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 UCSBXylar Asay-Davis – UC Berkeley

Sushil Shetty – UC Berkeley

Numerical Simulations of

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 PictureWhat 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 eddy

* Replace the nonlinear, and difficult-to calculate advective term – (V ¢ r)V with a ficticious, linear, easy-to-calculate diffusion r ¢ eddyr V.

* Set eddy = cs H0 (Shakura & Sunyaev 1973)

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

Origin of turbulence? Shear instabilities, convection, MHD instabilities … etc.

* Parameterize angular mom. transfer and/or rate of mass accretion in terms of

Does 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

• 2 Competing Theories– Binary Agglomeration: Sticking vs. Disruption?

– Gravitational Instability: Settling vs. Turbulence?

Toomre criterion: vd < πGΣd/ΩK ≈ 10 cm/s

mm grains km planetesimals

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 = – T/cool 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 ~ σk 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

sh ´ 2/ ¼ 8 d. ¼ 1 y.

rot ´ 2/ ¼ 26 h. ¼ 1 y.

bv ´ 2/bv ¼ 6 m. ¼ 1 y.

Rossby Ro ´ rot/sh ¼ 0.13 ¼ 1

Froude Fr ´ bv/sh ¼ 5 £ 10-

4

¼ 1

Richardson ´ 1/Fr2 ¼ 4 £ 106 ¼ 1Wave 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 makevortices due to limited freedom, conservation of and inverse cascade of energy

Computational Method

• Cartesian Domain: (r,,z) (x,y,z)– Valid when ≡ H0 /r0 << 1 and r/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

• 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)

r

t=Lx/ rt=0

Lx

Open Boundaries or 1 Domain

• Mapping - 1 < z < 1 ! 0 < < • Arbitrary truncation to finite domain

impose arbitrary boundary conditions

• Embedding

-Ldomain < -Lphysical < z < Lphysical < Ldomain

Mappings

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

Others: Matsushima & Marcus JCP 1999 Rational Legendre for cylindrical coordinate at origin and harmonic at 1

Embedding

Domain of physical interest

Buffer region

Buffer region

Lembedding

-Lembedding

Lphysical

Lphysical

Lose 1/3 of points in buffer regions

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: hi / exp{-z2/2H0

2} (Gaussian)• Energy flux ~ hi 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 =

• Vy / t =

• Vz / t = - h / z + g T/hTi

• T / t = - (d hSi / dz) (hTi/cp) Vz

- T/cool

2 Vz / t2 = - (d hSi / dz) (g/cp) Vz

Gravity Wave Damping 1

• Vx / t =

• Vy / t =

• Vz / t = - h / z + g T/hTi – T/ t

• T / t = - (d hSi / dz) (hTi/cp) Vz – Vz/ t

- T/cool

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

+ [(d hSi / dz) (hTi/cp) (g/hTi)] Vz/ t

What does 2D mean?

Columns or pancakes?

Tall Columnar Vortex

Vortex in the Midplane of PPD

BlueBlue = Anticyclonic vorticity, Red = Cyclonic vorticity

Ro = 0.3125

à r0 = 4H0 !

Ã

r

= 2

H0

!

z planeat r=r0

Ro = 0.3125BlueBlue = Anticyclonic vorticityRed = Cyclonic vorticity

Ã

z =

8H

0

!

à r0 = 4H0 !

rz planeat =0

Ro = 0.3125BlueBlue = Anticyclonic vorticityRed = Cyclonic vorticity

Ã

z =

8H

0

!

à r = 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.

HighPressure

Coriolis Force

v?

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

v?

z

vz

rate, the faster of cool 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

• Energy provided by “step-functioning” the linearly stable background shear flow

arcus JFM

Coughlin & arcus PRL

Radial Cascade of Vortices Create Radial bands of z

• Vx / t =

• Vy / t =

• D z / D t = (baroclinic) - (2 + + z) (vz /H)

• Vz / t =

• T / t =

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

Fore-Aft symmetry:

Transport / h Vr V i

r

Vr() = -Vr(-)

V() = +V(-)

t/ORB

t/ORB 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 DynamicsAttracting Regions

ConclusionsDisk 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 dust ~ gas

Future work• Simulate back-reaction of dust on gas

– Dust can be modeled as continuous fluid without pressure

• Simulate particles in full turbulent flow

Questions & Future WorkTime scales for MERGE and FORM (isolated vortices?)

Understand why time-average h Vr V 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/{mixing}

• This simulation: rmax ~ 1 mm

• Stable orbits: fixed point or limit cycles

-1 -0.5 0 0.5 1 1.5 2

-0.2

0

0.2

0.4

x

y

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 gas cs rgrain v– Epstein: D = CD, S gas cs r2

grain v

• For R = 1 AU ~ 150 Million km:– ~ 1 cm– rgrain < 1 cm ! Epstein regime

• Size parameterized by stopping time

s ´ gas cs / (grain r)

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: visc ¼ R2/ » 1012 years!– Couple orders of magnitude longer than age of universe!

• Shakura-Sunyaev (1973): Re = VL/ » 1014… perhaps turbulence enhances the effective viscosity:– turb = H0 cs

– < 1 because turbulent eddies are probably subsonic and no larger than a scale height in extent.

• Origin of turbulence? Nonlinear shear instabilities, convection, MHD 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– Particles establish orbits where Drag ¼ –Weight– vz / s(z) = g = const. ¢ z– Dust sorted vertically by size (weight)

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.

• Vortices in H2 disk around a protostar.

vz ¿ v?

L?

Lz vz»v?Lz

L?L? À Lz L? < Lz

z = 0 z = H

• Keplerian gas and dust around a Protostar• Gravity approximately linear in z:

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

• In rotating frame local shear = -3/2

Protoplanetary disk