Effect of turbulence on collisions of dust particles with planetesimals in protoplanetary disks

Planetesimals in gaseous protoplanetary disks may grow by collecting dust particles. Hydrodynamical studies show that small particles generally avoid collisions with the planetesimals because they are entrained by the flow around them. This occurs when $St$, the Stokes number, defined as the ratio of the dust stopping time to the planetesimal crossing time, becomes much smaller than unity. However, these studies have been limited to the laminar case, whereas these disks are believed to be turbulent. We want to estimate the influence of gas turbulence on the dust-planetesimal collision rate and on the impact speeds. We used three-dimensional direct numerical simulations of a fixed sphere (planetesimal) facing a laminar and turbulent flow seeded with small inertial particles (dust) subject to a Stokes drag. A no-slip boundary condition on the planetesimal surface is modeled via a penalty method. We find that turbulence can significantly increase the collision rate of dust particles with planetesimals. For a high turbulence case (when the amplitude of turbulent fluctuations is similar to the headwind velocity), we find that the collision probability remains equal to the geometrical rate or even higher for $St\geq 0.1$, i.e., for dust sizes an order of magnitude smaller than in the laminar case. We derive expressions to calculate impact probabilities as a function of dust and planetesimal size and turbulent intensity

Universality of Velocity Gradients in Forced Burgers Turbulence

It is demonstrated that Burgers turbulence subject to large-scale white-noise-in-time random forcing has a universal power-law tail with exponent -7/2 in the probability density function of negative velocity gradients, as predicted by E, Khanin, Mazel and Sinai (1997, Phys. Rev. Lett. 78, 1904). A particle and shock tracking numerical method gives about five decades of scaling. Using a Lagrangian approach, the -7/2 law is related to the shape of the unstable manifold associated to the global minimizer.Comment: 4 pages, 2 figures, RevTex4, published versio

Dynamics and statistics of heavy particles in turbulent flows

We present the results of Direct Numerical Simulations (DNS) of turbulent flows seeded with millions of passive inertial particles. The maximum Taylor's Reynolds number is around 200. We consider particles much heavier than the carrier flow in the limit when the Stokes drag force dominates their dynamical evolution. We discuss both the transient and the stationary regimes. In the transient regime, we study the growt of inhomogeneities in the particle spatial distribution driven by the preferential concentration out of intense vortex filaments. In the stationary regime, we study the acceleration fluctuations as a function of the Stokes number in the range [0.16:3.3]. We also compare our results with those of pure fluid tracers (St=0) and we find a critical behavior of inertia for small Stokes values. Starting from the pure monodisperse statistics we also characterize polydisperse suspensions with a given mean Stokes.Comment: 13 pages, 10 figures, 2 table

Ergodic properties of a model for turbulent dispersion of inertial particles

We study a simple stochastic differential equation that models the dispersion of close heavy particles moving in a turbulent flow. In one and two dimensions, the model is closely related to the one-dimensional stationary Schroedinger equation in a random delta-correlated potential. The ergodic properties of the dispersion process are investigated by proving that its generator is hypoelliptic and using control theory

Caustics in turbulent aerosols

Networks of caustics can occur in the distribution of particles suspended in a randomly moving gas. These can facilitate coagulation of particles by bringing them into close proximity, even in cases where the trajectories do not coalesce. We show that the long-time morphology of these caustic patterns is determined by the Lyapunov exponents lambda_1, lambda_2 of the suspended particles, as well as the rate J at which particles encounter caustics. We develop a theory determining the quantities J, lambda_1, lambda_2 from the statistical properties of the gas flow, in the limit of short correlation times.Comment: 4 pages, 3 figure

The analytic structure of 2D Euler flow at short times

Using a very high precision spectral calculation applied to the incompressible and inviscid flow with initial condition $\psi_0(x_1, x_2) = \cos x_1+\cos 2x_2$, we find that the width $\delta(t)$ of its analyticity strip follows a $\ln(1/t)$ law at short times over eight decades. The asymptotic equation governing the structure of spatial complex-space singularities at short times (Frisch, Matsumoto and Bec 2003, J.Stat.Phys. 113, 761--781) is solved by a high-precision expansion method. Strong numerical evidence is obtained that singularities have infinite vorticity and lie on a complex manifold which is constructed explicitly as an envelope of analyticity disks.Comment: 19 pages, 14 figures, published versio

Superdiffusion of massive particles induced by multi-scale velocity fields

We study drag-induced diffusion of massive particles in scale-free velocity fields, where superdiffusive behavior emerges due to the scale-free size distribution of the vortices of the underlying velocity field. The results show qualitative resemblance to what is observed in fluid systems, namely the diffusive exponent for the mean square separation of pairs of particles and the preferential concentration of the particles, both as a function of the response time.Comment: 5 pages, 5 figures. Accepted for publication in EP

Turbulent dissipation in the ISM: the coexistence of forced and decaying regimes and implications for galaxy formation and evolution

We discuss the dissipation of turbulent kinetic energy Ek in the global ISM by means of 2-D, MHD, non-isothermal simulations in the presence of model radiative heating and cooling. We argue that dissipation in 2D is representative of that in three dimensions as long as it is dominated by shocks rather than by a turbulent cascade. Energy is injected at a few isolated sites in space, over relatively small scales, and over short time periods. This leads to the coexistence of forced and decaying regimes in the same flow. We find that the ISM-like flow dissipates its turbulent energy rapidly. In simulations with forcing, the input parameters are the radius l_f of the forcing region, the total kinetic energy e_k each source deposits into the flow, and the rate of formation of those regions, sfr_OB. The global dissipation time t_d depends mainly on l_f. In terms of measurable properties of the ISM, t_d >= Sigma_g u_rms^2/(e_k sfr_OB), where Sigma_g is the average gas surface density and u_rms is the rms velocity dispersion. For the solar neighborhood, t_d >= 1.5x10^7 yr. The global dissipation time is consistently smaller than the crossing time of the largest energy-containing scales. In decaying simulations, Ek decreases with time as t^-n, where n~0.8-0.9. This suggests a decay with distance d as Ek\propto d^{-2n/(2-n)} in the mixed forced+decaying case. If applicable to the vertical direction, our results support models of galaxy evolution in which stellar energy injection provides significant support for the gas disk thickness, but not models of galaxy formation in which this energy injection is supposed to reheat an intra-halo medium at distances of up to 10-20 times the optical galaxy size, as the dissipation occurs on distances comparable to the disk height.Comment: 23 pages, including figures. To appear in ApJ. Abstract abridge

Population dynamics in compressible flows

Organisms often grow, migrate and compete in liquid environments, as well as on solid surfaces. However, relatively little is known about what happens when competing species are mixed and compressed by fluid turbulence. In these lectures we review our recent work on population dynamics and population genetics in compressible velocity fields of one and two dimensions. We discuss why compressible turbulence is relevant for population dynamics in the ocean and we consider cases both where the velocity field is turbulent and when it is static. Furthermore, we investigate populations in terms of a continuos density field and when the populations are treated via discrete particles. In the last case we focus on the competition and fixation of one species compared to anotherComment: 16 pages, talk delivered at the Geilo Winter School 201