  We present an one-dimensional numerical study of Alfven waves propagating
along a radial magnetic field. Neglecting losses, any spherical Alfven wave, no
matter how small its initial amplitude is, becomes nonlinear at sufficiently
large radii. From previous simulations of Alfven waves in plane parallel
atmospheres we did expect the waves to steepen and produce current sheets in
the nonlinear regime, which was confirmed by our new calculations. On the other
hand we did find that even the least nonlinear waves were damped out almost
completely before 10 solar radii. A damping of that kind is required by models
of Alfven wave-driven winds from old low-mass stars as these winds are mainly
accelerated within a few stellar radii.
