Production and damping of turbulence by particles in the solar nebula

Abstract

Abstract. We have developed a "one-equation" model for the turbulent boundary layer surrounding the midplane of the solar nebula and compared it with the "zero-equation" (Prandtl) model used in our previous work. Unlike the Prandtl model, our current model includes local generation, transport, and dissipation of turbulence. In particular, our derivation of the equation governing the kinetic energy of turbulence explicitly includes the damping of turbulence by particle drag. We have also included the kinetic pressure of the particles in both models, analogous to the turbulent pressure of the gas phase, and refined certain dimensionless constants of the flow. We present numerical results for particles 60 cm in radius. Both the switch to the one-equation model and the inclusion of turbulence damping by particles tend to increase particle concentrations at the midplane of the nebula, but the particle pressure and the improved flow constants both decrease the particle concentration there. These effects nearly offset one another, supporting our previous conclusion that settling of particles to the midplane of the solar nebula is selflimiting, so that direct accretion of planetesimals by gravitational instability is inhibited.

Shear instabilities in the dust layer of the solar nebula III. Effects of the Coriolis force

“…Thus, the density at the midplane cannot exceed the critical density of the gravitational instability (Weidenschilling, 1980). Subsequently, many authors have investigated this issue (Weidenschilling, 1984;Cuzzi et al, 1993;Weidenschilling and Cuzzi, 1993;Champney et al, 1995;Sekiya, 1998;Dobrovolskis et al, 1999). They also concluded that the turbulence prevents the dust from settling and that the formation model of planetesimals through the gravitational fragmentation of the dust layer is denied.…”

Shear instabilities in the dust layer of the solar nebula II. Different unperturbed states

“…Thus the dust layer would be stirred by turbulence induced by the shear instability. The dust density distribution is considered to have converged to a certain state where dust settling and the turbulent diffusion balanced each other Champney et al, 1995;Sekiya, 1998;Dobrovolskis et al, 1999). Figures 8 and 9 show the shear rates of the unperturbed state normalized by the Keplerian angular velocity, (dv 0 /dz)/ K for models (1A) and (1B), and (2A) and (2B), respectively.…”

“…The results show that the growth rate is much larger than the Keplerian angular velocity, as long as the dust density on the midplane is much larger than the gas density, in contrast to the previous result (Paper I). The dust layer is considered to evolve to a turbulent state, and the dust density distribution would converge to a certain state where dust settling and the turbulent diffusion balanced each other Champney et al, 1995;Sekiya, 1998;Dobrovolskis et al, 1999). The dust density in this state is much lower than the critical density of the gravitational instability.…”

“…The gravitational instability theories (Safronov, 1969;Goldreich and Ward 1973;Coradini et al, 1981;Sekiya, 1983) are mathematically neat and were preferred previously. It was claimed, however, that the shear induced turbulence might have prevented the dust from settling sufficiently to be gravitationally unstable (Weidenschilling, 1980(Weidenschilling, , 1984Cuzzi et al, 1993;Weidenschilling and Cuzzi, 1993;Champney et al, 1995;Sekiya, 1998;Dobrovolskis et al, 1999).…”

On the pressure of collisionless particle fluids

“…Consequently, this induces a velocity dispersion inside the particle flow and, for similar reasons to those described in this paper, the particle fluid develops a pressure. As a side note, when turbulence modelling is considered, turbulent pressure terms can arise from fluctuating or subgrid velocity dispersions (Dobrovolskis et al 1999). However, turbulent pressure terms are a consequence of Reynolds averaging or spatial filtering and disappear when turbulence is directly simulated instead of modelled.…”