  The streaming instability is a popular candidate for planetesimal formation
by concentrating dust particles to trigger gravitational collapse. However, its
robustness against physical conditions expected in protoplanetary disks is
unclear. In particular, particle stirring by turbulence may impede the
instability. To quantify this effect, we develop the linear theory of the
streaming instability with external turbulence modelled by gas viscosity and
particle diffusion. We find the streaming instability is sensitive to
turbulence, with growth rates becoming negligible for alpha-viscosity
parameters $\alpha \gtrsim \mathrm{St} ^{1.5}$, where $\mathrm{St}$ is the
particle Stokes number. We explore the effect of non-linear drag laws, which
may be applicable to porous dust particles, and find growth rates are modestly
reduced. We also find that gas compressibility increase growth rates by
reducing the effect of diffusion. We then apply linear theory to global models
of viscous protoplanetary disks. For minimum-mass Solar nebula disk models, we
find the streaming instability only grows within disk lifetimes beyond $\sim
10$s of AU, even for cm-sized particles and weak turbulence ($\alpha\sim
10^{-4}$). Our results suggest it is rather difficult to trigger the streaming
instability in non-laminar protoplanetary disks, especially for small
particles.
