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Astron. Astrophys. 339, 215-224 (1998)

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6. Conclusions

The main aim of this paper is to show that in low [FORMULA] solar plasmas even a slow bulk flow, can have a strong effect on resonant MHD wave absorption and reflection from localized inhomogeneous layers. Numerical calculations were performed for two values of the plasma [FORMULA] ([FORMULA] and [FORMULA] relevant to the lower and the higher parts of the solar atmosphere, respectively) and for two values of the temperature ratio [FORMULA] and [FORMULA].

The efficiency of absorption of slow MHD waves is the same for the slow and the Alfvén resonance. The fast MHD waves are absorbed much more efficiently at the Alfvén resonance than at the cusp resonance in our models. Additionally, the flow has a strong influence on the variation of the absorption coefficient for slow MHD waves. The variation for the fast MHD waves is much smoother. Finally, in the four cases considered here the peak values are much higher for the slow MHD waves than for the fast MHD waves.

Our results can be applied to processes in the solar atmosphere in a broader sense and with certain restrictions, as commonly done in literature. This means that the physical properties of regions 1 and 2 may in fact apply to local domains of at least several wavelengths on both sides of the nonuniform layer and not necessarily to the whole atmosphere.

Thus, regions 1 and 2 can, in principle, be considered uniform only locally if so needed in applications. Uniformity of regions 1 and 2 is therefore not required on length scales much larger than the wavelength [FORMULA]. The same holds for the uniform flow meaning that the flow is present throughout the transitional layer but it may be of only local extent in the region 1 where the medium is considered uniform.

In addition, as the effects of the spherical shape of the solar atmosphere and the gravity stratification are not taken into account in our study, the wavelength [FORMULA] of the plane waves should be much smaller than the solar radius and the typical stratification scale length.

We turn now to applications of our results to solar conditions. Let us take a local inhomogenity in the high chromosphere, which is embraced by two uniform media with temperatures [FORMULA] and [FORMULA] (for example the lower parts of an arcade, where the magnetic field lines are approximately perpendicular to the solar surface). Model IIa is suitable for this case, since [FORMULA] is a proper value for the plasma beta at this height and the ratio of the temperatures is [FORMULA].

These temperatures correspond to sound speeds of [FORMULA] km s-1 and [FORMULA] km s-1 respectively, which yield [FORMULA] 493 km s-1 and [FORMULA] 350 km s-1 in the two regions. These values fall in the interval (102-104) km s-1 typical for the Alfvén speed in the higher solar atmosphere (Priest, 1985).

The speed of flow [FORMULA] is scaled to [FORMULA] and the dimensional values [FORMULA] are then


In solar conditions, realistic values for [FORMULA] are up to [FORMULA] (Durrant, 1988) which corresponds to [FORMULA].

In Figs. 7ab we present zooms made from the central parts of Fig. 5a. Figs. 7a and 7b show the absorption and the over-reflection for an incoming slow wave, respectively. In model IIa, the choice of [FORMULA] gives [FORMULA] as the value of the absorption coefficient for the static case. Fig. 7a shows, that in the realistic flow interval mentioned above, the absorption can slightly increase up to [FORMULA], and it also can decrease to zero due to the variation of the equilibrium flow.

[FIGURE] Fig. 7a and b. Model IIa, slow MHD wave: the dependence of the absorption coefficient on [FORMULA]. [FORMULA], [FORMULA].

It is important to show, that in the presented realistic model the slow magnetosonic waves can easily undergo over-reflection. In Fig. 7b it is shown that a relatively weak flow present in the solar atmosphere is sufficient to increase the energy of the incoming slow wave due to over-reflection. Indeed, a plasma flow in the direction of the magnetic field with speed of 60 km s-1 can cause an 80 [FORMULA] increase of the energy of the incoming slow wave. We would like to point out that the effect of over-reflection can be detected in the solar atmosphere, and it could be a challenging task for the observers community to observe such an interesting phenomenon.

We can now estimate the oscillation period of the wave and the thickness L of the transitional layer. As the dimensional frequency is normalized to the value of [FORMULA], the corresponding oscillation period [FORMULA] is [FORMULA][FORMULA] where [FORMULA] is the dimensionless frequency used in the calculations. If L is given in [km] then [FORMULA][s] in our case. For example, [FORMULA] km gives oscillation periods of [FORMULA]s.

The conclusion is that the plasma flows expected to be present in the solar atmosphere, can significantly affect the resonant absorption of both types of magnetosonic waves in the observed frequency range. The considered resonant processes occurring in nonuniform layers, can contribute to atmospheric heating. Additionally, we showed that under solar conditions the slow magnetosonic waves can gain significant amount of energy from the existing bulk plasma flows. Later on this energy can be deposited by a following dissipative process, giving a rise to the heating of the solar atmosphere.

Finally, the following remark on over-reflection is in order. The increase of the wave amplitude after the reflection from the layer may also be related to eventual flow instabilities induced by the incident wave. Our basic state model allows, in principle, for two types of instabilities, the Kelvin-Helmholtz instability modified by the presence of the nonuniform layer, and resonant flow instabilities when the conditions (29) and (30) are satisfied. These instabilities were considered by Hollweg, Yang, ade & Gakovic (1990) in the limit of incompressible fluid. To fully understand the process of over-reflection it is therefore necessary to investigate stability properties of the considered basic state by solving the related eigenvalue problem.

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© European Southern Observatory (ESO) 1998

Online publication: September 30, 1998