マルチスケール手法を用いた高温降着流における乱流加熱研究
川面洋平 (東北大学)
Collaborators:
Michael Barnes (Oxford), Alexander Schekochihin (Oxford),
Steven Balbus (Oxford), Bill Dorland (Maryland), Y. Tong (Berkely),
J. M. TenBarge (Princeton), Kristopher Klein (Arizona)

高エネルギー宇宙物理学研究会, 2019/12/6@蔵王アストリアホテル

Plan of the talk
‣ Research background‣ Three key questions of turbulent heating in collisionless plasma
1. Partition of Alfvenic fluctuations between ion and electron
heating [Kawazura+ PNAS 2019]
2. Partition of compressive fluctuations between ion andelectron heating [Kawazura+ in prep]
3. Partition of Alfvenic and compressive fluctuation in MRI
driven turbulence [Kawazura+ in prep]‣ SummaryIon vs electron heating in hot accretion disks‣ Hot accretion disks (e.g M87, Sgr A*)‣ Collisionless plasma → Ti ≫ Te
‣ Only electrons emit radiation→ What we can measure is only Te→ Ion vs electrons heating (Qi/Qe) is important[EHT collaboration 2019]‣ GRMHD reproduced the observation [EHT 2019-V]‣ MHD can’t deal two temperature→ They assumed Ti/Te

We need to determine
Qi/Qe physically1

“Two temperature” MHD simulation<latexit 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Radiative GRMHD

Entropy evolution for ions &electrons, respectively‣ How we prescribe Qi/Qe changes the result‣ left : Qi/Qe = increasing func of β
‣ right : Qi/Qe = decreasing func of β

🤔 Which is correct?
[Chael+ 2018]

Kinetic theory is necessary to study plasma heating<latexit sha1_base64="LE3femd+iQ6Y7dtq7DVWxSO7G0g=">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</latexit>

!

!

@ fs
qs
v ⇥ B @ fs
@ fs
+ v · r fs +
E+
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@t
ms
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@t c

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f s = f s (r, v) : 6D phase space

Vlasov eq‣ To get turbulent heating, “in principle,” we must resolve
Inertial range ~ ion gyroscale ~ electron gyroscale
ex. 0.1ρi-1 ≦ k⊥ ≦ 2ρe-1 ~ 100ρi-1 → grid number ~ 1000→ nx×ny×nz×nv3 ~ 1000×1000×200×163

No way!

2

Gyrokinetics[Rutherford & Frieman 1968; Catto 1982; Howes+ 2006; Brizard & Halm 2007]‣ Average gyro-motions‣ Solve distribution func of gyro-centers‣ (r, v) → (R, v⊥, v||) : 5D phase space‣ Popular in magnetic confinement fusion‣ Has been used in space plasma[Howes+ 2006]‣ Never been used for accretion disks

GK simulation

Solar wind (Cluster)

Earth’s magnetosphere (MMS)3

Gyrokinetic ordering
1. Cyclotron motion is much faster than turbulent fluctuations!
⌧1⌦<latexit sha1_base64="M2r6gxinMFweuIseoTVPCedFp3o=">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</latexit>

2. Anisotropic [Goldreich & Sridhar 1995]<latexit sha1_base64="kK9MtBJxj2pqx/sZMVbsRDjEBPw=">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</latexit>

kk
⌧1
k?

[Sundar+ 2017]

3. Small fluctuation amplitude<latexit sha1_base64="efq5/TI7c2WugyAOuC4ITlOJ7c0=">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</latexit>

