Baroclinic Vorticity Production in Protoplanetary Disks. I. Vortex Formation

Abstract

The formation of vortices in protoplanetary disks is explored via pseudospectral numerical simulations of an anelasticgas model. This model is a coupled set of equations for vorticity and temperature in two dimensions that includes baroclinic vorticity production and radiative cooling. Vortex formation is unambiguously shown to be caused by baroclinicity, because (1) these simulations have zero initial perturbation vorticity and a nonzero initial temperature distribution, and (2) turning off the baroclinic term halts vortex formation, as shown by an immediate drop in kinetic energy and vorticity. Vortex strength increases with larger background temperature gradients, warmer background temperatures, larger initial temperature perturbations, higher Reynolds number, and higher resolution. In the simulations presented here, vortices form when the background temperatures are $200 K and vary radially as r À0:25 , the initial vorticity perturbations are zero, the initial temperature perturbations are 5% of the background, and the Reynolds number is 10 9 . A sensitivity study consisting of 74 simulations showed that as resolution and Reynolds number increase, vortices can form with smaller initial temperature perturbations, lower background temperatures, and smaller background temperature gradients. For the parameter ranges of these simulations, the disk is shown to be convectively stable by the Solberg-Høiland criteria.

The baroclinic instability in the context of layered accretion

“…The extreme cases of an adiabatic run (run E) and a nearly isothermal one (τ c = 0.1 orbit, run B) are also presented. In agreement with Petersen et al (2007a), the runs with longer thermal times allow for a stronger increase in enstrophy in the first orbits, also seen in the adiabatic case. This is because the initial thermal perturbations disperse slowly without thermal relaxation, thus remaining tight (strong gradients) and allowing for a stronger baroclinic amplification.…”

“…The enstrophy is then amplified by the local baroclinic vector via the positive feedback described in the introduction. We witness the same general phenomena as Petersen et al (2007a), even though the details of implementating the entropy gradient are different. In global simulations, the vortex swings gas parcels back and forth from cold to hot, which causes baroclinicity and vortex growth.…”

“…Another way is of course diffusion. Petersen et al (2007a) and Lesur & Papaloizou (2010) report sustained baroclinic growth using thermal diffusion, and this is also why the 3D simulations of Klahr & Bodenheimer (2003) with flux-limited diffusion also experienced baroclinic growth. In the 2D simulations by Klahr & Bodenheimer (2003), the thermal relaxation was numerical and a result of low resolution and a dispersive numerical scheme.…”

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

“…Klahr & Bodenheimer (2006) further argue that anticyclonic vortices would be less turbulent than the ambient gas, which in turn would lead to velocity dispersions that are low enough to prevent fragmentation. Vortices in disks can be the result of the baroclinic instability (Klahr & Bodenheimer 2003;Klahr 2004;Petersen et al 2007), the Rossby wave instability (Lovelace et al 1999;Li et al 2000;Li et al 2001) or, perhaps, the MRI (Fromang & Nelson 2005).…”

Disk Weather

“…Nevertheless Klahr (2004) [2] showed that there is no linear instability for baroclinic disks. Finally Petersen et al (2007a,b) [3,4] showed in their global incompressible simulations that in addition to radial entropy gradients a disk needs the right amount of thermal readjustment in order to let vortices grow. The combined condition of strong enough perturbations and sufficiently large Reynolds numbers led them to the conclusion that the Baroclinic Instability in accretions disks is actually a non-linear feature.…”