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A simple phenomenological model for grain clustering in turbulence

Hopkins, Philip F. (2016) A simple phenomenological model for grain clustering in turbulence. Monthly Notices of the Royal Astronomical Society, 455 (1). pp. 89-111. ISSN 0035-8711.

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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 ρ̃ ρ~, strength of turbulence α, 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 Ω ∼ 10^(−4)–10, ρ̃ ∼0−3ρ~∼0−3, α ∼ 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 ≳1000. We provide simple analytic expressions for the predictions, and discuss implications for planetesimal formation, grain growth, and the structure of turbulence.

Item Type:Article
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Hopkins, Philip F.0000-0003-3729-1684
Additional Information:© 2015 The Author Published by Oxford University Press on behalf of the Royal Astronomical Society. Accepted 2015 September 23. Received 2015 September 18; in original form 2013 July 16. First published online November 2, 2015. We thank Jessie Christiansen and Eugene Chiang for many helpful discussions during the development of this work. We also thank Andrew Youdin, Anders Johansen, and Jeff Cuzzi for several suggestions. We also thank our referee for a number of constructive and useful suggestions, especially pointing out the means to replace some earlier simplified approximations with exact solutions. Support for PFH was provided by NASA through Einstein Postdoctoral Fellowship Award Number PF1-120083 issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the NASA under contract NAS8-03060.
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NASA Einstein FellowshipPF1-120083
Subject Keywords:accretion, accretion discs – hydrodynamics – instabilities – turbulence – planets and satellites: formation – protoplanetary discs
Issue or Number:1
Record Number:CaltechAUTHORS:20160204-155313897
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Official Citation:Philip F. Hopkins A simple phenomenological model for grain clustering in turbulence MNRAS (January 01, 2016) Vol. 455 89-111 doi:10.1093/mnras/stv2226 First published online November 2, 2015
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:64250
Deposited By: George Porter
Deposited On:08 Feb 2016 21:26
Last Modified:03 Oct 2019 09:36

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