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

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8. Discussion

The results obtained in the present paper show that the process of phase-mixing, which is inevitable in the non-uniform solar corona, can switch on the parametric decay of AWs into a spectrum of secondary AWs. As a result, the waves generated at the base of a coronal magnetic structure undergo a transition from laminar to nonlinear (or even turbulent) propagation at a height [FORMULA], determined by the plasma and wave parameters. Consequently, the whole picture of the AW dynamics in a coronal plasma may be considerably modified, and not only at heights [FORMULA], but also at [FORMULA], because a part of the wave flux can reverse its direction of propagation if the time of parametric decay into counterstreaming AWs is shorter than the time of wave dissipation. The energy transferred to the inward propagating AWs will not directly contribute to accelerating the solar wind; but it can again couple nonlinearly to upward propagating waves, increasing the transverse wavenumbers.

We found that the collisional wave dissipation of the phase-mixed AW, represented by the Ohmic dissipation of the parallel wave current, is much weaker than the nonlinear damping due to parametric decay. The linear damping rate due to the ion-ion collisions (shear ion viscosity) can in some cases be comparable to Ohmic dissipation (Hasegawa & Chen 1976), but can hardly exceed it. Under coronal hole conditions [FORMULA].

In Sect. 4 our results have been applied to the phase-mixed Alfvén waves in a coronal hole. The influence of vertical inhomogeneity in coronal holes on the AW phase mixing has been investigated in great detail in the linear approximation (Ruderman et al. 1998; De Moortel et al. 1999). The nonlinear generation of fast MHD waves by phase-mixed AWs has been considered by Nakariakov et al. (1997), but this process seems to be less effective for waves with amplitudes [FORMULA], as suggested by observations. It is therefore important to check if a nonlinear process can influence phase mixed AWs in the corona, and in coronal holes in particular.

The length-scale of the field-aligned inhomogeneity in a coronal hole is primarily determined by the characteristic scale of the density variation with height, [FORMULA] [FORMULA] km. The transverse inhomogeneity length-scale is uncertain. If [FORMULA] is determined by the visible plume sizes, then one can take [FORMULA] km. However, it is quite possible that the corona is much more structured in the horizontal direction, up to [FORMULA] km (Woo 1996).

The wave frequency is also not an accurately known parameter in the problem. From the point of view of the power present in the photospheric motions, and in the magnetic fields of the chromosphere and corona, it can range from [FORMULA] s-1, for the waves excited by granular motions, up to [FORMULA] s-1, for waves excited by the small-scale magnetic activity in the chromospheric network (Axford & McKenzie 1992). Persistent nonthermal plasma motions in the corona have been confidently detected by means of spectroscopic observations, (Saba & Strong 1991; Tu et al. 1998; Hara & Ichimoto 1999, and references therein). Wave amplitudes estimated from these data range from 0.01 to 0.03 for [FORMULA], and energy requirements can be satisfied.

However, because of the lack of observational evidence that the low-frequency global modes supply the energy for the high-temperature corona, one can suppose that either the wave periods fall below the available time resolution, [FORMULA] s [FORMULA], or the wave coherence lengths can not yet be spatially resolved, i.e., at least, wavelength [FORMULA] m. This last condition in turn maps onto the frequency scale as [FORMULA] for Alfvén waves if [FORMULA] is the parallel wavelength. In principle, Alfvén waves with [FORMULA] m can be highly localized in planes perpendicular to [FORMULA], which makes them unresolved. But the direct footpoint excitation of short-scale waves requires too much power in the photosphere concentrated at small scales, and any alternative process involving an evolutional structurization in the corona should be observed at the initial large-scale stage if [FORMULA] s-1.

In the closed magnetic configurations, like coronal loops, the random global photospheric motions can excite short-scale standing AWs localized around resonant field lines, where the Alfvén travel time is equal to the photospheric time-scale at which the photospheric motions contain enough energy (De Groof et al. 1998). The problem here is that the setup time seems to be long compared to [FORMULA]-minute variations of the fine-scale chromospheric activity, repulsing the dynamics of the coronal heating process (Berger et al. 1999).

The above consideration suggests that the waves responsible for coronal heating have frequencies [FORMULA], or even higher (Tu et al. 1998; Wilhelm et al. 1998). Moreover, this suggestion can help to circumvent some difficulties with in situ spacecraft observations of waves in the solar wind. Using the [FORMULA] spectra derived from the Helios (and later Ulysses) observations, Roberts (1989) has shown that the wave energy contained in those spectra, is not sufficient to accelerate the solar wind when extrapolated to the corona. This difficulty can be circumvented by a spectrum of AWs with higher frequencies, which can be effectively dissipated by cyclotron damping when the waves reach the height where the cyclotron frequency equals the wave frequency (Axford & McKenzie 1992; Tu et al. 1998).

