Mixing in Supersonic Turbulence

Abstract

In many astrophysical environments, mixing of heavy elements occurs in the presence of a supersonic turbulent velocity field. Here we carry out the first systematic numerical study of such passive scalar mixing in isothermal supersonic turbulence. Our simulations show that the ratio of the scalar mixing timescale, τ c , to the flow dynamical time, τ dyn (defined as the flow driving scale divided by the rms velocity), increases with the Mach number, M , for M ∼ < 3, and becomes essentially constant for M ∼ > 3. This trend suggests that compressible modes are less efficient in enhancing mixing than solenoidal modes. However, since the majority of kinetic energy is contained in solenoidal modes at all Mach numbers, the overall change in τ c /τ dyn is less than 20% over the range 1 ∼ < M ∼ < 6. At all Mach numbers, if pollutants are injected at around the flow driving scale, τ c is close to τ dyn . This suggests that scalar mixing is driven by a cascade process similar to that of the velocity field. The dependence of τ c on the length scale at which pollutants are injected into flow is also consistent with this cascade picture. Similar behavior is found for the variance decay timescales for scalars without continuing sources. Extension of the scalar cascade picture to the supersonic regime predicts a relation between the scaling exponents of the velocity and the scalar structure functions, with the scalar structure function becoming flatter as the velocity scaling steepens with Mach number. Our measurements of the volume-weighted velocity and scalar structure functions confirm this relation for M ∼ < 2, but show discrepancies at M ∼ > 3, which arise probably because strong expansions and compressions tend to make scalar structure functions steeper.

Turbulent Clustering of Protoplanetary Dust and Planetesimal Formation

“…First, at Mach numbers close to unity, the density fluctuations are weak, with the rms amplitude δρ 2 g 1/2 /ρ g at a level of ∼10%. Second, the velocity structures in a transonic flow are very close to those in incompressible flows (Porter et al 2002;Padoan et al 2004;Pan & Scannapieco 2010). In Section 4.1, we find that the clustering properties in our transonic flow are indeed in good agreement with the results from direct numerical simulations (DNS) for incompressible flows by Collins & Keswani (2004).…”

“…We calculate ν eff from the equation¯ = ν eff ω 2 , because solenoidal modes dominate the kinetic energy dissipation even in a transonic flow (Pan & Scannapieco 2010). The energy dissipation rate,¯ , can be derived either from Kolmogorov's 4/5 law (which also applies to transonic flows; see Pan & Scannapieco 2010 and also Benzi et al 2008), or from the relation,¯ = Du 3 /L 1 , established by DNS, where u and L 1 are the one-dimensional velocity dispersion and the integral length scale, and the coefficient D 0.4 (Ishihara et al 2009). The dissipation rate values derived from the two approaches are consistent with each other.…”

Atomic Chemistry in Turbulent Astrophysical Media. Ii. Effect of the Redshift Zero Metagalactic Background

“…As shown in Pan & Scannapieco (2010), most of the turbulent kinetic energy is found in the solenoidal modes at smaller scales over a wide range of mach numbers. This justifies our choice of solenoidal only driving.…”

“…Each of these simulations was carried out in a 128 3 periodic box, a resolution which is low compared to most current turbulence simulations (e.g. Ostriker et al 2001;Haugen et al 2004;Kritsuk et al 2007;Federrath et al 2008;Pan & Scannapieco 2010;Federrath et al 2010;Downes 2012;Gazol & Kim 2013;Folini et al 2014;Sur et al 2014;Federrath & Banerjee 2015) but nevertheless sufficient to obtain accurate results for the species abundances we are interested in here, as shown in Paper 1. This allows us to significantly reduce computational costs and span a large parameter space.…”

Mapping the core mass function to the initial mass function

“…7. For verification of evolution timescale in simulations, see Pan & Scannapieco (2010). Note that, as the sub-regions collapse the total ensemble density distribution -even for isothermal gas -will deviate significantly from a lognormal.…”