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3D Vortices in Protoplanetary Disks

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					     Numerical Simulations of

3D Vortices in Stratified,
  Rotating, Shearing
 Protoplanetary Disks
        April 8, 2005 - I PAM
Workshop I: Astrophysical Fluid Dynamics


 Philip Marcus – UC Berkeley
  Joe Barranco – KITP UCSB
       Xylar Asay-Davis – UC Berkeley
        Sushil Shetty – UC Berkeley
    Observations of Protoplanetary
                Disks
Mass 0.01 – 0.1 Msun
Diameter ≈ 100 – 1000 AU
Age ≤ 10 million years
Fluid Dynamics (along with radiation)
Determines Transport Properties
Angular momentum
Dust Grains
Migration of Planetesimals and Planets

    No observations of turbulence or fluid structures (yet)
           Big Picture
    What Does a PPD Look Like?
•   Is it laminar or turbulent – is it both?
•   Is the mid-plane filled with vortices?
•   Are vortices long-lived or transient?
•   Is there an energy cascade?
•   Is it a 2D environment?
•   Is a Keplerian disk stable?
•   Can energy be extracted from the mean shear?
•   Are there pure hydro mechanisms at work?
              Eddy Viscosity neddy
* Replace the nonlinear, and difficult-to calculate
  advective term – (V ¢ r)V with a ficticious,
  linear, easy-to-calculate diffusion r ¢ neddyr V.
* Set neddy = a cs H0 (Shakura & Sunyaev 1973)

      a< 1 because turbulent eddies are probably subsonic and
      not larger than a scale height H0 in extent.

* Parameterize angular mom. transfer and/or rate
  of mass accretion in terms of a.
      Origin of turbulence? Shear instabilities,
      convection, MHD instabilities … etc.
        Does a work for Angular
     Momentum Transport (or anything else)?
• Perhaps with heat transfer in non-rotating, (r ¢ V),
  non-shearing flows such as thermal convection,
    e.g., Prandlt mixing-length theory.
• Runs into problems when used for transporting
  vectors quantities and when there is competition
  among “advectively conserved” quantities.
     e.g., Spherical Couette Flow
• Do not violate Fick’s Law.
    Angular momentum angular must be transported radially
    outward from a forming protostar; yet the disk’s ang. mom.
    increases with radius.
    Formation of Planetesimals



                 mm grains  km planetesimals




• 2 Competing Theories
  – Binary Agglomeration: Sticking vs. Disruption?
  – Gravitational Instability: Settling vs. Turbulence?
    Toomre criterion: vd < πGΣd/ΩK ≈ 10 cm/s
Transport Implied by
   “Hot Jupiters”
       Formed farther out in the disk
       where it was cooler, and then
       migrated to present location.

       Type I Migration: Small
       protoplanet raises tides in the
       disk, which exert torques on the
       planet.
       Type II Migration: Large
       protoplanet opens gap in the
       disk; both gap and protoplanet
       migrate inward on slow viscous
       timescale.
Vortices in Protoplanetary Disks?



                      Anticyclonic
                      shear leads to
                      anticyclonic
                      vortices
Vortices in Protoplanetary Disks?




                     Recipe for vortices:
                      Rapid rotation
                      Intense shear
                      Strong stratification
          Assumptions for PPD Flow
              (being, or about to be, dropped)

Disk is Cold:

• Pressure small, so is radial pressure gradient

• Gas is un-ionized and therefore not coupled to
  magnetic field

• Sound speed cs ¿ Keplerian velocity VK


Cooling modeled as  T / t = L – T/tcool where is fixed.
   Base Flow in Protoplanetary Disk
Near balance between gravity and
centrifugal force:




                                    Anticyclonic
                                    Shear



  No hydrostatic balance in the radial direction!
   Base Flow in Protoplanetary Disk
            (and pre-computational bias)




