The Axisymmetric Streaming Instability in Protoplanetary Disks

We reexamine the streaming instability in protoplanetary disks under the axisymmetric geometry. We identify the roles of all the relevant physical processes and clarify the clustering mechanism of dust, offering a new physical interpretation. Through back-reaction, the dust density fluctuations in combination with the mean relative drift velocity provide a source for the gas velocity, which in turn drives the velocity and divergence of the dust flow. In the limit where the dust-to-gas ratio (epsilon) over bar << 1, the action of the Coriolis force on the radial gradient of the azimuthal dust velocity generates the divergence of the dust. In the opposite limit (epsilon) over bar >> 1, the vertical gradient of the vertical dust velocity makes the main contribution to the dust divergence. The different dust clustering mechanisms at (epsilon) over bar << 1 and (epsilon) over bar >> 1 are referred to as Mode I and Mode II, respectively. In both cases, the dust divergence further enhances the dust density fluctuations, resulting in a positive feedback loop. In Mode I (or Mode II), the growth rate is contributed by the mean azimuthal (or radial) drag force in the gas equation of motion, while the mean radial (or azimuthal) drag tends to reduce it. The instability makes a transition from Mode I to Mode II when the coupling between the perturbed gas and dust velocities is stronger than the Coriolis force in the gas equations, which occurs at (epsilon) over bar similar or equal to 1 and (epsilon) over bar similar or equal to St for Stokes numbers St < 1 and St > 1, respectively.