Give a streaming wave two directions to grow

Choose a uniform drifting background

Take the local rotating coordinates and peculiar velocities from the drift chapter. Now the gas background is uniform and incompressible; an imposed constant radial pressure acceleration maintains the equilibrium drift. There is no prepared pressure bump. Let $\mathbf U_{\rm g}$ and $\mathbf U_{\rm d}$ be the coupled steady velocities and $\mathbf W=\mathbf U_{\rm d}-\mathbf U_{\rm g}$ their relative drift.

Gas velocity perturbation is $\mathbf a$ , dust velocity perturbation $\mathbf b$ and relative dust-density perturbation $\delta=\delta\rho_{\rm d}/\rho_{{\rm d},0}$ . The gas density does not change in this incompressible model. The stopping time is fixed; density-dependent Epstein stopping is outside this calculation. No vertical gravity, settling or self-gravity is present.

Reading: Youdin & Goodman 2005, Streaming Instabilities, arXiv v1 · Youdin & Johansen 2007, Linear Evolution and Numerical Methods, v1

Keep the density contribution to feedback

Expand the gas drag acceleration $\epsilon(1+\delta)(\mathbf W+\mathbf b-\mathbf a)/t_{\rm stop}$ . Subtract its equilibrium value and drop the product of two perturbations. The remaining force contains $\mathbf b-\mathbf a+\delta\mathbf W$ . The last term is important: a density enhancement exerts a different drag on the gas even before its velocity changes.

Define $D_j=\partial_t+U_{j,x}\partial_x$ for advection by each background, and $\mathcal C\mathbf q=(2\Omega q_y,-\Omega q_x/2,0)$ for Coriolis plus Kepler-shear coupling. The linearized equations are then:

D_{\rm g}\mathbf a=\mathcal C\mathbf a-\nabla h+\frac\epsilon{t_{\rm stop}}(\mathbf b-\mathbf a+\delta\mathbf W),\qquad\nabla\cdot\mathbf a=0

The pressure perturbation h enforces incompressible gas motion. Its background pressure acceleration has already been removed with the equilibrium.

D_{\rm d}\mathbf b=\mathcal C\mathbf b-\frac{\mathbf b-\mathbf a}{t_{\rm stop}},\qquad D_{\rm d}\delta=-\nabla\cdot\mathbf b

Dust is pressureless. Its convergence increases density; its velocity is coupled back to the gas through drag.

Reading: Youdin & Goodman 2005, Streaming Instabilities, arXiv v1 · Benítez-Llambay, Krapp & Pessah 2019, FARGO3D multifluid tests, v3

Turn a spatial wave into a small matrix

Write each real disturbance as the real part of $\widehat f\exp[\sigma t+i(k_xx+k_zz)]$ . A complex amplitude can be thought of as two real numbers, one multiplying a cosine and one a sine. Spatial derivatives become multiplication by $ik_x$ or $ik_z$ , and the time derivative becomes $\sigma$ . Its real part is the growth rate.

Pressure need not become another evolving variable. For $\mathbf k=(k_x,0,k_z)$ , project the gas acceleration perpendicular to $\mathbf k$ . That enforces $\mathbf k\cdot\mathbf a=0$ . Use one perpendicular direction in the radial–vertical plane and the independent azimuthal direction. Gas then has two amplitudes, dust three velocities, and dust density one: six evolving states.

\mathsf P=\mathsf I-\frac{\mathbf k\mathbf k^{\mathsf T}}{k^2},\qquad\mathbf e_\perp=\frac{(k_z,0,-k_x)}k,\qquad\sigma\widehat{\mathbf q}=\mathsf L\widehat{\mathbf q}

Apply this projector to the gas equation after collecting rotation, advection and feedback. The resulting eigenvalues give six growth/frequency modes.

