Vortices in accretion discs: formation process and dynamical...

Frontiers In Computational Astrophysics

Geoffroy LesurDAMTP (Cambridge UK)

LAOG (Grenoble)

John PapaloizouSijme-Jan Paardekooper

Vortices in accretion discs: formation process and dynamical evolution

Giant vortex in Naruto straight (Japan)

Geoffroy Lesur Frontiers in Computational Astrophysics October 2010

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


Vortices in nature Well known in planetary atmospheres

3

Cyclones on EarthGreat red spot

Generally associated with quasi 2D configuration and rotation/shear


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


Equilibrium of an anticyclonic vortex

5

Ω

Vortexstreamline

Pressuregradient

Coriolis force

2Ω × v = −∇P

ρ

Anticyclones are associated with high pressure regions. Only true for «slow» vortices ( ). Lagrangian particles tend to accumulate in the centre (they only feel

the Coriolis force).

v/r Ω

Geostrophic balance:


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?

6

Dust captured in a vortex(Barge & Sommeria 1995)

Geoffroy Lesur Frontiers in Computational Astrophysics October 2010 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»

shear stress with its associated outward transport of angular mo-mentum. Previous results using a local model have assumed in-compressible flow and either report no angular momentumtransport (Umurhan& Regev 2004) or report a value (! ! 10"5;Barranco & Marcus 2005) that is 2 orders of magnitude lowerthan what we find when we include the effects of compress-ibility. Global simulations (Godon & Livio 1999, 2000; Li et al.2001) have a difficult time accessing the high resolution that we

have shown is required for a significant shear stress due tocompressibility.Our work leaves open the key question of what happens in

three dimensions. Our vortices, which have radial and azimuthalextent PH, are inherently three-dimensional. Three-dimensionalvortices are susceptible to the elliptical instability (Kerswell 2002)and are likely to be destroyed on a dynamical timescale. The factthat vortices persist in our two-dimensional simulations and not inthe local (three-dimensional) shearing-box calculations of Balbuset al. (1996) is likely due to dimensionality. The recent numericalresults of Barranco & Marcus (2005) indicate that vortices nearthe disk midplane are quickly destroyed, whereas vortices sur-vive if they are a couple of scale heights away from the mid-plane. Strong vertical stratification away from the midplane may

Fig. 6.—Snapshot of the velocity divergence superimposed on the potentialvorticity in a medium-resolution (2562) version of the fiducial run. The thin (redin electronic edition) contours indicate negative divergence and are associatedwith shocks. The thick (blue in electronic edition) contours indicate negativepotential vorticity. [See the electronic edition of the Journal for a color versionof this figure.]

Fig. 7.—Evolution of a vortical shwave amplitude in a low-resolution (642)run, in units of " . The initial shwave vector (kx0; ky) is ("16#/Lx; 4#/Ly), cor-responding to "0 # "4. The interval between successive peaks (a numerical ef-fect due to aliasing) is " # Nx/ny, where ny # 2 is the azimuthal wavenumber.

Fig. 8.—Evolution of the shear stress ! in a set of runs with varying initial$. Apparently for low values of $ the shear stress is reduced, but for an initialMach number near 1 the stress saturates. All runs have L # 4H .

Fig. 9.—Evolution of the shear stress ! in a set of runs with varying initial Lbut the same initial Mach number $. [See the electronic edition of the Journal fora color version of this figure.]

JOHNSON & GAMMIE154 Vol. 635

Density waves produced by vortices (Johnson & Gammie 2005)

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)


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 (2003). Many numerical problems (boundary conditions, numerical convergence)

Local linear and numerical studies (Johnson & Gammie 2005, 2006) did not find anything.

8

No shear

No shear

No shear

with shear

change, we could also drop the damping of the radial com-ponent of the velocity close to radial boundaries. The den-sity distribution again was ! / R!1 (constant ! " 300 gcm!2), and the temperature distribution was T / R!1; thus,we have a baroclinically unstable situation as in model 3,which results from H=R # 0:055. The model was first runinto a stable one-dimensional axisymmetric state, where theresidual velocities were less than 10!4 cm s!1. Without asymmetry-breaking instability and turbulence generation,this disk cannot evolve and would stay perfectly laminarforever, as in the dead zone described byGammie (1996).

