The Streaming Instability in Two-dimensional Protoplanetary Disks

Abstract

Abstract We conduct a two-dimensional (2D) stability analysis for the aerodynamically coupled system of gas and dust in protoplanetary disks using the WKB approximation. The 2D analysis may apply to a vertically integrated disk or a thin dust layer at the disk midplane. We show that the streaming instability exists in 2D and examine the growth rate and its dependence on the mean dust-to-gas ratio, ϵ ¯ , and the Stokes number, St. For small ϵ ¯ , the dust back reaction may be viewed as a perturbation to the wave system, and analytical results are successfully derived by the method of perturbation with respect to ϵ ¯ . At a given azimuthal wavenumber, m, the growth rate as a function of the radial wavenumber, kr, shows a peak corresponding to the resonance between the relative drift velocity and the Rossby/trivial waves in the gas. For small ϵ ¯ , the peak growth rate at resonant modes increases as m 1/2 when the Rossby wave effect is negligible and the gas wave becomes trivial. With increasing St, the peak growth rate increases as St 1/2 at St ≪ 1, peaks at St ∼ 1, and finally decreases as St −1/2 for St ≫ 1. We find that the growth rate scales with ϵ ¯ as ϵ ¯ 1 / 2 at ϵ ¯ ≪ 1 and peaks at ϵ ¯ ∼ 1 . The instability is physically interpreted as a runaway process of dust trapping in relative drift-induced anticyclonic vortices. The effects of the compressibility and viscosity of the gas, the applicability of the WKB application, and the effects of the Keplerian shear are discussed.