PHYSICS OF FLUIDS VOLUME 11, NUMBER 8 AUGUST 1999 Particle aggregation in a turbulent Keplerian flow A. Bracco, P. H. Chavanis, and A. Provenzale Istituto di Cosmogeofisica, Corso Fiume, 10133 Torino, Italy E. A. Spiegela) Astronomy Department, Columbia University, New York, New York 10027 ~Received 21 October 1998; accepted 9 April 1999! For the problem of planetary formation one seeks a mechanism to gather small dust particles together into larger solid objects. Here we describe a scenario in which turbulence mediates this process by aggregating particles into anticyclonic regions. If, as our simulations suggest, anticyclonic vortices form as long-lived coherent structures, the process becomes more powerful because such vortices trap particles effectively. Even if the turbulence is decaying, following the upheaval that formed the disk, there is enough time to make the dust distribution quite lumpy. © 1999 American Institute of Physics. @S1070-6631~99!00208-1# I. INTRODUCTION Laplace disks was somehow mediated by the interaction of ¨ 8 To rationalize the distribution and the motions of the turbulence and rotation was adumbrated by von Weizsaker. planets and their satellites in the solar system, Kant1 and Subsequently, experiments showed how a field of vortices of 2 opposite senses can produce inhomogeneous particle Laplace separately introduced forms of the nebular hypoth- 9,10 esis, a conjecture that the solar system formed from a flat- distributions. The experimental evidence that anticyclonic tened gas cloud or disk. Their prescience has been con- vortices ~whose vorticity is in the sense counter to that of the firmed: disks do occur in many places in the universe. Since prevailing global motions! can trap particles is now 11,12 much of the interesting behavior in disks has been ascribed extensive. to turbulence, a problem motivated by them is appropriate in An application of the mechanism of anomalous diffusion 13,14 a volume dedicated to Robert Kraichnan. That problem is to by anticyclonic vortices is to the formation of planets, as understand how the planets formed in the primitive solar seen in several numerical studies of Langrangian orbits in 15 nebula. prescribed vorticity fields. In this paper, we consider flows Stars condense from an interstellar medium consisting of the kind that would arise in a Keplerian disk and follow mostly of gas with an admixture of solid particles called the aggregation process they produce. Our purpose is to interstellar dust. Both observational and numerical studies show how this aggregation process can work despite some suggest that, as the central star contracts, it leaves around it objections to the notion ~to be discussed below! that pro- material that contains a good share of the initial angular mo- toplanetary disks can sustain turbulence.16,17 Without turbu- mentum of the whole system. In this nebula, the centrifugal lence, it is unlikely that vortices on disks form at all. force balances the stellar gravity in the radial direction and a Our way around this impasse is to note that the initial disk is formed. As the dust settles toward the midplane of aggregation of dust could take place in the decaying turbu- this disk it is somehow accumulated into protoplanetary ob- lence that was likely to have occured in the formation of the jects. nebula. Once this has happened, the usual sequel of gravita- The general picture has been reviewed by Lissauer3 who tional clumping can be presumed to take over. We shall has provided an extensive bibliography. Other summaries present simulations to show how anticyclonic vortices form exist ~such as Ref. 4! and there are a number of introductory from sufficiently large initial perturbations of Keplerian expositions ~for example, Ref. 5!. Two main possibilities disks ~see also Refs. 18–20!, just as in earlier simulations have been considered. If the disk is not turbulent, solid dust with mild shears,21,22 and strongly promote the aggregation particles settle into a very thin layer and then can accumulate of dust. The question we then investigate is whether the vor- 6,7 into larger solid objects by gravitational instability. On the tices have time before they die away to strongly aggregate other hand, if the disk sustains turbulence, it has been the particles. thought that one must rely on a slower process of adhesion or In Sec. II, we write down the equations of motion to set growth by two-body collisions. We examine here an alterna- the notation in order. We turn then to a brief summary of tive possibility in which turbulence in the early history of a relevant aspects of disks, after which we exhibit results of protoplanetary disk leads to a rapid aggregation of particles simulations of a Keplerian flow. These show how long-lived by anomalous diffusion. Even if the turbulence is decaying, anticyclonic vortices form from initial finite disturbances. there is time for the particles to aggregate significantly. Then, from a study of the Lagrangian orbits of dust in this The idea that the formation of planets in the Kant– flow, we illustrate how an initially homogeneous distribution of particles is redistributed into strong local accumulations of a!Electronic mail: [email protected] particles. 