  We obtain an equation for the density profile in a self-gravitating
polytropic spherically symmetric turbulent fluid with an equation of state
$p_{\rm gas}\propto \rho^\Gamma$. This is done in the framework of ensembles of
molecular clouds represented by single abstract objects as introduced by Donkov
et al. (2017). The adopted physical picture is appropriate to describe the
conditions near to the cloud core where the equation of state changes from
isothermal (in the outer cloud layers) with $\Gamma=1$ to one of `hard
polytrope' with exponent $\Gamma>1$. On the assumption of steady state, as the
accreting matter passes through all spatial scales, we show that the total
energy per unit mass is an invariant with respect to the fluid flow. The
obtained equation reproduces the Bernoulli equation for the proposed model and
describes the balance of the kinetic, thermal and gravitational energy of a
fluid element. We propose as well a method to obtain approximate solutions in a
power-law form which results in four solutions corresponding to different
density profiles, polytropic exponents and energy balance equations for a fluid
element. One of them, a density profile with slope $-3$ and polytropic exponent
$\Gamma=4/3$, matches with observations and numerical works and, in particular,
leads to a second power-law tail of the density distribution function in dense,
self-gravitating cloud regions.
