Quasi-steady vortices in protoplanetary disks (original) (raw)
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Vortex generation in protoplanetary disks with an embedded giant planet
Astronomy & Astrophysics, 2007
Context: Vortices in protoplanetary disks can capture solid particles and form planetary cores within shorter timescales than those involved in the standard core-accretion model. Aims: We investigate vortex generation in thin unmagnetized protoplanetary disks with an embedded giant planet with planet to star mass ratio 10-4 and 10-3. Methods: Two-dimensional hydrodynamical simulations of a protoplanetary disk with a planet are performed using two different numerical methods. The results of the non-linear simulations are compared with a time-resolved modal analysis of the azimuthally averaged surface density profiles using linear perturbation theory. Results: Finite-difference methods implemented in polar coordinates generate vortices moving along the gap created by Neptune-mass to Jupiter-mass planets. The modal analysis shows that unstable modes are generated with growth rate of order 0.3 ΩK for azimuthal numbers m=4,5,6, where ΩK is the local Keplerian frequency. Shock-capturing Cartesian-grid codes do not generate very much vorticity around a giant planet in a standard protoplanetary disk. Modal calculations confirm that the obtained radial profiles of density are less susceptible to the growth of linear modes on timescales of several hundreds of orbital periods. Navier-Stokes viscosity of the order ν=10-5 (in units of a2 Ωp) is found to have a stabilizing effect and prevents the formation of vortices. This result holds at high resolution runs and using different types of boundary conditions. Conclusions: Giant protoplanets of Neptune-mass to Jupiter-mass can excite the Rossby wave instability and generate vortices in thin disks. The presence of vortices in protoplanetary disks has implications for planet formation, orbital migration, and angular momentum transport in disks.
Structure, stability, and evolution of 3D Rossby vortices in protoplanetary disks
Astronomy & Astrophysics, 2013
Context. Large-scale persistent vortices could play a key role in the evolution of protoplanetary disks, particularly in the dead zone where no turbulence associated with a magnetic field is expected. These vortices are known to form easily in 2D disks via the Rossby wave or the baroclinic instability. In three dimensions, however, their formation and stability is a complex problem and still a matter of debate. Aims. We study the formation of vortices by the Rossby wave instability in a stratified inviscid disk and describe their 3D structure, stability, and long-term evolution. Methods. Numerical simulations were performed using a fully compressible hydrodynamical code based on a second-order finite volume method. We assumed a perfect-gas law and a non-homentropic adiabatic flow. Results. The Rossby wave instability is found to proceed in 3D in a similar way as in 2D. Vortices produced by the instability look like columns of vorticity in the whole disk thickness; the weak vertical motions are related to the weak inclination of the vortex axis that appears during the development of the RWI. Vortices with aspect ratios higher than 6 are unaffected by the elliptical instability. They relax into a quasi-steady columnar structure that survives hundreds of rotations while slowly migrating inward toward the star at a rate that reduces with the vortex aspect ratio. Vortices with a lower aspect ratio are by contrast affected by the elliptic instability. Short aspect ratio vortices (χ < 4) are completely destroyed in a few orbital periods. Vortices with an intermediate aspect ratio (4 < χ < 6) are partially destroyed by the elliptical instability in a region away from the midplane where the disk stratification is sufficiently strong. Conclusions. Elongated Rossby vortices can survive many orbital periods in protoplanetary disks in the form of vorticity columns. They could play a significant role in the evolution of the gas and the gathering of solid particles to form planetesimals or planetary cores, a possibility that receives a renewed interest with the recent discovery of a particle trap in the disk of Oph IRS 48.
Surveys in Geophysics, 1994
Large, long-lived vortices are abundant in the atmospheres of the giant planets. Some of them survive a few orders of magnitude longer than the dispersive linear Rossby wave packets, e.g. the Great Red Spot (GRS), Little Red Spot (LRS) and White Ovals (WO) of Jupiter, Big Bertha, Brown Spot and Anne's Spot of Saturn, the Great Dark Spot (GDS) of Neptune, etc. Nonlinear effects which prevent their dispersion spreading are the main subject of our consideration. Particular emphasis is placed on determining the dynamical processes which may explain the remarkable properties of observed vortices such as anticyclonic rotation in preference to cyclonic one and the uniqueness of the GRS, the largest coherent vortex, along the perimeter of Jupiter at corresponding latitude. We review recent experimental and theoretical studies of steadily translating solitary Rossby vortices (anticyclones) in a rotating shallow fluid. Two-dimensional monopolar solitary vortices trap fluid which is transported westward. These dualistic structures appear to be vortices, on the one hand, and solitary "waves", on the other hand. Owing to the presence of the trapped fluid, such solitary structures collide inelastically and have a memory of the initial disturbance which is responsible for the formation of the structure. As a consequence, they have no definite relationship between the amplitude and characteristic size. Their vortical properties are connected with geostrophic advection of local vorticity. Their solitary properties (nonspreading and stationary translation) are due to a balance between Rossby wave dispersion and nonlinear effects which allow the anticyclones, with an elevation of a free surface, to propagate faster than the linear waves, without a resonance with linear waves, i.e. without wave radiation. On the other hand, cyclones, with a depression of a free surface, are dispersive and nonstationary features. This asymmetry in dispersion-nonlinear properties of cyclones and anticyclones is thought to be one of the essential reasons for the observed predominance of anticyclones among the long-lived vortices in the atmospheres of the giant planets and also among the intrathermocline oceanic eddies. The effects of shear flows and differences between the properties of monopolar vortices in planetary flows and various laboratory experiments are discussed. General geostrophic (GG) theory of Rossby vortices is presented. It differs essentially from the traditional quasi-geostrophic (QG) and intermediategeostrophic (IG) approximations by the account of (i) all scales between the deformation radius and the planetary scale and (ii) the arbitrary amplitudes of vortices. It is shown that, unlike QG-and IGmodels, the GG-model allows for explaining the mentioned cyclonic-anticyclonic asymmetry not only in planetary flows, but also in laboratory modeling with vessels of near paraboloidal form.
