martin e. pessah colin mcnally · martin e. pessah colin mcnally nbia workshop on protoplanetary...

Martin E. Pessah Colin McNally

NBIA Workshop on Protoplanetary Disks, Copenhagen, Aug 4 - 8, 2014

Niels Bohr Institute

Vertically Global, Radially Local Models for Astrophysical Disks

[arXiv:1406.4864]

Local Models of Astrophysical Disks

Essence of the local approximation: expand the equations of motion around a point corotating with the disk

Beckwith, Armitage and Simon, 2011

•Write down equations of motion

•Find an equilibrium solution

•Expand the equations around fiducial point

•Define boundary conditions

•Solve problems!

Deriving Local Disks Models

Equations for an ideal MHD Fluid@⇢

@t+r · (⇢v) = 0

continuity

@v

@t+ (v ·r)v = ⌦2

Fr � 2⌦F ⇥ v �r�� rP

⇢+

1

⇢J ⇥B

momentum

@B

@t+ (v ·r)B = (B ·r)v �B (r · v) induction

@e

@t+r · (ev) = �P (r · v) energy

The Local Approximation

V (x) ' V0 + S0 xV (r) = r⌦(r)

V (x) ' r0

⌦0 +

d⌦

dr

��0

(r � r0)

�

x

y

r0 r

V (r) = r⌦(r)�

⌦2(r) ⌘ 1

r

@�

@r=

GM

r31

⇢h

@P (⇢h)

@z= �@�

@z

⌦(x) ⌘ ⌦0 +@⌦(r)

@r

��r=r0

x

S0 ⌘ r0@⌦(r)

@r

��r=r0

V (x) ⌘ V0 + S0x•2. Taylor expand in r

w ⌘ v � V (x)y•3. Define departures

•1. Bulk disk flow

Vertically Local & Radially Local

Isothermal, thin disks pressure support is negligible and angular frequency is height-independent

w ⌘ v � S0xy D0 ⌘ @t + S0x@y

(D0 +w ·r)w = �2⌦0z ⇥w � S0wx

y

� rP

⇢� @�0(z)

@zz +

1

⇢J ⇥B

(D0 +w ·r)B = S0Bx

y + (B ·r)w �B (r ·w)

Standard Shearing Box - Model

Shearing Periodic Boundaries

DSSB0 ⌘ @t + xS0@y y

0 = y � xS0 t

f(x0, y

0, z

0, t

0) = f(x0 + L

x

, y

0, z

0, t

0)

f(x, y, z, t) = f(x+ L

x

, y + S0Lx

t, z, t)

DSSB0 = @0

t

x

y

y + S0Lx

t

SSB

@�

@z' 0

@�

@z' z⌦2

0

from J. Simon’s website

The Standard Shearing Box (SSB)

Pros:•Offers a controlled environment

•Can study small-scale turbulent dynamics

Cons:•Unable to address any global disc dynamics

•Framework valid for isothermal, thin disks

The Standard Shearing Box (SSB)

What if we care about z ~ R?

M82

�(r, z) =�GMpr2 + z2

A. Tchekhovskoy

@�

@z= z⌦2

0

✓1 +

z2

r20

◆�3/2

Beyond Vertically Local Models

•Non-trivial thermal structure implies

•Thick disks have pressure support

d⌦

dz6= 0

rP 6= 0

•Isothermal disks are barotropic P = P (⇢)

•Barotropic fluids rotate on cylinders d⌦

dz= 0

•1. Bulk disk flow V (r, z) = r [⌦(r, z)� ⌦F] �

1

⇢h

@P (⇢h, eh)

@z= �@�

@z⌦2(r, z) ⌘ 1

r

@�

@r+

1

r⇢h

@P (⇢h, eh)

@r

•3. Define departures w ⌘ v � [V0(z) + S0(z)x] y

•2. Taylor expand in r

⌦(x, z) ⌘ ⌦0(z) +@⌦(r, z)

@r

��r=r0

x S0(z) ⌘ r0@⌦(r, z)

@r

��r=r0

V (x, z) ⌘ V0(z) + S0(z)x

Vertically Global & Radially Local

What is New?

