  To evaluate the effect of turbulent heating in the thermal balance of
interstellar clouds, we develop an extension of the log-Poisson intermittency
model to supersonic turbulence. The model depends on a parameter, d,
interpreted as the dimension of the most dissipative structures. By comparing
the model with the probability distribution of the turbulent dissipation rate
in a simulation of supersonic and super-Alfvenic turbulence, we find a best-fit
value of d=1.64. We apply this intermittency model to the computation of the
mass-weighted probability distribution of the gas temperature of molecular
clouds, high-mass star-forming cores, and cold diffuse HI clouds. Our main
results are: i) The mean gas temperature in molecular clouds can be explained
as the effect of turbulent heating alone, while cosmic ray heating may dominate
only in regions where the turbulent heating is low; ii) The mean gas
temperature in high-mass star-forming cores with typical FWHM of ~6 km/s
(corresponding to a 1D rms velocity of 2.5 km/s) may be completely controlled
by turbulent heating, which predicts a mean value of approximately 36 K, two to
three times larger than the mean gas temperature in the absence of turbulent
heating; iii) The intermittency of the turbulent heating can generate enough
hot regions in cold diffuse HI clouds to explain the observed CH+ abundance, if
the rms velocity on a scale of 1 pc is at least 3 km/s, in agreement with
previous results based on incompressible turbulence. Because of its importance
in the thermal balance of molecular clouds and high-mass star-forming cores,
the process of turbulent heating may be central in setting the characteristic
stellar mass and in regulating molecular chemical reactions.
