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Astron. Astrophys. 337, L9-L12 (1998)

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3. Relaxation along the minimum state: the source of "nanoflares"

The dynamics of solar magnetic loops is defined by the rate of energy input and energy loss. The velocity of photospheric motions V, twisting flux tubes inside a loop, is relatively small (1 km/s) compared to the Alfvén speed (100 km/s or higher in the low corona). For twisting a thin flux tube of radius [FORMULA] however, the characteristic time [FORMULA] can be comparable with the characteristic magnetic relaxation time [FORMULA]. The relaxation typically involves reconnection which proceeds at the rate of tens or more Alfvén times.

Consider the dynamics of a loop (braid) in the minimum state (Fig. 2). If the crossing number of the loop structure is high enough then internal reconnections will move the configuration down the minimum curve releasing the energy in small portions. According to estimates by Berger (1994) for a three string braid, the reconnections become effective when the ratio between the transverse and axial components of the field in the braid, [FORMULA], exceeds 0.3, which corresponds to about a 30 degree angle between the directions of neighboring field lines. If the transverse field is smaller there is enough time between reconnections for random convective motions to entangle the braid, thus increasing the crossing number and moving the structure up along the minimum state. This leads to a stationary situation in which the rate of magnetic energy input through the random motions is balanced by the rate of energy release through the reconnections. The energy per unit time released in a loop due to moving along the minimum state can be estimated from Eq. (1): [FORMULA]. Due to convective motions of a random-walk type (with root-mean-square velocity V) the flux tubes become more entangled. If the stepsize [FORMULA] is large enough, e.g. [FORMULA], then the entanglement prevents cancellation between subsequent steps going in opposite directions. Thus the rate of change of [FORMULA], and hence µ and K, will be proportional to V. From Eq. (2), [FORMULA], where [FORMULA] measures the efficiency of braiding due to the random motions. Berger (1994) found [FORMULA] for a three-string braid. The value of µ at which energy loss through reconnection balances energy supply from the twisting will be assumed to be 0.3. If N is the total number of such loops on the solar surface then the power per unit area for the whole Sun is

[EQUATION]

Note that it does not depend on n. To evaluate the power (3) we identify the braids with closed loops evolving in the solar magnetic network. The network magnetic field is mostly open, the only polarity mixing (closed loops) is provided by small bipoles of typical flux [FORMULA] Mx called ephemeral regions. Ephemeral regions have been extensively studied by K. Harvey (Ph.D. thesis Utrecht Univ., 1993) and others whose ground-based results are summarized and extended by the new SOHO/MDI results in a recent paper by Schrijver et al. 1997. The emergence rate of ephemeral regions estimated from the ground observations is about [FORMULA] per day on the entire Sun. The rate estimated from the MDI magnetograms is about 10 times higher due to the improved spatial resolution. The mean life-time of the ephemeral regions is 4.4 hours (the dispersion is high, there are ephemeral regions living 12 hours). Hence about [FORMULA] of these regions, which we identify with our braided loops are present on the Sun at any time. An ephemeral region is composed of many unresolved flux tubes. We conservatively estimate the radius associated with a loop to be about the width of the network, [FORMULA] km. The speed of motions in the network is about 1 km/s. Substituting these numbers into Eq. (3) we obtain [FORMULA]. This power is released in small portions, intermittently, and may be associated with "nanoflares" envisaged by Parker (1990). Although this power is insufficient to heat the whole corona it can be important source of heating of the lower part of the corona, see the next section, and can explain the phenomenon of "blinkers" recently observed by SOHO/CDS (see http://solg2.bnsc.rl.ac.uk/cds/main.html).

[FIGURE] Fig. 2. A magnetic loop emerging with arbitrary values of [FORMULA] and [FORMULA] evolves fast (in a reconnection time) to the minimum state defined by the magnetic energy and crossing number. Reconnections destroy crossings and thus move the configuration down along the minimum state curve. Random photospheric motions increase topological complexity and thus move the configuration up along the curve.

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998
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