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Astron. Astrophys. 363, 1186-1194 (2000)

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6. Conclusion

In a transversally inhomogeneous plasma, Alfvén waves are subject to phase mixing (Heyvaerts & Priest 1983). This phenomenon leads to the linear growth of transversal gradients in Alfvén waves, producing a varying magnetic pressure which is the source of fast magnetosonic waves. Phase mixing of a harmonic Alfvén wave excites fast magnetosonic waves at double the frequency of the Alfvén wave (Nakariakov et al. 1997). This analytical result was confirmed by the numerical simulation of a plane Alfvén wave in a two-dimensional domain. It was shown that the fast magnetosonic wave amplitude saturates and experiences a slow modulation, which is a function of the background Alfvén speed gradient and the frequency of the driven wave.

The fast magnetosonic waves, generated continuously by Alfvén wave phase mixing, propagate across the magnetic field, away from the layer of phase mixing. There is therefore a permanent leakage of energy away from the phase mixing layer. This can cause indirect heating of the plasma through phase mixing when the obliquely propagating fast waves, excited by Alfvén wave phase mixing, are dissipated at some distance from the layer of phase mixing. This mechanism leads to the spreading out of the heated plasma region from the inhomogeneous layer. However, the results in this paper have shown that the nonlinearly generated fast wave component saturates at an amplitude proportional to the square of the Alfvén wave amplitude. Thus for Alfvén waves generated continuously by photospheric motions, which would have a typical amplitude of [FORMULA] in the normalization of this paper, only a small fraction of the Alfvén wave energy is radiated away as fast waves. We thus conclude that the classical model of phase mixing is valid despite its ignoring nonlinear wave coupling. The picture is different for Alfvén waves with a large amplitude, as might be excited by the shocks emanating from flares. With Alfvén amplitudes of 0.1 the fast mode amplitude would saturate at approximately 0.01. Since the phase mixing mechanism requires the Alfvén waves to propagate out until they are about 10 full wavelengths out of phase before significant phase mixing damping occurs, the transfer of energy into fast waves would be a significant sink of energy in this case. Also, the saturation level of the compressive perturbations generated by the transversal gradients in the Alfvén wave ([FORMULA]) is 2 to 3 times higher than the longitudinal compressive components of the Alfvén wave ([FORMULA]). This can strongly affect self-interaction of the Alfvén wave through its generation of compressive perturbations, as discussed in Sect. 2.

The model we have used for phase mixing in this paper is greatly simplified compared to the reality of coronal plumes. We have assumed 2D geometry, a uniform magnetic field, simple density structure and a monochromatic source of Alfvén waves. These are all clear limitations of the model. However, this set of simplifying assumptions is entirely consistent with the great majority of work on phase mixing and thus we were able to directly comment on the applicability of simpler models which ignore nonlinear wave coupling. We have also presented a simplified model as a possible explanation of the results. The saturation of fast waves was then interpreted as destructive interference from incoherent sources. This mechanism is effective to a large extent due to the restrictive geometry of these simulations. If the y coordinate in this work were not ignorable the possibility would exist for constructive interference in other directions. Thus while this work can be definitive in its support for classical phase mixing theories in this simplified geometry a conclusive answer for general frequency drivers and realistic geometries must wait until more detailed study of geometric effects, nonharmonic drivers etc. has been completed.

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© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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