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 2 r Ω GM/r 2 r 0 , Ω 0 Central Star r, Ω

Ang. Mom. Transport & Mass Accretion • If Ang. Mom. conserved: Ω = Ω 0 ( r 0 r ) 2 2 r Ω GM/r 2 r 0 , Ω 0 Central Star r, Ω

Ang. Mom. Transport & Mass Accretion • If Ang. Mom. conserved: Ω = Ω 0 ( r 0 r ) 2 • Centrifugal force at r : 2 r Ω 0 ( r 0 r Ω 2 = r 0 Ω 2 r ) 3 GM/r 2 r 0 , Ω 0 Central Star r, Ω

Ang. Mom. Transport & Mass Accretion • If Ang. Mom. conserved: Ω = Ω 0 ( r 0 r ) 2 • Centrifugal force at r : 2 r Ω 0 ( r 0 r Ω 2 = r 0 Ω 2 r ) 3 GM/r 2 r 0 , Ω 0 • Gravity at r : Central ( r 0 GM r ) 2 Star r, Ω r 2 0

Ang. Mom. Transport & Mass Accretion • If Ang. Mom. conserved: Ω = Ω 0 ( r 0 r ) 2 • Centrifugal force at r : 2 r Ω 0 ( r 0 r Ω 2 = r 0 Ω 2 r ) 3 GM/r 2 r 0 , Ω 0 • Gravity at r : Central ( r 0 GM r ) 2 Star r, Ω r 2 0 • If r 0 Ω 2 0 = GM / r 2 0 , ⇒ r Ω 2 > GM / r 2

Ang. Mom. Transport & Mass Accretion • If Ang. Mom. conserved: Ω = Ω 0 ( r 0 r ) 2 • Centrifugal force at r : 2 r Ω 0 ( r 0 r Ω 2 = r 0 Ω 2 r ) 3 GM/r 2 r 0 , Ω 0 • Gravity at r : Central ( r 0 GM r ) 2 Star r, Ω r 2 0 • If r 0 Ω 2 0 = GM / r 2 0 , ⇒ r Ω 2 > GM / r 2 • 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 8 π ) − ( B · ∇ ) B ρ ∇ ( p + B 2 d � dt + 1 + ∇ Φ = 0 4 πρ ∂ B ∂ t = ∇ × ( � × B − η ∇ × B ) dt = − p ∇ · � + η ρ de 4 π | ∇ × B | 2 • axisymmetric perturbation: ∝ exp( − i ω t + ik r r + ik z z ) • Gravity by a central star ∇ Φ ≈ ( GM r 2 , 0 , GMz r 3 ) • Assuming B 0 = (0 , 0 , B z , 0 ) , ideal MHD ( η = 0 ), & incompressive ( k r δ � r + k z δ � z = 0 ) Dispersion relation : z + κ 2 k 2 k 4 ω 4 − (2 � 2 z + ( κ 2 − 4 Ω 2 ) � 2 A , z k 2 k 2 ) ω 2 + � 4 z A , z k 4 z k 2 = 0 A , z where κ : epicycle frequency ( = Ω for Kepler rotation) � � A , z = B z , 0 / 4 πρ

MRI –Dispersion Relation– Sano & Miyama 1999 • Always unstable for the weak B ( β = 8 π p B 2 � 1 ) • The growth rate ∼ Ω − 1

Magneto-Rotational Instability (MRI) A fluid element moves outward (Connection through B-field) Center The Fluid element rotates faster than A.M. conservation Centrifugal F. > Gravity (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 Ω 0 . (neglect curvature) Hawley, Gammie, & Balbus 1995

MHD in Cartesian Shearing Box (CaSB) • Local Cartesian coordinate with co-rotating with Ω 0 . (neglect curvature) • x = r − r 0 ; y ↔ φ -direction Hawley, Gammie, & Balbus 1995

MHD in Cartesian Shearing Box (CaSB) • Local Cartesian coordinate with co-rotating with Ω 0 . (neglect curvature) • x = r − r 0 ; y ↔ φ -direction � GM / r 3 ) • Basic equations for Keplerian rotation ( Ω 0 = ∂ρ ∂ t + ∇ · ( ρ � ) = 0 8 π ) + ( B · ∇ ) B x ρ ∇ x ( p + B 2 ∂ � x ∂ t = − 1 + 2 Ω 0 � y + 3 Ω 2 0 x 4 πρ ( B · ∇ ) B y ∂ � y ρ ∇ y ( p + B 2 ∂ t = − 1 Hawley, Gammie, & Balbus 1995 8 π ) + − 2 Ω 0 � x 4 πρ 8 π ) + ( B · ∇ ) B z ∂ � z ρ ∇ z ( p + B 2 ∂ t = − 1 − Ω 2 0 z 4 πρ ∂ B ∂ t = ∇ × ( � × B − η ∇ × B ) ∇ · B = 0

