  We present a one-dimensional model of the formation and viscous evolution of
protoplanetary disks. The formation of the early disk is modeled as the result
of the gravitational collapse of an isothermal molecular cloud. The disk's
viscous evolution is integrated according to two parameterizations of
turbulence: The classical $\alpha$ representation and a $\beta$
parameterization, representative of non-linear turbulence driven by the
keplerian shear. We apply the model to DM Tau and GM Aur, two classical T-Tauri
stars with relatively well-characterized disks, retrieving the evolution of
their surface density with time. We perform a systematic Monte-Carlo
exploration of the parameter space (i.e. values of the $\alpha$-$\beta$
parameters, and of the temperature and rotation rate in the molecular cloud) to
find the values that are compatible with the observed disk surface density
distribution, star and disk mass, age and present accretion rate. We find that
the observations for DM Tau require $0.001<\alpha<0.1$ or $2\times
10^{-5}<\beta<5\times 10^{-4}$. For GM Aur, we find that the turbulent
viscosity is such that $4\times 10^{-4}<\alpha<0.01$ or $2\times
10^{-6}<\beta<8\times 10^{-5}$. These relatively large values show that an
efficient turbulent diffusion mechanism is present at distances larger than
$\sim 10 $AU. This is to be compared to studies of the variations of accretion
rates of T-Tauri stars versus age that mostly probe the inner disks, but also
yield values of $\alpha\sim 0.01$. We show that the mechanism responsible for
turbulent diffusion at large orbital distances most probably cannot be
convection because of its suppression at low optical depths.
