Nonlinear evolution of torsional Alfven waves

Aims. We study the efficiency of the energy transfer to shorter scales in the field-aligned direction - the parallel nonlinear cascade - that accompanies the propagation of torsional Alfven waves along open magnetic fields in the solar and stellar coronae, and compare it with the same effects for the shear Alfven wave. The evolution of the torsional Alfven wave is caused by the back reaction of nonlinearly induced compressive perturbations on the Alfven wave. Methods. The evolution of upwardly propagating torsional Alfven waves is considered in terms of the second-order thin flux-tube approximation in a straight untwisted and non-rotating magnetic flux-tube. The Cohen-Kulsrud equation for weakly nonlinear torsional waves is derived. In the model, the effect of the cubic nonlinearity on the propagation of long-wavelength axisymmetric torsional waves is compared with the similar effect that accompanies the propagation of plane linearly-polarised (shear) Alfven waves of small amplitude. Results. The solution to the Cohen-Kulsrud type equation for torsional waves shows that their evolution is independent of the plasma-beta, which is in contrast to the shear Alfven wave. In a finite-beta plasma, the nonlinear evolution of torsional Alfven waves is slower and the parallel nonlinear cascade is less efficient than those of shear Alfven waves. These results have important implications for the analysis of possible heating of the plasma and its acceleration in the upper layers of solar and stellar coronae. In particular, one-dimensional models of coronal heating and wave acceleration, which use shear Alfven waves instead of torsional Alfven waves, over-estimate the efficiency of these processes.