1307.7147 (Philip F. Hopkins)

A Theory of Grain Clustering in Turbulence: The Origin and Nature of Large Density Fluctuations    [PDF]

Philip F. Hopkins

We propose a theory for the density fluctuations of aerodynamic grains embedded in a turbulent, gravitating gas disk. The theory combines calculations for the average behavior of grains encountering a single turbulent eddy, with a hierarchical description of the eddy velocity statistics. We show that this makes analytic predictions for a wide range of quantities, including: the distribution of volume-average grain densities, the power spectrum and correlation functions of grain density fluctuations, and the maximum volume density of grains reached. For each, we predict how these scale as a function of grain stopping/friction time (t_stop), spatial scale, grain-to-gas mass ratio, strength of the turbulence (alpha), and detailed disk properties (orbital frequency, sound speed). We test these against numerical simulations and find good agreement over a huge parameter space. Results from 'turbulent concentration' simulations and laboratory experiments are also predicted as a special case. We predict that vortices on a wide range of scales act to disperse and concentrate grains hierarchically (even if the gas is incompressible). For small grains this is most efficient in eddies with turnover time comparable to the stopping time. But for large grains, shear and gravity are important and lead to a broad range of eddy scales driving fluctuations, with most power on the largest scales. The grain density distribution is driven to a log-Poisson shape, with fluctuations for large grains up to >1000 times the mean density. We predict much smaller grains will also experience large fluctuations, but on small scales (not resolved in most simulations). We provide simple analytic expressions for the important predictions, and discuss implications for planetesimal formation, grain growth, and the structure of turbulence.

View original: http://arxiv.org/abs/1307.7147