Title: A self-consistent model for dust settling and the vertical shear instability in protoplanetary disks

Abstract: The spatial distribution of dust particles in protoplanetary disks affects dust evolution and planetesimal formation processes. The vertical shear instability (VSI) is one of the candidate hydrodynamic mechanisms that can generate turbulence in the outer disk region and affect dust diffusion. Turbulence driven by the VSI has a predominant vertical motion that can prevent dust settling. On the other hand, the dust distribution controls the spatial distribution of the gas cooling rate, thereby affecting the strength of VSI-driven turbulence. Here, we present a semi-analytic model that determines the vertical dust distribution and the strength of VSI-driven turbulence in a self-consistent manner. The model uses an empirical formula for the vertical diffusion coefficient in VSI-driven turbulence obtained from our recent hydrodynamical simulations. The formula returns the vertical diffusion coefficient as a function of the vertical profile of the cooling rate, which is determined by the vertical dust distribution. We use this model to search for an equilibrium vertical dust profile where settling balances with turbulent diffusion for a given maximum grain size. We find that if the grains are sufficiently small, there exists a stable equilibrium dust distribution where VSI-driven turbulence is sustained at a level of alpha_z ~ 10^{-3}, where alpha_z is the dimensionless vertical diffusion coefficient. However, as the maximum grain size increases, the equilibrium solution vanishes because the VSI can no longer stop the settling of the grains. This runaway settling may explain highly settled dust rings found in the outer part of some protoplanetary disks.