Nonlinear long-wavelength torsional Alfvén waves

Aims. We investigate the nonlinear phenomena accompanying long-wavelength torsional waves in solar and stellar coronae. Methods. The second order thin flux-tube approximation is used to determine perturbations of a straight untwisted and non-rotating magnetic flux-tube, nonlinearly induced by long-wavelength axisymmetric magnetohydrodynamic waves of small, but finite amplitude. Results. Propagating torsional waves induce compressible perturbations oscillating with double the frequency of the torsional waves. In contrast with plane shear Alfvén waves, the amplitude of compressible perturbations is independent of the plasma-β and is proportional to the torsional wave amplitude squared. Standing torsional waves induce compressible perturbations of two kinds, that grow with the characteristic time inversely proportional to the sound speed, and that oscillate at double the frequency of the inducing torsional wave. The growing density perturbation saturates at the level, inversely proportional to the sound speed