Vertical hydrostatic balance:   Cool, thin disk:
Vortices in Protoplanetary Disks
• Shear will tear a vortex apart unless:
  – Vortex rotates in same sense as shear. In PPD,
    vortices must be ANTICYCLONES.
  – Strength of the vortex is at least of the same
    order as the strength of the shear.
• ωV ~ σk ~ Ωk
• Ro ≡ ωz/2Ωk ~ 1
Vortices in Protoplanetary Disks
• Velocity across vortex likely to be subsonic;
  otherwise shocks would rapidly dissipate
  kinetic energy of vortex:
  –   V ~ σ k L < cs
  –   V ~ Ωk L < cs
  –   ε ≡ V/cs ~ (Ωk/cs) L < 1
  –   But from hydrostatic balance: cs ~ ΩkH
  –   ε ≡ V/cs ~ L/H < 1
                PPD vs. GRS
Timescales             GRS        PPD
tsh ´ 2/             ¼ 8 d.     ¼ 1 y.
trot ´ 2/            ¼ 26 h.    ¼ 1 y.
tbv ´ 2/bv           ¼ 6 m.     ¼ 1 y.
Rossby Ro ´ trot/tsh   ¼ 0.13     ¼1
Froude Fr ´ tbv/tsh    ¼ 5 £ 10-4 ¼ 1
Richardson ´ 1/Fr2     ¼ 4 £ 106 ¼ 1
Wave speeds:                      cg / cs ¼ 1
         Equations of Motion
• Momentum: With Coriolis and Buoyancy



• Divergence: Anelastic Approximation

• Temperature: With Pressure-Volume Work
   Two-Dimensional Approx. is not
       correct and misleading
Too easy to make
vortices due to
limited freedom,
conservation of 
and inverse
cascade of energy
        Computational Method

• Cartesian Domain: (r,f,z)  (x,y,z)
   – Valid when d ≡ H0 /r0 << 1 and Dr/r0 << 1
• 3D Spectral Method
   – Horizontal basis functions: Fourier-Fourier basis for
     shearing box; Chebyshev otherwise
   – Vertical basis functions: Chebyshev polynomials for
     truncated domain OR Cotangent mapped Chebyshev
     functions for infinite domain
• Parallelizes and scales
    Spectral Methods Are Not L
•   Limited to Cartesian boxes
•   Limited to Fourier series
•   Limited to linear problems
•   Impractical without fast transforms
           Spectral Methods
• Beat 2nd order f.d. by ~ 4 per spatial
  dimension for 1% accuracy, (4)2 for 0.1%
• Often have diagonalizable elliptic operators
• Non-dissipative & dispersive – must
  explicitly put in
• Have derivative operators that commute
• 2 /  x2 = ( /  x) ( /  x)
• Should not be used with discontinuities
              Tests of the Codes
•   Energy, momentum, enstrophy and p.v. balance
•   Linear eigenmodes
•   Agrees with 2D solutions – Taylor Columns
•   Agrees with 3D asymptotics (equilibria)
•   Different codes (mapped, embedded and truncated)
    agree with each other
• Gave unexpected results for which
  the algorithms were not “tuned”
Sliding box: shear  =(-3/2)
          t=0          t=Lx/ d r




                            dr




            Lx
  Open Boundaries or 1 Domain
• Mapping - 1 < z < 1 ! 0 < q < 
• Arbitrary truncation to finite domain
   impose arbitrary boundary conditions
• Embedding
    -Ldomain < -Lphysical < z < Lphysical < Ldomain
                       Mappings



             q
                                            q




   z = L cos(q)
   Chebyshev                          z = L cot(q)

Others: Matsushima & Marcus JCP 1999
  Rational Legendre for cylindrical coordinate at origin
  and harmonic at 1
                      Embedding
               Lose 1/3 of points in buffer regions

  Lembedding
                          Buffer region

  Lphysical

                  Domain of physical interest

  Lphysical
                          Buffer region

-Lembedding
             No Free Lunch
• Mappings lose 1/3 of the points for |z| > L
      advantage is (maybe) realistic b.c.
• Truncations waste 1/3 of the points due to
     clustering at the artificial boundary
• Embeddings lose 1/3 of the points
     at Lphysical < |z| < Lembedding
        Internal Gravity Waves
• Breaking: hri / exp{-z2/2H02} (Gaussian)
• Energy flux ~ hri V3 /2
• Huge energy source (locally makes slug flow of
  Keplerian, differential velocity)
• Source of perturbations is oscillating vortices
• Fills disk with gravity waves
• Waves are neutrally stable
• Gravity / z, stratification / z ! Brunt-Vaisalla
  frequency / z
• For finite z domain (-L < z < L) there are wall-
  modes / exp{+z2/2H02}
           Gravity Wave Damping 1
•    Vx /  t = L
•    Vy /  t = L
•    Vz /  t = L -  h /  z + g T/hTi
•    T /  t = L - (d hSi / dz) (hTi/cp) Vz
       - T/tcool

       2 Vz /  t2 = - (d hSi / dz) (g/cp) Vz
           Gravity Wave Damping 1
•    Vx /  t = L
•    Vy /  t = L
•    Vz /  t = L -  h /  z + g T/hTi – a1  T/ t
•    T /  t = L - (d hSi / dz) (hTi/cp) Vz – a2  Vz/ t
       - T/tcool

        2 Vz /  t2 = - (d hSi / dz) (g/cp) Vz/(1 - a1 a2)

         + [a1(d hSi / dz) (hTi/cp) - a2(g/hTi)]  Vz/ t
What does 2D mean?