Reading: Youdin & Goodman 2005, Streaming Instabilities, arXiv v1

Why a radial traffic jam is insufficient

Set $k_z=0$ in the incompressible system. The gas constraint now forces its radial perturbation to vanish. That removes motion needed for the growing secular streaming mode. The full equations have no growing mode in this limit, as the original paper shows in Section 5.1.

A radial dust concentration can still arise because a pre-existing pressure maximum draws particles inward from either side. That is the separate finite trap experiment. It does not establish growth of an infinitesimal perturbation about a uniform drifting state. In particular, replacing the full velocity equations by a density-dependent steady drift gives transport and wave steepening, not the complete streaming instability.

Keeping vertical variation restores radial gas motion while respecting incompressibility. Azimuthal velocities remain essential because they exchange angular momentum through the rotating dynamics. Two-dimensional spatial variation therefore still carries three-dimensional velocity vectors.

Reading: Youdin & Goodman 2005, Streaming Instabilities, arXiv v1 · Youdin & Johansen 2007, Linear Evolution and Numerical Methods, v1

Work a growing-mode example

Use $\mathrm{St}=0.1$ and $\epsilon=3$ with $K_x=K_z=30$ , where $K_j=k_j\eta r$ . Normalize time by $1/\Omega$ and velocity by $\eta v_{\rm K}$ . Solving the full incompressible six-state system gives $\sigma/\Omega=0.41903111-0.34801854i$ .

A relative dust amplitude $10^{-6}$ grows to $6.6043\times10^{-5}$ after $10/\Omega$ , a factor of 66.043. Its e-folding time is $1/(0.41903111\Omega)=2.3865/\Omega$ , or about 0.380 orbital periods. An orbital period is $2\pi/\Omega$ ; confusing it with the inverse angular frequency changes the quoted time by a factor of $2\pi$ .

The browser starts a larger but still small amplitude of $0.005$ for visibility, evolves the equations, and stops around $0.1$ . For the default growing eigenmode this boundary is reached after about $7.149/\Omega$ from its seed. The amplitude stop marks the limit of the linear demonstration.

Reading: Youdin & Goodman 2005, Streaming Instabilities, arXiv v1 · Youdin & Johansen 2007, Linear Evolution and Numerical Methods, v1

Compare the right published systems

Youdin & Goodman’s original calculation is incompressible. Youdin & Johansen 2007 Table 1 adds weak gas compression and quotes a nearby growing rate. Benítez-Llambay, Krapp & Pessah 2019 Table 4 and Appendix E give the full weakly compressible system, retaining the perturbation of $\rho_{\rm d}/\rho_{\rm g}$ . Its LinA growth is $0.4190091323\Omega$ and LinB growth $0.0154862262\Omega$ . The tutorial reproduces both independently.

The 2019 paper’s Section 3.5.3, equation 86, explains the small difference from the 2007 values through two gas-density terms in the drag feedback. Keep the systems and their conventions separate. In 2019 the time factor is $e^{-\omega t}$ , so its eigenvalue is $-\sigma$ in the convention used here. Eigenvectors must also be normalized to the same absolute or relative dust-density amplitude before comparing their coefficients.

Reading: Benítez-Llambay, Krapp & Pessah 2019, FARGO3D multifluid tests, v3 · Youdin & Johansen 2007, Linear Evolution and Numerical Methods, v1

Add mixing as an explicit new assumption

A simple teaching extension puts the same term $D\nabla^2$ on every evolved perturbation. For a Fourier mode this subtracts $Dk^2$ from every eigenvalue. In the units above, $D=10^{-3}$ with both wave numbers 30 subtracts $1.8\Omega$ , making the LinA mode decay.

Real turbulent models need a physical closure for gas viscosity, dust diffusion, dust velocity dispersion and sometimes how settling changes the local loading. These ingredients need not share one coefficient. The common-diffusion exercise illustrates scale-dependent smoothing; it is not a reproduction of a full alpha-disk model or a planetesimal-formation threshold.

