Three‐dimensional Compressible Hydrodynamic Simulations of Vortices in Disks

Abstract

We carry out three-dimensional, high-resolution (up to 1024 2 ; 256) hydrodynamic simulations of the evolution of vortices in vertically unstratified Keplerian disks using the shearing sheet approximation. The transient amplification of incompressible, linear amplitude leading waves (which has been proposed as a possible route to nonlinear hydrodynamic turbulence in disks) is used as one test of our algorithms; our methods accurately capture the predicted amplification, converge at second order, and are free from aliasing. Waves that are expected to reach nonlinear amplitude at peak amplification become unstable to Kelvin-Helmholtz modes when j W max jk (where W max is the local maximum of vorticity and the angular velocity). We study the evolution of a power-law distribution of vorticity consistent with Kolmogorov turbulence; in two dimensions long-lived vortices emerge and decay slowly, similar to previous studies. In three dimensions, however, vortices are unstable to bending modes, leading to rapid decay. Only vortices with a length-to-width ratio smaller than 1 survive; in three dimensions the residual kinetic energy and shear stress is at least 1 order of magnitude smaller than in two dimensions. No evidence for sustained hydrodynamic turbulence and transport is observed in three dimensions. Instead, at late times the residual transport is determined by the amplitude of slowly decaying, large-scale vortices (with horizontal extent comparable to the scale height of the disk), with additional contributions from nearly incompressible inertial waves possible. Evaluating the role that large-scale vortices play in astrophysical accretion disks will require understanding the mechanisms that generate and destroy them.

The Dead Zones of Protoplanetary Disks are Not Dead

“…Rayleigh's criterion for centrifugal stability applies only to fluids with constant density and therefore is not relevant to PPDs where the density falls off approximately like a Gaussian away from the midplane. One stability study of PPDs [4] argues that the stability of a PPD is governed by Rayleigh's centrifugal criterion, and another initial-value study of PPDs [5] with a very high spatial resolution did not show our new instability. However, both studies initialized the flow with constant density even though they both used an ideal gas equation of state.…”

The baroclinic instability in the context of layered accretion

“…Lower order codes can be diffusive enough that the energy introduced by aliasing may be immediately dissipated. Indeed, Shen et al (2006) highlight that they do not see any aliasing happening, and suggest that this is due to the high degree of numerical diffusivity in their code. This is of course a case of two negative features canceling each other.…”

“…The solenoidality of the wave is guaranteed by initializing the velocity field through a streamfunction u = ∇×(ψẑ), with ψ = Ak −1 y sin k x x + k y y . We use the setup of Shen et al (2006), k x,0 = −16π/L x , k y = 4π/L y , with L x = L y = 0.5, A = 10 −4 , c s = Ω = ρ 0 = 1, γ = 7/5, and q = 3/2.…”

Planet formation bursts at the borders of the dead zone in 2D numerical simulations of circumstellar disks

“…Once excited, anticyclonic vortices are easily maintained in 2D simulations where, unless viscosity is present, they cannot decay and will instead merge, growing in size in a cascade of energy towards the largest scale of box (e.g., Johnson & Gammie 2005). However, three-dimensional studies in the context of protoplanetary disks found that tall vortex columns are destroyed, both in non-stratified (Shen et al 2006) and in stratified (Barranco & Marcus 2005) local boxes. This phenomenon is understood as a result of the elliptic instability (Crow 1970;Gledzel et al 1975;Kerswell 2002), by which the stretching term (ω · ∇)u, absent in 2D, breaks down elliptical streamlines such as vortical flow.…”