Length and Velocity Scales in Protoplanetary Disk Turbulence

In the theory of protoplanetary disk turbulence, a widely adopted \emph{ansatz}, or assumption, is that the turnover frequency of the largest turbulent eddy, $\Omega_L$, is the local Keplerian frequency $\Omega_K$. In terms of the standard dimensionless Shakura-Sunyaev $\alpha$ parameter that quantifies turbulent viscosity or diffusivity, this assumption leads to characteristic length and velocity scales given respectively by $\sqrt{\alpha}H$ and $\sqrt{\alpha}c$, in which $H$ and $c$ are the local gas scale height and sound speed. However, this assumption is not applicable in cases when turbulence is forced numerically or driven by some natural processes such as Vertical Shear Instability. Here we explore the more general case where $\Omega_L\ge\Omega_K$ and show that under these conditions, the characteristic length and velocity scales are respectively $\sqrt{\alpha/R'}H$ and $\sqrt{\alpha R'}c$, where $R'\equiv \Omega_L/\Omega_K$ is twice the Rossby number. It follows that $\alpha=\alphat/R'$, where $\sqrt{\alphat} c$ is the root-mean-square average of the turbulent velocities. Properly allowing for this effect naturally explains the reduced particle scale heights produced in shearing box simulations of particles in forced turbulence, and may help with interpreting recent edge-on disk observations; more general implications for observations are also presented. For $R'>1$ the effective particle Stokes numbers are increased, which has implications for particle collision dynamics and growth, as well as for planetesimal formation.