The vertical shear instability ( VSI ) offers a potential hydrodynamic mechanism for angular momentum transport in protoplanetary disks ( PPDs ) .
The VSI is driven by a weak vertical gradient in the disk ’ s orbital motion , but must overcome vertical buoyancy , a strongly stabilizing influence in cold disks , where heating is dominated by external irradiation .
Rapid radiative cooling reduces the effective buoyancy and allows the VSI to operate .
We quantify the cooling timescale t _ { c } needed for efficient VSI growth , through a linear analysis of the VSI with cooling in vertically global , radially local disk models .
We find the VSI is most vigorous for rapid cooling with t _ { c } < \Omega _ { K } ^ { -1 } h|q| / ( \gamma - 1 ) in terms of the Keplerian orbital frequency , \Omega _ { K } ; the disk ’ s aspect-ratio , h \ll 1 ; the radial power-law temperature gradient , q ; and the adiabatic index , \gamma .
For longer t _ { c } , the VSI is much less effective because growth slows and shifts to smaller length scales , which are more prone to viscous or turbulent decay .
We apply our results to PPD models where t _ { c } is determined by the opacity of dust grains .
We find that the VSI is most effective at intermediate radii , from \sim 5 AU to \sim 50 AU with a characteristic growth time of \sim 30 local orbital periods .
Growth is suppressed by long cooling times both in the opaque inner disk and the optically thin outer disk .
Reducing the dust opacity by a factor of 10 increases cooling times enough to quench the VSI at all disk radii .
Thus the formation of solid protoplanets , a sink for dust grains , can impede the VSI .