  Self-similar and semi-analytical solutions are found for the height-averaged
equations govern the dynamical behavior of a polytropic, self-gravitating disk
under the effects of winds, around the nascent object. In order to describe
time evolution of the system, we adopt a radius dependent mass loss rate, then
highlight its importance on both the traditional $\alpha$ and innovative
$\beta$ models of viscosity prescription. In agreement with some other studies,
our solutions represent that Toomre parameter is less than one in most regions
on the $\beta$-disk which indicate that in such disks gravitational
instabilities can occur in various distances from the central accretor and so
the $\beta$-disk model might provide a good explanation of how the planetary
systems form. The purpose of the present work is twofold. First, examining the
structure of disk with wind in comparison to no-wind solution; and second, to
see if the adopted viscosity prescription affects significantly the dynamical
behavior of the disk-wind system. We also considered the temperature
distribution in our disk by a polytropic condition. The solutions imply that,
under our boundary conditions, the radial velocity is larger for $\alpha$-disks
and increases as wind becomes stronger in both viscosity models. Also, we
noticed that the disk thickness increases by amplifying the wind or adopting
larger values for polytropic exponent $\gamma$. It also may globally decrease
if one prescribe $\beta$-model for the viscosity. Moreover, in both viscosity
models, surface density and mass accretion rate reduce as wind gets stronger or
$\gamma$ increases.
