High Resolution Parameter Study of the Vertical Shear Instability II: Dependence on temperature gradient and cooling time

A certain appeal to the alpha model for turbulence and related viscosity in accretion disks was that one scales the Reynolds stresses simply on the thermal pressure, assuming that turbulence driven by a certain mechanism will attain a characteristic Mach number in its velocity fluctuations. Besides the notion that there are different mechanism driving turbulence and angular momentum transport in a disk, we also find that within a single instability mechanism, here the Vertical Shear Instability, stresses do not linearly scale with thermal pressure. Here we demonstrate in numerical simulations the effect of the gas temperature gradient and the thermal relaxation time on the average stresses generated in the non-linear stage of the instability. We find that the stresses scale with the square of the exponent of the radial temperature profile at least for a range of $d \log T /d \log R = [-0.5, -1]$, beyond which the pressure scale height varies too much over the simulation domain, to provide clear results. Stresses are also dependent on thermal relaxation times, provided they are longer than $10^{-3}$ orbital periods. The strong dependence of viscous transport of angular momentum on the local conditions in the disk (especially temperature, temperature gradient, and surface density/optical depth) challenges the ideas of viscosity leading to smooth density distributions, opening a route for structure (ring) formation and time variable mass accretion.