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Astron. Astrophys. 357, 1073-1085 (2000)

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1. Introduction

The physical mechanisms responsible for the heating of the solar corona still lack unambiguous identification. Resonant absorption and phase mixing (see Goossens 1994 and references therein) are two popular theories of Alfvén wave heating which involve spectral energy transfer towards small length-scales. Both are due to the transverse plasma inhomogeneity, typical for the corona. Phase mixing of Alfvén waves which leads to enhanced wave dissipation and consequent plasma heating, has been first proposed as a coronal heating mechanism by Heyvaerts & Priest (1983). Since then, phase mixing has been studied both for open and closed magnetic configurations (see recent papers by Ofman & Davila 1995; Hood et al. 1997a,b; Ruderman et al. 1998; De Moortel et al. 1999). The first conclusions on the asymptotic behaviour of the wave amplitudes with height, [FORMULA], or time, [FORMULA], have been recovered for the typical coronal conditions, when the length scale of the plasma inhomogeneity along the magnetic field lines, [FORMULA], is very large compared to that of the plasma inhomogeneity transverse to the magnetic field lines, [FORMULA], [FORMULA] (e.g., Hood et al. 1997a). The only notable deviation from this behaviour was found by Ruderman et al. (1998), who presumed ad hoc that the scale height of wave damping is comparable to the vertical inhomogeneity scale.

Phase mixing of AWs induced by plasma flows has been studied by Ryutova & Habbal (1995). However, their results only apply to sub-Alfvénic flows, [FORMULA], as has been noted by Ruderman et al. (1999). Ruderman et al. (1999) relaxed the restriction of sub-Alfvénic flows and their investigation shows a good agreement between the linear analytical solutions and the nonlinear numerical simulations in the model of one-fluid nonlinear resistive MHD.

These results were obtained in the one-fluid MHD approximation. However, with the creation of short transverse length-scales in Alfvén waves, the Alfvén waves become essentially two-dimensional in the sense that they have long-wavelengths along the magnetic field and short-wavelengths across it. In this situation the ion polarization drift in the perpendicular direction creates a charge separation across [FORMULA], while field-aligned electron flows tend to cancel this charge separation, and thus the motions of the ions and electrons decouple from other. The finite temperature and/or electron inertia effects prevent complete charge cancellation, and a longitudinal wave electric field [FORMULA] arises. The presence of the longitudinal wave electric field, [FORMULA], and current, [FORMULA], bring about many new properties for Alfvén waves, including the effective interaction of waves with plasma particles (either via kinetic effects at Cherenkov resonance, or via collisions), as well as the nonlinear interaction with other modes and among Alfvén waves themselves. Since an adequate description of these effects requires kinetic plasma theory, these short-scale Alfvén waves are called kinetic Alfvén waves (KAWs).

Linear and nonlinear properties of KAWs have been extensively studied in the astrophysical and geophysical context. Examples are plasma heating (de Azevedo et al. 1994; Voitenko 1994), and current drive (Elfimov et al. 1996) in coronal loops in the solar atmosphere, instability of the interstellar plasma (Shukla et al. 1989), particles acceleration in galactic radio jets (Bodo & Ferrari 1982), impulsive energy release in solar flares (Voitenko 1996a,b, 1998c), wave instability (Voitenko et al. 1990; Hasegawa & Chen 1992) and plasma turbulence (Voitenko 1996c) in the Earth's magnetosphere, anomalous magnetic diffusion in coronal current layers (Voitenko 1995). The nonlinear properties of KAWs are of great importance (Shukla & Stenflo 1995). In fusion devices, the efficiency of the energy exchange between waves and plasma particles caused by kinetic properties of KAWs has been proven recently both by theory and by experiment (Jaun et al. 1997).

It is thus surprising, that almost no attention has been paid to these properties of the KAWs created by the phase-mixing process in the non-uniform corona. Indeed, there are several papers studying the creation of short transverse length-scales in Alfvén waves and the eventual wave dissipation. There are also a few papers which concentrate on nonlinear effects in phase mixing (Nakariakov et al. 1997), and resonant absorption (Poedts & Goedbloed 1997). Ofman & Davila (1997) studied both the nonlinear excitation of sound waves and plasma heating by AWs in a nonuniform plasma in coronal holes. But these investigations have been carried out in the one-fluid MHD approximation, missing important properties of Alfvén waves with large transverse gradients developed by phase mixing.

