Astron. Astrophys. 353, 741-748 (2000)

5. Discussion

We have shown that dynamics of nonlinear spherical linearly polarized, small amplitude Alfvén waves in a stratified and dissipative plasma of coronal holes is described by spherical scalar Cohen-Kulsrud-Burgers Eq. (19). Analysis of the equation allows us to investigate an interplay of the effects of nonlinearity, stratification and dissipation on the wave dynamics. We found that linearly polarized Alfvén waves of weak amplitude (2-3% of the background Alfvén speed at the base of the corona) and long periods (up to 300 s) are subject to nonlinear steepening and efficient nonlinear dissipation, which is almost independent of the value of the shear viscosity (when ), in the low corona (less than 10 ). These results confirm previous numerical findings (e.g. Ofman & Davila 1997a, 1998; Torkelsson & Boynton 1998) and provide us with a powerful tool for parametric studies of the Alfvén wave dynamics, allowing us to extract the main physical mechanisms responsible for the dynamics.

Domains of applicability of the theory developed can be estimated as follows. The wavelength of the waves considered is smaller than the density scale height H, . Consequently, the theory works for waves with periods less then . For  MK and  km s^-1, the period has to be much less than 65 s. (Practically, according to Tu & Marsch (1995), the criteria can be much weaker, , and, consequently, waves of much longer periods can be described too). Also, full MHD numerical simulations did not show any significant reflection for wave periods shorter than 300 s, so the application of the WKB approximation for waves with these periods is justified.

Eq. (19) obviously does not work when the Alfvén speed is approaching the sound speed . The single wave approximation brakes down at this distance and interaction between Alfvén and sound waves has to be considered in this case. Consequently, equation (19) is applicable in the low- parts of coronal holes only, at distances less than 15-20 solar radii.

The value of the viscosity remains an unknown parameter. According to Braginskii's theory, for the typical coronal hole conditions: the concentration 10⁸ cm^-3, the temperature  MK and the magnetic field 5 G, the dynamic shear viscosity is  g (cm s)^-1, which gives us the kinematic viscosity  cm² s^-1 and . According to our results, this number would lead to extremely sharp gradients in the wave fronts. Nevertheless, the wave evolution and nonlinear dissipation turn out to be nearly independent of the dissipation (see Fig. 2) and the results obtained above with much higher dissipation () may still be valid. We note that, the viscosity and the resistivity can be drastically enhanced by plasma turbulence (Nakariakov et al. 1999).

In this study, we neglected an alternative nonlinear damping process which affects Alfvén waves. This is the decay of the Alfvén waves into another Alfvén wave, traveling in the opposite direction, and a slow magnetoacoustic wave. Slow waves are subject to much stronger dissipation and, consequently, can be an indirect sink for Alfvén wave energy. However, according to Cohen & Dewar (1974), efficiency of such a process is low. Consequently, the process can be neglected.

The results obtained above show that nonlinear dissipation of the Alfvén waves can significantly contribute to heating of the coronal hole plasma and solar wind acceleration at distances less than 10 solar radii. The thermodynamic aspects of these studies will be discussed elsewhere in more detail. Here, we discuss implications of the theory developed for coronal seismology. Propagation of the Alfvén waves outward from the Sun is accompanied by two effects which can be observed: (a) the increase of the wave amplitude, contributing to non-thermal broadening of emission lines by the line-of-sight Doppler broadening, with distance from the Sun, and (b) appearance of the breaking point, corresponding to the maximum wave amplitude (after this point the wave is subject to very efficient nonlinear dissipation). Figs. 8 and 9 show the dependence of the breaking point position upon the wave period. It is seen that for waves with periods less than 400 s and amplitudes at the base of the corona over 25 km s^-1, the breaking point is closer than 10 solar radii to the Sun. Determination of the position of the breaking point by measurement of distance of the strongest non-thermal broadening would provide us with a unique tool for the determination of the unresolved spectrum of the Alfvén waves. Such an information would be of crucial importance for the coronal physics.

Fig. 8. Dependence of the breaking distance of Alfvén waves on the wave period. The curve with triangles corresponds to the amplitude 25 km s-1 near the surface and with diamonds to 35 km s-1. The temperature is 1.4 MK and the Alfvén speed is 1000 km s-1 near the base of the corona.

Fig. 9. Dependence of the maximum amplitude of an Alfvén wave upon the wave period for parameters of Fig. 8.

According to Fig. 8, if all the other parameters of coronal holes are fixed, the breaking distance is determined by the amplitude and the period of the wave. Waves of shorter periods break closer to the base of the corona. Alfvén waves with short (less than 10 s) periods break strongly and are dissipated within 1-3 solar radii.

In addition, we would like to note that our results are applicable not only for the physics of coronal holes, but also for other astrophysical situations, such as the problem of the support of molecular clouds by Alfvén waves. This subject will be discussed elsewhere.

© European Southern Observatory (ESO) 2000

Online publication: December 17, 1999
helpdesk.link@springer.de