Two saturated states of the vertical shear instability in protoplanetary disks with vertically varying cooling times

Turbulence in protoplanetary disks plays an important role in dust evolution and planetesimal formation. The vertical shear instability (VSI) is one of the candidate hydrodynamic mechanisms that can generate turbulence in the outer disk regions. The VSI requires rapid gas cooling in addition to vertical shear. A linear stability analysis suggests that the VSI may not operate around the midplane where gas cooling is inefficient. In this study, we investigate the nonlinear outcome of the VSI in disks with a linearly VSI-stable midplane region. We perform two-dimensional global hydrodynamical simulations of an axisymmetric disk with vertically varying cooling times. The vertical cooling time profile determines the thicknesses of the linearly VSI-stable midplane layer and unstable layers above and below the midplane. We find that the thickness of the midplane stable layer determines the vertical structure of VSI-driven turbulence in the nonlinear saturated state. We identify two types of final saturated state: (i) T states, characterized by vertical turbulent motion penetrating into the VSI-stable midplane layer, and (ii) pT states, characterized by turbulent motion confined in the unstable layers. The pT states are realized when the midplane VSI-stable layer is thicker than two gas scale heights. We also find that the VSI-driven turbulence is largely suppressed at all heights when the VSI-unstable region lying above and below the midplane is thinner than two gas scale heights. We present empirical formulas that predict the strength of VSI-driven turbulence as a function of the thicknesses of the unstable and stable layers. These formulas will be useful for studying how VSI-driven turbulence and dust grains controlling the disk cooling efficiency evolve simultaneously.