Linear and non-linear evolution of the vertical shear instability in accretion discs

By R. Nelson, Oliver Gressel, O. Umurhan. 2012

311 citations recorded by the source corpus.

We analyse the stability and non-linear dynamics of power-law accretion disc models. These have mid-plane densities that follow radial power laws and have either temperature or entropy distributions that are strict power-law functions of cylindrical radius, R. We employ two different hydrodynamic codes to perform high-resolution 2D axisymmetric and 3D simulations that examine the long-term evolution of the disc models as a function of the power-law indices of the temperature or entropy, the disc scaleheight, the thermal relaxation time of the fluid and the disc viscosity. We present an accompanying stability analysis of the problem, based on asymptotic methods, that we use to guide our interpretation of the simulation results. We find that axisymmetric disc models whose temperature or entropy profiles cause the equilibrium angular velocity to vary with height are unstable to the growth of perturbations whose most obvious character is modes with horizontal and vertical wavenumbers that satisfy vertical bar k(R)/k(Z)vertical bar 1. Instability occurs only when the thermodynamic response of the fluid is isothermal, or the thermal evolution time is comparable to or shorter than the local dynamical time-scale. These discs appear to exhibit the Goldreich-Schubert-Fricke or 'vertical shear' linear instability. Closer inspection of the simulation results uncovers the growth of two distinct modes. The first are characterized by very short radial wavelength perturbations that grow rapidly at high latitudes in the disc, and descend down towards the mid-plane on longer time-scales. We refer to these as 'finger modes' because they display k(R)/k(Z) 1. The second appear at slightly later times in the main body of the disc, including near the mid-plane. These 'body modes' have somewhat longer radial wavelengths. Early on they manifest themselves as fundamental breathing modes, but quickly become corrugation modes as symmetry about the mid-plane is broken. The corrugation modes are a prominent feature of the non-linear saturated state, leading to strong vertical oscillation of the disc mid-plane. In a viscous disc with aspect ratio H/r = 0.05, instability is found to operate when the viscosity parameter alpha < 4 x 10(-4). In three dimensions the instability generates a quasi-turbulent flow, and the associated Reynolds stress produces a fluctuating effective viscosity coefficient whose mean value reaches alpha similar to 10(-3) by the end of the simulation. The evolution and saturation of the vertical shear instability in astrophysical disc models which include realistic treatments of the thermal physics has yet to be examined. Should it occur on either global or local scales, however, our results suggest that it will have significant consequences for their internal dynamics, transport properties and observational appearance.

Subfields: Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/f14b1c84e2d0e93fcb95d75df4596b214461a7b1

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Citations in the corpus

The Effect of Jupiter's Formation on the Distribution of Refractory Elements and Inclusions in Meteorites (2017, background): “At the same time, one of the greatest recent advances has been the discovery that purely hydroydnamic turbulence is capable of transporting angular momentum in disks, via instabilities such as convective overstability (Klahr & Hubbard 2014; Lyra 2014) and the vertical shear instability, or VSI (Nels”
Global Simulations of the Inner Regions of Protoplanetary Disks with Comprehensive Disk Microphysics (2017, methodology): “In the absence of magnetic fields, a number of mechanisms have been studied, such as the vertical shear instability (Nelson et al. 2013; Stoll & Kley 2014; Lin & Youdin 2015), the Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.”
The great dichotomy of the Solar System: Small terrestrial embryos and massive giant planet cores (2015, background): “In fact, some turbulence can be sustained in the dead zone due to a number of effects, e.g. diffusion of the magnetic accretion stress (Turner et al., 2007), baroclynic instability (Klahr and Bodenheimer, 2003), vertical shear instability (Nelson et al., 2013).”
OBSERVATIONAL SIGNATURES OF PLANETS IN PROTOPLANETARY DISKS: SPIRAL ARMS OBSERVED IN SCATTERED LIGHT IMAGING CAN BE INDUCED BY PLANETS (2015, background): “The disk density and azimuthal velocity are set to maintain hydrostatic equilibrium (e.g., Nelson et al. 2013).”
Turbulence in the TW Hya Disk (2018, background): “A number of turbulence generating mechanisms have been studied, including the Vertical Shear Instability (Nelson et al. 2013; Lin & Youdin 2015), the Zombie Vortex Instability (Marcus et al. 2015; Lesur & Latter 2016), Gravitational Instabilities (Gammie 2001; Forgan et al. 2012; Hirose & Shi 2017),”
Dust Evolution and the Formation of Planetesimals (2016, background): “Alternatively, other hydrodynamical concentration mechanisms may be are able to bridge the gap, e.g. the baroclinic instability/convective overstability (Klahr & Bodenheimer 2003) or the vertical shear instability (Nelson et al. 2013).”
The structure of protoplanetary discs around evolving young stars (2014, background): “Additionally, hydrodynamical instabilities, such as the baroclinic instability (Klahr & Bodenheimer 2003) or the vertical shear instability (Nelson et al. 2013; Stoll & Kley 2014), can act as a source of turbulence in the weakly ionised regions of the disc.”
A Three-dimensional View of Turbulence: Constraints on Turbulent Motions in the HD 163296 Protoplanetary Disk Using DCO+ (2017, background): “The vertical shear instability may also produce turbulence at the level of a few percent of the sound speed (Nelson et al. 2013), although the vertical structure of the turbulence arising from this instability has not been fully characterized in detail.”
Pebble-isolation mass: Scaling law and implications for the formation of super-Earths and gas giants (2018, background): “Our hydrodynamical simulations do not include any turbulent motion of the gas, such as those seen in magnetohydrodynamics (MHD) simulations (Bai 2017) or in simulations with the vertical shear instability (Nelson et al. 2013; Stoll & Kley 2016).”
WEAK TURBULENCE IN THE HD 163296 PROTOPLANETARY DISK REVEALED BY ALMA CO OBSERVATIONS (2015, background): “In regions of weak MRI, a vertical shear instability may develop in the presence of varying angular velocity with height resulting in Reynolds stresses that drive α∼ 10−3 (Nelson et al. 2013).”
Global simulations of protoplanetary disks with net magnetic flux: I. Non-ideal MHD case (2016, background): “It was shown by Nelson et al. (2013) that this instability could be strongly attenuated by letting the entropy distribution reorganize, and then relaxing the temperature distribution over a fraction of orbital period.”