A simple phenomenological model for grain clustering in turbulence

作者：Hopkins Philip F^*

来源：Monthly Notices of the Royal Astronomical Society, 2016, 455(1): 89-111.

DOI：10.1093/mnras/stv2226

• accretion, accretion discs
• hydrodynamics
• instabilities
• turbulence
• planets and satellites: formation
• protoplanetary discs

We propose a simple model for density fluctuations of aerodynamic grains, embedded in a turbulent, gravitating gas disc. The model combines a calculation for the behaviour of a group of grains encountering a single turbulent eddy, with a hierarchical approximation of the eddy statistics. This makes analytic predictions for a range of quantities including: distributions of grain densities, power spectra and correlation functions of fluctuations, and maximum grain densities reached. We predict how these scale as a function of grain drag time t(s), spatial scale, grain-to-gas mass ratio (rho) over tilde, strength of turbulence alpha, and detailed disc properties. We test these against numerical simulations with various turbulence-driving mechanisms. The simulations agree well with the predictions, spanning t(s) Omega similar to 10(-4)-10, (rho) over tilde similar to 0-3, alpha similar to 10(-10)-10(-2). Results from 'turbulent concentration' simulations and laboratory experiments are also predicted as a special case. Vortices on a wide range of scales disperse and concentrate grains hierarchically. For small grains this is most efficient in eddies with turnover time comparable to the stopping time, but fluctuations are also damped by local gas-grain drift. For large grains, shear and gravity lead to a much broader range of eddy scales driving fluctuations, with most power on the largest scales. The grain density distribution has a log-Poisson shape, with fluctuations for large grains up to factors greater than or similar to 1000. We provide simple analytic expressions for the predictions, and discuss implications for planetesimal formation, grain growth, and the structure of turbulence.

• 出版日期2016-1-1