  Studying the driving modes of turbulence is important for characterizing the
impact of turbulence in various astrophysical environments. The driving mode of
turbulence is parameterized by $b$, which relates the width of the gas density
PDF to the turbulent Mach number; $b\approx 1/3$, $1$, and $0.4$ correspond to
driving that is solenoidal, compressive, and a natural mixture of the two,
respectively. In this work, we use high-resolution (sub-pc) ALMA $^{12}$CO ($J$
= $2-1$), $^{13}$CO ($J$ = $2-1$), and C$^{18}$O ($J$ = $2-1$) observations of
filamentary molecular clouds in the star-forming region N159E (the Papillon
Nebula) in the Large Magellanic Cloud (LMC) to provide the first measurement of
turbulence driving parameter in an extragalactic region. We use a non-local
thermodynamic equilibrium (NLTE) analysis of the CO isotopologues to construct
a gas density PDF, which we find to be largely log-normal in shape with some
intermittent features indicating deviations from lognormality. We find that the
width of the log-normal part of the density PDF is comparable to the supersonic
turbulent Mach number, resulting in $b \approx 0.9$. This implies that the
driving mode of turbulence in N159E is primarily compressive. We speculate that
the compressive turbulence could have been powered by gravo-turbulent
fragmentation of the molecular gas, or due to compression powered by H I flows
that led to the development of the molecular filaments observed by ALMA in the
region. Our analysis can be easily applied to study the nature of turbulence
driving in resolved star-forming regions in the local as well as the
high-redshift Universe.