•Rotation rate depends on z •Shear rate depends on z

McNally & MEP, 2014

⌦(r) ' ⌦0 +@⌦(r)

@r

��r0

x⌦(r, z) ' ⌦0(z) +@⌦(r, z)

@r

��r0

x

•4. Taylor-expand background equilibrium in r


⇢h(r, z) = ⇢h0(z) + O

✓x

r0

◆eh(r, z) = eh0(z) + O

✓x

r0

◆

rP (⇢h, eh)

⇢h=

rP (⇢h0, eh0)

⇢h0+O

✓x

r0

◆1

⇢h0

@P (⇢h0, eh0)

@z= �@�0(z)

@z

This eliminates radial gradients, which is necessary for domains with shearing-periodic boundaries

0.95 1.05

x

0.0

0.2

0.4

0.6

0.8

1.0

z

Global

0.95 1.05

x

VGSB

0.95 1.05

x

SSB

�1.08

�1.07

�1.06

�1.05

�1.04

�1.03

�1.02

�1.01

�1.00

�0.99

log(

T)

Temperature

0.95 1.05

x

0.0

0.2

0.4

0.6

0.8

1.0

z

Global

0.95 1.05

x

VGSB

0.95 1.05

x

SSB

�20.0

�17.5

�15.0

�12.5

�10.0

�7.5

�5.0

�2.5

0.0

log(

⇢)

Density

0.95 1.05

x

0.0

0.2

0.4

0.6

0.8

1.0

z

Global

0.95 1.05

x

VGSB

0.95 1.05

x

SSB

0.80

0.84

0.88

0.92

0.96

1.00

1.04

⌦

Angular Frequency

McNally & MEP, 2014


w ⌘ v � [V0(z) + S0(z)x] y

Vertically Global, Radially Local

D0 ⌘ @t + [V0(z) + S0(z)x] @y

(D0 +w ·r)w + w

z

@V (x, z)

@z

y =

� 2⌦0(z)z ⇥w � S0(z)wx

y � rP

⇢

� @�0(z)

@z

z +1

⇢

J ⇥B

momentum

induction(D0 +w ·r)B �B

z

@V (x, z)

@z

y =

S0(z)Bx

y + (B ·r)w �B (r ·w)

Not-consistent with shearing-periodic boundaries!

D0 ⌘ @t + [V0(z) + S0(z)x] @y

Shearing Periodic Boundaries(z)

y

0 = y � [V0(z) + S0(z)x] t

f(x, y, z, t) = f(x+ L

x

, y + S0(z)Lx

t, z, t)

f(x0, y

0, z

0, t

0) = f(x0 + L

x

, y

0, z

0, t

0)

D0 = @0t

x

y

y + S0(z)Lx

t@zV (x, z) = @zV0(z) + x @zS0(z)

@zV (x, z) ' @zV0(z)

For this to work we only need the approx.

VGSB

Vertically Global Shearing Box

w ⌘ v � [V0(z) + S0(z)x] y D0 ⌘ @t + [V0(z) + S0(z)x] @y

(D0 +w ·r)w + wz

@V0(z)

@zy =

� 2⌦0(z)z ⇥w � S0(z)wx

y � rP

⇢� @�0(z)

@zz +

1

⇢J ⇥B

momentum

(D0 +w ·r)B �Bz

@V0(z)

@zy =

S0(z)Bx

y + (B ·r)w �B (r ·w)

induction

Amenable to shearing-periodic boundary conditions !!

Conserved Fluid Properties

[@t + v ·r](r · (r⇥ v)) = 0

� =

Iv · dl

•Kelvin’s circulation theorem

@t(r⇥ v) = r⇥ [v ⇥ (r⇥ v)]

[@t + v ·r]� = 0

•Alfven’s frozen-in theorem

@tB = r⇥ [v ⇥ (r⇥B)]

[@t + v ·r](r ·B) = 0

�B =

ZB · dS

[@t + v ·r]�B = 0

•Alfven’s frozen-in theorem

•Kelvin’s circulation theorem

Conserved Fluid Properties in VGSB

(D0 +w ·r)� =

Z

S

r⇥

✓xwz

@S0(z)

@z

y

◆�+ . . .

