Astron. Astrophys. 319, 1007-1019 (1997)

2. Basic design of our model

The evolution of both the gas and solid particles is described by equations of continuity

and momentum conservation

Terms associated with external sources of mass are absent in these equations, as we restrict ourselves to considering only a dissipative stage of a protoplanetary disk. Note that the equation of momentum conservation is written in its conservative form. The preference for the conservative form over the more familiar force form [which can be easily obtained by combining (2) and (1)] will become clear momentarily. Both the gas and solid particles are considered to be perfect fluids. Thus, for the gas, P is simply the thermodynamic pressure, but for the particles . The last term on the right-hand side of (2) represents frictional coupling between particles and the gas. It is neglected in the computation of gas evolution, but preserved in the computation of particle behavior. We neglect the disk's self-gravity and assume that despite the growing mass of the central star its gravitational potential remains constant. The symbol denotes the dyadic product of two velocity vectors.

The gas is explicitly assumed to be turbulent; thus quantities such as gas and particle velocities and densities are broken up into average and fluctuating parts

Fluctuations of the gas are conveyed to particles by frictional coupling.

Cylindrical polar coordinates are used, and an axial symmetry () is assumed throughout the paper. The basic procedure is identical for both the gas and the particles; representation (3) is substituted into Eq. (2), which is then expanded out and the Reynolds averaging technique is applied to isolate the relationship between average quantities characterizing the state of the gas and the particles. The effects of turbulence manifest themselves by the presence of the correlation terms, , , and in the averaged equation. Note that because the averaging was applied to the equation of motion in the conservative form no correlation terms involving derivatives of fluctuating quantities appear. As there are no acknowledged models of such correlations, the preference for using the conservative form of the equation of motion becomes evident. Nonvanishing correlations are modeled in terms of averaged quantities, and the equation of motion is solved under the thin disk approximation. The evolution of the mass in the disk is derived by vertically integrating the continuity equation (1) supplemented by the equation of motion.

In the remainder of this paper subscript denotes quantities describing the state of solids, whereas quantities describing gas have no subscript.

2.1. Evolution of gas

Applying the procedure described above to the gaseous component of the disk we obtain

where is the stress tensor resulting from interactions among the fluctuations in the flow field

The presumption that makes it possible to neglect the triple correlations in definition (5). The remaining correlations , are modeled in terms of averaged quantities. The diagonal components of the tensor are expressed in terms of a turbulent kinetic energy per unit mass

The off-diagonal elements of the symmetric tensor are specified to mimic the corresponding elements of stress associated with molecular viscosity

Finally, the correlations of the form are modeled by the gradient diffusion hypothesis

Turbulent stress, , is characterized by three quantities, turbulent viscosity , turbulent diffusivity D, and turbulent velocity given by

where H is the disk's vertical scale-height (used as a measure of the disk's thickness) and is the speed of sound. The character of turbulence is encapsulated into three dimensionless parameters: the Shakura-Sunyaev dimensionless viscosity parameter , the Rossby number for turbulent motions , and K, which can be identified with the inverse of the turbulent Prandtl number. In our calculations we assume these parameters to be constant and uniform and use , , and in the range from to . Note that in order for our model of [Eqs. (6) to (8)] to be self-consistent and physically meaningful, is required. Otherwise, the Schwarz inequality applied to the Reynolds averaging operation is not satisfied, which leads to the averages of the square of some real fluctuating quantities to be negative. This detail is worth mentioning because models of , identical or similar to ours are often used, but rarely checked for consistency.

With such a model of turbulence and the stress tensor resulting from it, the self-consistent solution to Eq. (4) can be found under the thin disk () approximation. The r -component of Eq. (4) determines the average tangential velocity . Under the assumption of vertical equilibrium () and vertical isothermicity, the z -component of Eq. (4) yields the vertical profile of the disk's density and sets the scale-height, , where is the keplerian angular velocity. The average vertical velocity, , is obtained directly from the condition of vertical equilibrium. Finally, the -component of Eq. (4) gives the average radial velocity . Putting everything together, we have

It's worth pointing out that ; even so vertical equilibrium is assumed. This is because the equilibrium requires the entire mass flux to vanish, and must be nonzero in order for to balance .

The Reynolds averaged continuity equation (1) is

Integrating over the z coordinate and substituting we obtain the familiar equation for time evolution of the surface density () of the gas

Notice that the last term on the left-hand side of Eq. (11) vanishes due to vertical integration, regardless of whether vanishes or not. Because is not an explicit function of time, but instead depends only on the local disk's quantities [see Eq. 9)], it can be expressed as and Eq. (12) 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.

