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The Role of The Equation of State in Resistive Relativistic Magnetohydrodynamics

The Role of The Equation of State in Resistive Relativistic Magnetohydrodynamics. Yosuke Mizuno Institute of Astronomy National Tsing-Hua University. Mizuno 2013, ApJS, 205, 7. ASIAA CompAS Seminar, March 19, 2013. Contents. Introduction: Relativistic Objects, Magnetic reconnection

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The Role of The Equation of State in Resistive Relativistic Magnetohydrodynamics

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  1. The Role of The Equation of State in Resistive Relativistic Magnetohydrodynamics Yosuke Mizuno Institute of Astronomy National Tsing-Hua University Mizuno 2013, ApJS, 205, 7 ASIAA CompAS Seminar, March 19, 2013

  2. Contents • Introduction: Relativistic Objects, Magnetic reconnection • Difference between Ideal RMHD and resistive RMHD • How to solve RRMHD equations numerically • General Equations of States • Test simulation results (code capability in RRMHD, effect of EoS) • Summery

  3. Relativistic Regime • Kinetic energy >> rest-mass energy • Fluid velocity ~ light speed • Lorentz factor g>> 1 • Relativistic jets/ejecta/wind/blast waves (shocks) in AGNs, GRBs, Pulsars • Thermal energy >> rest-mass energy • Plasma temperature >> ion rest mass energy • p/r c2~ kBT/mc2 >> 1 • GRBs, magnetar flare?, Pulsar wind nebulae • Magnetic energy >> rest-mass energy • Magnetization parameter s >> 1 • s = Poyniting to kinetic energy ratio = B2/4pr c2g2 • Pulsars magnetosphere, Magnetars • Gravitational energy >> rest-mass energy • GMm/rmc2 = rg/r > 1 • Black hole, Neutron star • Radiation energy >> rest-mass energy • E’r /rc2>>1 • Supercritical accretion flow

  4. Relativistic Jets Radio observation of M87 jet • Relativistic jets: outflow of highly collimated plasma • Microquasars, Active Galactic Nuclei, Gamma-Ray Bursts, Jet velocity ～c • Generic systems: Compact object（White Dwarf, Neutron Star, Black Hole）+ Accretion Disk • Key Issues of Relativistic Jets • Acceleration & Collimation • Propagation & Stability • Modeling for Jet Production • Magnetohydrodynamics (MHD) • Relativity (SR or GR) • Modeling of Jet Emission • Particle Acceleration • Radiation mechanism

  5. Relativistic Jets in Universe Mirabel & Rodoriguez 1998

  6. Ultra-Fast TeV Flare in Blazars • Ultra-Fast TeV flares are observed in some Blazars. • Vary on timescale as sort as • tv~3min << Rs/c ~ 3M9hour • For the TeV emission to escape pair creation Γem>50is required (Begelman, Fabian & Rees 2008) • But PKS 2155-304, Mrk 501 show “moderately” superluminal ejections (vapp ~several c) • Emitter must be compact and extremely fast • Model for the Fast TeV flaring • Internal: Magnetic Reconnection inside jet (Giannios et al. 2009) • External: Recollimation shock (Bromberg & Levinson 2009) PKS2155-304 (Aharonian et al. 2007) See also Mrk501, PKS1222+21 Giannios et al.(2009)

  7. Magnetic Reconnection in Relativistic Astrophysical Objects • Pulsar Magnetosphere & Striped pulsar wind • obliquely rotating magnetosphere forms stripes of opposite magnetic polarity in equatorial belt • magnetic dissipation via magnetic reconnection would be main energy conversion mechanism • Magnetar Flares • May be triggered by magnetic reconnection at equatorial current sheet Spitkovsky (2006)

  8. Purpose of Study • Quite often numerical simulations using ideal RMHD exhibit violent magnetic reconnection. • The magnetic reconnection observed in ideal RMHD simulations is due to purely numerical resistivity, occurs as a result of truncation errors • Fully depends on the numerical scheme and resolution. • Therefore, to allow the control of magnetic reconnection according to a physical model of resistivity, numerical codes solving the resistive RMHD (RRMHD) equations are highly desirable. • We have newly developed RRMHD code and investigated the role of the equation of state in RRMHD regime.

