  We study the impulsively generated non-linear Alfv\'en waves in the solar
atmosphere, and describe their most likely role in the observed non-thermal
broadening of some spectral lines in solar coronal holes. We solve numerically
the time-dependent magnetohydrodynamic equations to find temporal signatures of
large-amplitude Alfv\'en waves in the model atmosphere of open and expanding
magnetic field configuration, with a realistic temperature distribution. We
calculate the temporally and spatially averaged, instantaneous transversal
velocity of non-linear Alfv\'en waves at different heights of the model
atmosphere, and estimate its contribution to the unresolved non-thermal motions
caused by the waves. We find that the pulse-driven nonlinear Alfv\'en waves
with the amplitude $A_{\rm v}$=50 km s$^{-1}$ are the most likely candidates
for the non-thermal broadening of Si VIII $\lambda$1445.75 \AA\ line profiles
in the polar coronal hole as reported by Banerjee et al. (1998). We also
demonstrate that the Alfv\'en waves driven by comparatively smaller velocity
pulse with its amplitude $A_{\rm v}$=25 km s$^{-1}$ may contribute to the
spectral line width of the same line at various heights in coronal hole without
any significant broadening. The main conclusion of this paper is that
non-linear Alfv\'en waves excited impulsively in the lower solar atmosphere are
responsible for the observed spectral line broadening in polar coronal holes.
This is an important result as it allows us to conclude that such large
amplitude and pulse-driven Alfv\'en waves do indeed exist in solar coronal
holes. The existence of these waves and their undamped growth may impart the
required momentum to accelerate the solar wind.
