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?
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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
Wave speeds:
¼ 4 £ 106 ¼ 1
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
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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
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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
z = L cos(q)
Chebyshev
q
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 wallmodes / exp{+z2/2H02}
Gravity Wave Damping 1
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 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
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 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
!
D z = 8H0
f-z plane
at r=r0
Ro = 0.3125
Ã
Blue = Anticyclonic vorticity
Red = Cyclonic vorticity
à r0Df = 4H0 !
!
D z = 8H0
r-z plane
at f=0
Ro = 0.3125
Ã
Blue = Anticyclonic vorticity
Red = Cyclonic vorticity
à 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
equation:
• For Ro · 1, Geostophic
balance between gradient of
pressure and the Coriolis
force.
• Anticyclones have high
pressure centers .
• For Ro À 1, low pressure
centers.
v?
Coriolis
Force
High
Pressure
z
Role of vz
• In sub-adiabatic flow: rising cools the
fluid while sinking warms it.
• This in turn creates cold, heavy top
lids and warm, buoyant bottom lids.
• This balances the vertical pressure
force (and has horizontal temperature
gradients in accord with the thermal
wind equation.
• Numerical calculations show that after
lids are created, vz ! 0.
• Magnitude of vz is set by dissipation
rate, the faster of tcool or “advective cooling”.

v?
vz
z
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 oppositesigned 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?
τS
Grain Trapping in 2D Vortices
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
Work
• Simulate Questions
back-reaction&
of Future
dust on gas
– Dustscales
can befor
modeled
as continuous
fluid
withoutvortices?)
Time
t
and
t
(isolated
* pressure
MERGE
FORM
Understand
why time-average
h Vrflow
Vf i > 0
•* Simulate
particles
in full turbulent
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
y
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Vortex traps grains with rgrain < rmax
rmax / vz,max / 1/t{mixing}
This
simulation:
r
~
1
mm
max
0.4
0.2
Stable orbits: fixed point or limit cycles
0
-0.2
-1
-0.5
0
0.5
x
1
1.5
2
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.
– Jupiter’s Great Red
Spot, White Ovals, etc.
– Earth’s Antarctic Polar
Vortex.
Lz
vz ¿ v?
L?
L? À Lz
• Vortices in H2 disk
around a protostar.
vz»v? Lz
L?
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))