Streaming Instability in Protoplanetary Disks

• Streaming instability (SI) is a drag-driven collective instability in protoplanetary disks where interactions between dust and gas generate turbulent clumping.
• Thresholds for strong clumping depend critically on particle stopping times and local dust-to-gas ratios, which shape the efficiency of planetesimal formation.
• Nonlinear SI dynamics create dense, radially narrow filaments and promote coagulation, enabling the retention of primordial pebbles in emerging planetesimals.

Searching arXiv for recent and foundational papers on streaming instability, thresholds, morphology, and planetesimal formation. Streaming instability (SI) is a drag-driven collective instability of solids and gas in protoplanetary disks. It arises because pressure-supported gas orbits slightly sub-Keplerian while solids tend to orbit closer to Keplerian speed, so particles feel a headwind, drift, and exchange momentum with the gas. In its nonlinear state, SI concentrates solids into dense filaments and clumps that can exceed Hill- or Roche-like collapse thresholds and seed planetesimal formation; in local collapse models, the resulting bodies can retain primordial mm–cm pebbles and develop radially sorted pebble-pile interiors (Magnan et al., 2024, Lim et al., 22 Sep 2025, Visser et al., 2021).

1. Dynamical origin and two-fluid mechanism

The standard control parameters are the solid-to-gas column density ratio $Z=\Sigma_d/\Sigma_g$, the dimensionless stopping time $\tau_s=\Omega_K t_{\rm stop}$, the midplane dust-to-gas density ratio $\epsilon=\rho_d/\rho_g$, and the pressure-gradient parameter $\Pi=\eta v_K/c_s$. In the Nakagawa–Sekiya–Hayashi equilibrium, the dust radial drift speed is

$$v_{p,x}=-\frac{2\,\eta v_K\,\tau_s}{(1+\epsilon)^2+\tau_s^2},$$

so increasing $\epsilon$ slows radial drift by reducing the headwind through dust backreaction. This headwind-driven free energy is the classical SI energy source, and it sets both the characteristic drift and the nonlinear tendency toward filament formation (Lim et al., 22 Sep 2025).

A recent physical interpretation casts SI as a resonant drag instability involving inertial waves. In that picture, the gas supports inertial waves with

$\omega_{\rm IW}=\pm \kappa \frac{k_z}{k},$

and resonance occurs when the dust advection frequency matches the inertial-wave frequency,

$\omega-k\!\cdot\! w_s=\omega_{\rm IW}.$

The feedback loop has a forward action, in which an inertial wave concentrates dust, and a backward reaction, in which the drifting dust clump excites an inertial wave. Each step contains a fast and a slow mechanism; the fast–fast loop is stable, while the mixed fast–slow and slow–fast loops are unstable. This formulation clarifies why SI requires both rotation and pressure-supported drift, and why fully three-dimensional geometry is intrinsic to the mechanism (Magnan et al., 2024).

The same dynamical framework explains several later extensions. In vertically stratified calculations, SI growth and saturation are controlled not only by $Z$ and $\tau_s$, but also by the thickness of the dust layer, because $\tau_s=\Omega_K t_{\rm stop}$0 links column enrichment to the midplane mass loading that directly enters the drag feedback (Lim et al., 22 Sep 2025).

2. Thresholds for strong clumping

Threshold studies commonly define “strong clumping” by requiring peak particle density to exceed a Hill- or Roche-like threshold. In stratified simulations without particle self-gravity, a widely used proxy is $\tau_s=\Omega_K t_{\rm stop}$1 with $\tau_s=\Omega_K t_{\rm stop}$2 for the adopted normalization. The principal result of threshold work is that the critical solid abundance is a strong function of stopping time: it is lowest for moderately coupled particles and rises sharply for small solids (Li et al., 2021).

