Local Behavior of MRI in Accretion Disks

1
Magnetic Field in the Universe (Angra dos Reis,
Brazil) Nov 28-Dec 3 2004
Local Behavior of MRI in Accretion Disks
Takayoshi SANO (Osaka University, Japan)
J. Stone (Princeton) S. Inutsuka (Kyoto) N.
Turner (JPL)
2
Motivation

• Angular Momentum Transport in Accretion Disks
• Kepler Disk Weak B Field gt MRI
• Balbus Hawley 1991, Chandrasekhar 1961, Velikov
  1959
• Maxwell Stress in MHD Turbulence
• Open Questions
• What determines the saturation level of the
  stress?
• Is the stress proportional to gas pressure?
• Shakura Sunyaev 1973
• Saturation Mechanism
• Dependence of the Saturation Level on Pressure, B
  Field, Disk Thickness, etc.

3
Method

• Local Shearing-Box Calculations
• Advantage
• Higher Resolution Longer Time Integration

• Kepler Disk Vertical Field
• Ignore the Vertical Gravity
• Constant Magnetic Diffusivity
• Numerical Scheme
• 2nd Order Godunov
• MoC-CT

4
Basic Equations
Coriolis Force Effective Potential
Ohmic Dissipation
5
Nonlinear Evolution
Sano et al. 2004

• Time Evolution of Magnetic Fields
• Time-Averaged Maxwell Stress

Magnetic Field

• Gas Pressure
• Joule Heating
• Fluctuation-Dissipation Relation

Gas Pressure
Orbit
6
Saturation Level
Sano et al. 2004
Sano et al. in prep

• Dependence on Physical Quantities
• Gas Pressure
• Field Strength
• Vertical Box Size
• Resolution

Maxwell Stress

• Origin of these dependences are still unclear.
• Characteristics of Turbulent State

Gas Pressure
7
Time Variability
Sano Inutsuka 2001

• Maxwell Stress in the Saturated State
• Highly Fluctuating
• Spike-Shaped Excursions

Maxwell Stress
Orbit
8
Spike-Shaped Variation

• Exponential Growth of MRI (Channel Flow)
• Exponential Decay through Reconnection

Sano Inutsuka 2001
9
2D Channel Flow

• Exact Solution for Nonlinear MHD Equations
• Shear Flow
• Thin Current Sheet

10
Parasitic Instability

• Shear Flow gt KH-type Instability
• Goodman Xu 1994
• Unstable Wavelength
• Growth Rate
• Good Agreement with Linear Analysis

Mode Amplitude
Orbit
11
Sweet-Parker Reconnection

• After Parasitic Instability gt Reconnection
• Decay Time Sweet-Parker Model
• MRI Reconnection gt Spike
• No Gas-Pressure Dependence

Magnetic Energy
Orbit
12
Summary

• Local Simulations of MRI
• Saturation level of Maxwell stress
• Depends on Gas pressure, Box Size, Resolution
• Not Proportional to P?
• Saturation mechanism
• Time Variability Channel Flow gt Parasitic
  Instability gt Sweet-Parker Reconnection
• Resistive MHD
• Magnetic Reconnection
• Protoplanetary Disks (with Ambipolar Diffusion
  and Hall Effect)

13
(No Transcript)
14
TITLE

• Item

15
??????????????????

• ???????(MRI) Balbus Hawley (1991)
• ??????????????????
• ????????
• ???????
• ???????
• ????????
• ? ?20AU????????
• ???????
• ??????????????
• ???????????
• ????????????
• ? ??????????

??????????? MRI?????? Sano et al. (2000)
16
??????????

• ??????
• ?????????? ? ???????
• ??????? ? ????????????
• ??????
• ????????????? Sano Inutsuka (2001)
• MRI???????? ? ?????????
• ?????????
• ???????
• ??????????
• ???????????

???????????
?????????????? ??????????????? ? ??????????????
17
??????

• ????????????????????
• ?????? Goodman Xu (1994)
• ?????????????????
• ???MHD???????
• ?????????? ? ??????????
• ????????????????

??????? ?????
18
??????????

• ???????
• ??????????
• ????(????)
• ?????????
• ???????? ??
• ?????? ??
• ???
• ? ?????????????
• ?????????
• ????
• ??????????

????
19
??1?????????

• Parasitic???(KH????) Goodman Xu (1994)
• ?????
• ???
• ???????????
• ? ???????

??????
MRI???(?)?Parasitic???(?) ???????? ????????????
??
20
(No Transcript)
21
??2?????????

• ?????????
• ????? ? ????????
• ??????????
• ???????????
• Sweet-Parker????????

??
??
22
??3??????

• ??????????????? Sano el al. (2004)
• ????????????????????
• ? ?????
• ????????
• ????????
• ?????????
• Sweet-Parker??????
• ? ?????????
• ???????????

????????
?????????(?)? ????????(?)???????
23
???

• ????????????????????????
• ?????????????????????????
• MRI???????Parasitic??????????
• Parasitic????????????
• ??????????Sweet-Parker??????
• ?????????????
• ?????????????????????
• ?????? ? Parasitic??? ? ?????????
• ?????????????????
• ?) ???????????? ?
• ???MRI?Parasitic???????????

24
(No Transcript)
25
??????????
?????????????? ??????????????? ? ??????????????