The initial density distribution was then perturbed byrandom noise of amplitude only 0.1%. The initial state ispractically axisymmetric. Figure 19 illustrates the evolutionof the flow in two space dimensions, over the full 360$. Afterthe first orbit (30 yr at 10 AU; Fig. 19a), only little structurehas evolved, but with time, a prominent anticyclonic vortexforms, which reflects the assumption that m # 1 is the pre-ferred mode. Intermittently, a second vortex also forms,and we assume that their number is limited only by the nar-rowness of our disk and a lack of matter. The vortex growsin mass and propagates radially outward, possibly as aresult of the gradient of background vorticity and the fact

Fig. 17.—Model 5: quantities plotted have the samemeaning as those in Fig. 12. This calculation was run over 230 orbits.

Fig. 18.—Model 5: spectral density distribution of the velocities at themidplane computed along the ’-direction and averaged over radius. Theslope for isotropic, incompressible turbulence (i.e., a Kolmogorov spec-trum) is indicated by the dashed line, and the spectrum for two-dimensionalgeostrophic flows by the dotted line.

886 KLAHR & BODENHEIMER Vol. 582



9


Petersen et al. (2007) revived the idea, with anelastic spectral simulations showing vortex amplification.

They also included a new ingredient: thermal diffusion.

Questions: is this instability real? impact on disc structure & dynamic?


10

Petersen et al. 2007


Outline




Conclusions

11


The shearing box model

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

H x

y z


(some) equations

Incompressible equations in 2D (x,y)=(r,φ) Stratification in the Boussinesq approximation Introduces the buoyancy frequency

In 2D, stratification is a source of vertical vorticity through the baroclinic term

13

∂tω + u · ∇ω = ΛN2∂yθ + ν∆ω

∂tθ + u · ∇(θ + x/Λ) = µ∆θ

N2 = − 1γΣ

∂P

∂R

∂

∂Rln

P

Σγ

Non axisymmetric temperature perturbations can locally produce vorticity


The Snoopy code

14

HD/MHD 3D spectral scheme. Available online (http://www.damtp.cam.ac.uk/user/glesur/).

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)

http://www.damtp.cam.ac.uk/user/glesur/

http://www.damtp.cam.ac.uk/user/glesur/


The effect of stratification

Vortex amplification is due to the stratification.Requires N2<0 (or equivalently )

15

(no stratification)

t=0.1 orbits t=10 orbits t=50 orbits

x

y

!0.5 0 0.5!0.5

0

0.5

!2

!1.5

!1

!0.5

0

0.5

1

1.5

2

x

y

!0.5 0 0.5!0.5

0

0.5

!1.2

!1

!0.8

!0.6

!0.4

!0.2

0

0.2

0.4

x

y

!0.5 0 0.5!0.5

0

0.5

!1

!0.5

0

0.5

x

y

!0.5 0 0.5!0.5

0

0.5

!2

!1.5

!1

!0.5

0

0.5

1

1.5

2

x

y

!0.5 0 0.5!0.5

0

0.5

!1.2

!1

!0.8

!0.6

!0.4

!0.2

0

0.2

0.4

x

y

!0.5 0 0.5!0.5

0

0.5

!1

!0.5

0

0.5

N2

Ω2= −0.022

N2

Ω2= 0

T ∼ rd with d < −0.5


A nonlinear instability

The instability appears for finite amplitude disturbances.Explains Johnson & Gammie (2005) negative result.