For pure SI at $\tau_s=\Omega_K t_{\rm stop}$3, one survey found a minimum threshold $\tau_s=\Omega_K t_{\rm stop}$4 at $\tau_s=\Omega_K t_{\rm stop}$5, with $\tau_s=\Omega_K t_{\rm stop}$6–$\tau_s=\Omega_K t_{\rm stop}$7 for $\tau_s=\Omega_K t_{\rm stop}$8–$\tau_s=\Omega_K t_{\rm stop}$9, followed by a sharp rise to $\epsilon=\rho_d/\rho_g$0 at $\epsilon=\rho_d/\rho_g$1 and $\epsilon=\rho_d/\rho_g$2 at $\epsilon=\rho_d/\rho_g$3. The same work emphasized that the relevant criterion is not simply $\epsilon=\rho_d/\rho_g$4: the fitted midplane threshold reaches $\epsilon=\rho_d/\rho_g$5 for $\epsilon=\rho_d/\rho_g$6 and is substantially smaller for intermediate stopping times (Li et al., 2021).

Three-dimensional threshold mapping revised this picture in two ways. First, for $\epsilon=\rho_d/\rho_g$7 the empirical 3D fit is

$\epsilon=\rho_d/\rho_g$8

with $\epsilon=\rho_d/\rho_g$9 at $\Pi=\eta v_K/c_s$0 and $\Pi=\eta v_K/c_s$1 at $\Pi=\eta v_K/c_s$2–$\Pi=\eta v_K/c_s$3. Second, 3D introduces a sharp transition between $\Pi=\eta v_K/c_s$4 and $\Pi=\eta v_K/c_s$5: $\Pi=\eta v_K/c_s$6 rises from $\Pi=\eta v_K/c_s$7–$\Pi=\eta v_K/c_s$8 at $\Pi=\eta v_K/c_s$9 to $$v_{p,x}=-\frac{2\,\eta v_K\,\tau_s}{(1+\epsilon)^2+\tau_s^2},$$0–$$v_{p,x}=-\frac{2\,\eta v_K\,\tau_s}{(1+\epsilon)^2+\tau_s^2},$$1 at $$v_{p,x}=-\frac{2\,\eta v_K\,\tau_s}{(1+\epsilon)^2+\tau_s^2},$$2, and exceeds $$v_{p,x}=-\frac{2\,\eta v_K\,\tau_s}{(1+\epsilon)^2+\tau_s^2},$$3 at $$v_{p,x}=-\frac{2\,\eta v_K\,\tau_s}{(1+\epsilon)^2+\tau_s^2},$$4. The sign of $$v_{p,x}=-\frac{2\,\eta v_K\,\tau_s}{(1+\epsilon)^2+\tau_s^2},$$5 correlates with this bifurcation: for $$v_{p,x}=-\frac{2\,\eta v_K\,\tau_s}{(1+\epsilon)^2+\tau_s^2},$$6, $$v_{p,x}=-\frac{2\,\eta v_K\,\tau_s}{(1+\epsilon)^2+\tau_s^2},$$7 and 3D lowers thresholds relative to 2D; for $$v_{p,x}=-\frac{2\,\eta v_K\,\tau_s}{(1+\epsilon)^2+\tau_s^2},$$8, $$v_{p,x}=-\frac{2\,\eta v_K\,\tau_s}{(1+\epsilon)^2+\tau_s^2},$$9 and 3D raises them (Lim et al., 22 Sep 2025).

High-resolution axisymmetric calculations complicate the small-grain regime further. For $\epsilon$0, one re-analysis found strong clumping already at $\epsilon$1 and $\epsilon$2, but not at $\epsilon$3, revising the earlier 2D threshold downward to roughly interstellar metallicity. The revised fit was

$\epsilon$4

and the authors argued that large radial domains and high resolution were essential to capture the necessary filament interactions (Lim et al., 2024).

