Linear Hydrodynamics of Protoplanetary Disks: a Crash Course

Linear hydrodynamics of protoplanetary disks: a crash course

Min-Kai Lin

@linminkai

July 2018 Outline

Fluid instabilities in protoplanetary disks Linear stability analysis

Aim: concepts and methods through examples

M-K. Lin (ASIAA) July 2018 2 / 25 Planets form in accretion disks around young stars

M-K. Lin (ASIAA) July 2018 3 / 25 Birthsites of planets: protoplanetary disks

HL Tau

Yen et al., 2016 (ASIAA)

M-K. Lin (ASIAA) July 2018 4 / 25 Birthsites of planets: protoplanetary disks

AB Aurigae

Gas spirals Dust ring

Tang et al. 2017 (ASIAA)

M-K. Lin (ASIAA) July 2018 4 / 25 Protoplanetary disks are dusty MIR

Scattered Light

(sub-)mm

1 2 3

4 a b

d c

Distance in AU

1 10 100 1 Turbulent Mixing (radial or vertical) 2 Vertical Settling 0.35 mm 3.0 mm ALMA

3 Radial Drift 10 µm VLTI/MATISSE 4 a) Sticking b) Bouncing 2 µm 10 µm EELT c) Fragmentation with mass transfer d) Fragmentation JWST/MIRI (Testi et al., 2014) Protoplanetary discs are ∼ 99% gas, ∼ 1% dust Planets form from the solids (at least in core accretion)

M-K. Lin (ASIAA) July 2018 5 / 25 Instabilities and PPD evolution

Fluid dynamical instabilties provies a means to Develop turbulence for angular momentum transport and mass accretion Develop large-scale, potentially observable sub-structures (rings, gaps, asymmetires) Transport/collecting dust particles for planetesimal formation

M-K. Lin (ASIAA) July 2018 6 / 25 Instabilities and PPD evolution

Examples in PPDs Gravitational instability Magneto-rotational instability (see M. Pessah’s talk) Vertical shear instability (see later) Rossby wave instability Convective overstability/baroclinic instability Elliptic instability Viscous overstability Zombie vortex instability Streaming instability

M-K. Lin (ASIAA) July 2018 6 / 25 Rossby wave instability and vortex formation

Disk-planet interaction → gap opening → unstable → vortices

M-K. Lin (ASIAA) July 2018 7 / 25 Rossby wave instability and vortex formation

Origin of dust-traps (OpH IRS 48, van der Marel, 2013)?

M-K. Lin (ASIAA) July 2018 7 / 25 Vertical shear instability and dust settling

Initially well-mixed dust

Laminar VSI-turbulent

ρdust/ρgas shown Particles settle to mid-plane if the gas is laminar vertical shear instability → hydro. turbulence → particles stirred up

M-K. Lin (ASIAA) July 2018 8 / 25 Vertical shear instability and dust settling

ALMA Band 6+7 no settling h1mm = 2.15 au h1mm = 0.70 au α = 3 10−3 α = 3 10−4 0.5 SS SS

0.0 Dec (") ∆

−0.5

−0.5 0.0 0.5 ∆ Ra (") (HL Tau, Pinte et al. 2015) ) −1

Observed thin dust10−3 layers consistent with weak turbulence (Jy.beam ν F

10−4

−50 0 50 −50 0 50 −50 0 50 Distance from star (AU) Distance from star (AU) Distance from star (AU)

M-K. Lin (ASIAA) 2 2 July 20182 8 / 25 40 χ =2379 χ =1856 χ =1282

Real (mJy) 0

Band 7 −20

500.1000. 2000. 5000. 500.1000. 2000. 5000. 500.1000. 2000. 5000. Baseline (kλ) Baseline (kλ) Baseline (kλ) Streaming instability and planetesimal formation

streaming instability?

