martin e. pessah colin mcnally · martin e. pessah colin mcnally nbia workshop on protoplanetary...
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![Page 1: Martin E. Pessah Colin McNally · Martin E. Pessah Colin McNally NBIA Workshop on Protoplanetary Disks, Copenhagen, Aug 4 - 8, 2014 Niels Bohr Institute Vertically Global, Radially](https://reader035.vdocuments.net/reader035/viewer/2022063003/5f6d1fed539dbd2de2019d05/html5/thumbnails/1.jpg)
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]
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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
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•Write down equations of motion
•Find an equilibrium solution
•Expand the equations around fiducial point
•Define boundary conditions
•Solve problems!
Deriving Local Disks Models
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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
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The Local Approximation
V (x) ' V0 + S0 xV (r) = r⌦(r)
V (x) ' r0
⌦0 +
d⌦
dr
����0
(r � r0)
�
x
y
r0 r
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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
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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
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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
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@�
@z' 0
@�
@z' z⌦2
0
from J. Simon’s website
The Standard Shearing Box (SSB)
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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)
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What if we care about z ~ R?
M82
�(r, z) =�GMpr2 + z2
A. Tchekhovskoy
@�
@z= z⌦2
0
✓1 +
z2
r20
◆�3/2
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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
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•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
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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
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•4. Taylor-expand background equilibrium in r
Vertically Global & Radially Local
⇢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
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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
Vertically Global & Radially Local
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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!
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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
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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 !!
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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
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•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
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•Hydrodynamic Disk Instabilities
•Disk Convection
•Disk Coronae and Thick Disks
•Disk Winds
•Interstellar Medium and Galactic Disks
When Will It Matter?
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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
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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
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�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!
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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
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•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
Nelson, Gressel, & Umurhan 2013
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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
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Hydrodynamic Disk Instabilities
Nelson, Gressel, & Umurhan 2013
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A First Implementation of the VGSB
McNally & MEP, 2014q = �1.5 p = �1.0 c = 0.05 vK
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McNally & MEP, 2014
A First Implementation of the VGSB
q = �1.5 p = �1.0 c = 0.05 vK
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Thank you!