MHD in a Cylindrical Shearing Box Takeru K. Suzuki School of Arts - - PowerPoint PPT Presentation

MHD in a Cylindrical Shearing Box

Takeru K. Suzuki

School of Arts & Science, U. Tokyo

September 6th, 2018 Thanks to XC40@YITP & ATERUI@CfCA/NaOJ

Accretion Disks

Some Examples

• Accretion Disks around Black Holes
• Active Galactic Nuclei (SMBH)
• Galactic Disks
• Protoplanetary Disks

credit: NASA

• Ang. Mom. Transport & Mass Accretion

GM/r2 rΩ

2

r,Ω r0,Ω0 Central Star

• Ang. Mom. Transport & Mass Accretion

GM/r2 rΩ

2

r,Ω r0,Ω0 Central Star

• If Ang. Mom. conserved:

Ω = Ω0( r0

r )2

• Ang. Mom. Transport & Mass Accretion

GM/r2 rΩ

2

r,Ω r0,Ω0 Central Star

• If Ang. Mom. conserved:

Ω = Ω0( r0

r )2

• Centrifugal force at r :

rΩ2 = r0Ω2

0( r0 r )3

• Ang. Mom. Transport & Mass Accretion

GM/r2 rΩ

2

r,Ω r0,Ω0 Central Star

• If Ang. Mom. conserved:

Ω = Ω0( r0

r )2

• Centrifugal force at r :

rΩ2 = r0Ω2

0( r0 r )3

• Gravity at r :

GM r2

( r0

r )2

• Ang. Mom. Transport & Mass Accretion

GM/r2 rΩ

2

r,Ω r0,Ω0 Central Star

• If Ang. Mom. conserved:

Ω = Ω0( r0

r )2

• Centrifugal force at r :

rΩ2 = r0Ω2

0( r0 r )3

• Gravity at r :

GM r2

( r0

r )2

• If r0Ω2

0 = GM/r2 0,

⇒ rΩ2 > GM/r2

• Ang. Mom. Transport & Mass Accretion

GM/r2 rΩ

2

r,Ω r0,Ω0 Central Star

• If Ang. Mom. conserved:

Ω = Ω0( r0

r )2

• Centrifugal force at r :

rΩ2 = r0Ω2

0( r0 r )3

• Gravity at r :

GM r2

( r0

r )2

• If r0Ω2

0 = GM/r2 0,

⇒ rΩ2 > GM/r2

• Mass does NOT accrete.

(Rayleigh Criterion)

Turbulence in Accretion Disks

Turbulence ⇒ Macroscopic (effective) Viscosity

• Outward Transport of Angular Momentum
• Inward Accretion of Matters

Exchange fluid elements by ‘‘stirring with a spoon’’

MHD in an Accretion Disk

Suzuki & Inutsuka 2014

Magneto-Rotational Instability (MRI) –linear analyses–

Balbus & Hawley 1991

∂ρ ∂t + ∇ · (ρ) = 0 d dt + 1 ρ∇(p+ B2 8π) − (B·∇)B 4πρ

+ ∇Φ = 0

∂B ∂t = ∇ × ( × B − η∇ × B)

ρ de

dt = −p∇ · + η 4π|∇ × B|2

• axisymmetric perturbation: ∝ exp(−iωt + ikrr + ikzz)
• Gravity by a central star ∇Φ ≈ (GM

r2 ,0, GMz r3 )

• Assuming B0 = (0,0, Bz,0), ideal MHD (η = 0),

& incompressive (krδr + kzδz = 0) Dispersion relation :

ω4 − (22

A,zk2 z + κ2 k2

z

k2)ω2 + 4 A,zk4 z + (κ2 − 4Ω2)2 A,z k4

z

k2 = 0 where κ : epicycle frequency (= Ω for Kepler rotation)

A,z = Bz,0/

• 4πρ

MRI –Dispersion Relation–

Sano & Miyama 1999

• Always unstable for the weak B (β = 8πp

B2 1)

• The growth rate ∼ Ω−1

Magneto-Rotational Instability (MRI)

Center The Fluid element rotates faster than A.M. conservation Centrifugal F. > Gravity (Connection through B-field) A fluid element moves outward (Unstable)

Unstable under

• Weak B-fields
• (inner-fast) Differential Rotation

Velikov (1959); Chandrasekhar (1960); Balbus & Hawley (1991)

MHD in Cartesian Shearing Box (CaSB)

Hawley, Gammie, & Balbus 1995

MHD in Cartesian Shearing Box (CaSB)

• Local Cartesian coordinate with co-rotating with Ω0.

