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Astron. Astrophys. 335, 1085-1092 (1998)

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5. Discussion: the context of solar flares

We have shown that the CA of LH (Sect. 2) can be interpreted as the solution to a diffusion equation plus a source term. Here now, we will give some examples to demonstrate the benefit gained through this alternative description: the somewhat neutral rules of the CA can be interpreted (or modified) on the basis of our understanding of MHD equations, as related to solar flares.

Generally, flares are considered to be made up by a large number of reconnection events distributed somehow over an active region (Parker 1988; Parker 1989). In MHD, the processes in the active region are described by the induction equation


plus a momentum equation for the evolution of the velocity field [FORMULA] (the currents and the electric field can be considered as secondary quantities). In general then, the evolution of the magnetic field is governed by the convective term (2nd term on the r.h.s. of Eq. 39), since [FORMULA] is very small, mostly. Accidentally, this convective evolution may create small scale structures where [FORMULA] is not small anymore, and the diffusive term dominates the evolution of the magnetic field (1st term on the r.h.s. of Eq. 39). This diffusive regime is characterized by its spatial scale [FORMULA] and its temporal scale [FORMULA]. Both scales are bigger than the respective ones of the current sheet and the reconnection process, they characterize the volume and the time in which the magnetic field has been reconnected and the free magnetic energy has been released (for details see Biskamp 1994, and references therein).

Having the described picture of the flare scenario in mind, we can interpret the CA of LH (Sect. 2), not by considering the CA rules, however, but by looking at the continuous version of the CA model (Sect. 4):

  • The PDE corresponding to the CA of LH (Eq. 33) has two modes, the stable and the unstable one:

    • In the stable mode, Eq. (33) reduces to


      (see Eq. 34). Therewith, it mimics the induction equation (Eq. 39) in the convective regime (i.e. outside a reconnection region, where the diffusive term is negligible), describing in a simplifying way the convective term ([FORMULA]), which actually should reflect the turbulent motion in the active region and plasma inflows from the photosphere, through a simple random function, neglecting thus completely any structures which would be due to organized fluid motions.

    • If the Laplacian [FORMULA] of the magnetic field exceeds a certain threshold (Eq. 34), then Eq. (33) reduces to


      The loading term [FORMULA] can be neglected since its time scale is much slower than the one of the diffusive process (which is one of the assumptions of the CA model). Eq. (41) corresponds to the induction equation in the diffusive regime, i.e. there where [FORMULA] is so large that the convective term can be neglected in Eq. (39). According to Eq. (34), this diffusion is bounded to a region of radius [FORMULA] around the point where the instability criterion is met. Obviously, from the point of view of MHD, [FORMULA] is the length scale of the diffusive region, which is naturally assumed to be bounded. LH assume this [FORMULA] to be the same for every possibly occurring reconnection event, and moreover, they assume all these reconnection events to have the same diffusive time [FORMULA].

  • The amount of released energy during one diffusion (reconnection) event is assumed by LH to be [FORMULA] (Eqs. 7 and 10). This is a rough approximation, which can be put on more physical grounds: From the physical point of view, the released energy is the difference between the initial and the final magnetic energy: [FORMULA], where the volume integral is over the diffusive region [FORMULA]. As stated in Sect. 4, [FORMULA] is (more or less) flattened in the diffusive volume, i.e. [FORMULA], the spatial mean value of the initial field in [FORMULA]. Whence, we get for the released energy


    where we have identified the expression for the variance of the initial [FORMULA] field in [FORMULA]. In discrete language, this turns into


    where the sums extend over the central point and its nearest neighbours (N is the number of nearest neighbours plus one). This expression differs from the one proposed by LH (Eq. 7) in some 'weighting' factors and signs.

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© European Southern Observatory (ESO) 1998

Online publication: June 26, 1998