  9. Ideal / Resistive RMHD Eqs Ideal RMHD Resistive RMHD Solve 11 equations (8 in ideal MHD) Need a closure relation between J and E => Ohm’s law

  10. Ohm’s law • Relativistic Ohm’s law (Blackman & Field 1993 etc.) isotropic diffusion in comoving frame (most simple one) Lorentz transformation in lab frame Relativistic Ohm’s law with istoropic diffusion • ideal MHD limit (s => infinity) Charge current disappear in the Ohm’s law (degeneracy of equations, EM wave is decupled)

  11. Numerical Integration Resistive RMHD Constraint Hyperbolic equations Solve Relativistic Resistive MHD equations by taking care of 1. stiff equations appeared in Ampere’s law 2. constraints ( no monopole, Gauss’s law) 3. Courant conditions (the largest characteristic wave speed is always light speed.) Source term Stiff term

  12. For Numerical Simulations Physical quantities Basic Equations for RRMHD Primitive Variables Conserved Variables Flux Source term Operator splitting (Strang’s method) to divide for stiff term

  13. Basics of Numerical RMHD Code Conservative form System of Conservation Equations U=U(P) - conserved variables, P – primitive variables F- numerical flux of U, S - source of U • Merit: • Numerically well maintain conserved variables • High resolution shock-capturing method (Godonuv scheme) can be applied to RMHD equations • Demerit: • These schemes must recover primitive variables P by numerically solving the system of equations after each step (because the schemes evolve conservative variables U)

  14. Finite Difference (Volume) Method Conservative form of wave equation flux Finite difference FTCS scheme Upwind scheme Lax-Wendroff scheme

  15. Difficulty of Handling Shock Wave Numerical oscillation (overshoot) Diffuse shock surface In numerical hydrodynamic simulations, we need • sharp shock structure (less diffusivity around discontinuity) • no numerical oscillation around discontinuity • higher-order resolution at smooth region • handling extreme case (strong shock, strong magnetic field, high Lorentz factor) • Divergence-free magnetic field (MHD) • Time evolution of wave equation with discontinuity using Lax-Wendroff scheme (2nd order) initial

  16. Flow Chart for Calculation • Reconstruction • (Pn : cell-center to cell-surface) • 2. Calculation of Flux at cell-surface • 3. Integrate hyperbolic equations => Un+1 • 4. Integrate stiff term (E field) • 5. Convert from Un+1 to Pn+1 Primitive Variables Conserved Variables Flux Ui Fi-1/2 Fi+1/2 Pi-1 Pi+1 Pi

  17. Reconstruction • Minmod & MCSlope-limited Piecewise linear Method • 2nd order at smooth region • Convex ENO (Liu & Osher 1998) • 3rd order at smooth region • Piecewise Parabolic Method (Marti & Muller 1996) • 4th order at smooth region • Weighted ENO, WENO-Z, WENO-M (Jiang & Shu 1996; Borges et al. 2008) • 5th order at smooth region • Monotonicity Preserving (Suresh & Huynh 1997) • 5th order at smooth region • MPWENO5 (Balsara & Shu 2000) • Logarithmic 3rd order limiter (Cada & Torrilhon 2009) Cell-centered variables (Pi) → right and left side of Cell-interface variables(PLi+1/2, PRi+1/2) Piecewise linear interpolation PLi+1/2 PRi+1/2 Pni-1 Pni Pni+1

  18. Approximate Riemann Solver lR, lL: fastest characteristic speed Primitive Variables Conserved Variables Flux HLL flux Hyperbolic equations lL t lR M Ui Fi-1/2 Fi+1/2 R L x Pi-1 Pi+1 Pi If lL >0 FHLL=FL lL < 0 < lR , FHLL=FM lR < 0 FHLL=FR