Later disk-evolution studies often recast the threshold in terms of the midplane ratio,

$\epsilon$5

This gives $\epsilon$6 at $\epsilon$7, $\epsilon$8 at $\epsilon$9, and $\omega_{\rm IW}=\pm \kappa \frac{k_z}{k},$0 at $\omega_{\rm IW}=\pm \kappa \frac{k_z}{k},$1. Those values are substantially more restrictive than laminar $\omega_{\rm IW}=\pm \kappa \frac{k_z}{k},$2-based thresholds, and they are used in recent one-dimensional disk models that include turbulence, settling, and evolving pressure gradients (Gonzalez et al., 27 Jan 2026, Ying et al., 26 Jan 2026).

3. Nonlinear morphology, dimensionality, and numerical structure

In the nonlinear regime, SI organizes solids into radially narrow, azimuthally extended filaments. Three-dimensional simulations show that the morphology depends sharply on $\omega_{\rm IW}=\pm \kappa \frac{k_z}{k},$3. For $\omega_{\rm IW}=\pm \kappa \frac{k_z}{k},$4, 3D runs develop a “filaments-in-filaments” structure in which a non-axisymmetric parent filament hosts multiple closely spaced subfilaments that merge stochastically and drive peak densities above 2D values. For $\omega_{\rm IW}=\pm \kappa \frac{k_z}{k},$5 and $\omega_{\rm IW}=\pm \kappa \frac{k_z}{k},$6, by contrast, the filament pattern becomes nearly stationary in 3D: individual particles drift inward, but the filament network does not, suppressing filament mergers and raising $\omega_{\rm IW}=\pm \kappa \frac{k_z}{k},$7 relative to 2D (Lim et al., 22 Sep 2025).

This dimensional sensitivity is one reason that linear, unstratified SI growth rates are a poor predictor of strong clumping in stratified simulations. A detailed threshold survey found that even when the fastest resolvable unstratified modes are comparable across runs, clumping outcomes differ sharply once stratification, diffusion, and nonlinear evolution are included. In particular, the measured damped linear rates can all be negative while nonlinear stratified runs still develop strong clumping, so linear growth alone does not supply a reliable collapse criterion (Li et al., 2021).

At $\omega_{\rm IW}=\pm \kappa \frac{k_z}{k},$8, however, the pre-collapse saturated turbulence at the midplane of stratified disks is quantitatively close to that in unstratified simulations. In that regime, velocity dispersions scale approximately linearly with $\omega_{\rm IW}=\pm \kappa \frac{k_z}{k},$9, dust and gas diffusion scale as $\omega-k\!\cdot\! w_s=\omega_{\rm IW}.$0, and the filament scale is $\omega-k\!\cdot\! w_s=\omega_{\rm IW}.$1. This agreement holds only before strong clumping sets in; once collapse-proximate overdensities appear, stratification and vertical gravity become dynamically essential (Lim et al., 29 May 2025).

A persistent misconception is that SI works by generating particle-trapping pressure bumps. Numerical robustness studies show otherwise. SI does excite zonal flows, but their amplitudes are too small to halt radial drift; in one set of vertically stratified 3D runs, the strongest zonal flows had $\omega-k\!\cdot\! w_s=\omega_{\rm IW}.$2 and $\omega-k\!\cdot\! w_s=\omega_{\rm IW}.$3, far below the background headwind $\omega-k\!\cdot\! w_s=\omega_{\rm IW}.$4. Filament spacing remained robust at $\omega-k\!\cdot\! w_s=\omega_{\rm IW}.$5 across vertical boundary conditions, indicating that the dominant structure is set by SI dynamics, not by pressure trapping (Li et al., 2018).

Resolution can modify the apparent relation between settling and clumping. In high-resolution $\omega-k\!\cdot\! w_s=\omega_{\rm IW}.$6 axisymmetric runs, the dust layer became thicker and $\omega-k\!\cdot\! w_s=\omega_{\rm IW}.$7 smaller, possibly because vertical-shearing streaming instability–like motions were resolved more effectively, yet strong clumping still emerged when $\omega-k\!\cdot\! w_s=\omega_{\rm IW}.$8. This suggests that large-scale filament interaction and merger can matter as much as instantaneous $\omega-k\!\cdot\! w_s=\omega_{\rm IW}.$9 in marginal regimes (Lim et al., 2024).

