Streaming Instability Growth Rates

• Streaming instability growth rates are defined by the imaginary part of the eigenfrequency from coupled fluid and kinetic equations in systems such as protoplanetary disks and cosmic-ray plasmas.
• Detailed analyses reveal that drag forces, pressure gradients, and turbulence critically shape the instability growth, with thresholds determined by dust-to-gas ratios and particle size distributions.
• Comparisons between mono- and multi-species regimes illustrate how specific scaling laws and environmental conditions influence key outcomes like planetesimal formation and cosmic-ray scattering.

The streaming instability (SI) encompasses a range of kinetic and fluid-dynamical instabilities driven by the relative streaming of different particle populations or fluids—most notably, solids and gas in protoplanetary disks, or cosmic rays and background plasma in astrophysical and laboratory contexts. The quantification and comparison of linear growth rates for SI in various physical regimes are critical for understanding particle concentration, turbulence, magnetic-field amplification, and subsequent nonlinear outcomes such as planetesimal formation or cosmic-ray scattering.

1. General Framework for Streaming Instability Growth Rates

The linear growth rate $\gamma$ of a streaming instability is defined by the imaginary part of the eigenfrequency $\omega$ of the linearized perturbation equations, $\gamma = \mathrm{Im}(\omega)$. The dispersion relation for the system, typically an algebraic polynomial in $\omega$, is derived from the coupled evolution equations for multiple fluids or particle species.

For the canonical drag-induced SI in a dusty-gas protoplanetary disk, the governing equations combine incompressible gas dynamics with pressureless dust particles and mutual drag. The SI growth rate emerges from the resonance between the dust drift and epicyclic oscillations of the gas, generalized as a resonant drag instability (RDI) framework (Squire et al., 2017, Squire et al., 2020).

For cosmic-ray-driven instabilities, the interaction of streaming CRs with magnetized plasma leads to both gyro-resonant modes (resonant streaming instability) and non-resonant (Bell/Buneman) modes, with growth rates controlled by drift velocities, density ratios, and ambient plasma parameters (Bai et al., 2019, Haggerty et al., 2019, Marret et al., 2020).

In all cases, growth rates are extracted as maximal values over the $(k_x,k_z)$ plane, where $k$ components are wavenumbers parallel and perpendicular to the mean flow or field. The instability is characterized by distinct parameter thresholds and limiting regimes that sharply separate fast and slow growth.

2. Drag-Induced Streaming Instability in Protoplanetary Disks

a. Mono- and Few-Species Growth Rate Scaling

For a single dust species (mono-disperse limit), the fastest SI growth rate at small Stokes number $\tau_s\ll1$, and dust-to-gas mass ratio $\mu\ll 1$, is (Squire et al., 2017, Squire et al., 2020): $$\gamma_\mathrm{max} \simeq \frac12\,\Omega\,\sqrt{\mu\,\tau_s}$$ where $\Omega$ is the Keplerian frequency, and the resonant wavenumber is $k_\mathrm{res}\sim1/(2\,\eta^{1/2}\,\tau_s\, r)$ with $\eta$ the dimensionless radial pressure gradient. This scaling is well supported by numerical solutions of the sixth-order SI dispersion relation (Squire et al., 2017).

For $\mu\rightarrow 1$, a transition occurs: for $\mu<1$, growth is oscillatory and $\gamma\propto\mu$; for $\mu>1$, pure growing modes emerge, with (Squire et al., 2020): $\gamma_\mathrm{max} \simeq \Omega\,\sqrt{\mu-1}$

b. Pressure Bumps and Viscous Disks

In turbulent or viscous disks, canonical SI modes are strongly suppressed, with growth rates declining sharply with increasing viscosity parameter $\alpha$ such that SI is negligible for $\alpha\gtrsim \mathrm{St}^{1.5}$, where $\mathrm{St}$ is the Stokes number (Chen et al., 2020, Umurhan et al., 2019). The only robust SI growth in high-viscosity disks occurs at local pressure maxima (“pressure bumps”) where a novel “bump mode” with growth rate (Auffinger et al., 2017): $$\gamma_\mathrm{bump} \approx \sqrt{\Gamma}\,\Omega_0$$ remains unsuppressed even for large $\alpha$; here, $\Gamma$ characterizes the logarithmic curvature of the pressure bump.

c. Turbulence and Diffusion Effects

With isotropic $\alpha$-disk turbulence, the SI fastest-growth rate is (for $\epsilon\equiv\rho_p/\rho_g\ll1$, $\tau_s\approx0.01-1$) (Umurhan et al., 2019): $$\gamma_\mathrm{max} \sim \Omega\left(\frac{\epsilon\,\tau_s}{\alpha}\right)^{1/2}(\Delta U/v_K)^{1/2}$$ with $\Delta U$ the mean drift. Turbulence shifts the wavenumbers of maximally growing SI modes to larger scales and reduces growth rates by factors $>10$ compared to laminar disks.