  19. Difficulty of RRMHD 1. Constraint should be satisfied bothconstraint numerically 2. Ampere’s law Equation becomes stiff at high conductivity

  20. Constraints Approaching Divergence cleaning method (Dedner et al. 2002, Komissarov 2007) Introduce additional field F & Y (for numerical noise) advect & decay in time

  21. Stiff Equation Komissarov (2007) Problem comes from difference between dynamical time scale and diffusive time scale => analytical solution Ampere’s law diffusion (stiff) term Operator splitting method Hyperbolic + source term Solve by HLL method Analytical solution source term (stiff part) Solve (ordinary differential) eqaution

  22. Flow Chart for Calculation (RRMHD) Strang Splitting Method Step1: integrate diffusion term in half-time step Step2: integrate advection term in half-time step Un=(En+1/2, Bn) Step3: integrate advection term in full-time step Step4: integrate diffusion term in full-time step (En+1, Bn+1)=Un+1

  23. General (Approximate) EoS Mignone & McKinney 2007 • In the theory of relativistic perfect single gases, specific enthalpy is a function of temperature alone (Synge 1957) Q: temperature p/r K2, K3: the order 2 and 3 of modified Bessel functions • Constant G-law EoS (ideal EoS) : • G: constant specific heat ratio • Taub’s fundamental inequality(Taub 1948) Q → 0, Geq → 5/3, Q → ∞, Geq → 4/3 Solid: Synge EoS Dotted: ideal + G=5/3 Dashed: ideal+ G=4/3 Dash-dotted: TM EoS • TM EoS (approximate Synge’s EoS) • (Mignone et al. 2005) c/sqrt(3)

  24. Numerical Tests

  25. 1D CP Alfven wave propagation test • Aim: Recover of ideal RMHD regime in high conductivity • Propagation of large amplitude circular-polarized Alfven wave along uniform magnetic field • Exact solution: Del Zanna et al.(2007) in ideal RMHD limit Bx=B０, vx=0, k: wave number, zA: amplitude of wave • =p=1, B0=0.46188 => vA=0.25c, ideal EoS with G=2 Using high conductivity s=105

  26. 1D CP Alfven wave propagation test Numerical results at t=4 (one Alfven crossing time) Solid: exact solution Dotted: Nx=50 Dashed: Nx=100 Dash-dotted: Nx=200 New RRMHD code reproduces ideal RMHD solution when conductivity is high L1 norm errors of magnetic field By almost 2nd order accuracy

  27. 1D Shock-Tube Test (Brio & Wu) • Aim: Check the effect of resistivity (conductivity) • Simple MHD version of Brio & Wu test • (rL, pL, ByL) = (1, 1, 0.5), (rR, pR, ByR)=(0.125, 0.1, -0.5) • Ideal EoS with G=2 Orange solid: s=0 Green dash-two-dotted: s=10 Red dash-dotted: s=102 Purple dashed: s=103 Blue dotted: s=105 Black solid: exact solution in ideal RMHD Smooth change from a wave-like solution (s=0) to ideal-MHD solution (s=105)

  28. 1D Shock-Tube Test (Balsara 2) • Aim: check the effect of choosing EoS in RRMHD • Balsara Test 2 • Using ideal EoS (G=5/3) & approximate TM EoS • Changing conductivity from s=0 to 103 • Mildly relativistic blast wave propagates with 1.3 < g < 1.4

  29. 1D Shock-Tube Test (Balsara 2) SS & FS CD FR SR Purple dash-two-dotted: s=0 Green dash-dotted: s=10 Red dashed: s=102 Blue dotted: s=103 Black solid: exact solution in ideal RMHD The solutions: Fast Rarefaction, Slow Rarefaction, Contact Discontinuity, Slow Shock and Fast Shock.

