## 7. Collisional dissipation versus parametric decay into counterstreaming wavesTo find the linear/nonlinear damping of the AW with height, we proceed further using a WKB-ansatz (32) for the AW amplitude. The damping rate includes linear collisional damping and nonlinear damping due to parametric decay: Taking integrals in the exponent we find the same behaviour of the amplitude with height as in the case of collisional damping of the phase-mixed wave, but now with the cumulative dissipation distance : where is determined by the relation and the nonlinear damping height is given by (64). The collisional damping height is determined by (41), which we rewrite in the form In a coronal hole it can be estimated as , which corresponds to the transverse wavelengths m. We see that the characteristic height of the wave damping, , can be very different from that predicted by resistive MHD, , if , which gives the condition for the nonlinearly-dominated phase-mixing: The corresponding threshold amplitude does not depend on the wave parameters and is almost the same as the collisional threshold for the parametric decay itself (60), which is a consequence of the local character of the AW parametric decay in -space. If condition (69) is satisfied, the initial Alfvén waves, excited at the base of the solar corona, are damped at heights . For typical coronal hole conditions discussed above, we obtain amplitudes of AWs, leading to the nonlinearly-dominated regime of phase-mixing: If the nonthermal broadening known from the spectral observations is due to AWs, then the wave amplitudes , and condition (70 ) is well satisfied. Then the damping height in this nonlinearly-dominated regime . The competition of the linear and nonlinear damping mechanisms for the phase-mixed AWs, including parametric decay into both parallel-propagating and counterstreaming waves, collisional dissipation and Landau damping, is studied in more detail in the accompanying paper (Voitenko & Goossens 2000). © European Southern Observatory (ESO) 2000 Online publication: June 5, 2000 |