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General Relativistic MHD Simulations with Finite Conductivity

General Relativistic MHD Simulations with Finite Conductivity. Shinji Koide (Kumamoto University) Kazunari Shibata (Kyoto University) Takahiro Kudoh (NAOJ). EANAM2006 @KASI, Daejeon, Korea, 2006.11.3(Fri). Outline.

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General Relativistic MHD Simulations with Finite Conductivity

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  1. General Relativistic MHD Simulations with Finite Conductivity Shinji Koide (Kumamoto University) Kazunari Shibata (Kyoto University) Takahiro Kudoh(NAOJ) EANAM2006 @KASI, Daejeon, Korea, 2006.11.3(Fri)

  2. Outline • Numerical results of Ideal general relativistic MHD (ideal GRMHD) simulation: Jet formation by magnetic bridges between the ergosphere and disk around a rapidly rotating black hole. Anti-parallel magnetic field is formed along the jet ⇒ Magnetic reconnection • Numerical method of GRMHD with finite conductivity (σGRMHD): Numerical algorithm and simple tests • Essential role of implicit method

  3. Gamma-ray burst AGN ~ Light years X-ray, optical, radio emission g-rays γ>10 γ～ 3 ~ ~ 1AU Relativistic jet ~ ~ 1 km Gravitational collapse Forming Spinning Black hole (?) Motivation: Relativistic Jets in the Universe γ>100 Several lys Several M lys Mirabel, Rodriguez 1998

  4. Models: Relativistic Jets ~ • Active galactic nuclei, Quasars: γ>10, Ljet~ several M pc • Stellar mass black hole binaries (Microquasars): γ~ 3, Ljet ~ several pc • Gamma-ray bursts: γ> 100, Ljet ~ 1AU-several pc ~ The jet formation mechanism may be common. These relativistic jets are formed by drastic phenomena around black holes. However, the confirmed model has not yet shown. Acceleration of plasma/gas Collimation of plasma/gas outflow 1) Magnetic field 2) Radiation pressure 3) Gas pressure Points of models

  5. Black Hole Magnetosphere （Corona） Closed magnetic field linesbetween ergosphere and disk Magnetic Field Lines Plasma Disk Black Hole Ergosphere Plasma Black Hole Magnetosphere

  6. Magnetic Field induced by Current Loop around Black Hole Magnetic Field Lines Magnetic bridges Black Hole Plasma Disk Current loop R0 Ergosphere Plasma

  7. Nonrelativistic MHD Simulation with Dipole-Magnetic Field and Disk Hayashi, Shibata, and Matsumoto (1996) Magnetic bridge Anomalous resistivity: Magnetic island (Plasmoid)

  8. Twist of magnetic bridge by ergosphere Magnetic bridge Twist by frame-dragging effect Rapidly Rotating Black Hole Disk rotation Current loop Frame-dragging effect Ergosphere Plasma

  9. Ideal General Relativistic Magnetohydrodynamics (Ideal GRMHD) To investigate dynamics of the magnetic bridge between the ergosphere and the disk, we have to consider the interaction of the plasma and magnetic field near the black hole. Simplest approximation for it is given by ideal GRMHD where electric conductivity σis infinite (σ→∞).

  10. 3+1 Formalism of Ideal GRMHD Equation ～ similar to nonrelativistic ideal MHD (conservative form) Special relativistic mass density, gr (conservation of particle number) general relativistic effect Special relativistic total momentum density (equation of motion) special relativistic effect Special relativistic total energy density (equation of energy) No coupling with other Eqs. (Maxwell equations) (ideal MHD condition) (equation of state) : (Lapse function), : (shift vector) where

  11. Numerical Method • The ideal GRMHD equations are similar to those of nonrelativistic ideal MHD. Therefore, we can use the numerical techniques developed for nonrelativistic MHD calculations. In this study, we use simplified TVD method. • Simplified TVD method • This method is developed by Davis (1984) as a simplest shock capturing scheme for hydrodynamics. • Merit: We don’t need eigen-vector of Jacobian matrix of equations like primary TVD scheme. Just maximum of eigen-value of the Jacobian is used. It is easily applied for complex equations like GRMHD equations.

  12. Results of Ideal GRMHD Simulations Physical Review D 74, 044005 (Aug., 2006) http://link.aps.org/abstract/PRD/v74/e044005

  13. Initial condition of Ideal GRMHD simulation 4 t =0 Corona: hydrostatic +background pressure 2 Solid white line: Magnetic field line Color: (Specific-heat ratio: ) 0 Ergosphere -2 Magnetic bridge -4 Disk: Kepler rotation -6 Almost maximally rotating Black hole

  14. Condition of Ideal GRMHD simulation 4 Calculation region: t =0 2 Solid white line: Magnetic field line Color: Axisymmetry 0 ( 210 × 70 mesh2) -2 -4 Mirror symmetry -6

  15. Time evolution:Mass density, magnetic configuration Solid white line: Magnetic field surface Color: Arrow: velocity

  16. Solid line: Magnetic field surface Color: Arrow: Velocity

  17. Mass density, velocity, magnetic pressure at Magnetic pressure, 1 4 0 2 -1 -2 0 -3 -2 -4 -4 -5 Solid line: Magnetic field line, Arrow: Velocity vmax : 0.4c -0.6c

