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

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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:

[EQUATION]

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 [FORMULA]:

[EQUATION]

where [FORMULA] is determined by the relation

[EQUATION]

and the nonlinear damping height is given by (64). The collisional damping height [FORMULA] is determined by (41), which we rewrite in the form

[EQUATION]

In a coronal hole it can be estimated as [FORMULA], which corresponds to the transverse wavelengths [FORMULA] m.

We see that the characteristic height of the wave damping, [FORMULA], can be very different from that predicted by resistive MHD, [FORMULA], if [FORMULA], which gives the condition for the nonlinearly-dominated phase-mixing:

[EQUATION]

The corresponding threshold amplitude [FORMULA] 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 [FORMULA]-space.

If condition (69) is satisfied, the initial Alfvén waves, excited at the base of the solar corona, are damped at heights [FORMULA]. For typical coronal hole conditions discussed above, we obtain amplitudes of AWs, leading to the nonlinearly-dominated regime of phase-mixing:

[EQUATION]

If the nonthermal broadening known from the spectral observations is due to AWs, then the wave amplitudes [FORMULA], and condition (70 ) is well satisfied. Then the damping height in this nonlinearly-dominated regime [FORMULA].

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).

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
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