Vortices in stratified protoplanetary disks
Astronomy & Astrophysics, 2016
Context. Large-scale vortices could play a key role in the evolution of protoplanetary disks, particularly in the dead-zone where no turbulence associated with magnetic field is expected. Their possible formation by the subcritical baroclinic instability is a complex issue because of the vertical structure of the disk and the elliptical instability. Aims. In 2D disks the baroclinic instability is studied as a function of the thermal transfer efficiency. In 3D disks we explore the importance of radial and vertical stratification on the processes of vortex formation and amplification. Methods. Numerical simulations are performed using a fully compressible hydrodynamical code based on a second-order finite volume method. We assume a perfect gas law in inviscid disk models in which heat transfer is due to either relaxation or diffusion. Results. In 2D, the baroclinic instability with thermal relaxation leads to the formation of large-scale vortices, which are unstable with respect to the elliptic instability. In the presence of heat diffusion, hollow vortices are formed which evolve into vortical structures with a turbulent core. In 3D, the disk stratification is found to be unstable in a finite layer which can include the mid-plane or not. When the unstable layer contains the mid-plane, the 3D baroclinic instability with thermal relaxation is found to develop first in the unstable layer as in 2D, producing large-scale vortices. These vortices are then stretched out in the stable layer, creating long-lived columnar vortical structures extending through the width of the disk. They are also found to be the source of internal vortex layers that develop across the whole disk along baroclinic critical layer surfaces, and form new vortices in the upper region of the disk. Conclusions. In 3D disks, vortices can survive for a very long time if the production of vorticity by the baroclinic amplification balances the destruction of vorticity by the elliptical instability. However, this possibility is strongly dependent on the disk properties. Such baroclinic vortices could play a significant role in the global disk evolution and in participating to the decoupling of solids from the gas component. They could also contribute to the formation of new out-of-plane vortices by a critical layer excitation mechanism.
Baroclinic Vorticity Production in Protoplanetary Disks. I. Vortex Formation
The Astrophysical Journal, 2007
The factors affecting vortex growth in convectively stable protoplanetary disks are explored using numerical simulations of a two-dimensional anelastic-gas model that includes baroclinic vorticity production and radiative cooling. The baroclinic feedback, in which anomalous temperature gradients produce vorticity through the baroclinic term and vortices then reinforce these temperature gradients, is found to be an important process in the rate of growth of vortices in the disk. Factors that strengthen the baroclinic feedback include fast radiative cooling, high thermal diffusion, and large radial temperature gradients in the background temperature. When the baroclinic feedback is sufficiently strong, anticyclonic vortices form from initial random perturbations and maintain their strength for the duration of the simulation, for over 600 orbital periods. Based on both simulations and a simple vortex model, we find that the local angular momentum transport due to a single vortex may be inward or outward, depending on its orientation. The global angular momentum transport is highly variable in time and is sometimes negative and sometimes positive. This result is for an anelastic-gas model and does not include shocks that could affect angular momentum transport in a compressible-gas disk.
Stability and nonlinear adjustment of vortices in Keplerian flows
Astronomy & Astrophysics, 2007
We investigate the stability, nonlinear development and equilibrium structure of vortices in a background shearing Keplerian flow. We make use of high-resolution global two-dimensional compressible hydrodynamic simulations. We introduce the concept of nonlinear adjustment to describe the transition of unbalanced vortical fields to a long-lived configuration. We discuss the conditions under which vortical perturbations evolve into long-lived persistent structures and we describe the properties of these equilibrium vortices. The properties of equilibrium vortices appear to be independent from the initial conditions and depend only on the local disk parameters. In particular we find that the ratio of the vortex size to the local disk scale height increases with the decrease of the sound speed, reaching values well above the unity. The process of spiral density wave generation by the vortex, discussed in our previous work, appear to maintain its efficiency also at nonlinear amplitudes and we observe the formation of spiral shocks attached to the vortex. The shocks may have important consequences on the long term vortex evolution and possibly on the global disk dynamics. Our study strengthens the arguments in favor of anticyclonic vortices as the candidates for the promotion of planetary formation. Hydrodynamic shocks that are an intrinsic property of persistent vortices in compressible Keplerian flows are an important contributor to the overall balance. These shocks support vortices against viscous dissipation by generating local potential vorticity and should be responsible for the eventual fate of the persistent anticyclonic vortices. Numerical codes have be able to resolve shock waves to describe the vortex dynamics correctly.
