Streaming Instability in Protoplanetary Disks

• Streaming instability is a drag-driven process in protoplanetary disks where differential dust–gas drift, aided by pressure support, amplifies dust density via resonant coupling with inertial waves.
• The instability’s nonlinear evolution forms dense filaments and clumps, potentially collapsing under self-gravity to initiate planetesimal formation.
• Research shows SI is highly sensitive to factors like particle size distribution, turbulence, and pressure gradients, with ongoing debates over the dominant resonant branches and model approximations.

Streaming instability is a drag-driven instability of a gas–solid mixture in protoplanetary disks. It arises because gas is partially pressure-supported and therefore rotates slightly sub-Keplerian, while solids do not feel pressure and drift relative to the gas under aerodynamic drag. Through drag back-reaction, this differential drift can amplify dust-density perturbations into dense filaments and clumps, and those concentrations can become sufficiently dense to collapse under particle self-gravity and form planetesimals (Yang et al., 2016, Lim et al., 2023, Schäfer et al., 2024). In modern linear theory, the instability is understood as a Resonant Drag Instability associated with inertial or epicyclic waves in the rotating gas (Squire et al., 2017, Magnan et al., 2024).

1. Physical setting and basic variables

The standard setting is a local shearing box co-rotating with angular velocity $\Omega$, with Cartesian coordinates $x,y,z$ corresponding to radial, azimuthal, and vertical directions. Gas has density $\rho_g$ and velocity $\mathbf{u}$; dust has density $\rho_d$ and velocity $\mathbf{v}$; the local dust-to-gas ratio is $\epsilon=\rho_d/\rho_g$, and the vertically integrated solids-to-gas ratio is $Z=\Sigma_p/\Sigma_g$ (Yang et al., 2016, Auffinger et al., 2017). Aerodynamic coupling is measured by the stopping time $t_s$, or equivalently by the Stokes number

$\mathrm{St}=\Omega\,t_s .$

Tightly coupled particles satisfy $x,y,z$0, while $x,y,z$1–1 is the regime classically associated with rapid linear growth and strong nonlinear concentration (Yang et al., 2016, Squire et al., 2017).

The gas orbits at a speed slightly below Keplerian because of the radial pressure gradient. A common dimensionless measure of that pressure support is

$x,y,z$2

where $x,y,z$3 encodes the sub-Keplerian offset (Schäfer et al., 2022, Umurhan et al., 2019). The resulting headwind causes dust to lose angular momentum and drift inward. In the unperturbed state, gas and dust obey the Nakagawa–Sekiya–Hayashi equilibrium, in which gas drifts slightly outward, dust drifts inward, and both radial and azimuthal velocities depend on $x,y,z$4 and $x,y,z$5 (Auffinger et al., 2017, Yang et al., 2016).

A standard two-fluid description consists of gas continuity and momentum equations coupled to dust continuity and momentum equations through drag. In the local shearing box, the gas momentum equation contains pressure, Coriolis, tidal, viscosity, and drag back-reaction terms, while dust is pressureless and inviscid but feels drag, Coriolis, and tidal forces (Auffinger et al., 2017, Krapp et al., 2019). This framework is the basis of both classical linear stability analysis and nonlinear simulations.

2. Linear theory, resonance, and the physical mechanism

A major conceptual advance is the identification of the streaming instability as an epicyclic or inertial-wave Resonant Drag Instability. In a rotating disk, the gas supports epicyclic oscillations with frequency $x,y,z$6, and the dust possesses an advective mode with frequency $x,y,z$7, where $x,y,z$8 is the background dust–gas drift. The resonant condition

$x,y,z$9

picks out the fastest-growing branch of the classical streaming instability (Squire et al., 2017). In this view, SI is not an isolated peculiarity of disk drift equilibria but one member of the broader class of drag-driven resonant instabilities.

Recent work clarifies the feedback loop underlying this resonance. In the “forward action,” an inertial wave concentrates dust into clumps; in the “backward reaction,” those drifting dust clumps excite an inertial wave. Each of these directions separates into a fast and a slow mechanism. At resonance, each forward mechanism can couple with a backward mechanism to close a feedback loop. The fast–fast loop is stable, so the streaming instability uses the fast–slow and slow–fast loops instead (Magnan et al., 2024). In that formulation, pressure bumps and imperfect radial entrainment supply the forward concentration of dust, while radial and azimuthal drag back-reaction supply the backward forcing of the gas wave.