Reading: Umurhan, Estrada & Cuzzi 2020, Streaming Instability in Turbulent Disks, v4 · Li & Youdin 2021, Thresholds for Particle Clumping, v2

Exercises and explanations

Why must the gas drag retain the dust-density perturbation even when the gas density is fixed?
Read the explanation
The gas force per volume depends on how much dust is present. Expanding density times relative velocity leaves a first-order density-times-background-drift term. Removing it changes the instability mechanism.
A mode doubles every $3/\Omega$ . What is its growth rate?
Read the explanation
Solve $e^{3s/\Omega}=2$ , obtaining $s/\Omega=\ln2/3\approx0.231$ . The doubling time is not the e-folding time.
Can a larger linear growth rate alone establish that a stratified disk will form denser clumps?
Read the explanation
The linear rate concerns infinitesimal disturbances in a chosen background. Nonlinear saturation, settling, diffusion and available spatial modes alter clumping. The cited stratified simulations directly show that their clumping boundary does not follow a simple ranking of unstratified linear rates.

Try the prediction

Follow a small wave into its validity limit

Select A linear streaming wave. Watch the integrated density amplitude rise and note the finite stopping condition.
Choose Diffusion damps the wave and use the fixed wave numbers to predict the extra damping.
Explain why the radial trap’s rising peak and this linear mode’s rising amplitude require different interpretations.

Sources and calculation scope

Youdin & Goodman 2005, Streaming Instabilities, arXiv v1
10 September 2004 version 1; ApJ 620,459–469(2005). Full text read, Section 2 equations 1–7/12–17/24–31, Figures 1–3 and Section 4 mode geometry; Section 5 equation 39 and its restricted terminal-velocity domain; Section 6 distinguishes arbitrary linear amplitude from subsequent collapse. The PDF itself prints a later Draft Modified July 5, 2018 header; the archive version is explicitly frozen.
Youdin & Johansen 2007, Linear Evolution and Numerical Methods, v1
23 February 2007 version 1; ApJ 662,613–626. Sections 2.1–2.3 equations 1–14, Table 1 and Section 4/Figures 4–5. Fixed stopping time, two-way drag, three velocity components with radial–vertical dependence, finite-resolution eigenmode tests and weakly compressible published roots. Section 2.3 explicitly explains why steady radial traffic-jam arguments alone do not produce linear SI growth.
Benítez-Llambay, Krapp & Pessah 2019, FARGO3D multifluid tests, v3
Version 3, 10 June 2019; ApJS 241, 25. Read Sections 3.4–3.5, Table 4, equation 86 and Appendix E. The temporal exponential uses the negative of the paper's eigenvalue; the complex growth eigenvalue used here has the opposite sign. Table 4 gives the full eight-state LinA growth rate as 0.4190091323 and LinB as 0.0154862262, matching the independent matrix. Section 3.5.3 attributes the small difference from the 2007 table to gas-density drag-feedback terms; this distinction is already established in that paper.
Li & Youdin 2021, Thresholds for Particle Clumping, v2
14 October 2021 version 2; DOI 10.3847/1538-4357/ac0e9f. Full-text Sections 2,3.1–3.5 and 4: stratified simulation setup, surface metallicity versus volume loading, threshold map Figure 1, linear-growth maps Figure 7 and clumping comparison Figure 8. Equation 21 diffusion diagnostic fails to predict all nonlinear stratified clumping, explicitly cautioning against universal linear thresholds.
Umurhan, Estrada & Cuzzi 2020, Streaming Instability in Turbulent Disks, v4
25 May 2020 version 4, PDF header 26 May. Sections 3.1–3.3, equations 4–18 and Appendix A: explicit viscosity, dust diffusion/pressure and equilibrium forcing; Section 5 interprets growth with a turbulent dilution model. The Lab’s equal damping of all linear fields is a much simpler declared teaching closure, not a reproduction of this alpha-disk model.