On the stability of elliptical vortices in accretion discs
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK e-mail: email@example.com
Accepted: 3 March 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.
Key words: accretion, accretion disks / instabilities / hydrodynamics
© ESO, 2009