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Astron. Astrophys. 364, 785-792 (2000)

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1. Introduction

It 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 [FORMULA] with which the non-uniform magnetic field has the characteristic scale of [FORMULA], where c is the velocity of light and [FORMULA] is the plasma frequency (from their Fig. 1), which corresponds to [FORMULA] km in coronal active regions(here [FORMULA] [FORMULA]). The non-uniform magnetic field is chosen so that [FORMULA] is about 0.4 - 0.2 ([FORMULA] is the electron gyrofrequency) or [FORMULA] in coronal active regions. Obviously, however, the mechanism of convective instability (Parker 1978; Spruit 1979) is not at work.

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 [FORMULA], which is very small compared to the electron-ion mean free path in the corona (Rosner et al. 1986):


where [FORMULA] 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 [FORMULA] [FORMULA] 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, [FORMULA] is very large


where [FORMULA] 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 [FORMULA] 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.

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001