4. Generation of waves by reconnections of braided loops with open field lines
There is another, more sporadic, effect of reconnection: From time to time, randomly, a topologically complex loop meets an open field of opposite direction and reconnects, releasing a train of Alfvén waves up into the corona (Fig. 3, first suggested by Axford and McKenzie 1992). Because these waves arise from a twisted configuration, they are circularly polarized. Since K scales as , the number measures how many times one flux tube inside the loop wiggles about its neighbors. The size of this wiggling can be associated with the minimum wavelength of the emitted waves: . The corresponding upper-bound frequency is defined by the Alfvén speed and the typical radius of the flux tubes braided in the loop of radius r. The lower- bound frequency is about .
At present, the numerical values of these frequencies can only be estimated on model grounds. For example, Marsch and Tu (1997b) use G at km where the density is and G at km where the density is . At both levels it gives the Alfvén speed about km/s. Because the size is below the present spatial resolution of observations (0.2 arcsec or about 150 km), we estimate the minimum of this size by the order of magnitude as a thickness of a skin-layer determined by plasma resistivity, i.e. 1 km, where is the magnetic Reynolds number on the solar surface. With this minimum size and km/s the upper frequency reaches about a thousand Hz required in these models. The lower-bound frequency is about 1 Hz. It is worth noting, however, that even with this extreme value of , these estimates are very crude already because there are no measurements of magnetic field in the solar corona. In view of the importance of these frequency bounds, especially the upper one, a more detailed study is needed.
On the other hand, there is no special need for very high frequency waves damped close to the solar surface. This heat release closest to the surface can instead be provided by the relaxation along the minimum energy state, described in the previous section. The waves with lower frequencies, say Hz, can still be an effective source of heat and momentum higher in the solar atmosphere provided they carry a sufficient energy flux.
The energy flux of the waves depends on the energy content of the closed loop, when it is in the minimum state, and the rate of its energy release contacts with the open configuration. To evaluate the energy flux we identify, as above, the closed loops with small-scale bipoles (ephemeral regions) and assume that the loops are destroyed through reconnections with the open field. Then the minimum rate of contacts with the open field is the rate of emergence of ephemeral regions. However during its life-time, a closed loop can come into contact with open configurations many times acquiring energy due to twisting its footpoints in between. The power released into the waves for entire Sun can be estimated as
where is the rate of contact, is a portion of energy released and the numerical values are used. The estimate (4) reduces to the value used in the model by Marsch and Tu (1997) when . Because the mean life-time of ephemeral regions is 4.4 hours and the mean time of gaining energy through twisting, proportional to , is short, the number of times the energy can be released by a closed loop into the open configuration can be high. For example, if the relaxation to the level of balance between twisting and reconnections on the minimum energy state takes , the loop can come into contact with the open field up to times. In this case, the power is achieved when 10% of contacts occur with about 15% energy released in a single contact. These values are not unreasonable.
The estimated power (4) is increased when we take into account the energy released due to reconnection of open field with newly emerging loops that are not in the minimum energy state. In this case the energy is higher than that, , used in Eq. (4), at least in the first collision. The difference can be estimated as follows. Assume that the magnetic loops within the convection zone are in their minimum state. In fact, it follows from the equipartition argument that the velocity of motions and the Alfvén speed in the convection zone are of the same order. Hence, the balance between increase and decrease of topological complexity (twisting and reconnections), can easily be achieved and sustained. Because the magnetic flux and topological complexity K in Eq. (1) are conserved, the energy of an emerged loop is different from the energy the loop had inside the convection zone due only to the change of the loop size. Let be the loop size beneath the solar surface and L is the size above the surface, as used before. Then, it follows from Eq. (1), . The expansion factor , expected , is unknown, although in principle can be found from MHD models of magnetic field generation, emergence and subsequent expansion the emerged loops into the solar atmosphere. In any case the emerging loop (not in the minimum state) can release times larger power than the loop in the minimum state. After that the loop will relax to the minimum state and the estimate (4) is valid.
We conclude that our estimates, based on the use of the concept of the minimum energy state, support models of coronal heating and acceleration of the solar wind by high-frequency Alfvén waves. The severe requirements on the upper-bound frequency, used in these models, can be relaxed by taking into account the energy released due to reconnections within magnetic loops whose footpoints are twisted by surface convective motions.
© European Southern Observatory (ESO) 1998
Online publication: August 6, 1998