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; $bapprox 1/3$, $1$, 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 ion{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.
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