4. Collapse, planetesimal demographics, and internal structure

When SI-generated filaments become self-gravitating, they fragment into bound clumps whose initial mass distribution is not universal. A high-resolution 3D self-gravitating study found that different runs preferred different statistical forms, with breaks at masses corresponding to collapsed planetesimals of order tens to $Z$0 km, depending on the mapping from the simulation mass scale $Z$1 to physical radius. The same work reported evidence for a low-mass turnover in the high-resolution run, consistent with gravitational collapse imposing a minimum bound scale (Li et al., 2019).

The internal evolution of an already bound SI clump can be followed further. A local collapse study initialized a spherical pebble cloud inside its Hill radius,

$Z$2

and evolved gas drag, self-gravity, and mutual collisions. In the gas-coupled regime, pebbles settle at terminal velocity,

$Z$3

with a fall time $Z$4. The consequence is aerodynamic ordering: pebbles with $Z$5 reach the center first, smaller pebbles arrive later, and the final body becomes radially size-stratified, with larger pebbles concentrated in the interior and smaller pebbles toward the surface (Visser et al., 2021).

That collapse does not virialize in the usual sense. The same simulations found that virial equilibrium is never reached; instead, the energy ratio $Z$6 peaks sharply when the largest pebbles settle nearly simultaneously to form the first core, then drops as the active cloud is depleted. Early core formation and optical thickness prevent particles from passing through the center and re-virializing, so the collapse is terminal-settling-dominated rather than oscillatory (Visser et al., 2021).

Collisional processing during this local collapse is weak. At the accretion radius where the innermost fraction first becomes optically thick, the critical terminal velocity scales as

$Z$7

so $Z$8 for $Z$9 and $\tau_s$0 for $\tau_s$1, well below the adopted fragmentation threshold $\tau_s$2. The observed collisions are therefore rare and sticking, which supports the interpretation of comets and small planetesimals as pebble piles composed of primordial mm–cm building blocks. The model was noted to be consistent with Rosetta observations of comet 67P, including low bulk density of about $\tau_s$3, porosity up to about $\tau_s$4, and layered pebble-pile morphology in the mm–cm range (Visser et al., 2021).

5. Dust growth, fragmentation barriers, and polydispersity

A central difficulty for SI is that the fastest clumping often occurs for particles with $\tau_s$5, while collisional fragmentation and radial drift can prevent growth to that regime. Polydisperse linear theory shows that a pure interstellar-like power law tends to slow the instability relative to the monodisperse case, but realistic dust evolution that enhances the largest grains restores growth rates close to monodisperse SI. The practical condition identified there was a size distribution with a peak stopping time of order $\tau_s$6 orbit and local dust-to-gas volume density ratio of order unity (McNally et al., 2021).

Moderate SI clumping can itself help overcome the growth barriers. In unstratified simulations that measured collision statistics directly from particle trajectories, collision velocities were typically $\tau_s$7, while coagulation rates were $\tau_s$8–$\tau_s$9 for $\tau_s=\Omega_K t_{\rm stop}$00. These imply growth times of order ten Keplerian periods or shorter, well below radial-drift times, so SI-assisted coagulation can bridge from $\tau_s=\Omega_K t_{\rm stop}$01 toward the more SI-active $\tau_s=\Omega_K t_{\rm stop}$02–$\tau_s=\Omega_K t_{\rm stop}$03 regime. A simple vertical active-zone model then reduced the critical midplane dust-to-gas ratio for eventual planetesimal formation to about $\tau_s=\Omega_K t_{\rm stop}$04 for initial $\tau_s=\Omega_K t_{\rm stop}$05 (Tominaga et al., 3 Mar 2025).

