On the Stability of Elliptical Vortices in Accretion Discs
📝 Abstract
(Abriged) 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. 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. Aim: 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 derive the linearised equations governing the 3D perturbations in the core of an elliptical vortex, and we show that they can be reduced to a Floquet problem. We solve this problem numerically in the astrophysical regime and we present several analytical limits for which the mechanism responsible for the instability can be explained. Finally, we compare the results of the linear analysis to some high resolution simulations. Results: We show that most anticyclonic vortices are unstable due to a resonance between the turnover time and the local epicyclic oscillation period. In addition, we demonstrate 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.
💡 Analysis
(Abriged) 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. 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. Aim: 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 derive the linearised equations governing the 3D perturbations in the core of an elliptical vortex, and we show that they can be reduced to a Floquet problem. We solve this problem numerically in the astrophysical regime and we present several analytical limits for which the mechanism responsible for the instability can be explained. Finally, we compare the results of the linear analysis to some high resolution simulations. Results: We show that most anticyclonic vortices are unstable due to a resonance between the turnover time and the local epicyclic oscillation period. In addition, we demonstrate 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.
📄 Content
The existence of 2D long-lived vortices in accretion discs was first proposed by von Weizsäcker (1944) in an outmoded model of planet formation. This idea was revived by Barge & Sommeria (1995) to accelerate planetesimals formation by a dust trapping process. This kind of vortex is often observed in 2D simulations of discs (see e.g. Godon & Livio 1999;Umurhan & Regev 2004;Johnson & Gammie 2005b;Bodo et al. 2007), since 2D turbulence is known to generate an inverse cascade of energy leading to large 2D vortices (Onsager 1949). Vortices may also be generated by 2D instabilities such as Rossby wave instabilities (Lovelace et al. 1999) or baroclinic instabilities (Klahr & Bodenheimer 2003;Petersen et al. 2007), although the latter is still a matter of ongoing debate (Johnson & Gammie 2005a, 2006). These vortices may play at least two important roles regarding accretion disc dynamics. First, they could lead to an efficient angular momentum transport process in regions in which the magneto-rotational instability (Balbus & Hawley 1998) doesn’t operate, such as in dead zones (Gammie 1996). Second, they are a very efficient way to accelerate the planetesimal formation process in protoplanetary discs (Johansen et al. 2004). However, the stability of these vortices when small 3D disturbances are imposed is largely unknown.
In the astrophysics community, this issue has been investigated mainly numerically. Shen et al. (2006) examined the formation of 2D vortices starting from 2D turbulence in fully compressible simulations. According to their results, a small 3D noise added to their initialy 2D configuration destroys the coherent vortices in a few orbits, relaxing the flow to its laminar state. Barranco & Marcus (2005) also computed the evolution of 3D vortices using an anelastic code incorporating vertical stratification. As Shen et al. (2006), they found that midplane vortices were destroyed by 3D perturbations. However, they also showed that off-midplane vortices could survive for several hundreds of orbits, leading to the possibility of a stabilizing effect due to the stratification It is often assumed that these vortices are unstable because of the elliptical instability. The elliptical instability is a parametric instability appearing when a multiple of the vortex turnover frequency matches an inertial wave frequency, leading to a positive resonance. It is observed when the backgound flow follows closed streamlines, and being localized on individual streamlines is a local instability (in particular it doesn’t need to involve the vortex boundaries). This instability was first found numerically by Pierrehumbert (1986) and described using Craik & Criminale (1986) solutions by Bayly (1986) for pure elliptical flows. The rotating case was studied by Craik (1989), who showed that anticyclonic elliptical flows can be stable for some rotation rates. Interested readers may consult Kerswell (2002) for a more extensive discussion of the elliptical instability and its development in fluid mechanics.
In the present paper, we investigate the elliptical instability in the context of accretion disc vortices. We first present a steady 2D vortex model, which is a non-linear solution of the local disc equations. We then present the linearised equations governing 3D perturbations inside the vortex. A criterion for the instability is derived from these equations and a physical understanding of the mechanism responsible for the instability is provided. We briefly extend these results to a simplified stratified case, and we compare our findings to fully non-linear simulations of accretion discs vortices. Finally, we provide a discussion and a comparison with previous work.
In the following, we will assume a local model for the accretion disc, following the shearing-sheet approximation. The reader may consult Hawley et al. (1995), Balbus (2003) and Regev & Umurhan (2008) for an extensive discussion of the properties and limitations of this model. As a simplification, we will assume the flow is incompressible, consistently with the small shearing box model (Regev & Umurhan 2008). The shearing box equations are found by considering a Cartesian box centred at r = R 0 , rotating with the disc at angular velocity Ω = Ω(R 0 ). We define R 0 φ → x and r -R 0 → -y for consistency with the standard notation for plane Couette flows (e.g. Drazin & Reid 1981). Note that this definition differs from the standard notation used in shearing boxes (Hawley et al. 1995) with x → -y SB , y → x SB and z → z SB . In this rotating frame, one obtains the following set of governing equations
(2)
In these equations, we have defined the mean shear S = -r∂ r Ω, which is set to S = (3/2)Ω for a Keplerian disc. The generalised pressure Π = P/ρ 0 is calculated solving a Poisson equation with the incompressibility condition. One can check easely that the velocity field u = Sye x is a steady solution of these equations.
We want to study the stability of a steady
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