B
fs
⇠
⌧1
B0
Fs

hs : gyro-center dist func
χ = φ - v･A/c<latexit sha1_base64="UAu+KTvfZkCD9ryX/t5JDbB5naU=">AAAxTHicvZpbb9vIFce9abvdqLds+9gXotlgE6wS2OkWRQsEiC3f5Ltl+RoFAS8zEmMOSc9QtmQuv0O/Q79P3/s9+lYU6BlK9jnSGbSbBLsKEJP/3+HVPPOTPAryJDbF4uI/P3vwk5/+7POff/Gw8Ytf/urXv3n05W9PTDbUoTgOsyTTZ4FvRBKn4riIi0Sc5Vr4KkjEaXDZsvz0WmgTZ2m3GOfirfL7aSzj0C8gevflZ3/vSe2HXun1cl8XsZ94A+8drBqvgn8kLmD1G++6hr3vYOX7bnhbb3hXHdZwpS5drEt7iZBFDzZI/LSfCK8XDmKvpycr9dGUXwwCCUsdKMdjNMkRezruD4pe5b26P9LVzOl0yRo78w8+NLsz6/fwm+kFTXfZmqy+Yef69u6c///h3j16vPhisX55fGFpuvB4Yfo6ePfli7/1oiwcKpEWYeIb88bPTVOLXMdp0cxSAQ/OUKXNNKsfmdzvi2YK1Rqei6Rphjk8MaGO88KPIi2MgUKZQW0axMHbUovrQoy+rRq9oYFtw0vYvPSVMWMVND1YsFfRhHusRNSE1WKgqsYTj1abIh7NZ3AcSIPAncOJuQEc57JqNBpw7y6FD1dTFEI3eq9jOa1LMj8S0eSQ3hMvlt7Xdvlrb8qNFxtvUtQoG1XpOV/1hnC/7iof0jOxF6yNNNVsXIxklhammt9TN1bCPLdtG3l1BfyvPbinheenkWf35j0VV8O6R40XwK9LCa+wT8uzxsPZm5CKm2JkN23WS3bbynv4scdIssyIZw24n71YjnqB6g3TSEgYXKIGnHl9E2YSrweHhU0V7LSE+gr+y5LIPgpZUt/uwk/i0LMh1M9Xv1l6W9ItvLJuAa28x0tVZbdPs/Q53UdPJEbcn8snnIkWn34uMr577LJiAE9dg7bEm2Eah1kkXhV6KJp5JAHV7dYMsuxS+frSTND9KrRgILSIXkkfLhLzLBfpXQaD+yUM9rDtJID9BpmOhH5VwpjgLVbNAA6vhZzyWhGTDezB3pYDsIAGDu1bL8JJDfMyjAtRl74KEjglGSd01W5PVoc6wbXJfgYirZ+ksr7CauZGlHVxE5yVzI4ZJoZbVNRd2coUeAv2YuJr4T09aj/zLDMz9X3t54M4HFVPvHYaJsNIeDLuD7X9kYjZ2mgyxEHpchL3U6/wQYneJDRelsKjE8YKRu88gzFxbmTx9TgygySdH3H8MBS2oW28KmBk1WIXHpHlOp69ZD8NxdzlRrHfDzK43JlUDZMi1tnNbG2QXPpa++PZ1D6PUWZPoDHztOs6LHsDA5WifPFSjCrwSmwEHK9cgjWoyQq/qNefL72syq96kd3oK3iy4bF98vx7vZ54cOXktXvura6tt/fa3fb+3tHMIPcBu5y5lBxOdSrWapbI6s3Lt2Xt7hI6snz8spqrULpu4mnb2qK5gkTC3q2O53Id2xtkx9c5ENm7GsUmT/yxKcb2yZ7lETmpnlaRPbH6Jzu5PKeluVdXwg9HYSLnS3sDeMKT6QbTlflTEXSjSCSFPznGdJEdB3Lp2uT+WHcbuo+3Gl9DTQpt5fdCeJTmcAsGiXteWBnNFaTBPZ+/Ab72q7L87rv5Q9oRYfIbHviFHbPrX/FcUQCsmtSWQVXNPV63hN4yOiJ0xOiY0PEcrT0Jw2pS39D6ARTJZCHL7aPaS2IF49m7u1OefbdCD3PtG3uYYCXu91+X385f38kM/hPua6KgmdqDzo4RoR2XJ7dNqHwAJ/AC+g62+/4d+mEDRGt5p73RWT7YPPd21rrdtc6R93SYg2q8EN6OwNibjJ99/AABbwSXq8mdhcVyef7+QLhC+IqDtwhvOfgq4asOvkb4moOvE77u4BuEbzj4JuGbDt4mvO3gW4RvOfg24dsOvkP4joPvEr7r4HuE7zn4PuH7Dn5A+IGDHxJ+6OAdwjsOfkT4kYN3Ce86+DHhxw5+QviJg58SfurgZ4SfOfg54ecOfkH4xQ/Y6J531Oq0D7r/o80/4Z0AfAK9a3RY5I0O4QrhrNEhbBHOGh3CVcJZo0O4RjhrdAjXCWeNDuEG4azRIdwknDU6hG3CWaNDuEU4a3QItwlnjQ7hDuGs0SHcJZw1OoR7hLNGh3CfcNboEB4QzhodwkPCWaND2CGcNTqER4SzRoewSzhrdAiPCWeNDuEJ4azRITwlnDU6hGeEs0aH8Jxw1ugQXhD+oY0+8wEmyQd+IIrJ3vLbsCr3u/DesV6CT9LwtusDB4YLP5dea2A/Bemx6x3Bp3xEuA396v5US5+9gb4NA8IDBw8JDx08IjxycEG4cHBJuHTwPuF9Bx8QPnDwmPDYwd8T/t7BLwm/dPCEcPbOH0JFuHLwlPDUwTPCMwfPCc8d/IrwKwfXhGsHN4QbBy8ILxx8SPjQwa8Jv3bwG8JvHHxE+MjBx4SPHfyW8Ntq/s8FkC6TAmZWCFcIZ2aFsEU4MyuEq4Qzs0K4RjgzK4TrhDOzQrhBODMrhJuEM7NC2CacmRXCLcKZWSHcJpyZFcIdwplZIdwlnJkVwj3CmVkh3CecmRXCA8KZWSE8JJyZFcIO4cysEB4RzswKYZdwZlYIjwlnZoXwhHBmVghPCWdmhfCMcGZWCM8JZ2aF8ILwH/Yt9N1rvbPc6h53iCA//Y9pdm7kTpV2mbvSpgGtYLa0aUgrmC9tGtEKZkybClrBnGlTSSuYNW3apxXMmzYd0ApmTpvGtIK506bvaQWzp00vaQXzp00TWsEMalNFK5hDbZrSCmZRm2a0gnnUpjmtYCa16RWtYC61qaYVzKY2NbSC+dSmBa1gRrXpkFYwp9r0mlYwq9r0hlYwr9p0RCuYWW06phXMrTa9pRXcrjZepiXMrzZdoRXMsDZt0QrmWJuu0gpmWZuu0QrmWZuu0wpmWptu0ArmWptu0gpmW5u2aQXzrU23aAUzrk23aQVzrk13aAWzrk13aQXzrk33aAUzr033aQVzr00PaAWzr00PaQXzr007tIIZ2KZHtII52KZdWsEsbNNjWsE8bNMTWsFMbNNTWsFcbNMzWsFsbNNzWsF8bNMLWvHjGNnr7O8u77mE/PFG1urOx1pxG2sVIGUm1ipEyiysVYSUGVgrgZTZVyuJlJlXqz5SZl2tBkiZcbWKkTLbavUeKTOtVpdImWW1SpAyw2qlkDK7apUiZWbVKkNqrTr33QKtcuTMqVpdIWU+1UojZS7VyiBlHtWqQMocqtUQKfOnVtdImTu1ukHKvKnVCClzplZjpMyXWt0i5a7Uahkx86RWK0iZI7VqIWV+1GoVKXOjVmtImRe1WkfKnKjVBlLmQ602kTIXatVGyjyo1RZS5kCttpEy/2m1g5S5T6tdpMx7Wu0hZc7Tah8p851WB0iZ67Q6RMo8p1UHKXOcVkdImd+06iJlbtPqGCnzmlYnSJnTtDpFynym1RlS5jKtzpEyj2l1gfRHcph3BAo7Wuu019k32j56YkbeOcxI7jAjA6TMYUaGSJnDjIyQMocZKZAyhxkpkTKHGdlHyhxm5AApc5iRMVLmMCPfI2UOM/ISKXOYkQlS5jAjFVLmMCNTpMxhRmZI2SdDI3OkzGBGXiFlBjNSI2UGs1+CvKfMYEYWSJnBjBwiZQYz8hopM5iRN0iZwYwcIWUGM3KMlBnMyFuk3GBGLiPmc5RyBSmfoZQtpHx+Uq4i5bOTcg0pn5uU60j5zKTcQMrnJeUmUj4rKdtI+Zyk3ELKZyTlNlI+Hyl3kPLZSLmLlM9Fyj2kfCZS7iPl85DyACmfhZSHSPkcpOwg5TOQ8ggpn3+UXaR89lEeI+Vzj/IEKZ95lKdI+byjPEPKZx3lOVI+5ygvkH6Qwd49erw0/411vnDy8sXSH18sHn77+PXK9NvsXyz8fuEPC08Xlhb+vPB6YXPhYOF4IXzQeLD44C8P/rr4j8V/Lf578T+T0gefTbf53cLMa+nz/wIP026r</latexit>+ Maxwell’s equations