A complementary route is a feedback between clumping and fragmentation physics. Two-dimensional SI simulations with monodisperse growth and a mass-loading-modified collision model replaced the sound speed in the turbulent relative velocity by

$\tau_s=\Omega_K t_{\rm stop}$06

so that higher local dust loading reduces collision velocities and raises the fragmentation barrier. In a case with $\tau_s=\Omega_K t_{\rm stop}$07 and low fragmentation threshold, growth without mass loading stalled at $\tau_s=\Omega_K t_{\rm stop}$08 with $\tau_s=\Omega_K t_{\rm stop}$09, whereas mass loading shifted the barrier to $\tau_s=\Omega_K t_{\rm stop}$10 and raised $\tau_s=\Omega_K t_{\rm stop}$11 to about $\tau_s=\Omega_K t_{\rm stop}$12. For $\tau_s=\Omega_K t_{\rm stop}$13 and a higher fragmentation threshold, the same mechanism reached $\tau_s=\Omega_K t_{\rm stop}$14 and $\tau_s=\Omega_K t_{\rm stop}$15 (Vallucci-Goy et al., 27 Nov 2025).

This synergy is not unlimited. Once particles decouple with $\tau_s=\Omega_K t_{\rm stop}$16, further growth no longer strengthens SI. The most favorable regime is therefore the “tightly coupled” one with $\tau_s=\Omega_K t_{\rm stop}$17, where SI raises $\tau_s=\Omega_K t_{\rm stop}$18, lowered collision speeds permit growth toward $\tau_s=\Omega_K t_{\rm stop}$19, and the enhanced stopping time in turn strengthens clumping (Vallucci-Goy et al., 27 Nov 2025).

6. Environmental regulation, variants, and empirical constraints

Disk substructure can either enable or inhibit SI. Stellar X-ray photoevaporation, for example, opens an inner cavity and creates a pressure maximum at the cavity edge that traps drifting pebbles. In one DustPy model with $\tau_s=\Omega_K t_{\rm stop}$20, $\tau_s=\Omega_K t_{\rm stop}$21, $\tau_s=\Omega_K t_{\rm stop}$22, $\tau_s=\Omega_K t_{\rm stop}$23, $\tau_s=\Omega_K t_{\rm stop}$24, and $\tau_s=\Omega_K t_{\rm stop}$25, the cavity edge reached $\tau_s=\Omega_K t_{\rm stop}$26–$\tau_s=\Omega_K t_{\rm stop}$27 and $\tau_s=\Omega_K t_{\rm stop}$28, producing $\tau_s=\Omega_K t_{\rm stop}$29 of planetesimals with an initial dust-to-planetesimal conversion efficiency of $\tau_s=\Omega_K t_{\rm stop}$30. Larger disks, higher metallicity, lower viscosity, higher fragmentation thresholds, and higher X-ray luminosities all increased the yield (Ying et al., 26 Jan 2026).

Thermochemical structure can also matter. In porous-grain disk models, including the CO snow line and lowering the fragmentation threshold exterior to it produced extended regions with $\tau_s=\Omega_K t_{\rm stop}$31–$\tau_s=\Omega_K t_{\rm stop}$32 and $\tau_s=\Omega_K t_{\rm stop}$33 within the first $\tau_s=\Omega_K t_{\rm stop}$34–$\tau_s=\Omega_K t_{\rm stop}$35 kyr. With the CO snow line, strong-clumping conditions extended from the inner edge to about $\tau_s=\Omega_K t_{\rm stop}$36 au by $\tau_s=\Omega_K t_{\rm stop}$37 kyr, while neglecting the CO snow line restricted early clumping to the inner $\tau_s=\Omega_K t_{\rm stop}$38 au and delayed broad outer-disk clumping until after about $\tau_s=\Omega_K t_{\rm stop}$39 yr (Gonzalez et al., 27 Jan 2026).