Streaming instability → dust clumping mediated by mutual dust-gas drag

M-K. Lin (ASIAA) July 2018 9 / 25 Streaming instability and planetesimal formation

ρdust/ρgas

Streaming instability → dust clumping mediated by mutual dust-gas drag

M-K. Lin (ASIAA) July 2018 9 / 25 Streaming instability and planetesimal formation

Early phase of the instability can be described by a linear analysis

M-K. Lin (ASIAA) July 2018 9 / 25 Linear analysis

Full equations describing disk evolution is non-linear → need numerical simulations Linear analysis: evolution of small perturbations away from a known reference state, e.g.

Σ → Σref +δΣ(x, t)

δΣ: perturbations (Eulerian today) 2 Assume |δΣ/Σref| 1, ignore terms of O(|δΣ| ) Useful for: gaining physial insight, code-testing

M-K. Lin (ASIAA) July 2018 10 / 25 Generic procedure

1 Identify physics and geometry, write down the full governing equations

M-K. Lin (ASIAA) July 2018 11 / 25 Generic procedure

1 Identify physics and geometry, write down the full governing equations

2 Important : solve governing equations for a well-deﬁned basic state: often interested in steady equilibria with symmetries

M-K. Lin (ASIAA) July 2018 11 / 25 Generic procedure

1 Identify physics and geometry, write down the full governing equations

2 Important : solve governing equations for a well-deﬁned basic state: often interested in steady equilibria with symmetries

3 Insert Σ → Σref + δΣ etc. into governing equations and linearize

M-K. Lin (ASIAA) July 2018 11 / 25 Generic procedure

1 Identify physics and geometry, write down the full governing equations

2 Important : solve governing equations for a well-deﬁned basic state: often interested in steady equilibria with symmetries

3 Insert Σ → Σref + δΣ etc. into governing equations and linearize

4 Take Fourier dependence for co-ordinates in which the basic state is symmetric. For example, if ∂t ≡ 0,∂ φ ≡ 0 for the basic state, then take δΣ(r, φ, z, t) = Re [Σ0(x, z) exp (imφ−iωt)]

I Complex amplitude

I Can also take Fourier dependence when considering perturbations that vary much more rapidly than background, |∂r ln Σref| |kr |

I Fourier decomposition with non-uniform backgrounds possible for special perturbations

M-K. Lin (ASIAA) July 2018 11 / 25 Generic procedure

1 Identify physics and geometry, write down the full governing equations

2 Important : solve governing equations for a well-deﬁned basic state: often interested in steady equilibria with symmetries

3 Insert Σ → Σref + δΣ etc. into governing equations and linearize

I Complex amplitude 5 Simplify/combine perturbation equations and solve for {ω, δfΣ, δfv}.

M-K. Lin (ASIAA) July 2018 11 / 25 Basic equations

Physics

I Hydrodynamics

I Magneto-hydrodynamics

I Full energy equation with heating/cooling/radiation, or ﬁxed equation of state (polytropic, isothermal)

I Gravity: star, planet, disk

I Viscosity

I Dust Geometry

I Global: consider the entire disk

I Local: follow a small patch of the disk Dimensions

I 1D: azimuthally and vertically integrated

I 2D: axisymmetric

I 2D: razor-thin disk

I 3D: full treatment

M-K. Lin (ASIAA) July 2018 12 / 25 Basic equations

Physics

I Hydrodynamics

I Magneto-hydrodynamics

I Full energy equation with heating/cooling/radiation, or ﬁxed equation of state (polytropic, isothermal)

I Gravity: star, planet, disk

I Viscosity

I Dust Geometry

I Global: consider the entire disk

I Local: follow a small patch of the disk Dimensions

I 1D: azimuthally and vertically integrated

I 2D: axisymmetric

I 2D: razor-thin disk

I 3D: full treatment

M-K. Lin (ASIAA) July 2018 12 / 25 Isothermal, self-gravitating shearing sheet

Following a small patch of the disk as it rotates around the star (Goldreich & Lynden-Bell, 1965) Razor-thin disk model: material and velocity conﬁned to z = 0 (but disk potential is still 3D)

