  Theoretical and observational studies on the turbulence of the interstellar
medium developed fast in the past decades. The theory of supersonic magnetized
turbulence, as well as the understanding of projection effects of observed
quantities, are still in progress. In this work we explore the characterization
of the turbulent cascade and its damping from observational spectral line
profiles. We address the difference of ion and neutral velocities by clarifying
the nature of the turbulence damping in the partially ionized. We provide
theoretical arguments in favor of the explanation of the larger Doppler
broadening of lines arising from neutral species compared to ions as arising
from the turbulence damping of ions at larger scales. Also, we compute a number
of MHD numerical simulations for different turbulent regimes and explicit
turbulent damping, and compare both the 3-dimensional distributions of velocity
and the synthetic line profile distributions. From the numerical simulations,
we place constraints on the precision with which one can measure the 3D
dispersion depending on the turbulence sonic Mach number. We show that no
universal correspondence between the 3D velocity dispersions measured in the
turbulent volume and minima of the 2D velocity dispersions available through
observations exist. For instance, for subsonic turbulence the correspondence is
poor at scales much smaller than the turbulence injection scale, while for
supersonic turbulence the correspondence is poor for the scales comparable with
the injection scale. We provide a physical explanation of the existence of such
a 2D-3D correspondence and discuss the uncertainties in evaluating the damping
scale of ions that can be obtained from observations. However, we show that the
statistics of velocity dispersion from observed line profiles can provide the
spectral index and the energy transfer rate of turbulence. Also, comparing two
similar simulations with different viscous coefficients it was possible to
constrain the turbulent cut-off scale. This may especially prove useful since
it is believed that ambipolar diffusion may be one of the dominant dissipative
mechanism in star-forming regions. In this case, the determination of the
ambipolar diffusion scale may be used as a complementary method for the
determination of magnetic field intensity in collapsing cores. We discuss the
implications of our findings in terms of a new approach to magnetic field
measurement proposed by Li & Houde (2008).