16

Influence of the amplitude of the initial perturbation

0 2000 4000 6000 8000 10000

10−3

10−2

10−1

t

<!2 /

2 >

Ap=0.2Ap=0.4Ap=0.6Ap=0.8Ap=1.0


Phenomenological description

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

A B

CD

A B

CD

Convectively unstable radial temperature gradient


Outline




Conclusions

18


Compressibility, waves and transport

19

In fully compressible simulations, vortices produce density waves (see also Johnson & Gammie 2005 ;Bodo et al 2005, 2007 ; Heinemann & Papaloizou 2009a,b).

The SBI is still active in a compressible setup, and produces density waves.

vorticity density

Geo!roy Lesur and John C. B. Papaloizou: The subcritical baroclinic instability in local accretion disc models 9

Fig. 12. Vorticity map (left) and surface density map (right) for the large box simulation with applied viscosity andA = 0.5 at t=190.

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

50 100 150 200

< !

>

t

Fig. 13. Time history of the running means of the the spatially averagedvalue of ! for the small box (lower curve) and large box (upper curve)simulations with applied viscosity and with A = 0.5 and A = 0.25respectively.

with and without applied viscosity are shown. These lead to so-lutions for which anticyclonic vortices are sustained for longtimes. The case with no viscosity is more active as expected butotherwise looks similar to the case with applied viscosity. Bycontrast when the amplitude of the initial perturbation is reducedto A = 0.025 no sustained vortices are seen. This demonstratesthat our numerical setup is not subject to any linear instability,such as Rossby wave instabilities (Lovelace et al. 1999). The en-strophy attains a low level in the case with no applied viscositywhich is a consequence of a long wavelength linear axisymmet-ric disturbance that shows only very weak decay provided by

numerical viscosity in this run. Fig. 10 shows a vorticity map forthe small box simulation with applied viscosity and A = 0.25.Anticylconic vortices are clearly seen in this case supporting thefinding from the incompressible runs that a finite amplitude ini-tial kick is required to generate them.

The time history of the evolution of the enstrophy for largebox simulations with A = 0.5 with and without applied viscos-ity is shown in Fig. 11. Corresponding vorticity and surface den-sity maps for the case with applied viscosity are shown in Fig.12. Again the inviscid case is more active but nonetheless thecorresponding maps look very similar. In these cases the anticy-clonic vortices are present as in the small box case but there is in-creased activity from density waves as seen in the surface densitymaps. These waves could be generated by a process similar to theswing amplifier with vorticity source described by Heinemann& Papaloizou (2009a,b), although the structures we observe donot strictly correspond to a small scale turbulent flow. The den-sity waves are associated with some outward angular momen-tum transport. However, the value of ! measured from the vol-ume average of the Reynolds stress is always highly fluctuating.Accordingly we plot running means as a function of time for thesmall box and large box with applied viscosity in Fig. 13. In thesmall box case there is a small residual time average ! 10"4 butin the large box this increases to ! 3 # 10"3. This is clearly aconsequence of the fact that the small box is close to the incom-pressible regime, whereas the large box being e!ectively largerthan a scale height in radial width allows the vortices to becomelarge enough to become significantly more e!ective at excitingdensity waves.

Density waves transport angular momentum outward with α~10-3


Waves and vortex migration

20

The SBI still work in global simulations Asymmetric wave excitation Vortex migration ! (Paardekooper, Lesur & Papaloizou 2010)


3D instabilities and the SBI

21

Fig. 17. Vorticity map from a simulation starting with a 2D Kida vortex plus a weak 3D noise. We show a snapshot at t 660 (left), t 750

Time history of the maxima of the 3 components of the velocity3D noise. Once the SBI has formed vor-

nd balance

0 500 1000 1500

−0.2

0

0.2

v z

0 500 1000 1500

2

4

6

8

t

!

Accretion disc vorticesare unstable in 3D

(Lesur & Papaloizou 2009)


3D instabilities and the SBI (cont’d)

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Self-sustained turbulent vorticesSNOOPY (1024 x 512 x 128)

After some time, a quasi-equilibrium is reached...


Conclusions

23

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. 2010) Temperature profile in the disc?

efficient Numerical tools required

Thank you

Vortices in accretion discs: formation process and dynamical...

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