Dusty Vortices in Protoplanetary Disks
The Astrophysical Journal, 2006
Global two-dimensional simulations are used to study the coupled evolution of gas and solid particles in a Rossby unstable protoplanetary disk. The initial radial bump in density is unstable to the formation of Rossby waves, which roll up and break into anticyclonic vortices that gradually merge into a large-scale vortical structure persisting for more than 100 rotations. Conditions for the growth of such vortices may naturally appear at the outer edge of the ''dead zone'' of a protoplanetary disk where gas tends to pile up. We find that solid particles are captured by the vortices and change the evolution: (1) large particles rapidly sink toward the center of the vortices and increase the solid-togas ratio by an order of magnitude, (2) solid particles tend to reduce the lifetime of the vortices, and (3) solid particles are effectively confined in the vortices before they are dispersed by the Keplerian shear flow. These results confirm that in a minimum mass solar nebula, persistent vortices could be good places for the formation of the planetesimals or the rocky cores of gas giant planets as soon as particles reach boulder size.
On the stability of elliptical vortices in accretion discs
Astronomy & Astrophysics, 2009
Context. The existence of large-scale and long-lived 2D vortices in accretion discs has been debated for more than a decade. They appear spontaneously in several 2D disc simulations and they are known to accelerate planetesimal formation through a dust trapping process. In some cases, these vortices may even lead to an efficient way to transport angular momentum in protoplanetary discs when MHD instabilities are inoperative. However, the issue of the stability of these structures to the imposition of 3D disturbances is still not fully understood, and it casts doubts on their long term survival Aims. We present new results on the 3D stability of elliptical vortices embedded in accretion discs, based on a linear analysis and several non-linear simulations. Methods. We introduce a simple steady 2D vortex model which is a non-linear solution of the equations of motion, and we show that its core is made of elliptical streamlines. We then derive the linearised equations governing the 3D perturbations in the core of this vortex, and we show that they can be reduced to a Floquet problem. We solve this problem numerically in the astrophysical regime, including a simplified model to take into account vertical stratification effects. We present several analytical limits for which the mechanism responsible for instability can be explained. Finally, we compare the results of the linear analysis to some high resolution numerical simulations obtained with spectral and finite difference methods. A discussion is provided, emphasising the astrophysical consequences of our findings for the dynamics of vortices. Results. We show that most anticyclonic vortices are unstable due to a resonance between the turnover time and the local epicyclic oscillation period. A small linearly stable domain is found for vortex cores with an aspect-ratio of around 5. However, our simulations show that it is only the vortex core that is stable, with the instability still appearing on the vortex boundary. In addition, we find numerically that results obtained under the assumption of incompressibility are not affected by the introduction of a moderate compressibility. Finally, we show that a strong vertical stratification does not create any additional stable domain of aspect ratio, but it significantly reduces growth rates for relatively weak (and therefore elongated) vortices. Conclusions. Elliptical vortices are always unstable, whatever the horizontal or vertical aspect-ratio is. The instability can however be weak and is often found at small scales, making it difficult to detect in low-order finite-difference simulations.
Morphology and dynamical stability of self-gravitating vortices
Astronomy & Astrophysics
Context. Theoretical and numerical studies have shown that large-scale vortices in protoplanetary discs can result from various hydro-dynamical instabilities. Once produced, such vortices can survive nearly unchanged over a large number of rotation periods, slowly migrating towards the star. Lopsided asymmetries recently observed at sub-millimetre and millimetre wavelengths in a number of transition discs could be explained by the emission of the solid particles trapped by vortices in the outer disc. However, at such a distance from the star, disc self-gravity (SG) may affect the vortex evolution and must be included in models. Aims. Our first goal is to identify how vortex morphology is affected by its own gravity. Next, we look for conditions that a self-gravitating disc must satisfy in order to permit vortex survival at long timescales. Finally, we characterise as well as possible the persistent self-gravitating vortices we have found in isothermal and non-isothermal discs. Metho...
Structure and evolution of 3D Rossby Vortices
EPJ Web of Conferences, 2013
Three dimensional compressible simulations of the Rossby Wave Instability are presented in a non-homentropic model of protoplanetary disk. The instability develops like in the two dimensional case, gradually coming to the formation of a single big vortex. This 3D vortex has a quasi-2D structure which looks like a vorticity column with only tiny vertical motions. The vortex survives hundred of rotations in a quasi-steady evolution and slowly migrates inward toward the star.