Saturation Level of the Streaming Instability
In proto-planetary disks, gas turbulence can cluster dust grains and catalyze the formation of planetesimals. This process depends on the strength of the turbulence. Here, we evaluate the turbulence strength that is induced by the streaming instability. Based on the amount of free energy available, we estimate the viscous parameter to be alpha ~ St Pi^2 f(epsilon), where St (< 1) is the Stokes number of the streaming grains, Pi is the dimensionless pressure gradient, and f (epsilon) is an explicit function of epsilon, the local dust-to-gas density ratio. This estimate explains the results of many previous studies, and is also supported by our additional tailor-made simulations. We discuss two easy applications of our result, one is a self-consistent dust scale height, one is the astrophysical relevance of a pressure bump. Lastly, while this work allows one to characterize the streaming turbulence, we emphasize our current inability to connect the strength of turbulence to the formation of planetesimals.