  18. Final stage of calculation：Density, velocity, magnetic configuration 4 Solid line: Magnetic field line Color: Arrow: Velocity vmax : 0.4c -0.6c 2 0 -2 -4

  19. Magnetic configuration of final stage:Numerical magnetic island • Ideal GRMHD: • No magnetic • reconnection • Magnetic Island:Numerical • Anti-parallel magnetic field 4 Magnetic island(Plasmoid) : Numerical 2 0 -2 -4 Solid line: Magnetic flux surface Color: Arrow: Velocity

  20. Summary of Results of Ideal GRMHD and Expected Phenomena beyond Ideal case Schematic picture of phenomena caused by the magnetic bridge near the black hole Sub-relativistic jet Initial Magnetic bridge Magnetic surface Magnetic surface Accretion disk Accretion disk Kerr black hole Kerr black hole Current loop Ergosphere Ergosphere Ideal GRMHD result

  21. Ideal GRMHD result GRMHD with finite conductivity Intermittent Jet Flare of X-ray Magnetic surface Magnetic surface Magnetic reconnection Accretion disk Accretion disk Kerr black hole Kerr black hole heating Ergosphere Ergosphere Anti-parallel magnetic field is formed Mixture of hot and cool plasma: Constant polytropic index EoSis not good approximation

  22. Development of Numerical Method for GRMHD Simulation with Finite Conductivity Fairly new topic. But no new results of physics. Only explanation of new required method and preliminary tests.

  23. Previous GRMHD Simulations= ideal GRMHD with polytropic EoS • Koide, Shibata, Kudoh 1999 • Gammie 2003 • DeVillier & Hawley 2003 • Komissarov 2004 • McKinney 2005 σ=∞， Γ=5/3, 4/3 This assumptionneglect astrophysically important effects But no GRMHD simulation with finite conductivityand more appropriate EoS.

  24. GRMHD Equations with Finite Conductivity (σGRMHD) Special relativistic mass density, gr (conservation of particle number) general relativistic effect Special relativistic total momentum density (equation of motion) special relativistic effect Special relativistic total energy density (equation of energy) (Maxwell equations) no correspondence to non-relativistic MHD (Ohm’s law with finite conductivity) conductivity

  25. GRMHD Equations with Finite Conductivity (σGRMHD) Special relativistic mass density, gr (conservation of particle number) general relativistic effect Special relativistic total momentum density (equation of motion) special relativistic effect Special relativistic total energy density (equation of energy) (Maxwell equations) (Ohm’s law with finite conductivity) ～ N. Watanabe & T. Yokoyama, ApJ 647, pp. L123-L126(astro-ph/0607285)

  26. Numerical method ofσGRMHD: Tests • Electric conductivity → finite: Explicit(before improved EoS (Equation of State)) • Recalculation of dynamics of magnetic bridge with large conductivity (σ=100c2/τ) • Electric conductivity → finite: Implicit(before improved EoS (Equation of State)) • Recalculation of dynamics of magnetic bridge with large conductivity (σ=10,000c2/τ) • Improved EoS (Electric conductivity: finite) • Recalculation of dynamics of magnetic bridge with large conductivity (σ=100c2/τ)

  27. Explicit method: Comparison of results of ideal and finite s GRMHD simulations at (no anti-parallel magnetic field) Finite s: Ideal GRMHD: Color: 4 2 0 -2 -4 Solid line: Magnetic field line, Arrow: Velocity

  28. Explicit method: Comparison of results of ideal and finite s GRMHD simulations at (no anti-parallel magnetic field) Finite s: Ideal GRMHD: 4 Stop due to numericalinstability 2 0 -2 -4 Solid line: Magnetic field line, Arrow: Velocity

  29. Implicit method: Comparison of results of explicit and implicit methods with very large conductivity at σ=104/crS Color: Implicit (simplified) Explicit (ideal) 4 2 0 -2 -4 Solid line: Magnetic field line, Arrow: Velocity

  30. Equation of State (EoS) ― Comparison between different EoS’s ― Exact improved RP : Exact (Synge 1957) Γ=4/3, 5/3: Constant polytropic index TM : Mignone et al (2005) RC : Ryu et al (2006) Ryu, Chattopadhyay, & Choi 2006

  31. Comparison of results of finite s GRMHD simulations before/after improved EoS at (explicit) Color: Improved EoS (TM) Γ＝５／３ (before improvement) 4 2 0 -2 -4 Solid line: Magnetic field line, Arrow: Velocity

  32. Summary • Ideal GRMHD: • The magnetic bridges between the ergosphere and disk around rapidly rotating black hole can not be stationary and expand explosively to form a jet. • The anti-parallel magnetic field is formed along the jet where the magnetic reconnection will take place, which may influence the jet propagation. • GRMHD with finite conductivity (σGRMHD) is required to investigate the magnetic reconnection. We showed the new numerical method of σGRMHD and test calculations for it. • Implicit method is essential.

  33. Near future plan • Development of correct implicit σGRMHD code • σGRMHD simulations of magnetic bridge between the ergosphere and disk around rapidly rotating black hole;Importance of magnetic reconnection in the mechanism of relativistic jet formation.

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