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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 • 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 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 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 Relativistic Jets in Universe Mirabel & Rodoriguez 1998 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 ~ 3M9 hour • For the TeV emission to escape pair creation Γem>50 is required (Begelman, PKS2155-304 (Aharonian et al. 2007) See also Mrk501, PKS1222+21 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) Giannios et al.(2009) 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 Spitkovsky (2006) Magnetar Flares • May be triggered by magnetic reconnection at equatorial current sheet 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. 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 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) Numerical Integration Resistive RMHD Constraint Hyperbolic equations Solve Relativistic Resistive MHD equations by taking care of Source term Stiff term 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.) For Numerical Simulations Basic Equations for RRMHD Physical quantities Primitive Variables Conserved Variables Flux Source term Operator splitting (Strang’s method) to divide for stiff term 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) Finite Difference (Volume) Method Conservative form of wave equation Finite difference flux FTCS scheme Upwind scheme Lax-Wendroff scheme Difficulty of Handling Shock Wave Numerical oscillation (overshoot) initial Diffuse shock surface • Time evolution of wave equation with discontinuity using Lax-Wendroff scheme (2nd order) 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) Flow Chart for Calculation 1. Reconstruction (Pn : cell-center to cell-surface) 2. Calculation of Flux at cell-surface Primitive Variables Conserved Variables Flux 3. Integrate hyperbolic equations => Un+1 Fi-1/2 Ui Fi+1/2 Pi-1 Pi 4. Integrate stiff term (E field) Pi+1 5. Convert from Un+1 to Pn+1 Reconstruction 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 Pn i Pni+1 • Minmod & MC Slope-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, WENOM (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) Approximate Riemann Solver lR, lL: fastest characteristic speed Primitive Variables Conserved Variables Flux HLL flux Hyperbolic equations lL lR t M Fi-1/2 Ui Fi+1/2 Pi-1 Pi L Pi+1 R x If lL >0 FHLL=FL lL < 0 < lR , FHLL=FM lR < 0 FHLL=FR Difficulty of RRMHD 1. Constraint should be satisfied both constraint numerically 2. Ampere’s law Equation becomes stiff at high conductivity Constraints Approaching Divergence cleaning method (Dedner et al. 2002, Komissarov 2007) Introduce additional field F & Y (for numerical noise) advect & decay in time Komissarov (2007) Stiff Equation Problem comes from difference between dynamical time scale and diffusive time scale => analytical solution Ampere’s law Operator splitting method diffusion (stiff) term Hyperbolic + source term Solve by HLL method Analytical solution source term (stiff part) Solve (ordinary differential) eqaution 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 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 • TM EoS (approximate Synge’s EoS) (Mignone et al. 2005) Solid: Synge EoS Dotted: ideal + G=5/3 Dashed: ideal+ G=4/3 Dash-dotted: TM EoS c/sqrt(3) Numerical Tests 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 r =p=1, B0=0.46188 => vA=0.25c, ideal EoS with G=2 Using high conductivity s=105 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 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 wavelike solution (s=0) to idealMHD solution (s=105) 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 1D Shock-Tube Test (Balsara 2) SS & FS FR CD 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. 1D Shock-Tube Test (Balsara 2) SS & FS FR SR CD 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 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 2D Kelvin-Helmholtz Instability • Linear and nonlinear growth of 2D Kelvin-Helmholtz instability (KHI) & magnetic field amplification via KHI Initial condition • Shear velocity profile: a=0.01, characteristic thickness of shear layer vsh=0.5 => relative g=2.29 • 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: mp=0.5, mt=1.0 • Single mode perturbation: • Simulation box: -0.5 < x < 0.5, -1 < y < 1 A =0.1, a=0.1 0 Growth Rate of KHI Amplitude of perturbation Volume-averaged Poloidal field Purple dash-two-dotted: s=0 Green dash-dotted: s=10 Red dashed: s=102 Blue dotted: s=103 Black solid: s=105 • 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 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 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 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 2D Relativistic Magnetic Reconnection • Consider Pestchek-type reconnection • Initial condition: Harris-like model Density & gas pressure: Uniform density & gas pressure outside current sheet, rb=pb=0.1 Magnetic field: Current: Resistivity (anomalous resistivity in r<rh): Electric field: hb=1/sb=10-3, h0=1.0, rh=0.8 Global Structure of Relativistic MR Plasmoid Slow shock Strong current flow t=100 Time Evolution of Relativistic MR • Outflow gradually accelerates and saturates t~60 with vx~0.8c Reconnection outflow speed Solid: ideal EoS Dashed: TM EoS • TM EoS case slightly faster than ideal EoS case • Magnetic energy converted to thermal and kinetic energies (acceleration of outflow) Magnetic energy • TM EoS case has larger reconnection rate than ideal EoS. Reconnection rate time • Different EoSs lead to a quantitative difference in relativistic magnetic reconnection 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