Columns or pancakes?
Tall Columnar Vortex
  !
à D r = 2H0   Vortex in the Midplane of PPD




               Ã                 r0Df = 4H0                    !
               Blue = Anticyclonic vorticity, Red = Cyclonic vorticity
                                Ro = 0.3125
                                                  !
   f-z plane




                                                 D z = 8H0
   at r=r0

Ro = 0.3125
Blue = Anticyclonic vorticity
Red = Cyclonic vorticity




                                                  Ã
                                Ã r0Df = 4H0 !
                                                !
   r-z plane




                                               D z = 8H0
   at f=0

Ro = 0.3125
Blue = Anticyclonic vorticity
Red = Cyclonic vorticity




                                                Ã
                                Ã Dr = 2H0 !
z = 1.5 H0



z = 1.0 H0



z = 0 H0
Spontaneous Formation of
 Off-Midplane Vortices
3D Vortex in PPD
           Equilibrium in Horizontal
                                                    
• Horizontal momentum
                                                   v?
  equation:
• For Ro · 1, Geostophic        Coriolis
  balance between gradient of   Force       High
  pressure and the Coriolis                Pressure
  force.
• Anticyclones have high
  pressure centers .                           z

• For Ro À 1, low pressure
  centers.
                           Role of vz
• In sub-adiabatic flow: rising cools the                  
  fluid while sinking warms it.
                                                          v?
• This in turn creates cold, heavy top
  lids and warm, buoyant bottom lids.
• This balances the vertical pressure                      vz
  force (and has horizontal temperature
  gradients in accord with the thermal
  wind equation.
• Numerical calculations show that after
  lids are created, vz ! 0.                           z
• Magnitude of vz is set by dissipation
  rate, the faster of tcool or “advective cooling”.
    Radial Cascade of Vortices
• Internal gravity waves or small velocity
  perturbations with vz, or that create vz at
  outer edge of disk
    Radial Cascade of Vortices
• Anticlonic bands embedded in like-signed
  shear flow are unstable and break up into
  stable vorices
• Cyclonic bands embedded in the opposite-
  signed Keplerian shear are stable Marcus JFM 1990
• Energy provided by “step-functioning” the
  linearly stable background shear flow
                                 Coughlin & Marcus PRL 1997
       Radial Cascade of Vortices
       Create Radial bands of z

•  Vx /  t = L
•  Vy /  t = L
                   }
• D z / D t = (baroclinic) - (2 +  + z) (vz /H)

•  Vz /  t = L
• T/t =L
Radial Cascade of Vortices
             • New vortex wobbles
               in shear ambient flow
               (3D Kida ellipse)
             • Produces new gravity
               waves, characterized
               by T, and more
               importantly vz
        One Test of Picture
• Add gravity wave damping mechanism at
  just one thin radial band
• Stops Cascade
      Angular Momentum Transport
   Transport / h Vr Vf i

 Fore-Aft symmetry:




Vr(f) = -Vr(-f)
                                   r
Vf(f) = +Vf(-f)
                            f
t/tORB
t/tORB
         t/t ORB
    Grain Trapping in Vortices
• Grains feel Coriolis, centrifugal, gravity
  (from all sources) and gas drag forces.
• Stokes Drag Stopping Time (or use Epstein
  for rgrain < mean free path ' 1cm at 1AU):




• At 1 AU:
Why don’t grains centrifuge out of vortices?
       Grain Trapping in 2D Vortices
τS
0.01


