Relationship between turbulence energy and density variance in the Solar neighbourhood molecular clouds

[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.