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Large-scale clustering of inertial particles in a rotating, stratified and inhomogeneous turbulence

Nathan Kleeorin, Igor Rogachevskii

TL;DR

This paper develops a mean-field theory for large-scale clustering of inertial particles in rotating, density-stratified or inhomogeneous turbulence, describing clustering through the effective pumping velocity ${\bm V}^{\rm eff} = - \left\langle \tau {\bm V} {\rm div}{\bm V} \right\rangle$. It derives explicit expressions for five distinct pumping contributions ${\bm V}^{(k)}$ under slow and fast rotation, incorporating Stokes and Epstein drag regimes and the anisotropy induced by rotation, culminating in a fast-rotation form ${\bm V}^{(\rm eff)} = f_1(\omega) D_T {\bm \lambda}_{\perp} + f_2(\omega) D_T \hat{\bm \Omega} \times {\bm \lambda}_{\perp}$ with $\omega = 2 \tau_{p} \Omega$ and $\Omega_{\ast} = 4 \Omega \tau_0$. The results show clustering localizes in the plane perpendicular to the rotation axis, with two radial maxima corresponding to the Stokes and Epstein regimes, and they are applied to planetesimal formation in accretion discs, predicting size- and radius-dependent concentration timescales. These insights illuminate how rotation and stratification drive large-scale dust concentrations in astrophysical discs, offering a framework for connecting microphysical drag regimes to macroscopic clustering and planetesimal formation timescales.

Abstract

We develop a theory of various kinds of large-scale clustering of inertial particles in a rotating density stratified or inhomogeneous turbulent fluid flows. The large-scale particle clustering occurs in scales which are much larger than the integral scale of turbulence, and it is described in terms of the effective pumping velocity in a turbulent flux of particles. We show that for a fast rotating strongly anisotropic turbulence, the large-scale clustering occurs in the plane perpendicular to rotation axis in the direction of the fluid density stratification. We apply the theory of the large-scale particle clustering for explanation of the formation of planetesimals (progenitors of planets) in accretion protoplanetary discs. We determine the radial profiles of the radial and azimuthal components of the effective pumping velocity of particles which have two maxima corresponding to different regimes of the particle--fluid interactions: at the small radius it is the Stokes regime, while at the larger radius it is the Epstein regime. With the decrease the particle radius, the distance between the maxima increases. This implies that smaller-size particles are concentrated nearby the central body of the accretion disk, while larger-size particles are accumulated far from the central body. The dynamic time of the particle clustering is about $τ_{\rm dyn} \sim 10^5$--$10^6$ years, while the turbulent diffusion time is about $10^7$ years, that is much larger than the characteristic formation time of large-scale particle clusters ($\sim τ_{\rm dyn}$).

Large-scale clustering of inertial particles in a rotating, stratified and inhomogeneous turbulence

TL;DR

This paper develops a mean-field theory for large-scale clustering of inertial particles in rotating, density-stratified or inhomogeneous turbulence, describing clustering through the effective pumping velocity . It derives explicit expressions for five distinct pumping contributions under slow and fast rotation, incorporating Stokes and Epstein drag regimes and the anisotropy induced by rotation, culminating in a fast-rotation form with and . The results show clustering localizes in the plane perpendicular to the rotation axis, with two radial maxima corresponding to the Stokes and Epstein regimes, and they are applied to planetesimal formation in accretion discs, predicting size- and radius-dependent concentration timescales. These insights illuminate how rotation and stratification drive large-scale dust concentrations in astrophysical discs, offering a framework for connecting microphysical drag regimes to macroscopic clustering and planetesimal formation timescales.

Abstract

We develop a theory of various kinds of large-scale clustering of inertial particles in a rotating density stratified or inhomogeneous turbulent fluid flows. The large-scale particle clustering occurs in scales which are much larger than the integral scale of turbulence, and it is described in terms of the effective pumping velocity in a turbulent flux of particles. We show that for a fast rotating strongly anisotropic turbulence, the large-scale clustering occurs in the plane perpendicular to rotation axis in the direction of the fluid density stratification. We apply the theory of the large-scale particle clustering for explanation of the formation of planetesimals (progenitors of planets) in accretion protoplanetary discs. We determine the radial profiles of the radial and azimuthal components of the effective pumping velocity of particles which have two maxima corresponding to different regimes of the particle--fluid interactions: at the small radius it is the Stokes regime, while at the larger radius it is the Epstein regime. With the decrease the particle radius, the distance between the maxima increases. This implies that smaller-size particles are concentrated nearby the central body of the accretion disk, while larger-size particles are accumulated far from the central body. The dynamic time of the particle clustering is about -- years, while the turbulent diffusion time is about years, that is much larger than the characteristic formation time of large-scale particle clusters ().

Paper Structure

This paper contains 13 sections, 137 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Particles in a rotating turbulence
  3. Large-scale effects
  4. The effective pumping velocity ${\bm V}^{(1)}$ in inhomogeneous and density stratified turbulence
  5. The effective pumping velocity ${\bm V}^{(2)}$
  6. The effective pumping velocity ${\bm V}^{(3)}$
  7. The effective pumping velocity ${\bm V}^{(4)}$
  8. The effective pumping velocity ${\bm V}^{(5)}$
  9. Applications to formation of planetesimals in accretion protoplanetary discs
  10. Discussion and conclusions
  11. Total effective pumping velocity of particles
  12. Effect of rotation on turbulence
  13. The identities used for the integration in the ${\bm k}$ space

Figures (4)

  • Figure 1: The functions $f_1(\omega)$ (solid) and $f_2(\omega)$ (dashed) for $\Omega_\ast =60$.
  • Figure 2: The radial profiles of the parameter $\omega(r_{_{\rm AU}})$ for different particle radius: $a_{\rm p} =$ 5 cm (solid), 10 cm (dashed); 50 cm (dashed-dotted). Here $r_{_{\rm AU}}$ is the radial coordinate measured in the astronomical units $L_{_{\rm AU}}=1.5 \times 10^{13}$ cm.
  • Figure 3: The radial profiles of the azimuthal $V_\varphi^{(\rm eff)}(r_{_{\rm AU}})$ component of the effective velocity for different particle radius: $a_{\rm p} =$ 5 cm (solid), 10 cm (dashed-dotted); 50 cm (solid blue); and of the radial component $V_r^{(\rm eff)}(r_{_{\rm AU}})$ of the effective velocity for different particle size $a_{\rm p} =$ 5 cm (dashed), 10 cm (dotted); 50 cm (dashed blue) for $\Omega_\ast =60$ (that corresponds to $\alpha=1/225$). The velocity is measured in cm/s.
  • Figure 4: The equilibrium radial profile of the normalised mean particle number density $\overline{n}(r_{_{\rm AU}})$ for different particle radius: $a_{\rm p} =$ 5 cm (dashed-dotted), 10 cm (dashed); 50 cm (solid) and for $\Omega_\ast =60$.