0.1



1.0


        The attractors here are all limit cycles not points
 Dust Dynamics
Attracting Regions
            Conclusions
  Disk filled with off-mid-plane
       vortices and waves
• Vortices are unstable in the midplane (where the
  stratification vanishes)
• Vortices thrive off the midplane (where there is
  stratification)
• Vortices from spontaneously from “noise” and
  draw their energy from the Keplerian shear.
• Vortices transport angular momentum radially
  outward
• Vortices capture and concentrate dust grains
                  Future Work
• MRI
• Formation Mechanisms
  – Spin-up of temperature & density “lumps”
  – Inverse cascade (small vortices merging into larger
    vortices)
  – Turbulence between vortices or laminar?
• Dust dynamics
  – Grain collision rates & velocities
  – Gravitational settling & instability
  – 2-Fluid Model
• Mass, Angular Momentum, Planetesimal Transport
                   Conclusions
• Accretion Disk Vortices appear to be non-isolated
• Fore-Aft symmetry breaking important for
  transport of angular momentum
• Large Vz affects dust accumulaton
• Dust grains trap in 3D vortices
   – Dust densities can be enhances at least to rdust ~ rgas

Future work
           back-reaction of dust on Work
• Simulate Questions & Futuregas
    – Dust can be modeled as and t
   Time scales for tMERGE continuous fluid withoutvortices?)
 * pressure                       FORM (isolated
   Understand why time-average h Vflow
•* Simulate particles in full turbulent r Vf i > 0
                  Conclusions
• Scaling for         agrees with CFD for both
  planetary and accretion disk vortices over a wide
  range of parameters.

• vz is set by dissipation time scale(s).

• Aspect ratio (L?/H) for both planetary and disk
  vortices set by (break-up/)(/H)1/2.

• Scaling shows that planetary and accretion disk
  vortices obey same physics and scaling laws.
                     Particle Trapping
•   Vortex traps grains with rgrain < rmax
•   rmax / vz,max / 1/t{mixing}
•   This simulation: rmax ~ 1 mm
    0.4

•   Stable orbits: fixed point or limit cycles
    0.2
y




     0

    -0.2

         -1   -0.5     0   0.5   1   1.5   2
                           x
Dust Grains (Particle-laden Flow)
• Forces on the dust
   –   Coriolis
   –   Centrifugal
   –   Gravity
   –   Gas drag (steady state vortex)
• Drag laws for locally laminar flows
   – Stokes: D = CD, S rgas  cs rgrain D v
   – Epstein: D = CD, S rgas cs r2grain D v
• For R = 1 AU ~ 150 Million km:
   –  ~ 1 cm
   – rgrain < 1 cm ! Epstein regime
                  Eddy Viscosities
• Replace the nonlinear advective terms $-{\bf v} \cdot \nabla
  {\bf v}$ with a linear diffusion $\nabla \cdot nabla {\bf v}$

• ngular momentum problem: In order for mass to spirlal
  inward onto the growing protostar, angular momentum must
  be transported outward.
• Viscous torques in a differentially rotating disk?
   – Timescale: tvisc ¼ R2/n » 1012 years!
   – Couple orders of magnitude longer than age of universe!
• Shakura-Sunyaev (1973): Re = VL/n » 1014… perhaps
  turbulence enhances the effective viscosity:
   – nturb = a H0 cs
   – a < 1 because turbulent eddies are probably subsonic and no larger
     than a scale height in extent.
• Origin of turbulence? Nonlinear shear instabilities,
                Dust Dynamics
• Horizontal grain dynamics are complex
   – See, e.g. Barranco and Marcus (2000), Chavanis (1999)
   – Trapping if vortex isn’t rotating too fast and grains are
     not too heavy




• Vertical dynamics not previously studied
 Long-lived, Compact, 3-D Vortices
• Rare in laboratory and engineering flows.
   – Short lived in 3-D.
     (Kolmogorov ) coherence time = turn-around time).
   – Long-lived vortices are common in 2-D.
     (vortex mergers, inverse cascade, infinitely many conserved
     quantities).
• Important examples in Geophysical and Astrophysical
  Systems with
   – Strong rotation (Rossby Number Ro ´ z/ · 1).
   – Background shear (z/).
   – Stable stratification (N/).
         Two Disparate Examples
• Planetary vortices.              • Vortices in H2 disk
   – Jupiter’s Great Red             around a protostar.
     Spot, White Ovals, etc.
   – Earth’s Antarctic Polar
     Vortex.



          vz ¿ v?
                              Lz             vz»v? Lz

            L?                                L?
                    L? À Lz                         L? < Lz
z=0   z=H
           Protoplanetary disk
• Keplerian gas and dust around a Protostar
• Gravity approximately linear in z:




• Vertical hydrostatic balance – pressure and density
  / exp(-z2/(2 H2))

				
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