6. Model for prominence formation
Recently, quadrupolar configurations made of point sources have been used to model qualitatively the formation of solar prominences (Priest et al. 1996). In this section the same problem is addressed using the infinitely long, wide sources introduced before. One starts with a photospheric quadrupole made of two sources of strength and placed at and , respectively, flanking a second pair of sources of strength and positioned at and (), respectively. All four sources have the same half-width, L, and a background horizontal field is assumed. Next, the two magnetic elements at are allowed to approach one another so that a null point first forms on the photosphere and then raises into the corona (see Fig. 6). During this process the field line that first reconnects on the photosphere moves upwards and drags photospheric mass and magnetic flux to prominence heights. According to Priest et al. (1996), when the two inner sources overlap (), their magnetic flux cancels and the cold plasma captured from the photosphere remains in equilibrium at a typical height of some Mm.
Now, we want to assess the influence of the width of the photospheric sources on the process of flux and mass capture to form a quiescent prominence. Following the procedure outlined in the previous section it is simple to derive the expression for the total flux function,
Because of the inherent symmetry in this quadrupolar structure, both the centre of the field line that first reconnects and the null point that forms in the corona lie at . The height of the X-point, c, follows after solving the equation with given by Eq. (22) (see Fig. 7). Just like in the case of cancelling magnetic features, studied before, the results for coincide with those obtained using infinitely thin sources and are qualitatively similar to those for point sources (Priest et al. 1996). In addition, as the width of the elements is increased, the X-point first appears for wider separations of the inner sources, it does not rise so much as for and reaches down the photosphere before the flux cancellation is complete.
On the other hand, the evolution of the height, h, of the line that first reconnects and then rises when the inner sources approach one another can also be studied. The condition to compute h is that the magnetic flux per unit length in the y-direction, , vanishes up to this height, i.e.
which using Eq. reduces to
with given by Eq. (21). Fig. 8 shows that this magnetic field line starts rising when the sources are far apart, achieves its maximum height for and then goes down, reaching before the flux in the two inner sources disappears. In fact, for the magnetic flux of these two sources has barely started to cancel and already . Therefore, the plasma that rises to form a prominence cannot be maintained above the photosphere unless the flux cancellation process is stopped, because the field line that drags this plasma into the corona inevitably falls down towards the photosphere, as well as the mass supported by it. In the limit of infinitely thin sources the behaviour is similar to the curve for Mm in Fig. 8. As the inner sources approach h continuously increases and takes its largest value as , but when the sources overlap () the field line that first reconnects suddenly drops to the photosphere. This behaviour is a consequence of the condition , since when the inner sources cancel the coronal X-point disappears and the flux above the photosphere at has only one sign, so that can only be satisfied for . Consequently, it turns out that full flux cancellation with destroys the formed prominence, so its existence should be based on a continuous emergence and cancellation of magnetic flux, in order to maintain the supply of mass and flux. It must also be mentioned that the finite width of the sources represents another problem in this model for prominence formation. For example, if the two inner fragments are Mm wide, the cold material rises at a maximum height which is about 20% that attained for ( Mm compared to Mm, see Fig. 8). On the other hand, the basic trends found here agree with the results obtained by Démoulin & Priest (1993) for an inverse polarity prominence in a quadrupolar region. In such model, they already showed that to obtain low height dips one needs a corridor nearly free of magnetic field, i.e. large a and L not too big.
© European Southern Observatory (ESO) 1999
Online publication: November 3, 1999