Current redistribution model for flares

The model proposed in M97 is that solar flares occur when two current carrying flux loops interact. Current and flux are transferred between the loops by magnetic reconnection. The energy released is determined by the difference between the magnetic energy stored in the current configuration before and after the flare. It is important to note that the timescale of flares is such that the currents through the photosphere do not have time to change over the course of the flare. This implies that only the photospheric boundary conditions on the magnetic field and the current are fixed by the initial conditions, and that subphotospheric processes play no role during the flare.

The model

  
Figure 1: Sketch of model of current and flux redistribution during a solar flare. The energy released during the flare is identified with the change of magnetic energy associated with the redistribution of currents above the photosphere.

Consider the current geometry shown in Figure 1. Four footpoints are shown on the solar photosphere where current and magnetic flux either emerge or reenter. Before the flare, there are two current carrying loops between these footpoints with some point of intersection at which reconnection occurs. During the flare some of the current, , and some of the flux, is transferred from the initial loops to new loops connecting the footpoints. The energy released through the current transfer shown in Figure 1 is given by
 
where
 
with



and where denotes the mutual inductances of the current carrying loops, and are the self inductances of the current carrying loops. The indexing of the loops is given by , , , and where the footpoints are labeled as in Figure 1. The division of equation (1) into irreducible reconnection (IR) and like-current separation (LCS) terms is discussed further in M97.

Equation ({1) is basis of the model of M97. If is positive, the geometry of the situation is such that a current transfer leads to a net release of energy, and a flare is expected. However, if is negative, a redistribution of currents would lead to a net increase of energy and would need another source of energy to drive it. Such a change could not occur spontaneously, and such configurations should not produce a flare.

There is a maximum allowed current transfer, imposed by the requirement that the photospheric currents not change, equal to the minimum of the currents flowing in the initial loops. (Note that M97 is in error when it says that the maximum current transferred is .) As particular loop structures within active regions may undergo multiple flaring events, the maximum current need not be transferred in a single flare. However, for the configurations investigated here, scales monotonically with the magnitude of the current transfer. Thus, assuming that maximal current transfer does take place allows the geometries which produce flares to be identified, without introducing a parameter representing the fraction of the maximal current which is transferred. Maximal current transfer is assumed throughout unless otherwise stated.

Mutual inductances of the loops

The mutual inductances of the current carrying chromospheric loops are defined by the current distributions within the loops. Clearly some approximation must be made to these to treat flares in any generality. M97 assumed that the loops are approximated as half tori which are aligned vertically with respect to the photosphere. The self-inductance of a loop with major radius and minor radius is given by (Landau and Lifshitz 1960, p. 139)

with

The term 7/4 corresponds to a uniform current profile.

The mutual inductance between two loops has not been solved analytically for a general configuration of the loops. There are, however, a number of limiting cases, and M97 presents an interpolation formula between these known results. Consider two loops of major radii and , with centres separated by , and which are oriented with an angle between them. The interpolation formula derived in M97 is
 
The interpolation formula, equation (7) is used here in the calculation of the energy difference between the pre- and post-flare states.

© Copyright Astronomical Society of Australia 1997