Density distribution function of a self-gravitating isothermal
  compressible turbulent fluid in the context of molecular clouds ensembles --
  III. Virial analysis
  In the present work we apply virial analysis to the model of self-gravitating
turbulent cloud ensembles introduced by Donkov & Stefanov (2018) and extended
by Donkov & Stefanov (2019). Using the Eulerian virial theorem at an arbitrary
scale, far from the cloud core, we derive an equation for the density profile
and solve it in approximate way. The result confirms the solution
$\varrho(\ell)=\ell^{-2}$ found in the previous papers. At scales far from the
core, we obtain virial equilibrium between i) gravitational energy and
accretion kinetic energy, or ii) between gravitational energy and accretion
plus turbulent kinetic energy per unit mass -- depending on the adopted scaling
relation for the turbulent velocity. Thus the obtained solution for density
profile is dynamically stable and shall be observable. At scales near the core,
one cannot neglect the second derivative of the moment of inertia of the gas,
which prevents derivation of an analytic equation for the density profile.
However, we obtain the qualitative result that gas near the core is not
virialized and its state can be characterized as marginally bound since the
total energy of the fluid element is close to zero.