Governing equations

∂Σ + ∇ · (Σv) = 0, ∂t ∂v 1 + v · ∇v + 2Ωzˆ × v = −∇Φrot − ∇Φd,z=0 − ∇P, ∂t | {z } Σ Coriolis 2 ∇ Φd = 4πGΣδ(z), | {z } thin-disk Poisson equation with 2 2 P = cs Σ , Φrot = −ΩSx . | {z } | {z } isothermal gas star and centrifugal S= shear rate, S = 3Ω/2 for a Keplerian disk

M-K. Lin (ASIAA) July 2018 13 / 25 Isothermal, self-gravitating shearing sheet

Following a small patch of the disk as it rotates around the star (Goldreich & Lynden-Bell, 1965) Razor-thin disk model: material and velocity conﬁned to z = 0 (but disk potential is still 3D)

Basic state

∂t = 0, ∂y = 0, Σref = const., Pref = const., vref = −Sxyˆ.

Generally can’t perform standard Fourier analysis in x (need a modiﬁed Fourier analysis) Exceptions:

I Non-shearing disk with S = 0 (i.e. solid body rotation)

I Waves independent of y

M-K. Lin (ASIAA) July 2018 13 / 25 Linearizing the shearing sheet equations

Σ → Σ + δΣ ...etc. Example ∂ (Σ + δΣ) + ∇ · [(Σ + δΣ)(v + δv)] = 0. ∂t

2 Ignoring δ quantities and using ∂t Σ + ∇ · (Σv) = 0 for basic state:

∂δΣ + Σ∇ · δv + v · ∇δΣ = 0. ∂t

M-K. Lin (ASIAA) July 2018 14 / 25 Linearizing the shearing sheet equations Let δΣ(x, y, t) = Σ0(x)ei(ky y−ωt), get

dv 0 (x) i [k v (x) − ω]Σ0(x) + Σ x + ik v 0 (x) = 0. y y dx y y

ODE with x-dependent coeﬃcients → can’t Fourier analyze (think Y 00 + σ2Y = 0). But x−dependence vanishes for axisymmetric waves and/or non-shearing disk (ky vy = 0)

0 ikx x I Set Σ (x) = Σee ...etc.

M-K. Lin (ASIAA) July 2018 14 / 25 Axisymmetric waves in the shearing sheet

Axisymmetric linear equations

ikx Σvex = iωΣe, ! 2 Σe ikx Φed,z=0 + c − 2Ωvy = iωvx , s Σ e e

(2Ω − S) vex = iωvey ,

Perturbed Poisson equation for a thin disk (disk potential is still 3D)

2 ∂ Φed 2 − k Φed = 4πGΣeδ(z) ∂z2 x For z 6= 0, solve

2 ∂ Φed 2 − k Φed = 0 ∂z2 x

→ Φed = A exp (− |kx z|)

M-K. Lin (ASIAA) July 2018 15 / 25 Axisymmetric waves in the shearing sheet

Axisymmetric linear equations

ikx Σvex = iωΣe, ! 2 Σe ikx Φed,z=0 + c − 2Ωvy = iωvx , s Σ e e

(2Ω − S) vex = iωvey ,

Apply jump condition at z = 0 to get

2πGΣe Φed,z=0 = − . |kx |

M-K. Lin (ASIAA) July 2018 15 / 25 Axisymmetric waves in the shearing sheet

Eigenvlaue problem

  0 ikx 0 Σ Σ 2 e e cs 2πG i − 0 −2Ω vx = iω vx .  Σ |kx |  e  e  0 (2Ω − S) 0 vey vey

T eigenvector [Σe vex vey ] is the iω the eigenvalue If σ ≡ Im(ω) > 0, mode grows like eσt Oscillation frequency Re(ω)

M-K. Lin (ASIAA) July 2018 15 / 25 Dispersion relation and Toomre instability

Need det(M − iωI ) = 0 for non-trivial solution, get ω = ω(kx )

2 2 2 2 ω = κ + cs kx −2πGΣ |kx | |{z} |{z} | {z } rotation pressure self-gravity

κ2 = 2Ω(2Ω − S) Examine limits

I No pressure, no self-gravity: epicyclic oscillations with ω = ±κ (need κ2 > 0 for stability)