This RDI picture does not eliminate the older modal language; rather, it reorganizes it. A third-order expansion in the Stokes number shows that the classical secular mode can be stable, while epicycles can be unstable and are identified by Green’s function analysis as promising channels for planetesimals formation (Jaupart et al., 2020). A plausible implication is that the standard emphasis on the secular branch alone is incomplete, particularly in the small-$\rho_g$0 regime and in viscous environments.

3. Terminal-velocity approximations and the question of what SI remains

For tightly coupled particles, many studies adopt a terminal velocity approximation, in which dust–gas slip is assumed to relax instantaneously. That approximation is asymptotic and must be used with a consistent ordering in $\rho_g$1. According to the abstract of “Does the Streaming Instability exist within the Terminal Velocity Approximation?”, the linearised equations that have been commonly used to study the Streaming Instability within the terminal velocity approximation actually exceed the accuracy of this approximation; for that reason, the corresponding dispersion equation recovers the long wavelength branch of the resonant Streaming Instability caused by the stationary azimuthal drift of the dust, but the refined equations for gas-dust dynamics in the terminal velocity approximation does not lead to the resonant Streaming Instability (Zhuravlev, 2021).

A related conclusion appears in the first paper of the polydisperse-streaming-instability series. In a continuum of tightly coupled particle sizes treated under the terminal velocity approximation, unstable modes that grow exponentially on a dynamical time scale exist, but for dust to gas ratios much smaller than unity they are confined to radial wave numbers that are a factor $\rho_g$2 larger than where the monodisperse streaming instability growth rates peak (Paardekooper et al., 2020). For dust to gas ratios larger than unity, dynamically fast polydisperse modes also exist at similarly large wave numbers, while at smaller wave numbers, where the classical monodisperse streaming instability shows secular growth, no growing polydisperse modes are found under the terminal velocity approximation (Paardekooper et al., 2020). Outside the region of validity for the terminal velocity approximation, unstable epicyclic modes with growth on $\rho_g$3 dynamical time scales were found (Paardekooper et al., 2020).

Taken together, these results imply that “streaming instability under TVA” is not a single, unambiguous object. In strict TVA, some widely discussed resonant branches are absent, displaced to shorter scales, or reclassified as modes outside the approximation’s domain of validity.

4. Dependence on particle sizes, turbulence, and pressure structure

The simplest SI calculations assume a single particle size, but realistic disks contain distributions of stopping times. The first linear study of a particle-size distribution found that, for a given dust-to-gas mass ratio, the multi-species streaming instability grows on timescales much longer than those expected when only one dust species is involved; broad distributions extending to $\rho_g$4 can drive growth rates below $\rho_g$5 when the distribution is well resolved (Krapp et al., 2019). In that framework, the fastest modes can scale approximately as $\rho_g$6 with the number of species in broad distributions, and the low-$\rho_g$7 limit follows the expected $\rho_g$8 behavior of individual resonant-drag branches (Krapp et al., 2019).

That suppression is not universal. In the later polydisperse survey tied to collisional dust evolution, a simple interstellar power-law distribution often produced slower growth than a single-size model, but distributions with an enhancement of the largest dust sizes yielded instability behaviour similar to the monodisperse case (McNally et al., 2021). In that diffusive-turbulent model, fast linear growth required peak stopping times on the order of $\rho_g$9 orbits, an enhancement of the largest dust significantly above the single power-law distribution produced by a fragmentation cascade, and local dust to gas volume mass density ratio of order unity; the paper also concluded that $\mathbf{u}$0 is required if the adopted turbulent-diffusion model is applicable (McNally et al., 2021).

External turbulence generally raises thresholds and pushes unstable scales upward. In the $\mathbf{u}$1-disk treatment of isotropic turbulence, turbulence always reduces the growth rates of the streaming instability relative to globally laminar disks; for $\mathbf{u}$2–$\mathbf{u}$3, the maximally growing wavelengths shift toward larger scales approaching $\mathbf{u}$4, and for $\mathbf{u}$5 the viable region narrows to a triangular patch centered on $\mathbf{u}$6 and $\mathbf{u}$7 (Umurhan et al., 2019). In 3D stratified shearing-box simulations with both particle self-gravity and forced turbulence, turbulence can increase the threshold $\mathbf{u}$8 by up to an order of magnitude: for $\mathbf{u}$9, planetesimal formation occurs when $\rho_d$0, $\rho_d$1, and $\rho_d$2 at $\rho_d$3, $\rho_d$4, and $\rho_d$5, respectively (Lim et al., 2023).

