  The alkali metal sodium (Na) is one of the most commonly detected chemical
species in the upper atmospheres of giant exoplanets. In this work we conducted
a homogeneous survey of Na in a diverse sample of ten highly irradiated giant
exoplanets using high-resolution transmission spectroscopy. Our sample includes
nine planets with previous Na detections and one new detection. We confirm
previous detections and assess multiple approaches for deriving Na line
properties from high-resolution transmission spectra. The homogeneously
measured sodium line depths were used to constrain the atmospheric heights
($H_{\text{Na}}$) with respect to the planetary radii ($R_{\text{p}}$). We
assess an empirical trend describing the relative atmospheric height
($H_{\text{Na}}/R_{\text{p}}$) as a function of planetary equilibrium
temperature ($T_{\text{eq}}$) and surface gravity ($g$), in which
$H_{\text{Na}}/R_{\text{p}}$ decreases exponentially with $\xi \propto
gT_{\text{eq}}$, approaching a constant at large $\xi$. We also report the
sodium D2/D1 line ratios across our sample and find that seven targets have
line ratios that are consistent with unity. Finally, we measured net
blueshifted offsets of the sodium absorption lines from their rest frame
wavelengths for all ten planets, corresponding to day-night wind velocities of
a few km s$^{-1}$. This suggests that the broad sample of exoplanets share
common underlying processes that govern atmospheric dynamics. Our study
highlights a promising avenue for using high-resolution transmission
spectroscopy to further our understanding of how atmospheric characteristics
vary over a diverse sample of exoplanets.
