Streaming Instability in Turbulent Protoplanetary Disks

The streaming instability for solid particles in protoplanetary disks is re-examined assuming the familiar alpha ($\alpha$) model for isotropic turbulence. Turbulence always reduces the growth rates of the streaming instability relative to values calculated for globally laminar disks. While for small values of the turbulence parameter, $\alpha < 10^{-5}$, the wavelengths of the fastest-growing disturbances are small fractions of the local gas vertical scale height $H$, we find that for moderate values of the turbulence parameter, i.e., $\alpha \sim 10^{-5}-10^{-3}$, the lengthscales of maximally growing disturbances shift toward larger scales, approaching $H$. At these moderate turbulent intensities and for local particle to gas mass density ratios $\epsilon < 0.5$, the vertical scales of the most unstable modes begin to exceed the corresponding radial scales so that the instability appears in the form of vertically oriented sheets extending well beyond the particle scale height. We find that for hydrodynamical turbulent disk models reported in the literature, with $\alpha = 4\times 10^{-5} - 5\times 10^{-4}$, together with state of the art global evolution models of particle growth, the streaming instability is predicted to be viable within a narrow triangular patch of $\alpha$--$\tau_s$ parameter space centered on Stokes numbers, $\tau_s \sim 0.01$ and $\alpha \sim 4\times 10^{-5}$ and, further, exhibits growth rates on the order of several hundred to thousands of orbit times for disks with 1 percent ($Z= 0.01$) cosmic solids abundance or metallicity. Our results are consistent with, and place in context, published numerical studies of streaming instabilities.