## 5. DiscussionWe 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 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 10 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
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 |