7. Collisional dissipation versus parametric decay into counterstreaming waves
To 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 :
In a coronal hole it can be estimated as , which corresponds to the transverse wavelengths m.
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