3. Multi-Species Streaming Instability and Size Distribution Effects

When a continuous dust-size distribution is considered, SI growth rates are substantially reduced compared to the mono-disperse case (Krapp et al., 2019, Zhu et al., 2020, Paardekooper et al., 2020):

• For broad size distributions ($T_{s,\min}\lesssim10^{-3}$, $T_{s,\max}\sim1$), the fast-growth regime (with $\sigma_\mathrm{max}\gtrsim10^{-2}\Omega$) is limited to either high dust-to-gas ratios $\varepsilon\gtrsim1$ or large maximum stopping times $T_{s,\max}\gtrsim1$.
• In the slow-growth regime ( $\varepsilon\,T_{s,\max}\lesssim1$), $\sigma_\mathrm{max}\propto\varepsilon\,T_{s,\max}$ and vanishes as the number of species $N\to\infty$.
• Only when the total solid-to-gas ratio or the largest grains are sufficiently dominant does the system approach classical fast SI rates (see Table below; (Zhu et al., 2020)).

Regime	Condition	$\sigma_\mathrm{max}/\Omega$	Scaling
Fast (converged)	$\varepsilon\gtrsim1$ or $T_{s,\max}\gtrsim1$	$\gtrsim 10^{-2}$	$\sim$constant
Slow (not converged)	$\varepsilon\,T_{s,\max}\lesssim 1$	$\ll 10^{-2}$	$\propto \varepsilon T_{s,\max}$

Multi-species resonance provides only weak coupling in the slow regime, leading to extreme timescales ($t_\mathrm{grow}\sim10^2$–$10^3\,\Omega^{-1}$) that can exceed disk lifetimes (Krapp et al., 2019, Zhu et al., 2020).

4. Streaming Instabilities in Cosmic-Ray Plasmas

a. Gyro-Resonant and Bell Instabilities

• The gyro-resonant CR streaming instability (CRSI) growth rate for slab Alfvén waves is (Bai et al., 2019): $$\gamma_\mathrm{CRSI}(k) = \frac12\,\frac{n_\mathrm{CR}}{n_i}\Omega_c \left(\frac{v_D}{v_A}-1\right)\,Q_2(k)$$ where $n_\mathrm{CR}/n_i$ is the CR-to-ion density ratio, $v_D/v_A$ the CR drift speed in units of Alfvén speed, and $Q_2$ encodes the momentum-space resonance.
• Non-resonant, current-driven (Bell) instability has (Haggerty et al., 2019, Marret et al., 2020): $$\gamma_\mathrm{Bell}(k) = \sqrt{\frac{k\,j_\mathrm{CR}\,B_0}{\rho\,c} - k^2 v_A^2}$$ with peak at

$$k_\mathrm{max} = \frac{j_\mathrm{CR}\,B_0}{2\,\rho\,v_A^2\,c}, \qquad \gamma_\mathrm{max} = \frac12\,\Omega_i\,\frac{n_\mathrm{CR}}{n_i}\frac{v_D}{v_A}$$

Thermal effects suppress growth as $\gamma_\mathrm{max}\propto T^{-1/2}$ in the hot, demagnetized limit (Marret et al., 2020).

• Perpendicular CR streaming can drive even faster-growing modes, with (Nekrasov et al., 2014): $$\gamma_{\perp} \sim k\,U_\mathrm{CR}\,\sqrt{d}/(1+d)$$ where $d=(n_\mathrm{CR}m_\mathrm{CR})/(n_im_i)$. At short wavelengths, these rates can exceed both standard parallel and Bell modes by orders of magnitude.

b. Buoyancy and New CR-Driven Modes

Streaming CRs in stratified, high-$\beta$ astrophysical plasmas drive a compressible buoyancy instability (CRBI) with growth rate (Kempski et al., 2022): $$\Gamma_\mathrm{CRBI} \simeq \left(\frac{p_c}{p_g}\right)\,\beta^{1/2}\,\omega_\mathrm{ff}$$ where $p_c/p_g$ is the CR-to-gas pressure ratio, $\beta = 8\pi p_g/B^2$ the plasma beta, and $\omega_\mathrm{ff}$ the local free-fall frequency. CRBI dominates over HBI/MTI at high CR pressure or large $\beta$.