(D0 +w ·r) (r ·B) = �x

@S0(z)

@z

@Bz

@y

(D0 +w ·r)� = 0 (D0 +w ·r)�B = 0

•If barotropic or axisymmetric

•Hydrodynamic Disk Instabilities

•Disk Convection

•Disk Coronae and Thick Disks

•Disk Winds

•Interstellar Medium and Galactic Disks

When Will It Matter?

Disk Winds

Suzuki & Inutsuka 2009

@�

@z' z⌦2

0

makes it hard to launch winds

Usual assumption

Some works have used

without considering changes in rotation rate

@�

@z= z⌦2

0

✓1 +

z2

r20

◆�3/2

Spherical Temperature Disk Structure

T (r, z) ⌘ T0r0p

r2 + z2

⇢(r, z) = ⇢0

✓rp

r2 + z2

◆⌫ p

r2 + z2

r0

!1�µ

⌦(r, z) =p⌫cs0r

pr2 + z2

r0

!�1/2

⌫ + µ =v2K0

c2s0

r

z

Suzuki & Inutsuka, 2013

�1

0

1

Bx

and

By

�1

0

1

Bx

and

By

�1

0

1

Bx

and

By

�1

0

1

Bx

and

By

�1

0

1

Bx

and

By

�2 0 2

z/Hs

�1

0

1

Bx

and

By

�2 0 2

z/Hs

�1

0

1

Bx

and

By

�1

0

1

wx

and

wy

�1

0

1

wx

and

wy

�1

0

1

wx

and

wy

�1

0

1

wx

and

wy

�1

0

1

wx

and

wy

�2 0 2

z/Hs

�1

0

1

wx

and

wy

�2 0 2

z/Hs

�1

0

1

wx

and

wy

Magnetorotational Instability (MRI)

McNally & MEP, 2014

Growth Rates Velocity Field Magnetic FieldSSB SSBVGSB VGSB

so far MRI only studied in isothermal disks!

T (r) ⌘ T0

✓r

r0

◆q

⇢(r, z) = ⇢0

✓r

r0

◆p

exp

"�v2K

c2s

1� 1p

1 + (z/r)2

!#

⌦(r, z) = ⌦K

vuut1 + (p+ q)c2sv2K

+ q

1� 1p

1 + (z/r)2

!

Cylindrical Temperature Disk Structure

r

z

Nelson, Gressel, & Umurhan 2013

•Thin, isothermal hydro disks seem to be stable, but there are several instabilities that feed off vertical shear (GSF in 60’s!)

•These could be important in low-ionization protoplanetary disks

0.00 0.05 0.10 0.15 0.20 0.25

|�i|/⌦0(0)

0.00

0.05

0.10

0.15

0.20

|�r|/

⌦0(0

)

Vertical Shear Instability (VSI)

McNally & MEP, 2014

VGSB


Unstable Modes In Hydro Disks

�0.01

0.00

0.01

Surf

ace

⇧f

�0.1

0.0

0.1

wx

�0.5

0.0

0.5

wy

�0.5

0.0

0.5

wz

�0.02

0.00

0.02

Cor

ruga

tion

�0.05

0.00

0.05

�0.5

0.0

0.5

�0.2

0.0

�6 �3 0 3 6

z/Hc

0.00

0.01

0.02

Bre

athi

ng

�6 �3 0 3 6

z/Hc

�0.05

0.00

0.05

�6 �3 0 3 6

z/Hc

�1.0

�0.5

0.0

�6 �3 0 3 6

z/Hc

�0.2

0.0

0.2

McNally & MEP, 2014

Hydrodynamic Disk Instabilities


A First Implementation of the VGSB

McNally & MEP, 2014q = �1.5 p = �1.0 c = 0.05 vK

McNally & MEP, 2014

A First Implementation of the VGSB

q = �1.5 p = �1.0 c = 0.05 vK

Thank you!

martin e. pessah colin mcnally · martin e. pessah colin mcnally nbia workshop on protoplanetary...

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