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} $��$186.88, Fe {\sc xii} $��$193.51, Fe {\sc xii} $��$195.12 and Fe {\sc xiii} $��$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} $��$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��n waves off-limb in polar coronal hole up to $\simeq 0.4$ R$_\odot$. The computed Alfv��n 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 $��f_w / ��h \simeq -4.5 \cdot 10^{-5}$ erg cm$^{-3}$ s$^{-1}$. Hence, this result suggests energy deposition by Alfv��n waves in a polar coronal hole, thus providing a significant source for coronal heating.

Physical units of the Alfv\'en wave Energy Decay Rate $\Delta f_w / \Delta h$ revised with respect to the published version. Scientific results and conclusions unchanged