8 Torsional Modes

Alfvén waves, e.g., as torsional modes of coronal loops (described by the first factor in Equation (6

)) are incompressible and, therefore, do not perturb the density and, hence, do not modify the loop emission. In this section we restrict our attention to torsional modes of untwisted magnetic structures. If the loop is twisted, torsional modes are compressible (see Section 2.4 for details). The velocity perturbations of Alfvén waves, though, can be observed through variations of the Doppler shift. Whilst long period waves may show resolved Doppler shifts, short period waves, with periods less than a few minutes, can be observed only indirectly by measuring the additional broadening of coronal emission lines, i.e., non-thermal broadening. An effective temperature $Teff$ , associated with a coronal optically thin emission line, can be defined that combined the effects of thermal and non-thermal broadening (McClements et al., 1991 ):

where $Ti$ and $mi$ are the temperature and mass of the line forming ion, respectively, $kB$ is the Boltzmann constant, and $VLOS$ is the velocity in the line of sight. The constant $C$ is defined by polarisation and other effects, and its value lies in the interval $2/3 < C < 1$ .

There have been several attempts to study the evolution of the amplitude of non-thermal broadening along possible wave paths. Banerjee et al. (1998 ) and Doyle et al. (1999 ), using SOHO/SUMER, measured the evolution of non-thermal broadening of the emission lines Si VIII and O VI in inter-plume regions of polar coronal holes and found that the amplitude of unresolved thermal broadening (possibly, the Alfvén wave amplitude) is growing up to $1.2 R o.$ , then has a plateau to $1.5 R o.$ , and then grows sharply again. The authors suggested that this phenomenon could be associated with nonlinear overturning of the waves, but a rigorous theoretical modelling is still required. Harrison et al. (2002 ) examined the width of Mg X emission line and found out the emission line narrowing as a function of altitude. This result was interpreted as further evidence for coronal wave activity in closed field regions, and most likely the first evidence of the dissipation of Alfvén waves in the corona. A similar conclusion was reached by O’Shea et al. (2003 ) by the analysis of the same emission line in polar plumes and inter-plume lanes. However, the analysis of the widths and its height variation of the Mg X doublet lines, in both the quiet equatorial corona and in a polar hole performed by Wilhelm et al. (2004 ), gave the opposite result: The Doppler width broadened with height in both the equatorial plane and in a polar coronal hole.

An indirect evidence for the presence of unresolved Alfvén waves in the corona was obtained by Erdélyi et al. (1998 ), who found out that quiet sun line widths increase from the disk centre outwards in SUMER data. This anisotropy of the non-thermal broadening could be caused by transverse waves; however, some other physical processes such as anisotropic turbulence could produce this effect too.

Alfvén waves with sufficiently long wave lengths can be observed directly through the spatial variation of the Doppler shift. Zaqarashvili (2003 ) suggested that the global torsional oscillations may be observed as a periodical variation of the spectral line width along the loop, as the loop footpoints are likely to provide the waves with rigid wall boundary conditions. The amplitude of the variation must be maximal at the velocity antinodes and minimal at the nodes of the torsional oscillation. The resonant period of a standing torsional mode of $n$ -th order is given by the expression

For example, for $L = 200 Mm$ and $C = 1 Mm s-1 A0$ , there can be several standing torsional modes with periods and wave lengths resolvable with modern coronal spectrometers. Such observational detection can be obtained in the raster mode, as usual 1D slit spectral observations do not give the required spatial information.

Another possibility to observe torsional modes is connected with the gyrosynchrotron emission, which can be detected in the radio band. As the observed emission depends strongly upon the angle $z$ between the magnetic field and the LOS, torsional waves changing the local direction of the field can modulate the observed emission. According to the simple estimations made by Tapping (1983 ) with the use of Equation (29 ), in the presence of torsional perturbations a substantial fraction of the emission can be observed from the axial direction, e.g., when the perturbed segment of loop is viewed end-on.