Pressure structure can instead promote SI. Inside local pressure maxima, a novel unstable mode appears because the bump imposes strong differential advection between gas and dust. That mode persists even when gas viscosity is large, and pressure bumps are found to be the only places where streaming instability occurs in viscous discs (Auffinger et al., 2017). In two-dimensional global models where SI coexists with the vertical shear instability, VSI-generated pressure bumps seed SI and lower the nominal thresholds: strong clumping occurs for $\rho_d$6 with $\rho_d$7, or for $\rho_d$8 with $\rho_d$9, conditions under which SI alone is weaker (Schäfer et al., 2022).

5. Nonlinear evolution, filament morphology, and planetesimal formation

In the nonlinear stage, SI generates strong dust concentrations. High-resolution simulations of small particles found that $\mathbf{v}$0 and $\mathbf{v}$1 particles can concentrate themselves via the streaming instability at a solid abundance of a few percent, revising earlier critical-abundance estimates in the $\mathbf{v}$2 regime (Yang et al., 2016). For $\mathbf{v}$3, strong concentration occurs for

$\mathbf{v}$4

while for $\mathbf{v}$5,

$\mathbf{v}$6

(Yang et al., 2016). The resulting solid density can exceed the Roche density, implying that direct collapse of particles down to mm sizes into planetesimals is possible (Yang et al., 2016).

A complementary route couples coagulation and SI semi-analytically. In that framework, realistic dust growth can create a sufficient quantity of large aggregates outside the snow line, where sticky icy aggregates are present; the process depends strongly on local dust abundance and radial pressure gradient, requires a super-solar metallicity, and typically needs $\mathbf{v}$7 orbits to produce sufficiently large and settled grains, followed by another $\mathbf{v}$8 orbits of planetesimal formation (Drazkowska et al., 2014). This suggests that SI is not merely a local fluid instability but one stage in a broader pathway from coagulation-limited pebbles to self-gravitating planetesimals.

Very large computational domains change the geometric picture of nonlinear SI. Simulations with side lengths up to $\mathbf{v}$9 in the disk plane show that the azimuthal extent of filaments in the non-linear state of the streaming instability is limited to approximately one gas scale height (Schäfer et al., 2024). The instability therefore does not transform the pebble density field into axisymmetric rings; instead, the nonlinear state appears as a complex structure of loosely connected filaments (Schäfer et al., 2024). With particle self-gravity included, these runs form up to 4,000 planetesimals and show that the high-mass end of the initial mass function is well-described by a steep exponential tapering (Schäfer et al., 2024).

6. Open issues, model dependence, and current synthesis

Several points remain actively debated. One concerns which linear branch is dynamically decisive: the secular mode emphasized in early reduced theories, or unstable epicyclic branches that can be more resistant to viscosity (Jaupart et al., 2020). A second concerns approximation hierarchy: strict terminal-velocity treatments exclude some resonant branches that looser “TVA-based” linearizations recover (Zhuravlev, 2021). A third concerns the realism of monodisperse models: broad size distributions can strongly reduce growth, while top-heavy evolved distributions can nearly restore monodisperse behavior (Krapp et al., 2019, McNally et al., 2021).

Threshold criteria are also model dependent. In stratified shearing boxes with forced turbulence and self-gravity, planetesimal formation requires a mid-plane particle-to-gas density ratio that exceeds unity, with the critical value being independent of $\epsilon=\rho_d/\rho_g$0 (Lim et al., 2023). In contrast, two-dimensional global models in which SI acts in unison with VSI find that reaching a mid-plane density ratio of one is not necessary to trigger planetesimal formation via the streaming instability (Schäfer et al., 2022). This suggests that the operational threshold depends on how pressure bumps, vertical stirring, dimensionality, and self-gravity are modeled.

Despite those qualifications, several features are robust across the literature represented here. SI is a drag-mediated instability powered by differential dust–gas drift in a pressure-supported disk. It is naturally expressed in terms of stopping time, dust loading, and pressure-gradient strength; it can be reinterpreted as an inertial-wave RDI; it is strongly modified by particle-size distributions, turbulence, and pressure maxima; and in nonlinear regimes it produces dense filaments that can collapse into planetesimals when self-gravity is included (Squire et al., 2017, Magnan et al., 2024, Schäfer et al., 2024). The main unresolved question is not whether streaming instability exists in the full gas–dust system, but which branches dominate under realistic disk conditions and how those branches connect dust evolution, turbulence, and self-gravitating collapse into a unified theory of planetesimal formation.