  The ionizing radiation of massive stars sculpts the surrounding neutral gas
into pillar-like structures. Direct signatures of star formation through
outflows and jets are observed in these structures, typically at their tips.
Recent numerical simulations have suggested that this star formation could
potentially be triggered by photoionising radiation, driving compressive modes
of turbulence in the pillars. In this study we use recent high-resolution ALMA
observations of $^{12}\mathrm{CO}$, $^{13}\mathrm{CO}$, and $\mathrm{C}^{18}
\mathrm{O}, \; J = 2-1$ emission to test this hypothesis for pillars in the
Carina Nebula. We analyse column density and intensity-weighted velocity maps,
and subtract any large-scale bulk motions in the plane of the sky to isolate
the turbulent motions. We then reconstruct the dominant turbulence driving mode
in the pillars, by computing the turbulence driving parameter $b$,
characterised by the relation $\sigma_{\rho/\rho_0} = b \mathcal{M}$ between
the standard deviation of the density contrast $\sigma_{\rho/\rho_0}$ (with gas
density $\rho$ and its average $\rho_0$) and the turbulent Mach number
$\mathcal{M}$. We find values of \mbox{$b\sim0.7$--$1.0$} for most of the
pillars, suggesting that predominantly compressive modes of turbulence are
driven in the pillars by the ionising radiation from nearby massive stars. We
find that this range of $b$ values can produce star formation rates in the
pillars that are a factor $\sim 3$ greater than with $b \sim 0.5$, a typical
average value of $b$ for spiral-arm molecular clouds. Our results provide
further evidence for the potential triggering of star formation in pillars
through compressive turbulent motions.