2.2. Evolution of solid particles

Applying the procedure described at the beginning of this section to the solid particle component of the disk we obtain

where is the relative velocity between particles and the gas, and is the stopping time. When averaging the gas-solids coupling term we approximate that , where if and if . The tensor is given by

where again, like in the case of the gas [see Eq. (5)], we have neglected the triple correlation in definition (14). As we did in paper I, following the arguments given by Cuzzi et al. (1993) and Dubrulle et al. (1995), we model correlations as and as , where denotes the Schmidt number

We also define the quantities

The magnitude of the dimensionless quantity determines the aerodynamic regime. If , the stopping time is small in comparison with the period of orbital revolution and particles are strongly coupled to the gas. This happens usually, but not exclusively, when particles are small, their size, s, smaller than about 1 mm. If , the stopping time is very long in comparison with the period of orbital revolution and particles are decoupled from the gas. This happens for large particles with cm. The behavior of intermediate-sized particles is characterized by the regime.

We seek a self-consistent solution to Eq. (13) under the thin disk approximation. Such a solution is required to be valid and self-consistent for all aerodynamic regimes. This cannot be achieved by the approximation fully analogical to the one we have used for the gas. Specifically, vertical transport of the r -component of momentum is not negligible in the regime. However, as our primary goal is to calculate the global evolution of the mass residing in solid particles, we are not interested in details of solid material vertical distribution, with the sole exception of its scale-height. Therefore, instead of seeking a solution to Eq. (13) we can look for a solution to the system of equations consisting of the z -component of (13) and density-weighted vertical averages of r and components of (13). It turns out that the terms responsible for the vertical transport of the r -component of momentum vanish when averaged, and a self-consistent solution, valid for all aerodynamic regimes , can be found.

We start with the z -component of (13) and assume that particles are vertically in equilibrium so or

Because of vertical equilibrium, only two terms, both originating from , remain on the left-hand side of the z -component of (13). Of these we preserve a term containing but neglect a term containing . We preserve all terms on the right-hand side of this equation. Using (17) and (10) we transform the z -component of (13) into

with being the only unknown variable. In order to solve this equation we expand in a Taylor series at

The absence of even terms in expansion (19) follows from being an odd function of z. Substituting this expansion into (17) and integrating the resulting equation we obtain the vertical profile of

It's clear that the characteristic length of a particle's vertical distribution is determined by quantities . Substituting (19) into (18) and comparing the terms proportional to z and , respectively, we find and to be

where , and

For particles in the regime, and . The vertical distribution of solids and gas are the same, an expected result in the regime that stands for a perfect gas-solids coupling. For particles in the regime, , and for particles in the regime, , whereas . In all regimes we use as a measure of solid particles sub-disk thickness .

We now consider density-weighted, vertical averages of the r and components of Eq. (13). As our goal is to calculate the relative velocity between particles and gas, we also have to consider density-weighted, vertical averages of the r and components of the momentum conservation equation for the gas (4). Because the gas and particles have different vertical distributions of density we introduce two different averaging operators

We apply the operator to components of (13), to get:

Now we have to evaluate . When particles are small, in the regime, and the dust is well mixed with the gas and . In this case,

When particles are big, in the regime, the dust is concentrated in the midplane and . Thus, in this regime we get:

where the subscript o denotes the value of a vertically changing quantity evaluated at . For the intermediate-sized particles the values of and are somewhere midway between these given by Eqs. (26)-(27) and these given by Eqs. (28)-(29). Therefore, for all regimes we can use the following formulas

and

which are the result of interpolation between two regimes ( and ) for which the values of and have been expressed in an analytical form.

Note that the term proportional to on the right-hand side of Eq. (31) vanishes if as we have assumed. Thus, from the computational standpoint, is an especially convenient case ¹, and it is consistent with calculations by Canuto & Battaglia (1988). Under such an assumption, Eqs. (30), (31), and (19) combined with (10) form a closed system of equations from which , , and can be calculated. The system is, in general, neither linear, nor separable because is a function of , , and . The solution to this system is found numerically using the relevant drag laws outlined in Paper I.

The vertical integration of the equation of continuity (1) for particles yields

and the integral under the radial partial derivative can be substituted from the vertically integrated -component of (13) to obtain

This equation governs the time evolution of the surface density of solid particles, . Parameters , , and depend on the size of particles and therefore are implicit functions of r, t, and . In order to establish these relationships we need to model the coagulation process.

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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