The relationship between turbulence energy and gas density variance is a fundamental prediction for turbulence-dominated media and it is commonly used in analytic models of star formation .
We determine this relationship 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 _ { s } ) .
We find no significant correlation between \cal { M } _ { \mathrm { s } } and \sigma _ { s } among the studied clouds , however , we also can not rule out a weak correlation .
In the context of turbulence-dominated gas , the range of the \cal { M } _ { \mathrm { s } } and \sigma _ { s } values corresponds with the model predictions .
The data can not 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 .