At the same time, small perpendicular length scales do often develop; they are important for AWs and open various alternative dissipation channels (Matthaeus et al. 1999; Hollweg 1999). The structurization can be very efficient for the high-frequency AWs in a nonuniform coronal plasma because they undergo strong phase mixing and quickly develop strong transverse gradients that accelerate the nonlinear interaction. Nonlinear decay initially spreads the input wave spectrum out over short wavelengths across the magnetic field as shown on Figs. 2 and 3, and over long wavelength in parallel direction (not shown). With sufficient driver power, a further growth of the secondary waves gives rise to the nonlinear coupling among them resulting in the anisotropic turbulent cascade.

Another possible scenario is that anisotropic turbulent cascade is excited by the parametric instability of parallel-propagating AW (Ghosh & Goldstein 1994). Note that our approach differs from that in the papers (Viñas & Goldstein 1991; Ghosh et al. 1993; Ghosh & Goldstein 1994) in both the physical model and the origin of nonlinearity involved. While it is quite natural to take the parallel dispersion for the parallel-propagating waves into account (Viñas & Goldstein 1991), it is equally natural that the perpendicular dispersion is important for the highly oblique Alfvén wave. The planar model used in the above papers restricts all waves to propagate in the [FORMULA] plane. Since our leading nonlinearity contains the vector product of perpendicular wave vectors, the [FORMULA] D simulations reduce it to zero and hence can not account for it. Also, the use of one-fluid or reduced two-fluid equations rules out important nonlinear couplings of oblique AWs. This happens, in particular, when the nonlinear terms due to electron and ion electric drifts are assumed to be cancelled by each other.

It is not certain presently if the nonlinear interaction of AWs alone can reproduce the one-dimensional power [FORMULA] spectra with most (80%) fluctuation energy hidden in the small-scale transverse spectra as suggested by observations of the solar wind (Matthaeus et al. 1999and references therein). The analysis of possible causes of the appearance of anisotropic spectra suggests (Ghosh et al. 1998) that their most probable source is in the solar corona, where the solar wind originates. The anisotropic spectra can indeed be developed in the corona by the wave-wave interactions that we study, but with an even flatter parallel power spectral index [FORMULA] (the perpendicular power index is -2) (Voitenko 1998b). Such a flat spectrum can steepen towards an index -1 by the Landau damping which has a [FORMULA] dependence on the parallel wave number, but exact final spectra can be obtained via numerical simulations.

The dynamics of the spectrum of parallel-propagating AWs has been discussed by Agim et al. (1995) for waves generated by a proton beam. On the contrary, our input phase mixed spectra are essentially anisotropic. For example, if the initial one-dimensional spectrum of AWs is [FORMULA] at the base of corona, the two-dimensional spectrum created by phase mixing is [FORMULA], x is in the direction of inhomogeneity. This spectrum is restricted to the ([FORMULA]) plane and is thus parametrically unstable in the sense we are discussing, exciting waves with [FORMULA]. This is a feature of the vector nonlinearity ([FORMULA]), involving waves propagating in the different directions across [FORMULA].

As we restricted our analysis to the wavelengths [FORMULA] shorter than the vertical inhomogeneity length-scale, [FORMULA] km, [FORMULA]s-1. Also, the competition of the parametric decay into counterstreaming waves with collisional dissipation may be reduced in the low-frequency range, [FORMULA] s-1, where this decay channel is prohibited by the field-aligned plasma inhomogeneity and decay into parallel-propagating waves comes into play (see Voitenko & Goossens 2000). Although, as we discussed above, this low-frequency range tends to be excluded from the observational point of view, the corresponding linear and nonlinear wave damping is considered by Voitenko & Goossens (2000).

Here we neglected kinetic damping effects in comparison to collisional dissipation. To describe the dissipation of high frequency AWs, one has, in principle, to use the kinetic plasma model, accounting for the Landau damping. However, noting that the two-fluid MHD dispersion of AWs is close to the exact kinetic dispersion in the whole range of perpendicular wavenumbers and frequencies, one can use the two-fluid MHD model with the semi-empirically included Landau damping for the high-frequency AWs also. Moreover, the Landau damping itself may be reduced by the quasilinear relaxation of the velocity-space distribution, in which case the competition of the damping due to collisions and decay into counterstreaming waves can extend into high-frequency region, up to [FORMULA] s-1.

The complicated interplay of the different damping mechanisms in the different frequency ranges is studied by Voitenko & Goossens (2000).

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
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