Vortices in accretion discs: formation process and dynamical evolution

Transcription

1 Vortices in accretion discs: formation process and dynamical evolution Geoffroy Lesur DAMTP (Cambridge UK) LAOG (Grenoble) John Papaloizou Sijme-Jan Paardekooper Giant vortex in Naruto straight (Japan) Frontiers In Computational Astrophysics

2 Outline A brief introduction Vortices in accretion discs? Baroclinic instabilities: historical overview The baroclinic instability in 2D Simulation results Instability main properties Phenomenological description The instability in accretion discs Waves emission 3D stability Conclusions 2

3 Vortices in nature Well known in planetary atmospheres Cyclones on Earth Great red spot Generally associated with quasi 2D configuration and rotation/shear 3

4 Vortices in accretion discs? Initially suggested by von Weizsäcker (1944) to explain planetary formation. Reintroduced by Barge & Sommeria (1995) : dust accumulation. In discs, only anticyclonic (counter rotating) vortices can survive. 4

5 Equilibrium of an anticyclonic vortex Ω Coriolis force Pressure gradient Geostrophic balance: 2Ω v = P ρ Vortex streamline Anticyclones are associated with high pressure regions. Only true for «slow» vortices ( ). v/r Ω Lagrangian particles tend to accumulate in the centre (they only feel the Coriolis force). 5

6 Impact on disc dynamics: planet formation Vortices accumulate dust, as they are associated with pressure maxima. Faster collision rate leading to a faster sticking process? Dust becomes gravitationally unstable due to high densities? Dust captured in a vortex (Barge & Sommeria 1995) 6

7 Impact on disc dynamics: transport Rotating gas falls on the central star only if it looses angular momentum. One needs a way to transport angular momentum outward to have accretion: «angular momentum transport problem» Vortices can produce density waves, which then transports angular momentum outward. This is not exactly equivalent to a local turbulent process... (i.e. MHD turbulence, see S. Fromang s talk) Density waves produced by vortices (Johnson & Gammie 25) 7

8 Baroclinic instabilities: an overview (1) Baroclinic instabilities are driven by the radial entropy structure of the disc. Initially suggested in global simulations by Klahr & Bodenheimer (23). Many numerical problems (boundary conditions, numerical convergence) Local linear and numerical studies (Johnson & Gammie 25, 26) did not find anything. No shear No shear No shear with shear 8

9 Baroclinic instabilities: an overview (2) 9

10 Baroclinic instabilities: an overview (3) Petersen et al. (27) revived the idea, with anelastic spectral simulations showing vortex amplification. They also included a new ingredient: thermal diffusion. Petersen et al. 27 Questions: is this instability real? impact on disc structure & dynamic? 1

11 Outline A brief introduction Vortices in accretion discs? Baroclinic instabilities: historical overview The baroclinic instability in 2D Simulation results Instability main properties Phenomenological description The instability in accretion discs Waves emission 3D stability Conclusions 11

12 The shearing box model y z H x Local approximation: Neglect curvature effects Almost incompressible (incompressible approximation valid in first approximation) Have to include the radial stratification to take into account baroclinic effects. 12

13 (some) equations Incompressible equations in 2D (x,y)=(r,φ) Stratification in the Boussinesq approximation Introduces the buoyancy frequency N 2 = 1 γσ P R P R ln Σ γ In 2D, stratification is a source of vertical vorticity through the baroclinic term t ω + u ω = ΛN 2 y θ + ν ω t θ + u (θ + x/λ) = µ θ Non axisymmetric temperature perturbations can locally produce vorticity 13

14 The Snoopy code HD/MHD 3D spectral scheme. Available online ( Advantages: Sheared frame & incompressible approximation: no CFL constrain due to the background sheared flow/sound speed. Very weak numerical dissipation: tight control on physical dissipation processes Disadvantages: Shocks/discontinuities can t be treated spectrally (Gibbs oscillations) 14

15 The effect of stratification t=.1 orbits t=1 orbits t=5 orbits N 2 Ω 2 =.22 y x y x y x N 2 Ω 2 = (no stratification) y x y x y x Vortex amplification is due to the stratification. T r d with d<.5 Requires N 2 < (or equivalently ) 15

16 A nonlinear instability Influence of the amplitude of the initial perturbation 1 1 <! 2 / 2 > Ap=.2 Ap=.4 Ap=.6 Ap=.8 Ap= t The instability appears for finite amplitude disturbances. Explains Johnson & Gammie (25) negative result. 16

17 Phenomenological description A B A B D C D C Convectively unstable radial temperature gradient A to B: The fluid particle is cooler and heavier than the surrounding gas. It is accelerated by gravity toward the star. B to C: Background temperature is constant. The particle is reheated by thermal diffusion. C to D: Fluid particle hotter and lighter than the background: outward acceleration. D to A: Particle cooled by thermal diffusion. Fluid motion is amplified on the AB and CD branches. 17

18 Outline A brief introduction Vortices in accretion discs? Baroclinic instabilities: historical overview The baroclinic instability in 2D Simulation results Instability main properties Phenomenological description The instability in accretion discs Waves emission 3D stability Conclusions 18

19 Compressibility, waves and transport In fully compressible simulations, vortices produce density waves (see also Johnson & Gammie 25 ; Bodo et al 25, 27 ; Heinemann & Papaloizou 29a,b). The SBI is still active in a compressible setup, and produces density waves. vorticity density.4 Density waves transport angular momentum outward with α~1-3 < α > t 19

20 Waves and vortex migration The SBI still work in global simulations Asymmetric wave excitation Vortex migration! (Paardekooper, Lesur & Papaloizou 21) 2

21 3D instabilities and the SBI.2 Accretion disc vortices are unstable in 3D (Lesur & Papaloizou 29) v z! t 21

22 3D instabilities and the SBI (cont d) After some time, a quasi-equilibrium is reached... Self-sustained turbulent vortices SNOOPY (124 x 512 x 128) 22

23 Conclusions e t A «steep» temperature profile will generate vortices everywhere in a disc. Vortices are unstable in 3D, but are not totally destroyed. Vortices produce waves which transport angular momentum generate vortex migration Open questions: Magnetic fields? (magneto-elliptic instabilities, MRI turbulence, see H. Klahr s talk) 3D circulation? (cf Méheut et al. 21) Temperature profile in the disc? 23

24 Thank you