MHD in Cartesian Shearing Box (CaSB) • Local Cartesian coordinate with co-rotating with Ω 0 . (neglect curvature) • x = r − r 0 ; y ↔ φ -direction � GM / r 3 ) • Basic equations for Keplerian rotation ( Ω 0 = ∂ρ ∂ t + ∇ · ( ρ � ) = 0 8 π ) + ( B · ∇ ) B x ρ ∇ x ( p + B 2 ∂ � x ∂ t = − 1 + 2 Ω 0 � y + 3 Ω 2 0 x 4 πρ ( B · ∇ ) B y ∂ � y ρ ∇ y ( p + B 2 ∂ t = − 1 Hawley, Gammie, & Balbus 1995 8 π ) + − 2 Ω 0 � x 4 πρ 8 π ) + ( B · ∇ ) B z ∂ � z ρ ∇ z ( p + B 2 ∂ t = − 1 − Ω 2 0 z 4 πρ ∂ B ∂ t = ∇ × ( � × B − η ∇ × B ) ∇ · B = 0 • An Isothermal Equation of State

MHD in Cartesian Shearing Box (CaSB) • Local Cartesian coordinate with co-rotating with Ω 0 . (neglect curvature) • x = r − r 0 ; y ↔ φ -direction � GM / r 3 ) • Basic equations for Keplerian rotation ( Ω 0 = ∂ρ ∂ t + ∇ · ( ρ � ) = 0 8 π ) + ( B · ∇ ) B x ρ ∇ x ( p + B 2 ∂ � x ∂ t = − 1 + 2 Ω 0 � y + 3 Ω 2 0 x 4 πρ ( B · ∇ ) B y ∂ � y ρ ∇ y ( p + B 2 ∂ t = − 1 Hawley, Gammie, & Balbus 1995 8 π ) + − 2 Ω 0 � x 4 πρ 8 π ) + ( B · ∇ ) B z ∂ � z ρ ∇ z ( p + B 2 ∂ t = − 1 − Ω 2 0 z 4 πρ ∂ B ∂ t = ∇ × ( � × B − η ∇ × B ) ∇ · B = 0 • An Isothermal Equation of State • Steady-state solution • B = (0 , B y , B z ) & � = (0 , − 3 2 Ω 0 x , 0) • ρ = ρ 0 exp( − z 2 / H 2 ) ( H 2 ≡ 2 c 2 s / Ω 2 0 ): hydrostatic equilibrium

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 on either left or right side

Some Disadvantages of CaSB • Neglect the Curvature • Symmetry to the ± x direction The central star can be located on 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 of Mass+Momentum+(Energy)+ B

Equations

Equations • Mass: ∂ t ρ + R − 1 ∂ R ( ρ � R R ) + ∂ φ ( ··· ) + ∂ z ( ··· ) = 0

Equations • Mass: ∂ t ρ + R − 1 ∂ R ( ρ � R R ) + ∂ φ ( ··· ) + ∂ z ( ··· ) = 0 • Momentum– R : ∂ t ( ρ � R ) + R − 1 ∂ R ( ρ � 2 R R ) + ··· = 0

Equations • Mass: ∂ t ρ + R − 1 ∂ R ( ρ � R R ) + ∂ φ ( ··· ) + ∂ z ( ··· ) = 0 • Momentum– R : ∂ t ( ρ � R ) + R − 1 ∂ R ( ρ � 2 R R ) + ··· = 0 • Momentum– φ (Angular Momentum): ∂ t ( ρ � φ R ) + ∂ R [( ρ � R � φ + B R B φ / 4 π ) R 2 ] + ··· = 0

Equations • Mass: ∂ t ρ + R − 1 ∂ R ( ρ � R R ) + ∂ φ ( ··· ) + ∂ z ( ··· ) = 0 • Momentum– R : ∂ t ( ρ � R ) + R − 1 ∂ R ( ρ � 2 R R ) + ··· = 0 • Momentum– φ (Angular Momentum): ∂ t ( ρ � φ R ) + ∂ R [( ρ � R � φ + B R B φ / 4 π ) R 2 ] + ··· = 0 • Momentum– z : ∂ t ( ρ � z ) + R − 1 ∂ R ( ρ � R � z R ) + ··· = 0

Equations • Mass: ∂ t ρ + R − 1 ∂ R ( ρ � R R ) + ∂ φ ( ··· ) + ∂ z ( ··· ) = 0 • Momentum– R : ∂ t ( ρ � R ) + R − 1 ∂ R ( ρ � 2 R R ) + ··· = 0 • Momentum– φ (Angular Momentum): ∂ t ( ρ � φ R ) + ∂ R [( ρ � R � φ + B R B φ / 4 π ) R 2 ] + ··· = 0 • Momentum– z : ∂ t ( ρ � z ) + R − 1 ∂ R ( ρ � R � z R ) + ··· = 0 • Induction eq.– φ ∂ t B φ = ∂ z ( ··· ) − ∂ R ( � R B φ − � φ B R )

Equations • Mass: ∂ t ρ + R − 1 ∂ R ( ρ � R R ) + ∂ φ ( ··· ) + ∂ z ( ··· ) = 0 • Momentum– R : ∂ t ( ρ � R ) + R − 1 ∂ R ( ρ � 2 R R ) + ··· = 0 • Momentum– φ (Angular Momentum): ∂ t ( ρ � φ R ) + ∂ R [( ρ � R � φ + B R B φ / 4 π ) R 2 ] + ··· = 0 • Momentum– z : ∂ t ( ρ � z ) + R − 1 ∂ R ( ρ � R � z R ) + ··· = 0 • Induction eq.– φ ∂ t B φ = ∂ z ( ··· ) − ∂ R ( � R B φ − � φ B R ) • Induction eq.– z ∂ t B z = R − 1 ∂ R [( � z B R − � R B z ) R ] − ∂ φ ( ··· )