7.1 Propagating fast waves in coronal loops

Wave lengths of propagating waves should be much shorter than the size of the structure guiding the wave. The shorter wave lengths correspond to shorter periods of the waves, requiring high cadence observational tools. In particular, the typical cadence time of TRACE and EIT EUV imagers is $20 -30 s$ , which gives the minimal period of detected waves to be about $2 -3 min$ . For a typical coronal Alfvén speed of about $1000 km s-1$ , at least, this corresponds to wavelengths longer than $120- 180 Mm$ , which is comparable with the typical size of a coronal loop. Thus, the EUV imagers cannot normally be used for the detection of propagating fast waves guided by coronal loops. On the other hand, time resolution of a second or better can be achieved with ground based coronographs or radioheliographs. Recently, Williams et al. (2001

, 2002

) and Katsiyannis et al. (2003

) reported the observational discovery of rapidly propagating compressible wave trains in coronal loops with the SECIS instrument during a full solar eclipse. As the observed speed was estimated at about $-1 2100 km s$ , these waves were interpreted as fast magnetoacoustic modes. The waves were observed to have a quasi-periodic pattern with a mean period of about $6 s$ . Cooper et al. (2003

) found an encouraging agreement between the observed evolution of the wave amplitude along the loop with the theoretical prediction made for kink and sausage modes (see Figure 20

). Unfortunately, the high uncertainty in the measurement of the amplitude does not allow the authors to distinguish between the kink and sausage modes in this observation.

Figure 20:

A plot of the maximum amplitude at the most frequent scale of the points, analysed by Williams et al. (2001

), against their loop position. The $x$ axis error bars are the uncertainty in position and the $y$ axis error bars have been calculated statistically by taking the square root of the number of photons and normalising with respect to time. The figure also includes theoretically calculated dependences for kink (solid) and sausage (dashed) modes with the parameters (from Cooper et al., 2003 ).

The main difference of these works from a number of previous papers analysing eclipse observations is the application of the methods of the time-distance (stroboscopic) plot (Williams et al., 2001 , 2002 ) and of the wavelet transform (Katsiyannis et al., 2003

). Those techniques are certainly more preferable in the situation when the waves have modulation or form wave trains. According to dispersion relation (7

), fast waves are highly dispersive when their wave length is comparable or longer than the radius of the loop. It is well known that, in a dispersive medium, impulsively generated (or broadband because of another reason) waves evolve into a quasi-periodic wave train with a pronounced period modulation. In the coronal context, it was pointed out by Edwin and Roberts (1983 ) and Roberts et al. (1983 , 1984

) that periodicity of fast magnetoacoustic modes in coronal loops is not necessarily connected with the wave source, but can be created by the dispersive evolution of an impulsively generated signal. Studying the dispersive evolution, Roberts et al. (1984

) analytically predicted that the development of the propagating sausage pulse forms a characteristic quasi-periodic wave train with three distinct phases. Such evolution scenario is determined by the presence of minimum in the group speed dependence upon the wave number. That analysis was restricted to the case of the slab with sharp boundaries and to sausage modes only. The initial stage of the pulse evolution was numerically modelled by Murawski and Roberts (1993 , 1994 ), Murawski et al. (1998 ), and was found to be consistent with the analytical prediction.

Nakariakov and Roberts (1995b ) studied the case of a smooth density profile of a zero plasma- $b$ plasma,

(cf. Equation (19

) and Figure 8

). The power index $p$ determines the steepness of the profile. The cases when the power index $p$ equals to either unity or infinity correspond to the symmetric Epstein profile or to the step function profile, respectively, both with known analytical solutions in the eigenvalue problem. The group speed has a minimum for all profiles with the power index greater than unity, which are steeper than the symmetric Epstein profile. Thus, the steepness of the profile affects the shape of the wave train and, consequently, the analysis of wave trains can give us information about this profile.

Nakariakov et al. (2004a ) numerically simulated the developed stage of the dispersive evolution of a fast wave train in a smooth straight slab of a low plasma- $b$ plasma. It was found that development of an impulsively generated pulse leads to formation of a quasi-periodic wave train with the mean wavelength comparable with the slab width. In agreement with the analytical theory (Roberts et al., 1984 ), the wave train has a pronounced period modulation which was demonstrated with the wavelet transform technique (see Figure 21 ). In particular, it is found that the dispersive evolution of fast wave trains leads to the appearance of characteristic “tadpole” wavelet signatures. Similar signatures were found in the wavelet analysis of SECIS data (Katsiyannis et al., 2003 ), strengthening the interpretation of the SECIS waves as the fast magnetoacoustic wave trains.

Figure 21:

Numerical simulation of an impulsively generated fast magnetoacoustic wave train propagating along a coronal loop with a density contrast ratio of $5$ and profile steepness power index equal to $8$ . Upper panel: The characteristic time signature of the wave train at $z = 70 a$ , where $a$ is the loop semi-width, from the source point. The vertical lines show the pulse arrival time if the density was uniform, the dotted line using the external density, and the dashed line using the density at the centre of the structure. Lower panel: Wavelet transform analysis of the signal, demonstrating the characteristic “tadpole” wavelet signature (from Nakariakov et al., 2004a ).

This mechanism may also be responsible for the formation of quasi-periodic pulsations with the periods of about a second, observed in association with flares. However, the main property of this effect, which must be taken into account in interpretation, is the period modulation.