## 1. IntroductionIt is well known that there are cosmic magnetic fields in celestial
bodies. The fields observed in galaxies, stars or in planets are
large-scale macromagnetic fields. Besides the large-scale magnetic
field in cosmic bodies, there are small-scale, non-uniform fields
which cannot be resolved by present day instruments. Our sun is an
example. Magnetic structures on the sun, excluding sunspots, are too
small to explore directly (Stenflo 1989). It is shown that spatially
highly intermittent flux fragments can occur all over the sun. There
is now a considerable body of evidence suggesting that all scales of
structure in the solar corona, as well as other objects of
astrophysical interest, are coupled to small-scale processes
associated with intermittent magnetic fields. Microwave spike bursts
observed during solar flares exhibit characteristic properties of fine
temporal structure, narrow bandwidth, high degrees of circular
polarization and high brightness temperature (Slottje 1978; Benz 1986;
Crannel et al. 1988; Gary 1991). It is believed that these properties
are related to small-scale magnetic fields and their interactions.
Mckean et al. (1990, 1989) investigated the source of radio spike
bursts by using particle simulations to model the emission of maser
radiation and its propagation through the solar corona. In their
model, the formation of spike burst fine structure is dependent on the
existence of small-scale magnetic field inhomogeneities in both the
fundamental emission and the second-harmonic absorption layers. On the
basis of the results of their simulation model, the source regions of
the solar corona is located between
with which the non-uniform magnetic field has the characteristic scale
of , where c is the velocity of light
and is the plasma frequency (from
their Fig. 1), which corresponds to
It is well known that magnetohydrodynamic (MHD) and two-fluid equations in so- lar corona follow from taking the first three moments of the Boltzmann equation, together with the macroscopic Maxwell equations; in the limits of large-scale and low-frequency motion the constitutive relations can be derived and the corona may be properly regarded as a well-behaved, multicomponent fluid. However, if the macroscopic length-scales of interest in the problem are not large and the dynamic frequencies in the problem are not low, the choice of appropriate constitutive relations and the concept of a local description of the equations of motion becomes questionable. At present we have studied the genera- tion of localized magnetic flux with the characteristic scale , which is very small compared to the electron-ion mean free path in the corona (Rosner et al. 1986): where is the Coulomb logarithm, n is the density and T is the temperature; in such cases we may not define effectively the concept of a fluid element with the volume V and no rigorous perturbational expansion about the mean behavior will be possible under this circumstance. We must then generally attack the problem as one of kinetic theory rather than of fluid description. Can one describe this very small-scale phenomenon by the use of two-fluid equations, including electron pressure? The answer is negative. For such cases of we could still formally introduce the concept of a fluid element, but the definition would be of little practical service because the particles that constitute the matter of the fluid element would freely stream into and out of the volume as a result of their random velocities. In other words, in these cases, a fluid description fails: we have no way of defining effectively the concept of a fluid element. Thus, it is not appropriate to study the generation of very small magnetic field by magnetohydrodynamic (MHD) and two-fluid equations. On the other hand, we need to consider both high-frequency phenomena and interactions of plasma waves with particles in resonance, for example, the decay or coalescence processes of high-frequency fields and the modifications of particle densities by the low-frequency fields, as shown by Eqs. (14) and (19) in the next section. Hence, for the case of very localized regions during times of rapid changes, at the particle level, the kinetic theory would be required (Benz 1993): the fluid description for these processes is not valid. Indeed, in solar corona, the number of particles within a Debye sphere, is very large where is the Debye length; in other words the so called plasma approximation, required to make Debye shielding work and to justify a statistical treatment, is satisfied. Thus, the classical plasma kinetic theory is an appropriate description of the solar corona (Rosner et al. 1986). However, the processes involving rapid changes, small-scale and wave-particle interactions in resonance may be at the heart of some of the classical, unsolved coronal phenomena (Benz 1993). In addition, according to Mckean et al. (1989, 1990), the localized flux is attributed to a flare current system associated with the impulsive phase, i.e., to a neutral current sheet, where the background magnetic field is approximately zero with Maxwellian plasmas. And it has been shown analytically that the growth rate for linear instability of the self-generated magnetic field is much larger than the electron-ion collision frequency in hot and dilute corona (Li & Zhang 1996), implying that collisionless processes have turned out to control the situation. In summary, therefore, it is relevant to use the Vlasov equation and Maxwell equations, with vanishingly small background magnetic fields, to depict the wave-particle and wave-wave interactions of interest in the problem in the active region of corona in the next section. On the basis of the nonlinear equations derived by Li & Ma (1993), we try to examine numerically the self-generated magnetic fields with relevant initial values, resulting in the localized magnetic structures in coronal active regions in next sections. © European Southern Observatory (ESO) 2000 Online publication: January 29, 2001 |