C[hs] : collision operator〈…〉Rs : gyro-average‣ This ordering is OK for solar wind[Howes+ 2008], not sure for accretion disks

4

Simulation setting
Collisionless homegenius
slab plasma

Alfvén waves

Externally inject AWs to drive
turbulence [Tenbarge+ 2014]<latexit sha1_base64="lpF0/kc5cjklGKi8kBcClMCTg6o=">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</latexit>

k?0 A k 0
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k?0

Hybrid GK (Ion: GK, electron: fluid)
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Ion collisions

Qe

Hyper dissipation

|| ||

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+ hC[hi ]iRi

+ hyper resistivity+ hyper viscositydW=
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Z
|

a
J
d r || @A||3

V c @t
{z
}
=:PAW

+

Z
|

d3 v

Z

d3 Ri T 0i
hhiC[hi ]iRi
V F0i
{z
}
=:Qi

‣ Parameter scan 0.1 ≦ βi ≦ 100 & 0.1 ≦ Ti/Te ≦ 100(hyper res + hyper vis)
|
{z
}
=:Qe

5

Result : Qi/Qe‣ An incresing function of βi. But there is upper limit < 50
‣ At low βi, Qi/Qe → 0 : Consistent with theoretical prediction[Schekochihin, Kawazura, & Barnes 2019]‣ At the Hall regime (βi≪1, βe ~ 1), Qi/Qe ~ 0.5 : Consistent with thetheoretical prediction [Schekochihin, Kawazura, & Barnes 2019]6

Result : Qi/Qe‣ A prescription of EHT is not really bad :)6

Plan of the talk
‣ Research background‣ Three key questions of turbulent heating in collisionless plasma
1. Partition of Alfvenic fluctuations between ion and electron
heating [Kawazura+ PNAS 2019]
2. Partition of compressive fluctuations between ion andelectron heating [Kawazura+ in prep]
3. Partition of Alfvenic and compressive fluctuation in MRI
driven turbulence [Kawazura+ in prep]‣ SummaryIs accretion turbulence Alfvenic or compressive?
‣ In the previous GK simulation, turbulence was Alfvenically driven‣ This is based on the fact that solar wind is Alfvenic‣ But, we do not know if accretion disks are Alfvenic‣ We need to drive turbulence compressively

Alfvén waves

Collisionless homegenius
slab plasma
Qi
Qe

7

Alfvenic & compressive forcing
Collisionless homegenius
slab plasma

Alfvén waves

Slow waves

@hi
@hi
c
Ze @ h iRi+ v||+
h iRi , hi =
Fi
@t
@z
B0
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1 @Ak
T e ne+ rk = rk
c @t
e ne
d
dt
<latexit 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ne
ne

Bk
cT e
+ rk uke +
B0
eB0

dW
=
dt

Z
|

a
d r J|| @A||

V c @t
{z
}
=:PAW

+

Z
|

|| ||

c

+

Ri

v|| haext iRi
v2thi

Fi + hC[hi ]iRi

+ hyper resistivity!

3* v Aa +

3

d v

(

Z

ne Bk
,
ne B0
3

)

= 0 + hyper viscosityZd Ri v|| hi haext iRi
3
Ti
+
d
v
2
V
vthi
{z
} |
=:PcomprZ

d3 Ri T i
hhiC[hi ]iRi
V Fi
{z
}
=:Qi

(hyper res + hyper vis)
|
{z
}
=:Qe

8

Result : Qi/Qe‣ Qi/Qe is an increasing function of Pcompr/PAlfven‣ Qi/Qe ≃ Pcompr/PAlfven for βi = 0.1 ← Consistent with a theory [Schekochihin+ 2019]‣ When Pcompr/PAlfven ≫ 1, Qi/Qe ≃ Pcompr/PAlfven for any βi
9Result : Qi/Qe

9Plan of the talk
‣ Research background‣ Three key questions of turbulent heating in collisionless plasma
1. Partition of Alfvenic fluctuations between ion and electron
heating [Kawazura+ PNAS 2019]
2. Partition of compressive fluctuations between ion andelectron heating [Kawazura+ in prep]
3. Partition of Alfvenic and compressive fluctuation in MRI
driven turbulence [Kawazura+ in prep]‣ SummaryHow do we apply Qi/Qe to actual disks?

Qi/Qe = f(βi, Ti/Te, Pcompr/PAW)

1. Solve Global MHD and calculate B, T = Ti+Te,[Lessler+ 2015]

Pcompr/PAW at each grid
2. Use Qi/Qe = f(βi, Ti/Te, Pcompr/PAW) to solve thetransport equations for ions and electrons
3. One obtains Ti and Te respectively

How should we get Pcompr/PAW ?
→ A question of MRI driven MHD turbulence

10Compressive—Alfvenic decoupling at k||/k⊥ ≪ 1‣ In smaller scales, Alfven and slow fluctuations satisfy k||/k⊥ ≪ 1‣ Ideal MHD + k||/k⊥ ≪ 1 ＝ Reduced MHD→ Alfven and slow decouples [Schekochihin+ 2009]
d
@
= vA
,
dt
@z
du||
= v2A r||dt
<latexit sha1_base64="QVebpwtXL7Uel3CASe2Xl6ODYa0=">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</latexit>d 2
r? = vA r|| r2?
dt
0
!
2 1 1
vA CCCB||
d B|| BBB,
= B@1 + 2 CA r|| u||
B0
dt B0
cs

