Astron. Astrophys. 362, 1151-1157 (2000)

5. Conclusions

We present here a theoretical model for slow magnetoacoustic waves propagating along magnetic field lines in coronal loops. The model incorporates the effects of dissipation due to finite viscosity and thermal conduction, gravitational stratification and nonlinearity. It is shown that, in the short wave length limit, the evolution of the waves is described by evolutionary equation (27). This equation is of Burgers type, but with an additional "geometrical" term and with coefficients dependent of the evolutionary coordinate. Investigation of solutions of the evolutionary equation shows that the propagating compressive disturbances observed in coronal loops can be confidently interpreted as slow magnetoacoustic waves. This interpretation meets all observationally detected properties of the propagating disturbances.

The theory confirms that there can be longitudinally propagating slow magnetoacoustic waves, perturbing the density of the plasma in the loop. The speed of these waves is to be registered below (taking into account possible projection effects) the sound speed. The waves of observed periods (5-15 min) are strongly affected by the gravitational stratification and dissipation. For the estimated dissipation of the waves (for typical coronal loop conditions, dimensionless coefficient of dissipation (29) is greater than ), the typical scenario of the upwardly propagating wave evolution is the following: initially, the relative amplitude of the waves grows with height and reach a maximum somewhere near the loop apex, and then quickly dissipate. The fact that there are no downwardly propagating waves confirms this interpretation. Indeed, the waves of observed periods dissipate in the upper parts of the loops and in the descending stage, their amplitude is below the noise level. Waves of longer periods (10-15 min) are less affected by the dissipation, while the short period waves (5 min) are practically evanescent, because the dissipation length becomes comparable with the wavelength. In addition, the effective dissipation of the short period waves in the loops can be responsible for the absence of the short time periodicities in intensity variations on an active region, reported by Ireland et al. (1999).

According to our findings, the nonlinearity does not play an important role in the dynamics and dissipation of the waves. Indeed, only waves with initial amplitudes higher than 8-10% can be significantly distorted by the nonlinear generation of higher harmonics. Waves of lower amplitudes (e.g. of observed 1-2% in density) keep their initial shape. This is supported by the observations: Fig. 11 of Berghmans & Clette (1999) and Fig. 2 of De Moortel et al. (2000) do not show any signs of the wave distortion.

De Moortel et al. (2000) have deduced that the observed energy of the waves is insufficient for heating of coronal loops. However, the waves can be used as a tool for MHD coronal seismology. Indeed, combining the observationally measured properties of the waves with theoretical models, we can determine additional parameters of the coronal plasma (cf. Nakariakov at al. 1999). For example, accurate measurement of the wave amplitude as a function of the vertical coordinate and comparing this with the theoretical dependences (see Fig. 4), we can estimate the dissipative coefficient , connected with the coronal viscosity and thermal conduction (29).

Fig. 4. Evolution of amplitude of slow magnetoacoustic waves with the initial amplitude for three wave periods: 900 s (the solid curves), 600 s (the dotted curves) and 300 s (the dashed curves). The upper curve of each kind corresponds to the normalized dissipation coefficient , and the lower curve to . The amplitude of each wave is measured in units of the initial amplitude. Other parameters are as in Fig. 2 and Fig. 3.

The application of the method of MHD coronal seismology requires not only precise observations, but also elaborated theory. The theoretical model developed in this study is quite a simple one and neglects several physical mechanisms which can be important for the slow wave evolution. One of these neglected mechanisms is reflection of the waves from the density gradient. The WKB method used in the derivation of the evolutionary equation does not allow us to take into account the reflection. So, the waves of longer wavelengths, comparable with the scale height, can experience the reflection. In principle, this effect has to be taken into account. However, we can probably neglect this effect, according to results of Ofman et al. (1999), which show that the reflection of slow magnetoacoustic waves from density gradients in polar plumes is insignificant and the WKB approach works very well in the plume case. Anyway, the detailed study of the effect should be done in the future. Another effect neglected is dispersion of slow magnetoacoustic modes of a loop, connected with the finite radius of the loop cross-section. For example, it is well known that fast modes of coronal loops are strongly dispersive in the long wavelength limit, and their phase and group speeds are strongly influenced by the dispersion (Roberts et al. 1983, 1984). In contrast, slow modes are very weakly dispersive and the dispersion becomes important only in the nonlinear regime. The dispersion can slow down the nonlinear generation of higher harmonics and is very important on the nonlinear stage of the wave evolution (see, e.g. Zhugzhda & Nakariakov 1997a,b for slow body sausage modes of coronal loops). But, as the nonlinearity is found to be insignificant for the amplitude observed, the neglect of this effect also seems to be justified. Also, more developed models have to include the gradient of the temperature along the loop (see Aschwanden et al. 1999b) and effects of variable loop cross-section.

Thus, we believe that the model developed provides the correct qualitative interpretation of running intensity disturbances in coronal loops as slow magnetoacoustic waves and can be used as a basis for seismology of the coronal loops.

© European Southern Observatory (ESO) 2000

Online publication: October 30, 2000
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