The evolution of the gas in a viscous protoplanetary disk is
determined from the axisymmetric equations of continuity and motion.
Cylindrical polar coordinates 
are
used, with z=0 corresponding to a disk midplane (equator).
Because the disk is explicitely assumed to be turbulent,
quantities such as gas density and velocity are broken up into
mean and fluctuating parts:
The equations are expanded out and the Reynolds averaging technique
is applied to obtain equations describing the large-scale, mean
behavior of the gas. The general forms of the mean equations for
an axisymmetric disk with negligable molecular viscosity
are given by Cuzzi et al. (1993 [5]). The effects of turbulence
manifest themselves by the presence of correlation terms,
, 
, and
in the large-scale equations. The presumpting that 
makes it possible to neglect the triple correlations 
. The
gas velocity correlations are expressed in terms of turbulent
viscosity 
, turbulent velocity
, and gradients of the mean velocity:
and 
.
In expressing 
and 
in terms of the gas speed of
sound 
and disk half-thickness H we use the standard
-disk model and assume that the Rossby number is
of the order of unity. The correlations of the form 
are modeled by the gradient diffusion hypothesis: 
, with 
being the coefficient of turbulent diffusivity. Canuto & Battaglia (1988 [4]) argued that 
, where 
is the Kolmogorov constant.
As we are concerned here with the evolution of the radial distribution of the gas, we assume 
and ignore the vertical dependence of physical quantities describing the state of the disk.
Under such assumptions the only nonvanishing correlations are:
The r component of the equation of motion determines
the large-scale tangential velocity. We anticipate that a protoplanetary
disk is geometrically thin. In a thin accretion disk the large-scale velocities are such that 
and the r component of the equation of motion reduces to:
Here 
is the Keplerian velocity and M
is the mass of the central star. Of the five terms on the right-hand
side of equation (5) the first, due to centrifugal
acceleration, is the largest. The second, due to the gas pressure
gradient, is smaller by a factor of the order of 
. The
third, ``turbulent pressure" term, is smaller by a factor of the
order of 
. The two last terms, due
to turbulent diffusion, are smaller by a factor of the order of
. Thus the existence of sub-sonic turbulence
does not influence the mean tangential velocity of the gas,
which is slightly smaller than the Keplerian velocity because 
. This small difference is crucial for the
gas-solid interaction, but unimportant for the evolution of the
gas, where 
approximation is sufficient.
The mean radial velocity of the gas is computed from
the 
component of the equation of motion.
At the lowest order, this component can be written as
The convective mass flux based on an average radial velocity (the first term on the left-hand side of equation (6)) is, in general, of the same order of magnitude as the diffusive mass flux (the second term on the left-hand side of equation (6).) Together, they transport as much mass as the conservation of angular momentum allows (the right-hand side of equation (6)).
The Reynolds averaged continuity equation in the absence of sources is
Substituting 
from equation (6) into equation
(7) and integrating over the z coordinate we obtain the familiar equation for time evolution of the surface density ( 
) of the gas
Note that equation (8) does not depend on turbulent
diffusivity, as it only encapsulates conservation of mass and
angular momentum.
Because turbulent viscosity, 
, is not an explicit function
of time, but instead depends only on the local conditions
within a disk, 
can be expressed as 
and
equation (8) can be solved subject to boundary conditions
on the inner and outer edges of a disk and the opacity law. Given
, we can algebraically find all other disk variables.
In particular, 
can be calculated from equation
(6). Beware that the average radial velocity based on
Reynolds averaging does not give an accretion rate,
or 
. Instead,
the sum of convective and diffusive fluxes give the proper accretion
rate, 
.
The methods of how to solve equation
(8) and how to calculate other physical quantities
describing the state of the protoplanetary disk are given in papers
by Ruden & Pollack (1991 [18]) and Reyes-Ruiz & Stepinski (1995 [17]). Here
we follow the methodology of Reyes-Ruiz & Stepinski
.
In our numerical calculations, we start from the initial surface
density profile of the form
This distribution corresponds to 
being practically
uniform between the inner radius, which we have chosen to be
AU, and 
. The particular choice of 
is
not important as long as it is much smaller than the size of the
disk. Values of 
, 
, and s are calculated to correspond
to the desired initial disk mass and its angular momentum. For
calculations presented in this paper we started with a disk of
0.245 
and an angular momentum of 
g cm 
s 
. These correspond to an initial surface density
distribution characterized by 
g cm 
and
AU, initial conditions that correspond closely to those
considered ``standard" in protoplanetary disk evolution
calculations (Ruden & Pollack 1991 [18]). We examine disk evolution for
two values of 
: 
and 
.
The evolution of a gaseous disk proceeds according to well established principles: the surface density decreases as the mass is lost to the central star (see Figs. 4 and 5), the temperature decreases as well, and the disk spreads as a certain portion of the disk's mass moves outward, carrying angular momentum. The physical quantities describing the state of the gas change dramatically during the evolution, thus constantly changing the aerodynamic regime encountered by solid particles suspended in the gas.