1070-6631/99/11(8)/2280/8/$15.002280 © 1999 American Institute of Physics Downloaded 17 Jul 2001 to 128.128.249.17. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp Phys. Fluids, Vol. 11, No. 8, August 1999 Particle aggregation in a turbulent Keplerian flow 2281 II. EQUATIONS OF MOTION such as planet formation since many readers of this journal may not have previously encountered some of the issues To concentrate on vortex dynamics, we restrict ourselves raised. On the other hand, the occurrence of disks of many to incompressible flow with sizes throughout the universe is so well documented that our “–v50. ~1! discussion can safely start from the assumption that one of these once surrounded the protosun. We shall not go into the We also assume constant viscosity so that the equation of processes by which disks are thought to form but consider motion is only the simplest case of a disk that is axisymmetric in the Dv 1 large and symmetric about the midplane z50. In the radial 52 ¹p2¹F1nDv, ~2! Dt r direction, the principal support against the gravitational pull of the protosun is provided by centrifugal force, while in the where 52GM/r is the gravitational potential of the cen- F z direction the support is by a pressure gradient. If the ratio tral mass, the protosun. of these two support forces is large, the disk is quite thin, as The key quantity here is the vorticity, for which the evo- we shall presume here. lution equation is Let the rotational speed at distance r from the central Dv protosun be v. The centrifugal balance condition is 5v–“v1nDv. ~3! Dt v252GM/r, ~7! In a disk, the large scale flow is nearly two dimensional where M is the mass of the protosun. This describes a Keple- and we adopt this simplification here. Then we may express rian flow that is stable to small perturbations since it has no v in terms of a stream function c such that, in Cartesian inflection points and the Rayleigh discriminant, coordinates, the velocity field is given by D5r23d~r2V!2/dr, ~8! ]c ]c 2 , . ~4! S ]y ]x D is positive. In astronomy, where this stability condition is often met, it is custumary to introduce the epicyclic fre- The vorticity then has only one component, v, and it is re- quency k5AD, whose role in the dynamics is not wholly lated to the stream function by different from the buoyancy frequency of stably stratified v5¹2c. ~5! fluids. Since the disk is thin, we may identify r with the radius in cylindrical coordinates. In our two-dimensional Hence Eq. ~3! may be written as simulations, we retain this meaning of r and the form 2 ]v 2GM/r for the gravitational acceleration. 1J~c,v!5nDv, ~6! ]t If the flow remains laminar, there will be a slow viscous transport of angular momentum outward that allows a weak where J(c,v)5]xc]yv2]xv]yc. inward spiraling of the material onto the protosun. The two- In the absence of dissipation (n50), Eq. ~6! admits an dimensional version of such motion has been studied in the infinite number of conserved quantities, two of which are context of drag reduction theory30 where it is thought that a quadratic invariants; these are the kinetic energy E slow inflow can inhibit boundary layer separation and reduce 2 51/2*(¹c) dx dy and the enstrophy Z the drag on airfoils. This process is too slow for astrophysi- 2 2 51/2*(¹ c) dx dy. The conservation of these two quanti- cal purposes when it is laminar, but its turbulent analogue ties in the inviscid limit is what makes two-dimensional tur- has been invoked for the case of astrophysical disks,31 which bulence so different from three-dimensional turbulence at are called accretion disks because of this inflow. In the tur- 23,24 large Reynolds numbers. In two-dimensional turbulence bulent case, it is suggested that the dissipation rate could be there is a direct cascade of enstrophy from large to small high enough to render some disks quite bright. But can we scales and an inverse cascade from small to large scales of assume that disks are turbulent? 25–27 kinetic energy. Until recently, many took it for granted that Keplerian For freely decaying, barotropic turbulence in shear-free disks were turbulent because the Reynolds number in a disk environments, intense long-lived vorticity concentrations are is literally astronomical.