I Non-rotating, no self-gravity: sound waves with

ω = ±cs kx

I No self-gravity: sound waves modiﬁed by rotation √ 2 2 2 ω = ± κ + cs kx

M-K. Lin (ASIAA) July 2018 16 / 25 Dispersion relation and Toomre instability

Need det(M − iωI ) = 0 for non-trivial solution, get ω = ω(kx )

2 2 2 2 ω = κ + cs kx −2πGΣ |kx | |{z} |{z} | {z } rotation pressure self-gravity

κ2 = 2Ω(2Ω − S) 2 Instability requires ω (kx ) < 0. Consider ω(kx ) = 0 to ﬁnd

κc Q ≡ s < 1 for instability. πGΣ Critical wavenumbers: h i πGΣ p 2 |kx | = 2 1 ± 1 − Q cs

I Small kx : large-scales stabilized by rotation I Large kx : small-scales stabilized by pressure

M-K. Lin (ASIAA) July 2018 16 / 25 Axisymmetric waves in the 3D shearing box

1 Write down the basic equations (Keplerian disk)

∂ρ + ∇ · (ρv) = 0 ∂t ∂v 1 + v · ∇v = − ∇P − 2Ωzˆ × v + 3Ω2xxˆ −Ω2zzˆ , ∂t ρ | {z } approx. vertical stellar gravity 2 P = cs ρ.

M-K. Lin (ASIAA) July 2018 17 / 25 Axisymmetric waves in the 3D shearing box

2 Work out equilirbium c ρ(z) = ρ(0) exp −z2/2H2, H ≡ s Ω 3 v = − Ωxyˆ 2

I Stratiﬁed equilibrium depends on z → can’t Fourier analyze in z

M-K. Lin (ASIAA) July 2018 17 / 25 Axisymmetric waves in the 3D shearing box

3 Linearize, with δρ = ρe(z) exp (ikx x − iωt)...etc. W d ln ρ dv −iω = −ik v − v − ez , 2 x ex ez cs dz dz −iωvex = −ikx W + 2Ωvey , Ω −iωv = − v , ey 2 ex dW −iωv = − , ez dz

2 where W (z) ≡ cs ρe(z)/ρ(z).

M-K. Lin (ASIAA) July 2018 17 / 25 Axisymmetric waves in the 3D shearing box

4 Simplify/combine linearized equations

2 2 d W d ln ρ dW 2 1 kx 2 + + ω 2 + 2 2 W = 0. dz dz dz cs Ω − ω

M-K. Lin (ASIAA) July 2018 17 / 25 Axisymmetric waves in the 3D shearing box

4 Simplify/combine linearized equations

2 2 d W d ln ρ dW 2 1 kx 2 + + ω 2 + 2 2 W = 0. dz dz dz cs Ω − ω

∗ I Try to analyze equation without solving it. Multiply by ρW and integrate from z = z1 to z = z2. Integrate by parts and assume boundary terms vanish (no pert. there)

Z z2 2 2 Z z2 dW 2 1 kx 2 ρ dz = ω + ρ|W | dz . dz c2 Ω2 − ω2 z1 s z1 | {z } | {z } I1>0 I2>0

2 I Thus ω is real and > 0 (exercise) system is stable to axisymmetric perturbations

M-K. Lin (ASIAA) July 2018 17 / 25 Explicit solution Inserting the equilirbium density proﬁle, the governing equation becomes

d 2W dW K 2 − Z + ν2 1 + x W = 0. dZ 2 dZ 1 − ν2

Z = z/H, ν = ω/Ω, Kx = kx H

M-K. Lin (ASIAA) July 2018 18 / 25 Explicit solution Inserting the equilirbium density proﬁle, the governing equation becomes

d 2W dW K 2 − Z + ν2 1 + x W = 0. dZ 2 dZ 1 − ν2

Z = z/H, ν = ω/Ω, Kx = kx H Hermite diﬀerential equation, but if we didn’t know this:

M-K. Lin (ASIAA) July 2018 18 / 25 Explicit solution Inserting the equilirbium density proﬁle, the governing equation becomes

d 2W dW K 2 − Z + ν2 1 + x W = 0. dZ 2 dZ 1 − ν2

Z = z/H, ν = ω/Ω, Kx = kx H Series solution (check for singularities, may need Frobenius method)

∞ X n W (Z) = WnZ n=1 Recurrence relation K 2 (n + 2)(n + 1)W + ν2 1 + x − n W = 0. n+2 1 − ν2 n

2 Boundary condition |δρ| ∝ e−Z /2||W | → 0 as |z| → ∞ ⇒ W is a ﬁnite polynomial, say of order N (W ∝ HeN )

M-K. Lin (ASIAA) July 2018 18 / 25 Explicit solution Inserting the equilirbium density proﬁle, the governing equation becomes

d 2W dW K 2 − Z + ν2 1 + x W = 0. dZ 2 dZ 1 − ν2

Z = z/H, ν = ω/Ω, Kx = kx H Dispersion relation

K 2 ν2 1 + x = N. 1 − ν2

So low frequency waves with small wavelength (|ν| 1, Kx 1) obey

2 2 ν Kx = N (inertial waves)

Inertial waves can be destabilized by vertical shear

M-K. Lin (ASIAA) July 2018 18 / 25 Real life example: vertical shear instability

PPDs can exhibit vertical shear, ∂z Ω 6= 0,because ∇P × ∇ρ 6= 0 (baroclinic) Vertically isothermal thin-disk with T ∝ Rq,

∂Ω qz H R ' × Ω ∂z 2H R Kep

h ≡ H/R ∼ 0.05 in PPDs, so vertical shear is weak

M-K. Lin (ASIAA) July 2018 19 / 25 Real life example: vertical shear instability

∂z Ω 6= 0 ⇒free energy→instability?

Change in kinetic energy:

2 2 2 lz ∂Ω ∆E ∼ lR Ω + · R lR ∂z Vertical shear is weak, but

∆E < 0 if |lz | |lr |

Energy released for vertically elongated disturbances ⇒ instability

M-K. Lin (ASIAA) July 2018 19 / 25 Real life example: vertical shear instability

PLUTO simulation of the VSI, characteristic large-scale vertical velocities

M-K. Lin (ASIAA) July 2018 19 / 25 Vertical shear instability: the analysis

2 q System of interest : 3D, non-self-grav. disk with fixed temp. profile cs (R) ∝ R ∂ρ + ∇ · (ρv) = 0, ∂t ∂v 1 + v · ∇v = − ∇P − ∇Φ , ∂t ρ ∗ 2 P = cs (R)ρ. (global description for now) Equilibria : axisymmetric, stratified disk

∂t = ∂φ = 0, ρ = ρ(R, z), v = RΩ(R, z)φˆ Solve 1 ∂P ∂Φ + ∗ = 0, ρ ∂z ∂z 1 ∂P ∂Φ + ∗ = RΩ2. ρ ∂R ∂R

M-K. Lin (ASIAA) July 2018 20 / 25 Vertical shear instability: the analysis

2 q System of interest : 3D, non-self-grav. disk with ﬁxed temp. proﬁle cs (R) ∝ R

∂ρ + ∇ · (ρv) = 0, ∂t ∂v 1 + v · ∇v = − ∇P − ∇Φ , ∂t ρ ∗ 2 P = cs (R)ρ.

(global description for now) Equilibria : vertical shear

∂Ω2 qz Ω2 R = Kep . ∂z R (1 + z2/R2)3/2

M-K. Lin (ASIAA) July 2018 20 / 25 Vertical shear instability: the analysis

Axisymmetric linearized equations For δρ(R, z) = ρe(z) exp (ikx R − iωt) assuming large |kx R| 1 (motivated by sims.)

dv ∂ ln ρ iωW = ik v + ez + v , x ex dz ∂z ez iωvex = ikx W − 2Ωvey , ∂Ω κ2 iωv = R v + v , ey ∂z ez 2Ω ex dW iωv = . ez dz

T Solve for u(z; ω) = [W , vex , vey , vez ] subject to appropriate vertical boundary conditions (top and bottom) Eigenvalue problem because solution only exists for certain ω (think SHM).