5. Classical Two-Stream and Buneman Instabilities

For electron stream instabilities (laboratory or space plasma), the growth rate in a finite plasma of length $L$ is (Kaganovich et al., 2015): $$\gamma \approx \frac{1}{13}\,\omega_{pe}\,\frac{n_b}{n_p}\,\frac{L\,\omega_{pe}}{v_b}\,\ln \left(\frac{L\,\omega_{pe}}{v_b}\right)\,[1 - 0.18 \cos(L\,\omega_{pe}/v_b+\pi/2)]$$ The band structure of $\gamma$ as a function of $L$ and resonant conditions leads to strong enhancement or suppression depending on the system size.

In highly collisional regimes, a low-frequency evacuation mode exists whose growth rate is independent of collision frequency, $\nu_{ei}$, and given by (Zhou et al., 2023): $\gamma = k\,v_d\,\sqrt{\frac{m_e}{m_i}}$ with instability threshold $v_d > v_{Te}$, and governing the observed double-layer formation and rapid plasma depletion.

6. Physical Interpretation and Implications

Streaming instability growth rates are governed by:

• Relative drift velocities between species or fluids.
• Coupling mechanisms (drag, Lorentz force, pressure anisotropy).
• Number of species and distribution of physical parameters (size, charge, momentum).
• Environmental factors such as turbulence, pressure gradients, and plasma $\beta$.
• Thresholds in key parameters (e.g., $\epsilon\,T_{s,\max} \gtrsim 1$ for fast SI in disks (Zhu et al., 2020)).
• Boundary and size effects for resonant instabilities.

In protoplanetary disks, only zones with $\epsilon \gtrsim 1$ or $T_{s,\max} \gtrsim 1$ enable SI to grow on sufficiently short timescales ($\gamma \gtrsim 0.01\,\Omega$), and the presence of multiple dust species or turbulent stirring strongly suppresses SI elsewhere (Zhu et al., 2020, Chen et al., 2020, Krapp et al., 2019).

For cosmic-ray-driven instabilities, maximal growth rates scale linearly with the CR-to-background density and drift velocity, and can drive turbulence and field amplification on physically relevant timescales even at low $n_\mathrm{CR}/n_{i}$ provided that either local CR density or pressure is enhanced (Bai et al., 2019, Marret et al., 2020, Kempski et al., 2022).

Precise predictions require full solution of the relevant dispersion relations, accounting for all species, nonlinearity in drag or current coupling, and environmental conditions.

• Streaming Instability with Multiple Dust Species: I. Favourable Conditions for the Linear Growth (Zhu et al., 2020)
• Streaming Instability for Particle-Size Distributions (Krapp et al., 2019)
• Polydisperse Streaming Instability I. Tightly coupled particles and the terminal velocity approximation (Paardekooper et al., 2020)
• Resonant Drag Instabilities in protoplanetary disks: the streaming instability and new, faster-growing instabilities (Squire et al., 2017)
• Physical models of streaming instabilities in protoplanetary disks (Squire et al., 2020)
• How efficient is the streaming instability in viscous protoplanetary disks? (Chen et al., 2020)
• Streaming Instability in Turbulent Protoplanetary Disks (Umurhan et al., 2019)
• Linear growth of streaming instability in pressure bumps (Auffinger et al., 2017)
• Magnetohydrodynamic-Particle-in-Cell Simulations of the Cosmic-Ray Streaming Instability: Linear Growth and Quasi-linear Evolution (Bai et al., 2019)
• Hybrid Simulations of the Resonant and Non-Resonant Cosmic Ray Streaming Instability (Haggerty et al., 2019)
• On the growth of the thermally modified non-resonant streaming instability (Marret et al., 2020)
• Influence of the back-reaction of streaming cosmic rays on magnetic field generation and thermal instability (Nekrasov et al., 2014)
• A new buoyancy instability in galaxy clusters due to streaming cosmic rays (Kempski et al., 2022)
• Band Structure of the Growth Rate of the Two-Stream Instability of an Electron Beam Propagating in a Bounded Plasma (Kaganovich et al., 2015)
• Two-stream instability with a growth rate insensitive to collisions in a dissipative plasma jet (Zhou et al., 2023)