(neglect curvature)

Hawley, Gammie, & Balbus 1995

MHD in Cartesian Shearing Box (CaSB)

• Local Cartesian coordinate with co-rotating with Ω0.

(neglect curvature)

• x = r − r0; y ↔ φ-direction

Hawley, Gammie, & Balbus 1995

MHD in Cartesian Shearing Box (CaSB)

• Local Cartesian coordinate with co-rotating with Ω0.

(neglect curvature)

• x = r − r0; y ↔ φ-direction
• Basic equations for Keplerian rotation (Ω0 =
• GM/r3)

∂ρ ∂t + ∇ · (ρ) = 0 ∂x ∂t = − 1 ρ∇x(p+ B2 8π ) + (B·∇)Bx 4πρ

+ 2Ω0y + 3Ω2

0x ∂y ∂t = − 1 ρ∇y(p+ B2 8π ) + (B·∇)By 4πρ

− 2Ω0x

∂z ∂t = − 1 ρ∇z(p+ B2 8π ) + (B·∇)Bz 4πρ

− Ω2

0z ∂B ∂t = ∇ × ( × B − η∇ × B)

∇ · B = 0

Hawley, Gammie, & Balbus 1995

MHD in Cartesian Shearing Box (CaSB)

• Local Cartesian coordinate with co-rotating with Ω0.

(neglect curvature)

• x = r − r0; y ↔ φ-direction
• Basic equations for Keplerian rotation (Ω0 =
• GM/r3)

∂ρ ∂t + ∇ · (ρ) = 0 ∂x ∂t = − 1 ρ∇x(p+ B2 8π ) + (B·∇)Bx 4πρ

+ 2Ω0y + 3Ω2

0x ∂y ∂t = − 1 ρ∇y(p+ B2 8π ) + (B·∇)By 4πρ

− 2Ω0x

∂z ∂t = − 1 ρ∇z(p+ B2 8π ) + (B·∇)Bz 4πρ

− Ω2

0z ∂B ∂t = ∇ × ( × B − η∇ × B)

∇ · B = 0

• An Isothermal Equation of State

Hawley, Gammie, & Balbus 1995

MHD in Cartesian Shearing Box (CaSB)

• Local Cartesian coordinate with co-rotating with Ω0.

(neglect curvature)

• x = r − r0; y ↔ φ-direction
• Basic equations for Keplerian rotation (Ω0 =
• GM/r3)

∂ρ ∂t + ∇ · (ρ) = 0 ∂x ∂t = − 1 ρ∇x(p+ B2 8π ) + (B·∇)Bx 4πρ

+ 2Ω0y + 3Ω2

0x ∂y ∂t = − 1 ρ∇y(p+ B2 8π ) + (B·∇)By 4πρ

− 2Ω0x

∂z ∂t = − 1 ρ∇z(p+ B2 8π ) + (B·∇)Bz 4πρ

− Ω2

0z ∂B ∂t = ∇ × ( × B − η∇ × B)

∇ · B = 0

• An Isothermal Equation of State
• Steady-state solution
• B = (0, By, Bz) & = (0,−3

2Ω0x,0)

• ρ = ρ0 exp(−z2/H2) (H2 ≡ 2c2

s/Ω2 0):

hydrostatic equilibrium

Hawley, Gammie, & Balbus 1995

Cartesian Shearing Box Simulations

Hawley et al. 1995; Matsumoto & Tajima 1995; ...

Suzuki & Inutsuka 2009

Applications of CaSB

• PIC simulation in CaSB

Hoshino 2013; 2015; Shirakawa & Hoshino 2014

• MHD + non-thermal particles Kimura+ 2016

Some Disadvantages of CaSB

Some Disadvantages of CaSB

• Neglect the Curvature

Some Disadvantages of CaSB

• Neglect the Curvature
• Symmetry to the ±x direction

The central star can be located

• n either left or right side

Some Disadvantages of CaSB

• Neglect the Curvature
• Symmetry to the ±x direction

The central star can be located

• n either left or right side
• No Gas Accretion

A New Aproach

A New Aproach

• Break the Symmetry

A New Aproach

• Break the Symmetry
• Introduce the Curvature

A New Aproach

• Break the Symmetry
• Introduce the Curvature

⇒ can handle the net accretion ?

A New Aproach

• Break the Symmetry
• Introduce the Curvature

⇒ can handle the net accretion ? ⇒ Let’s try “Cylindrical Shearing Box (CySB)”

Cylindrical Shearing Box (CySB)

Cylindrical Shearing Box (CySB)

Key : Boundary Condition at R±

Cylindrical Shearing Box (CySB)

Key : Boundary Condition at R± Radial Boundary Condition

⇐ Conservation Laws

• f Mass+Momentum+(Energy)+B

Equations

Equations

• Mass:

∂tρ + R−1∂R(ρRR) + ∂φ(···) + ∂z(···) = 0

Equations

• Mass:

∂tρ + R−1∂R(ρRR) + ∂φ(···) + ∂z(···) = 0

• Momentum– R:

∂t(ρR) + R−1∂R(ρ2

RR) + ··· = 0

Equations

• Mass:

∂tρ + R−1∂R(ρRR) + ∂φ(···) + ∂z(···) = 0

• Momentum– R:

∂t(ρR) + R−1∂R(ρ2

RR) + ··· = 0

• Momentum– φ (Angular Momentum):

∂t(ρφR)+∂R[(ρRφ + BRBφ/4π)R2]+··· = 0

Equations

• Mass:

∂tρ + R−1∂R(ρRR) + ∂φ(···) + ∂z(···) = 0

• Momentum– R:

∂t(ρR) + R−1∂R(ρ2

RR) + ··· = 0

• Momentum– φ (Angular Momentum):

∂t(ρφR)+∂R[(ρRφ + BRBφ/4π)R2]+··· = 0

• Momentum– z:

∂t(ρz) + R−1∂R(ρRzR) + ··· = 0

Equations

• Mass:

∂tρ + R−1∂R(ρRR) + ∂φ(···) + ∂z(···) = 0

• Momentum– R:

∂t(ρR) + R−1∂R(ρ2

RR) + ··· = 0

• Momentum– φ (Angular Momentum):

∂t(ρφR)+∂R[(ρRφ + BRBφ/4π)R2]+··· = 0

• Momentum– z:

∂t(ρz) + R−1∂R(ρRzR) + ··· = 0

• Induction eq.– φ

∂tBφ = ∂z(···) − ∂R(RBφ − φBR)

Equations

• Mass:

∂tρ + R−1∂R(ρRR) + ∂φ(···) + ∂z(···) = 0

• Momentum– R:

∂t(ρR) + R−1∂R(ρ2

RR) + ··· = 0

• Momentum– φ (Angular Momentum):

∂t(ρφR)+∂R[(ρRφ + BRBφ/4π)R2]+··· = 0

• Momentum– z:

∂t(ρz) + R−1∂R(ρRzR) + ··· = 0

• Induction eq.– φ

∂tBφ = ∂z(···) − ∂R(RBφ − φBR)

• Induction eq.– z

∂tBz = R−1∂R[(zBR − RBz)R] − ∂φ(···)

Equations

• Mass:

∂tρ + R−1∂R(ρRR) + ∂φ(···) + ∂z(···) = 0

• Momentum– R:

∂t(ρR) + R−1∂R(ρ2

RR) + ··· = 0

• Momentum– φ (Angular Momentum):

∂t(ρφR)+∂R[(ρRφ + BRBφ/4π)R2]+··· = 0

• Momentum– z:

∂t(ρz) + R−1∂R(ρRzR) + ··· = 0

• Induction eq.– φ

∂tBφ = ∂z(···) − ∂R(RBφ − φBR)

• Induction eq.– z

∂tBz = R−1∂R[(zBR − RBz)R] − ∂φ(···)

• ∇ · B = 0

R−1∂R(BRR) + R−1∂φBφ + ∂zBz = 0

Cylindrical Shearing Condition

Cylindrical Shearing Condition

• Shear: A(R±,φ, z) = A(R∓,φ ± ∆Ωeqt, z)

where ∆Ωeq = Ωeq,− − Ωeq,+

Cylindrical Shearing Condition

• Shear: A(R±,φ, z) = A(R∓,φ ± ∆Ωeqt, z)

where ∆Ωeq = Ωeq,− − Ωeq,+

• Conserved quantities, A, at R− & R+

A = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ρRR ρ2

RR

(ρRφ + BφBR/4π)R2 ρRzR RBφ − φBR (zBR − RBz)R BRR

Cylindrical Shearing Box (CySB)

Cylindrical Shearing Box (CySB)

Time Evolution

(βz,0 = 103, T ∝ R−1/2, initial ρ ∝ R−1)

(LR, Lφ, Lz) = (0.4,π/6,0.1) ≈(4H,4H,H) resolved by (256,256,64)

αB = BRBφ/4πp

0.05 0.1 0.15 0.2 0.25 20 40 60 80 100 120 140 160 180 200 α t (rotation) αB at R0=1

Angular Momentum Flows

Under the steady-state condition:

∂ ∂R

• R2

ρRRΩeq + ρRδφ − BRBφ/4π

• = 0

Accretion & Bz Advection

∂ ∂t(RBz) = ∂ ∂R

R(zBR − RBz) ⇒ ⟨R,Bz⟩ = ⟨R(RBz − zBR)⟩/⟨RBz⟩

Zonal Flow

a long-lived ρ bump

Johansen+ 2009

• Physical ?
• Numerical Artifact ?

Summary

Cylindrical Shearing Box

Summary

Cylindrical Shearing Box Advantage to Cartesian SB

• can handle net mass accretion

Summary

Cylindrical Shearing Box Advantage to Cartesian SB

• can handle net mass accretion

Advantage to Global Simulations

• can resolve fine-scale turbulence
• can perform long-time simulations

Summary

Cylindrical Shearing Box Advantage to Cartesian SB

• can handle net mass accretion

Advantage to Global Simulations

• can resolve fine-scale turbulence
• can perform long-time simulations

Various potential applications

Summary

Cylindrical Shearing Box Advantage to Cartesian SB

• can handle net mass accretion

Advantage to Global Simulations

• can resolve fine-scale turbulence
• can perform long-time simulations

Various potential applications

• Accretion vs. Bz advection

Summary

Cylindrical Shearing Box Advantage to Cartesian SB

• can handle net mass accretion

Advantage to Global Simulations

• can resolve fine-scale turbulence
• can perform long-time simulations

Various potential applications

• Accretion vs. Bz advection
• Particle Acceleration

Summary

Cylindrical Shearing Box Advantage to Cartesian SB

• can handle net mass accretion

Advantage to Global Simulations

• can resolve fine-scale turbulence
• can perform long-time simulations

Various potential applications

• Accretion vs. Bz advection
• Particle Acceleration
• Dust grains in Prtoplanetary Disks

Summary

Cylindrical Shearing Box Advantage to Cartesian SB

• can handle net mass accretion

Advantage to Global Simulations

• can resolve fine-scale turbulence
• can perform long-time simulations

Various potential applications

• Accretion vs. Bz advection
• Particle Acceleration
• Dust grains in Prtoplanetary Disks
• ......