Φ, Ψ : Alfvenu||, δB|| : Slow

‣ The ratio of dissipation Dcompr/DAW at RMHD = Pcompr/PAW for GK

11

k||/k⊥ ≪ 1 is diﬃcult to be achieved in MRI turb
‣ B tends to be azimuthal in disks‣ The fastest growing mode resides in kZ → ∞ (Z : rotational axis)[Balbus&Hawley 1992, Hawley+ 1995, Matsumoto & Tajima 1995]‣ We cannot reach the compressive—Alfvenic decoupling scaleq ≡ kY2/kZ2ZYX

Suzuki+ 2019
Matsumoto & Tajima

12

k||/k⊥ ≪ 1 is diﬃcult to be achieved in MRI turb
‣ B tends to be azimuthal in disks‣ The fastest growing mode resides in kZ → ∞ (Z : rotational axis)[Balbus&Hawley 1992, Hawley+ 1995, Matsumoto & Tajima 1995]‣ We cannot reach the compressive—Alfvenic decoupling scaleq ≡ kY2/kZ2[Walker+ 2016]‣ When B is nearly azimuthal，kZ ~ k⊥, kY ~ k||Matsumoto & Tajima‣ kZ/kY → ∞ ⇔ k⊥/k|| → ∞ Reduced MHD limit

Our conjecture : Pcompr/PAW can be obtained by RMHD + shear
12Tilted coordinate‣ Let θ be angle between B0 and ϖ0 (azimuthalθ ≃ π/2)‣ Convenient to take z axis along B0 for considering k||/k⊥ ≪ 1

13

Shearing RMHD

RMHD ordering<latexit sha1_base64="(null)">(null)</latexit>

Φ, Ψ : Alfvenic
u||, δB|| : compressive<latexit sha1_base64="(null)">(null)</latexit>

14Linear analysis (MRI growth rate vs ky/k||)‣ Compare the growth rate between SHRMHD and ideal MHD tocheck if SHRMHD includes the fastest growing mode of ideal MHD
θ = 0.45π, β = 1

SHRMHD‣ Growth rate of ideal MHD : an increase func of ky/k||‣ Saturate at the growth rate of SHRMHD→ When θ ≃ π/2, it is enough to solve SHRMHD

15

Linear analysis (Icompr/IAW)‣ Substituting the eigenfunctions into

we obtaine the compressive-to-Alfven MRI injection
power ratio Icompr/IAW
Icompr/IAW ≃ 1 for any β16Nonlinear simulation of SHRMHD‣ Injection ratio Icompr/IAW and

RMHD( ϖ0 ≪ ωnl )dissipation ratio Dcompr/DAW shallbe diﬀerent

IAW

Icompr-5/3
SHRMHD( ϖ0 ≫ ωnl )Dcompr DAWk+ perp resitivity+ perp viscosity+ para resitivity+ para viscosity‣ Compute Dcompr/DAW at the MRI saturated state‣ Fix θ = 0.45π and scan β = 0.1, 1, 10
17

Time evolution

18

Dcompr/DAW vs βLinear & nonlinear calculations show Dcompr/DAW ≃ 1

withis the answer!
19k⊥-spectrum
Our SHRMHD (1024x1024x512 grids)‣ Both Alfven & slow look k⊥-5/3

Full-MHD (1024x1024x512 grids)
Reproduced from Walker+ 2016

‣ In full-MHD, k⊥-5/3 is not achieved→ RMHD regime is reached→ Alfven & slow are decoupled

20

Shear rotation vs nonlinear cascade

RMHD ( ϖ0 ≪ ωnl )AW and compressive aredecoupled

SHRMHD ( ϖ0 ≫ ωnl )AW and compressive arecoupled21

Summary‣ Ion vs electron heating is critical for interpreting the EHT data.
‣ In collisionless plasma, turbulent heating is multiscale in nature.
‣ We split the problem into three pieces① What is Pcompr/PAW for given PMRI?
② What is Qi/Qe for given PAW?
③ What is Qi/Qe for given Pcompr?

‣ For ② & ③, we used gyrokinetics•

GK allows us to solve kinetic turbulence with reasonable cost•Obtained a heating prescription‣ For ①, we developed SHRMHD because full-MHD is diﬃcult to use•

When θ≃π/2 (toroidal magnetic field), Pcompr/PAW ≃ 1
22