  [abridged] We determine the relationship between turbulence energy and gas
density variance for 15 molecular clouds in the Solar neighbourhood. We use the
linewidths of the CO molecule as the probe of the turbulence energy (sonic Mach
number, $\cal{M}_\mathrm{s}$) and three-dimensional models to reconstruct the
density probability distribution function ($\rho$-PDF) of the clouds, derived
using near-infrared extinction and Herschel dust emission data, as the probe of
the density variance ($\sigma_\mathrm{s}$). We find no significant correlation
between $\cal{M}_\mathrm{s}$ and $\sigma_\mathrm{s}$ among the studied clouds,
however, we also cannot rule out a weak correlation. In the context of
turbulence-dominated gas, the range of the $\cal{M}_\mathrm{s}$ and
$\sigma_\mathrm{s}$ values corresponds with the model predictions. The data
cannot constrain whether or not the turbulence driving parameter, $b$, and/or
thermal-to-magnetic pressure ratio, $\beta$, vary among the sample clouds. Most
clouds are not in agreement with field strengths stronger than given by $\beta
\lesssim 0.05$. A model with $b^2 \beta / (\beta+1) = 0.30 \pm 0.06$ provides
an adequate fit to the cloud sample as a whole. When considering the average
behaviour of the sample, we can rule out three regimes: (i) strong compression
combined with a weak magnetic field ($b \gtrsim 0.7$ and $\beta \gtrsim 3$),
(ii) weak compression ($b \lesssim 0.35$), and (iii) strong magnetic field
($\beta \lesssim 0.1$). Including independent magnetic field strength estimates
to the analysis, the data rule out solenoidal driving ($b < 0.4$) for the
majority of the Solar neighbourhood clouds. However, most clouds have $b$
parameters larger than unity, which indicates a discrepancy with the
turbulence-dominated picture; we discuss the possible reasons for this.
