## 2. Basic design of our modelThe evolution of both the gas and solid particles is described by equations of continuity 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, 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 gasApplying 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 where 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 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 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 particlesApplying 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 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,
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 We start with the Because of vertical equilibrium, only two terms, both originating
from , remain on the left-hand side of the
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 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
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 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 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
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 © European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |