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

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2. Review of the CA model of Lu and Hamilton

The vector of the magnetic field [FORMULA] is given on a 3-dimensional cubic grid. After assigning random initial values to the magnetic field at each grid site, the evolution is governed by two laws: In the quiet phases the magnetic field evolves according to

[EQUATION]

where [FORMULA] is an asymmetrically valued random source-term acting on a characteristic time scale which is large compared to the one of the instability that will be described in the following. The quantity

[EQUATION]

is used in an instability criterion ([FORMULA] labels the 6 nearest neighbours in the 3D cubic grid): If

[EQUATION]

at position [FORMULA] (for some given threshold [FORMULA]), the evolution changes into a fast burst mode: the source term is not acting anymore, and the magnetic field evolves as

[EQUATION]

for the point where the instability occurs, and

[EQUATION]

for its six nearest neighbours. Inserting these two equations into the definition of [FORMULA] (Eq. 2), it turns out that the latter quantity vanishes after one time step:

[EQUATION]

The energy released during one such burst-event is assumed to be

[EQUATION]

If all instabilities in the grid have been relaxed, then the evolution is again in the slow mode (Eq. 1).

The CA model of LH was the first one applied in the context of solar flares. Later developed CA models basically use the same set-up, just changing slightly the above rules. In the following, we concentrate on the LH model, but we emphasize that our approach can be applied to any CA model developed so far for solar flares.

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© European Southern Observatory (ESO) 1998

Online publication: June 26, 1998
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