The lack of interest in KAW in solar physics is due to an argument based on two characteristic transverse length scales, namely the Alfvén wave dissipation length scale [FORMULA] and the proton gyroradius [FORMULA], with [FORMULA] being much longer than [FORMULA]. The fact that [FORMULA] so that Alfvén waves are damped long before length scales of the order of [FORMULA] are set up by the phase mixed Alfvén waves leads to the intuitive conclusion that non-zero proton gyroradius effects are unimportant for Alfvén waves. However, this intuitive conclusion can be erroneous.

In the present article we shall show that the finite (ion) Larmor radius (FLR) effects in Alfvén waves can come into play at length scales which are much longer than both [FORMULA] and [FORMULA]. In this paper we study the parametric decay of a phase-mixed AW into two daughter AWs, induced by the combined action of the finite wave amplitude and FLR effects. This is an important nonlinear process which significantly modifies the wave spectral dynamics of phase mixing. So we are not concerned with the mechanisms that create phase mixed Alfvén waves. Instead, we try to find out how the nonlinearity induced by phase mixing affects the wave spectral dynamics.

A significant input of energy into a plasma, as observed in the solar corona, can only be achieved if the launched waves have sufficiently large amplitudes. E. g., AWs can balance the energy loss from the loop structures of active regions and from coronal holes, if the wave magnetic field amounts to 1-5% of the background magnetic field [FORMULA]. Nonlinear effects become important for waves of such amplitudes, and the ability of the waves to participate in different kinds of nonlinear interaction have to be examined.

It is well known that ideal MHD Alfvén waves cannot interact among themselves. Nonlinear coupling of AWs with other waves has been studied for a parallel-propagating pump wave (Galeev & Sagdeev 1979; Goldstein 1978; Viñas & Goldstein 1991; Jayanti & Hollweg 1993; Ghosh et al. 1993). In particular, the parametric decay into oblique waves was studied analytically (Viñas & Goldstein 1991) and numerically (Ghosh et al. 1993), and the consequent development of anisotropic turbulent cascade has been observed in numerical simulations (Ghosh & Goldstein 1994). In the Discussion we shall go into the details of how our approach differs from that used in the papers cited above. Here it suffices to note that we start with spectral anisotropy (due to phase mixing) which then initiates the nonlinear spectral dynamics. The papers cited above consider the reverse sequence.

We show that the influence of the three-wave nonlinear interaction becomes important with the creation of small transverse length-scales in the phase-mixed Alfvén waves, and can significantly change the picture of the phase mixing in the corona. In order to describe this process, we have to take into account that the motions of the electrons and the ions in AWs become decoupled from each another at small length-scales, and that the wave dynamics can be adequately described by kinetic theory. The nonlinear kinetic theory of KAWs, including nonlinear three-wave resonant interaction and linear Cherenkov wave-particle resonant interactions, has been developed by Voitenko (1998a). The collisional effects can be self-consistently taken into account in this theory by including collisional integrals into the Vlasov equations.

However, when we consider the low-frequency part of the Alfvén wave spectrum in the solar corona, [FORMULA]s-1, we note that irrespectively of the KAW wavelengths, the linear kinetic damping is weaker than the collisional dissipation in this frequency range (Voitenko & Goossens 2000). Neglecting kinetic wave-particle interaction, some properties of these waves, such as thermal wave dispersion, collisional dissipation, and nonlinear three-wave interaction, can be described in the framework of the much simpler two-fluid MHD model. As is pointed out in the Discussion, the high-frequency AWs with [FORMULA] s-1, which can be excited by the chromospheric magnetic activity (Axford & McKenzie 1992), can be also approximately described by two-fluid MHD with kinetic effects taken from kinetic theory.

The paper is arranged as follows. The basic equations are presented in Sect. 2. The second-order two-fluid MHD theory of the nonlinear plasma motions in short-scale Alfvén waves is developed, and the Alfvén wave eigenmode equation is derived in Sect. 3. The collisional (linear) damping of short-scale Alfvén waves results in the same asymptotic behaviour of the wave amplitude as was found by Heyvaerts & Priest (1983) (Sect. 4). The parametric decay of a pump oblique AW into two daughter oblique AWs is considered in Sect. 5, and the results are applied to a phase-mixed AW propagating upward in the solar corona in Sects. 6 and 7. We discuss possible consequences of this process in the solar corona in Sect. 8 and present our conclusions in the last section, Sect. 9.

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000