  Between February 24-25, 2009, the EIS spectrometer onboard the Hinode
spacecraft performed special "sit & stare" observations above the South polar
coronal hole continuously over more than 22 hours. Spectra were acquired with
the 1" slit placed off-limb covering altitudes up to 0.48 R$_\odot$
($3.34\times 10^2$ Mm) above the Sun surface, in order to study with EIS the
non-thermal spectral line broadenings. Spectral lines such as Fe {\sc xii}
$\lambda$186.88, Fe {\sc xii} $\lambda$193.51, Fe {\sc xii} $\lambda$195.12 and
Fe {\sc xiii} $\lambda$202.04 are observed with good statistics up to high
altitudes and they have been analyzed in this study. Results show that the FWHM
of Fe {\sc xii} $\lambda$195.12 line increases up to $\simeq 0.14$ R$_\odot$,
then decreases higher up. EIS stray light has been estimated and removed.
Derived electron density and non-thermal velocity profiles have been used to
estimate the total energy flux transported by Alfv\'en waves off-limb in polar
coronal hole up to $\simeq 0.4$ R$_\odot$. The computed Alfv\'en wave energy
flux density $f_w$ progressively decays with altitude from $f_w \simeq 1.2
\cdot 10^6$ erg cm$^{-2}$ s$^{-1}$ at 0.03 R$_\odot$ down to $f_w \simeq 8.5
\cdot 10^3$ erg cm$^{-2}$ s$^{-1}$ at 0.4 R$_\odot$, with an average energy
decay rate $\Delta f_w / \Delta h \simeq -4.5 \cdot 10^{-5}$ erg cm$^{-3}$
s$^{-1}$. Hence, this result suggests energy deposition by Alfv\'en waves in a
polar coronal hole, thus providing a significant source for coronal heating.
