Zombie Vortex Instability. II. Thresholds to Trigger Instability and the Properties of Zombie Turbulence in the Dead Zones of Protoplanetary Disks

Abstract

In Zombie Vortex Instability (ZVI), perturbations excite critical layers in stratified, rotating shear flow (as in protoplanetary disks), causing them to generate vortex layers, which roll-up into anticyclonic zombie vortices and cyclonic vortex sheets. The process is self-sustaining as zombie vortices perturb new critical layers, spawning a next generation of zombie vortices. Here, we focus on two issues: the minimum threshold of perturbations that trigger self-sustaining vortex generation, and the properties of the late-time zombie turbulence on large and small scales. The critical parameter that determines whether ZVI is triggered is the magnitude of the vorticity on the small scales (and not velocity), the minimum Rossby number needed for instability is \(Ro_{crit}\sim0.2\) for \(\beta\equiv N/\Omega = 2\), where \(N\) is the Brunt-V\"ais\"al\"a frequency. While the threshold is set by vorticity, it is useful to infer a criterion on the Mach number, for Kolmogorov noise, the critical Mach number scales with Reynolds number: \(Ma_{crit}\sim Ro_{crit}Re^{-1/2}\). In protoplanetary disks, this is \(Ma_{crit}\sim10^{-6}\). On large scales, zombie turbulence is characterized by anticyclones and cyclonic sheets with typical Rossby number \(\sim\)0.3. The spacing of the cyclonic sheets and anticyclones appears to have a "memory" of the spacing of the critical layers. On the small scales, zombie turbulence has no memory of the initial conditions and has a Kolmogorov-like energy spectrum. While our earlier work was in the limit of uniform stratification, we have demonstrated that ZVI works for non-uniform Brunt-V\"ais\"al\"a frequency profiles that may be found in protoplanetary disks.