  30. 1D Shock-Tube Test (Balsara 2) SS & FS FR CD SR • The solutions are same but quantitatively different. • rarefaction waves and shocks propagate with smaller velocities • <= lower sound speed in TM EoSs relatives to overestimated sound speed in ideal EoS • these properties are consistent with in ideal RMHD case Purple dash-two-dotted: s=0 Green dash-dotted: s=10 Red dashed: s=102 Blue dotted: s=103 Black solid: exact solution in ideal RMHD

  31. 2D Kelvin-Helmholtz Instability • Linear and nonlinear growth of 2D Kelvin-Helmholtz instability (KHI) & magnetic field amplification via KHI Initial condition • Shear velocity profile: • Uniform gas pressure p=1.0 • Density: r=1.0 in the region vsh=0.5, r=10-2 in the region vsh=-0.5 • Magnetic Field: • Single mode perturbation: • Simulation box: -0.5 < x < 0.5, -1 < y < 1 a=0.01, characteristic thickness of shear layer vsh=0.5 => relative g=2.29 mp=0.5, mt=1.0 A0=0.1, a=0.1

  32. Growth Rate of KHI Amplitude of perturbation Volume-averaged Poloidal field • Initial linear growth with almost same growth rate • Maximum amplitude; transition from linear to nonlinear • Poloidal field amplification via stretching due to main vortex developed by KHI • Larger poloidal field amplification occurs for TM EoS than for ideal EoS Purple dash-two-dotted: s=0 Green dash-dotted: s=10 Red dashed: s=102 Blue dotted: s=103 Black solid: s=105

  33. 2D KHI Global Structure (ideal EoS) • Formation of main vortex by growth of KHI in linear growth phase • secondary vortex? • main vortex is distorted and stretched in nonlinear phase • B-field amplified by shear in vortex in linear and stretching in nonlinear

  34. 2D KHI Global Structure (TM EoS) • Formation of main vortex by growth of KHI in linear growth phase • no secondary vortex • main vortex is distorted and stretched in nonlinear phase • vortex becomes strongly elongated in nonlinear phase • Created structure is very different in ideal and TM EoSs

  35. Field Amplification in KHI • Field amplification structure for different conductivities • Conductivity low, magnetic field amplification is weaker • Field amplification is a result of fluid motion in the vortex • B-field follows fluid motion, like ideal MHD, strongly twisted in high conductivity • conductivity decline, B-field is no longer strongly coupled to the fluid motion • Therefore B-field is not strongly twisted

  36. 2D Relativistic Magnetic Reconnection • Consider Pestchek-type reconnection • Initial condition: Harris-like model Uniform density & gas pressure outside current sheet, rb=pb=0.1 Density & gas pressure: Magnetic field: Current: Resistivity (anomalous resistivity in r<rh): hb=1/sb=10-3, h0=1.0, rh=0.8 Electric field:

  37. Global Structure of Relativistic MR Plasmoid t=100 Slow shock Strong current flow

  38. Time Evolution of Relativistic MR • Outflow gradually accelerates and saturates t~60 with vx~0.8c • TM EoS case slightly faster than ideal EoS case • Magnetic energy converted to thermal and kinetic energies (acceleration of outflow) • TM EoS case has larger reconnection rate than ideal EoS. • Different EoSs lead to a quantitative difference in relativistic magnetic reconnection Reconnection outflow speed Solid: ideal EoS Dashed: TM EoS Magnetic energy Reconnection rate time

  39. Summary • In 1D tests, new RRMHD code is stable and reproduces ideal RMHD solutions when the conductivity is high. • 1D shock tube tests show results obtained from approximate EoS are considerably different from ideal EoS. • In KHI tests, growth rate of KHI is independent of the conductivity • But magnetic field amplification via stretching of the main vortex and nonlinear behavior strongly depends on the conductivity and choice of EoSs • In reconnection test, approximate EoS case resulted in a faster reconnection outflow and larger reconnection rate than ideal EoS case

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