Magnetic reconnection is a phenomenon that is of particular importance in solar system plasmas. In the solar corona, it results in the rapid release to the plasma of energy stored in the large-scale structure of the coronal magnetic field, an effect that is thought to give rise to solar flares (Priest 1984). Small-scale reconnection may play a role in heating the corona, and, thereby, driving the outflow of the solar wind (Priest 1984). In the Earth's magnetosphere, magnetic reconnection in the magnetotail is thought to be the precursor for auroral sub-storms (Ratcliffe 1972).

The evolution of the magnetic field in a resistive-MHD plasma is governed by the following well-known equation [see Equation (7.102)]:

$$\frac{\partial{\bf B}}{\partial t} = \nabla\times({\bf V}\times{\bf B})+ \frac{\eta}{\mu_0}\,\nabla^2{\bf B}.$$ (7.169)

The first term on the right-hand side of this equation describes the convection of the magnetic field by the plasma flow. The second term describes the resistive diffusion of the field through the plasma. If the first term dominates, then magnetic flux is frozen into the plasma, and the topology of the magnetic field cannot change. On the other hand, if the second term dominates, then there is little coupling between the field and the plasma flow, and the topology of the magnetic field is free to change.

The relative magnitude of the two terms on the right-hand side of Equation (7.169) is conventionally measured in terms of magnetic Reynolds number, or Lundquist number:

$$S = \frac{\mu_0\,V\,L}{\eta} \simeq \frac{\vert\nabla\times({\bf V}\times {\bf B})\vert} {\vert(\eta/\mu_0)\,\nabla^{\,2}{\bf B}\vert},$$ (7.170)

where $V$ is the characteristic flow speed, and $L$ the characteristic lengthscale of the plasma. If $S$ is much larger than unity then convection dominates, and the frozen flux constraint prevails, whereas if $S$ is much less than unity then diffusion dominates, and the coupling between the plasma flow and the magnetic field is relatively weak.

It turns out that very large $S$ -values are virtually guaranteed to occur in the solar system because of the extremely large lengthscales of solar system plasmas. For instance, $S\sim 10^{\,8}$ for solar flares, whereas $S\sim 10^{\,11}$ is appropriate for the solar wind and the Earth's magnetosphere (Priest 1984). Of course, in calculating these values, we have identified the lengthscale $L$ with the characteristic size of the plasma under investigation.

On the basis of the previous discussion, it seems reasonable to neglect diffusive processes altogether in solar system plasmas. Of course, this leads to very strong constraints on the behavior of such plasmas, because, in this limit, all cross-field mixing of plasma elements is suppressed. Particles may freely mix along field-lines (within limitations imposed by magnetic mirroring, etc.), but are completely ordered perpendicular to the field, because they always remain tied to the same field-lines as they convect in the plasma flow.

Let us consider what happens when two initially separate plasma regions come into contact with one another, as occurs, for example, in the interaction between the solar wind and the Earth's magnetic field. Assuming that each plasma is frozen to its own magnetic field, and that cross-field diffusion is absent, we conclude that the two plasmas will not mix, but, instead, that a thin boundary layer will form between them, separating the two plasmas and their respective magnetic fields. In equilibrium, the location of the boundary layer will be determined by pressure balance. Because, in general, the frozen fields on either side of the boundary will have differing strengths, and differing orientations tangential to the boundary, the layer must also constitute a current sheet. Thus, flux freezing leads inevitably to the prediction that in plasma systems space becomes divided into separate cells, wholly containing the plasma and magnetic field from individual sources, and separated from each other by thin current sheets.

The ``separate cell'' picture constitutes an excellent zeroth-order approximation to the interaction of solar system plasmas, as witnessed, for example, by the well-defined planetary magnetospheres (Russell 1991). It must be noted, however, that the large $S$ -values upon which the applicability of the frozen flux constraint was justified were derived using the large overall spatial scales of the systems involved. However, strict application of this constraint to the problem of the interaction of separate plasma systems leads to the inevitable conclusion that structures will form having small spatial scales, at least in one dimension: that is, the thin current sheets constituting the cell boundaries. It is certainly not guaranteed, from the previous discussion, that the effects of diffusion can be neglected in these boundary layers. In fact, we shall demonstrate that the localized breakdown of the flux freezing constraint in the boundary regions, due to diffusion, not only has an impact on the properties of the boundary regions themselves, but can also have a decisive impact on the large lengthscale plasma regions in which the flux freezing constraint remains valid. This observation illustrates both the subtlety and the significance of the magnetic reconnection process.