M-K. Lin (ASIAA) July 2018 20 / 25 Vertical shear instability: the analysis

d 2W iqhK dW K 2 − 1 + x Z + ν2 1 + x W = 0, dZ 2 1 − ν2 dZ 1 − ν2

Can solve as before to get

2 1 + iqhKx ν = N 2 , N + 1 + Kx

for |ν| 1. ν2 is generally complex → there is an unstable solution with Im(ν) > 0.

Integral relations → growth rates σ < |R∂z Ω|

M-K. Lin (ASIAA) July 2018 20 / 25 Numerical solutions

Not always possible to solve linearized equations analytically Example: analytic solution assumes a boundary condition at inﬁnity, real disks may have ﬁnite thickness Numerical tools

I Discretizing the equations: ﬁnite-diﬀerence, pseudo-spectral

I Eigenvalue problem: ‘shooting’ method, matrix methods

M-K. Lin (ASIAA) July 2018 21 / 25 Numerical solutions Let

N Xz W (Z) = wk ψk (Z) k=1

where ψk are appropriate basis functions, e.g. Chebyshev polynomials Take vertical derivatives exactly.

Demand linearized equations to be satisﬁed at a (special) set of zj . P 0 P 0 W (Zj ) ≡ Wj = k wk ψk (Zj ) and Wj = k wk ψ (Zj )...etc., Linear equations becomes the generalized eigenvalue problem

Lu = iωTu

Modify matrix elements to account for boundary conditions Solve with standard matrix packages, e.g. lapack

M-K. Lin (ASIAA) July 2018 21 / 25 Numerical solutions

Example: the eigenvalue problem

d 2y = −k2y dx 2 is discretized to N N X 00 2 X ψij yj = −k ψij yj , j=1 |{z} j=1 |{z} L T

where ψij ≡ ψj (xi ), for interior points j = 2 to j = N − 1.

Suppose the boundary conditions are y(x1) = y(xN ) = 0. Can set L1j = ψ1j and LNj = ψNj

M-K. Lin (ASIAA) July 2018 21 / 25 Example VSI solutions

Eigenvalues

Roughly horizontal ‘body modes’ and nearly vertical ‘surface modes’ Here σ is growth rate, ω is real frequency Eigenvalues ‘oﬀ’ from analytical method related to ﬁnite domain size

M-K. Lin (ASIAA) July 2018 22 / 25 Example VSI solutions

Eigenvector vz (fundamental mode)

Large-scale vertical motions

M-K. Lin (ASIAA) July 2018 22 / 25 Summary

Fluid instabilities play a major role in PPD gas (and dust) dynamics Linear analysis reveals the properties of waves and instabilities Important to have a well-deﬁned basic state to perturb If possible, analyze linear equations ﬁrst to infer properties of solutions (can be used to check numerical results) If possible, solve equations analytically to obtain explicit dispersion relations If needed, solve equations numerically

M-K. Lin (ASIAA) July 2018 23 / 25 Potential projects

Stability of stratiﬁed, dusty disks with self-gravity, particle diﬀusion, viscosity Floquet stability analysis of exact vortex solutions in magnetized disks Stability of dusty vortices Global stability analysis of dusty disks Hydrodynamic instabilities with self-gravity Dusty, magnetized disks

M-K. Lin (ASIAA) July 2018 24 / 25 Further reading

Chandrasekhar, Hydrodynamic and Hydromagnetic Stability Pringle & King, Astrophysical Flows Clarke & Carswell, Principles of Astrophysical Fluid Dynamics Ogilvie, Lecture Notes on Astrophysical Fluid Dynamics Klahr, Pfeil & Schreiber, 2018 (review on hydrodynamic instabilities in PPDs)

M-K. Lin (ASIAA) July 2018 25 / 25