A second class of extensions concerns the free-energy source itself. In laminar accreting disks, an external azimuthal torque can drive the azimuthal-drift streaming instability (AdSI), which operates even when the global radial pressure gradient vanishes. Parameter surveys found that for $\tau_s=\Omega_K t_{\rm stop}$40 and $\tau_s=\Omega_K t_{\rm stop}$41, AdSI forms filaments down to $\tau_s=\Omega_K t_{\rm stop}$42, while for $\tau_s=\Omega_K t_{\rm stop}$43 it reaches peak dust-to-gas ratios of order $\tau_s=\Omega_K t_{\rm stop}$44. In mixed cases where classical SI would otherwise saturate in small-scale turbulence, adding the accretion torque promoted filament formation and stronger clumping (Wang et al., 2024).

The impact of turbulence depends strongly on how it is generated. Isotropically forced turbulence can raise clumping thresholds substantially, but self-consistent MRI turbulence with ambipolar diffusion appears less destructive because it also generates zonal flows that trap particles. In 3D stratified shearing boxes with $\tau_s=\Omega_K t_{\rm stop}$45, modest MRI turbulence with $\tau_s=\Omega_K t_{\rm stop}$46 yielded a clumping boundary similar to pure SI and permitted strong clumping down to $\tau_s=\Omega_K t_{\rm stop}$47 for $\tau_s=\Omega_K t_{\rm stop}$48; stronger MRI turbulence with $\tau_s=\Omega_K t_{\rm stop}$49 made the critical metallicity supersolar, but still several times lower than isotropically forced turbulence at comparable diffusion parameters (Eriksson et al., 17 Mar 2026).

Not every pressure bump is favorable. A large 3D shearing-box study of mm grains in an axisymmetric pressure bump at $\tau_s=\Omega_K t_{\rm stop}$50 au, with $\tau_s=\Omega_K t_{\rm stop}$51 and local $\tau_s=\Omega_K t_{\rm stop}$52 reaching $\tau_s=\Omega_K t_{\rm stop}$53, found no strong SI clumping even though those values far exceeded small-box threshold estimates. The stated reason was dynamical: the particle crossing time through the high-$\tau_s=\Omega_K t_{\rm stop}$54 region was shorter than the SI growth time, motivating an added criterion

$\tau_s=\Omega_K t_{\rm stop}$55

Slightly stronger bumps in the same setup reached the Roche density and formed planetesimals by gravitational instability instead, implying that mm grains in axisymmetric bumps likely need vortices or other long-residence-time structures if SI is to operate (Carrera et al., 2022).

Observationally, the outer ring of HD 163296 at $\tau_s=\Omega_K t_{\rm stop}$56 au has been argued to occupy an SI-favorable regime. Using a fragmentation-limited growth model, one study inferred $\tau_s=\Omega_K t_{\rm stop}$57, $\tau_s=\Omega_K t_{\rm stop}$58, and a strongly anisotropic turbulence field with $\tau_s=\Omega_K t_{\rm stop}$59, implying $\tau_s=\Omega_K t_{\rm stop}$60–$\tau_s=\Omega_K t_{\rm stop}$61 and $\tau_s=\Omega_K t_{\rm stop}$62–$\tau_s=\Omega_K t_{\rm stop}$63. That combination was interpreted as evidence that SI-driven clumping is plausible in the $\tau_s=\Omega_K t_{\rm stop}$64 au ring, whereas the $\tau_s=\Omega_K t_{\rm stop}$65 au ring with lower $\tau_s=\Omega_K t_{\rm stop}$66, smaller $\tau_s=\Omega_K t_{\rm stop}$67, and $\tau_s=\Omega_K t_{\rm stop}$68 was judged unfavorable (Zagaria et al., 2023).

Finally, SI-like behavior has been probed experimentally. Laboratory particle-stream experiments in a dilute gas with mean dust-to-gas mass density ratio approaching or greater than unity showed variations in particle concentration along the drag direction, a tendency for nearby particles to “catch up,” clumping on very small scales implying local enhancements by factors of several tens above the background ratio, and evidence for collective drag reduction at high local particle density in the continuum drag regime. Those measurements do not reproduce a full rotating disk, but they support the broader claim that drag-mediated collective concentration is a real hydrodynamic effect rather than a purely numerical construction (Capelo et al., 2018).