Astron. Astrophys. 351, 733-740 (1999)

5. Model for cancelling magnetic features

We first consider a dipolar magnetic configuration produced by a pair of photospheric magnetic elements of equal extent (), symmetrically placed about the origin () and with opposed flux strength (). This means that, for positive, the magnetic field emerges from the photosphere in the region around and sinks around . Moreover, is simply the distance between the centre of the sources.

A background horizontal magnetic field is added to the dipolar field, so the total flux function is given by (see Eq. [15])

The shape of magnetic field lines will then depend on L, a and , which are related to the linear extent of the photospheric magnetic elements, , their separation, , and the ratio of the background magnetic field to the vertical field at the centre of each of the sources, .

This configuration, with infinitely thin photospheric sources, was used by Priest et al. (1994) to model the process by which two photospheric magnetic elements of opposite polarity come together and disappear (cancelling magnetic features). The model starts with the sources widely separated and slowly approaching one another (pre-interaction phase). At this time there is no X-point above the photosphere (see Fig. 4i in Priest et al. 1994) but when the two elements come at a distance

a null point forms on the photosphere (see their Fig. 4ii). As the sources keep getting closer, this X-point first rises into the corona and then goes down, reaching the photosphere again when the two sources completely annihilate each other (). For source separations, , between and zero the magnetic configuration is that in Fig. 4iv of Priest et al. (1994) or our Fig. 2 (interaction phase). During this stage field lines reconnect at the null point and the energy released by this process results in the formation of an X-ray bright point. The interaction phase overlaps with the cancellation phase, during which the sources flux is slowly eroded until only the excess flux contained in the initially strongest element remains.

Fig. 2. Coronal magnetic field (solid lines) produced by a dipole with Mm in a horizontal background field. Other parameters are Mm and Mm-1. The separatrix (dotted line) marks the boundary between domains containing field lines of different origin. The X-point in the separatrix is a location where magnetic field reconnection can take place.

Because of the symmetry of the dipolar plus background field structure used here, the null point always lies at . To determine its height, , one needs the x-component of the magnetic field (cf. Eqs. [1] and [16]),

with . The height of the null point follows from the equation , i.e.

in which , L and a are free parameters. In the limit of infinitely thin sources this expression reduces to

We start by computing the positions of the magnetic elements for which the null point is on the photosphere, i.e. for which . Following Priest et al. (1994) we call d the particular value of a for which this happens and plot the results in Figs. 3a and 3b. To describe how to interpret these figures, let us consider Fig. 3a and the particular value Mm^-1. First, for the two values Mm and , which mark the beginning and end of the interaction phase, are recovered. Moreover, for Mm the two values of d become increasingly larger with respect to the case. In other words, the interaction phase begins (and magnetic reconnection starts to take place) when the two sources are at a distance larger than and finishes well before the total flux in the two photospheric elements completely cancels (). These are the logical consequences of dealing with extense photospheric magnetic sources.

Fig. 3a and b. Position of sources, , for which the X-point lies on the photosphere. a  d versus L taking as a parameter. b  d versus taking L as a parameter

More surprising, however, is the change of behaviour for Mm. A magnetic null point forms above the photosphere only for a small range of distances between the two elements and for Mm the null point does not form at all. Therefore, if the two sources are "too large" their magnetic flux cancels without the existence of coronal reconnection. Larger values of the ratio lead to similar conclusions, with the formation of the X-point up to a maximum width of photospheric sources, which is inversely proportional to . Fig. 3b shows the same results using the relative strength of the background field to the photospheric flux as the independent parameter. For infinitely thin sources () the interaction phase takes place between source positions and . For finite sources, however, reconnection occurs only when the background field is not "too strong" or the magnetic elements are not "too large".

Another matter of interest is the change of position of the X-point during the interaction phase. The height of the null point, c, can be obtained from Eq. (17) for extense sources or from Eq. (18) for infinitely thin photospheric sources. The results (see Fig. 4a) stress that coronal reconnection, characterised by , is less important as the ratio of the background magnetic field to the flux of each of the elements increases. Rising the background field intensity has the effect of reducing the range of positions for which the X-point exists and of lowering its height. In addition, Fig. 4b indicates that the null point reaches its maximum height for and illustrates the fact that, for , it goes down to the photosphere before the flux of the two elements has totally cancelled. Also, wide, weak sources (i.e. with large values) may result in flux cancellation without energy liberation in the corona.

Fig. 4a and b. Height of the X-point versus half the distance between sources for a a fixed photospheric element half-width ( Mm) and different values of ; and b  a fixed ratio of the background magnetic field strength to the dipole strength ( Mm-1) and different values of L.

The maximum height attained by the X-point, , follows from the condition

From Eq. (18), an analytical expression for can be obtained for ,

A similar formula cannot be derived for , since is known from Eq. (17) in implicit from. One can, however, take the partial derivative with respect to a of this expression and impose condition (19) to obtain

Then, Eqs. (17) and (20) provide and the value of a for which this maximum height of the X-point is attained as functions of the parameters L and . The results, plotted in Fig. 5, reinforce previous conclusions in that larger values of L or tend to bring down the null point and to shorten or even remove the interaction phase, in which the X-ray bright point forms.

Fig. 5. a Maximum height of the X-point versus L taking as a parameter. b  Maximum height of the X-point versus taking L as a parameter.

© European Southern Observatory (ESO) 1999

Online publication: November 3, 1999
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