2.4 Effects of twisting

It is clear that twisting of the magnetic field in the cylinder will lead to linear coupling of various MHD modes. However, modification of dispersion relation (7

) by the twisting is not well understood. Considering the thin flux tube limit, Zhugzhda and Nakariakov (1999 ) showed that long-wavelength ( $kza « 1$ ) torsional perturbations of a weakly twisted cylinder are described by the dispersion relation

where $K = a2a2/8$ is the twist parameter (here $K « 1$ ), with $a = 2J0/B0$ ( $J0$ is the current density) being the parameter of force-free magnetic fields and $a$ being the radius of the cylinder. The parameter $a = 2J /B 0 0$ , where $J 0$ is the current density. The twist results into the appearance of dispersion and modifies the phase speed of the mode. Actually, the considered mode cannot be referred to as a pure torsional mode as it perturbs the plasma density and its phase speed depends upon the plasma- $b$ . The compressibility of the torsional wave in a twisted cylinder may be demonstrated as follows. According to Zhugzhda (1996 ), the torsional components of the velocity, $V f$ , and the magnetic field, $B f$ , are connected with the longitudinal components of the magnetic field $Bz$ and the velocity $Vz$ by the equation

If the equilibrium value of the twist $Bf0r$ is non-zero, the torsional motions generate longitudinal flows and, consequently the density perturbations.

Bennett et al. (1999 ) considered the collective MHD modes of a straight uniformly twisted magnetic cylinder in the incompressible limit, and concluded that the twist leads to the appearance of body modes with phase speeds about the “longitudinal” Alfvén speed (calculated with the use of the longitudinal component of the field only). As well as in the untwisted case, there is a surface mode with the phase speed about the kink speed.

An alternative approach to the modelling of twisted and curved coronal loops was suggested by Cargill et al. (1994 ), which allowed the authors to take into account the hoop force - the feature missing from the straight cylinder model. The force is connected with both the loop twist and the curvature. The presence of the new restoring force was shown to give rise to a new oscillation mode manifested as the periodic change of the loop major radius and the loop density, whose frequency could be independent of the loop length. Oscillations of the loop minor radius were also found. The oscillation frequencies obtained were significantly different from the frequencies of straight cylinder eigenmodes. This approach certainly requires attention and further development.

Also, oscillations of current carrying loops can be described in terms of the LCR-model, see Section 3.5.