Density Fluctuations in a Polar Coronal Hole

We have measured the root-mean-square (RMS) amplitude of intensity fluctuations, $\Delta I$, in plume and interplume regions of a polar coronal hole. These intensity fluctuations correspond to density fluctuations. Using data from the \rev{Sun Watcher using Active Pixel System detector and Image Processing} (SWAP) on \textit{Project for Onboard Autonomy (Proba2)}, our results extend up to a height of about 1.35~$R_{\sun}$. One advantage of the RMS analysis is that it does not rely on a detailed evaluation of the power spectrum, which is limited by noise levels to low heights in the corona. The RMS approach can be performed up to larger heights where the noise level is greater, provided that the noise itself can be quantified. At low heights, both the absolute $\Delta I$, and the amplitude relative to the mean intensity, $\Delta I/I$, decrease with height. However, starting at about 1.2~$R_{\sun}$ $\Delta I/I$ increases, reaching 20--40\% by 1.35~$R_{\sun}$. This corresponds to density fluctuations of $\Delta n_{\mathrm{e}}/n_{\mathrm{e}} \approx$~10--20\%. The increasing relative amplitude implies that the density fluctuations are generated in the corona itself. One possibility is that the density fluctuations are generated by an instability of Alfv\'en waves. This generation mechanism is consistent with some theoretical models and with observations of Alfv\'en wave amplitudes in coronal holes. Although we find that the energy of the observed density fluctuations is small, these fluctuations are likely to play an important indirect role in coronal heating by promoting the reflection of Alfv\'en waves and driving turbulence.