Abstract. The propagation and the dissipation of small-amplitude Alfvén waves in an equilibrium configuration characterized by three-dimensional inhomogeneities is investigated. Disturbances are supposed to have a typical wavelength smaller than the scale of nonuniformity of the background structure, which allows us to use a WKB expansion technique. The approach we used is similar to that employed by Petkaki et al. (1998), who studied the case of an incompressible plasma. In the present paper a compressible cold plasma is considered, which is more suitable than an incompressible plasma to describe the situation of the solar Corona, where β 1. Small wavelengths allow to decouple Alfvén from magnetosonic fluctuations at the linear level. Considering small Alfvénic wavepackets, the evolution equations for the position, wavevector, and energy are derived. These equations are similar to those obtained by Petkaki et al. (1998), and they have the same form only if the background density ρ (0) is supposed to be constant. Then, the results found by Petkaki et al. (1998) for an incompressible plasma are valid also in a compressible cold plasma, provided that ρ (0) = const.: in particular, in the presence of regions of chaotic fieldlines the wavevector of Alfvénic perturbations exponentially grows and the dissipation time of the wave is t d ∝ log S , S being the Reynolds number. Thus, a fast dissipation is attained even with large values of S , as in the Corona. The equations derived in the present paper are more general than those of Petkaki et al. (1998), since they can be used also with a nonuniform background density. The present model is discussed with reference to the problem of coronal heating.
We study the propagation and dissipation of Alfvénic perturbation in a 3D equilibrium structure within a WKB model. We assume small amplitude and small wavelength of the perturbation. The generation of small scales in the perturbation is related to the property that nearby magnetic lines move apart from each other locally. This property is quantified by means of the Kolmogorov entropy H of magnetic lines. We numerically calculate the distribution of H for a 3D complex force-free equilibrium, which models the magnetic field above a quiet-Sun region, both for nonvanishing current and for a potential field. It is found that H decreases slightly with the altitude due to the decreasing complexity of the field, but it is relatively uniform except for the presence of sharp peaks, where H reaches much higher values than the average. These locations are those where Alfvén waves are preferentially dissipated. By analyzing the magnetic topology at these locations, we find that they correspond to separator lines which are intersections of separatrix surfaces. Then, in a high-Reynolds number plasma, such as in the solar corona, heating due to Alfvén wave dissipation takes place mainly at magnetic separatrices.
The propagation and dissipation of Alfvén waves in a three-dimensional inhomogeneous force-free equilibrium structure is considered. Assuming small wavelengths the wave propagation is described within the framework of a WKB theory. The equilibrium structure is a complex magnetic field, resulting from a superposition of several harmonics, which represents a model of the coronal magnetic field above a quiet-Sun region. The properties of wave dissipation are related to the topological complexity of the background magnetic field, and the typical scale law of fast dissipation is recovered. A partial reflection of waves is included in the model by calculating reflection and transmission coefficients of both Alfvén and magnetosonic waves. An energy balance is derived between the wave energy dissipated inside the coronal structure and the energy lost through the lower boundary. We find that even for large values of the Reynolds number, a non-negligible fraction of the Alfvén wave energy is dissipated inside the corona.
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