• ??????
• ?????????? ? ???????
• ??????? ? ????????????
• ??????
• ????????????? Sano Inutsuka (2001)
• MRI???????? ? ?????????
• ?????????
• ???????
• ????
• ??????
• ??????????
• Joung et al. (2004)

???????????
26
???????
???????
Tearing??? ? ?????????
27
??????
??????????
28
Parasitic???
29
???

• 20AU???????????????????
• ??????????????

30
????
31
???????
32
???????
???

• ????
• ????(????)
• ?????????
• ????? ??????????
• ??
• ??MHD?????????

????
????
????
???????
???????????
33
??????????

• ??MHD???????
• ???????????????
• ????????
• ??????????

Sano Miyama (1999)
???
??????????? ?????????
??
34
??????
????

• ???????????6??
• ??????
• Fast wave
• Alfven wave
• Slow wave
• ????????????Slow wave?????
• ????????????

???
?????
35
???????????????

• ?????????????????
• MHD???????????
• ????????(?????????)
• ????????
• ???????(???????)
• ?????????
• ?????
• ????????????
• a???????????
• (??????????)

???????????
Hawley, Balbus, Stone (2001)
36
?????

• ????????????
• ????
• ????lt?????
• ?????????
• ??????
• ????
• ????????
• ??????
• ???
• 2????????MoC-CT?

?????????? ???????
37
?????

• ????(???????)
• ???????

?????? ???????? (?????)
?????
38
?????

• ????ß????????????
• ????????102106??????10102
• ????????????????
• ?????????
• ?????????????????1??
• ???????????????(???)

39
???2D??(??1)

• ?????????????????
• ???????????????

?????
40
???2D??(??2)

• ?????????????????
• ???????????????????????????????????????

41
???2D??????

• ???RM???????

• ?????
• ?????????
• ?????????????
• ?????
• ?????????
• ?????????

??? (?????? ????)
42
3D?????

• ?????????????????
• 2D??????????????????????????MHD????

?????????????????
???????????
43
???????

• ?????????????

?????????

• Maxwell??
• Reynolds??

XY?? ??????RF??
??
44
?????????

• ??????
• (????????????)?(?????????)?????

????
??????? ?1????

??? ???Y?? ????
-
Sano Inutsuka (2001)
45
?????????

• ?????????????????
• MRI????????????????
• ???????????????????

46
????????????

• ??????????????
• ???
• ?????????
• ?????
• ????????
• ????????
• ???????(???)

Sano et al. (2004)
????????????

• ??????????????
• ?????????????????????????????
• ???????????????

???????????????
47
???????
????????
????????
48
Parasitic???
??????? ?????

• ???????????????????
• ??????????
• KH-like????(Parasitic???)
• ??????????????

Goodman Xu. (1994)
????
49
MRI????

• ????????????????(??????)????
• ????????????????????Kinematic Dynamo??????
• ??????????????????????????????
• ??????????
• ???????
• ????????????????????
• ??????????????
• ??????????????????????Shakura Sunyaev?a????????
• ?????????????????????????????????????
• ?????????????????????

50
Fluctuation vs. Dissipation
?????
??????
51
?????????????????? II. ???????

• ?? ?? (???????)
• ?? ??? (???)
• James M. Stone (Princeton)
• Neal J. Turner (JPL)

52
???????

• ??????????????????????????????????????????????????
  ?????????????
• ??????????????
• ???????????????????????????????
• ?????????1?????????????????
• ??????????a???????????????????(????????????????)?
• ????????????????????

53
?????????????

• ????????????????
• ?????
  ????????
• Balbus Hawley (1991)
• ?????????????
• ??????
• ??????(????) ??????(???)
• ???0.75O
• ?????VA/O

54
??

• ????????????????
• ??????????????
• ???????????
• ????????????????????
• ????????????????????????????
• ?????? ???MHD??????????????

55
??????????????

• ???????????????????????????

?????????MoC-CT

• ????????????????
• ?????????? ???????
• ????????? ???????????????????

?????Y
????Z
????X
56
?????

• ????????????????
• ???
• ???????(????????????0??)
• ???????
• ?????
• ????????(????????)
• ??????????
• ???????(???)
• ???????????????????????????????

57
??????
Sano, Inutsuka, Turner Stone (2004)

• ????????????????????
• (??????) ?(???)1/4
• ?????????????????
• ???????????(3???)????????

???????? ??????
Wmag?P1/4
58
??????????1

• ????????????????????
• (??????)? (???????)
• ????????????? ????????????????

??????????
Wmag?Lz
LxLyLz ?282 ?441 ?1161 144
???
59
??????????2

• ??????????????
• ?????????????????
• (141???)?(282??????141??)????????
  ???????(????)?????????

??282 ??141 ??282??????
??
60
??????????1

• ???????????????????
• (??????)? (???????)1/2
• ??????????????
• ????????????????????????(???????????)???????

??????????
Wmag??1/2
61
??????????2
?3212832 ?6425664 ?128512128
??????
??????
??
???
62
???????

• ????????????1???????????

????????
Wmag ?P1/4Lz?1/2
????????
63
??

• ???????Lz?????????????????
• Lz?HCS/O?P1/2
• ???????????3/4????
• a????????? aWmag/P0.08(Lz/
  H)(?/H)1/2
• ???????????
• ????????(???????????)???????????
• ?????????????????????????????????