The Role of The Equation of State in Resistive Relativistic

Magnetohydrodynamics

Yosuke MizunoInstitute 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 >> 1– Relativistic jets/ejecta/wind/blast waves (shocks) in AGNs, GRBs, Pulsars

• Thermal energy >> rest-mass energy– Plasma temperature >> ion rest mass energy– p/c2 ~ kBT/mc2 >> 1 – GRBs, magnetar flare?, Pulsar wind nebulae

• Magnetic energy >> rest-mass energy– Magnetization parameter >> 1– Poyniting to kinetic energy ratio = B2/4c22

– Pulsars magnetosphere, Magnetars

• Gravitational energy >> rest-mass energy– GMm/rmc2 = rg/r > 1– Black hole, Neutron star

• Radiation energy >> rest-mass energy– E’r /c2 >>1– Supercritical accretion flow

Relativistic Jets• 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

Radio observation of M87 jet

Relativistic Jets in Universe

Mirabel & Rodoriguez 1998

Ultra-Fast TeV Flare in Blazars

PKS2155-304 (Aharonian et al. 2007)See also Mrk501, PKS1222+21

• 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,

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 EqsIdeal 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 ( => infinity)

Charge current disappear in the Ohm’s law(degeneracy of equations, EM wave is decupled)

Numerical Integration Resistive RMHD

Hyperbolic equations

Source term

Stiff term

Constraint

Solve Relativistic Resistive MHD equations by taking care of 1. stiff equations appeared in Ampere’s law2. constraints ( no monopole, Gauss’s law)3. Courant conditions (the largest characteristic wave speed is always light speed.)

For Numerical SimulationsBasic Equations for RRMHD Physical quantities

Primitive VariablesConserved Variables

Flux

Source term

Operator splitting (Strang’s method) to divide for stiff term

U=U(P) - conserved variables,P – primitive variables F- numerical flux of U, S - source of U

System of Conservation Equations

Basics of Numerical RMHD CodeConservative form

Merit: • Numerically well maintain conserved variables• High resolution shock-capturing method (Godonuv scheme) can be applied to RMHD equationsDemerit: • 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

flux

Conservative form of wave equation

Finite difference

FTCS scheme

Upwind scheme

Lax-Wendroff scheme

Difficulty of Handling Shock Wave

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)

initial

Diffuse shock surface

• Time evolution of wave equation with discontinuity using Lax-Wendroff scheme (2nd order)

Numerical oscillation (overshoot)

Flow Chart for Calculation

Primitive VariablesConserved Variables

Flux

Pi Pi+1Pi-1

Fi-1/2 Fi+1/2Ui

1. 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

ReconstructionCell-centered variables (Pi)→ right and left side of Cell-interface variables(PL

i+1/2, PRi+1/2)

PLi+1/2 PR

i+1/2

Piecewise linear interpolation

• 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, 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)

Pni

Pni+1

Pni-1

Approximate Riemann Solver

Primitive VariablesConserved Variables

Flux

Pi Pi+1Pi-1

Fi-1/2 Fi+1/2Ui

Hyperbolic equations

x

t

HLL flux

RL: fastest characteristic speed

RL

If L >0 FHLL=FL

L < 0 < R , FHLL=FM

R < 0 FHLL=FR

L R

M

Difficulty of RRMHD1. Constraint

should be satisfied both

constraint numerically

2. Ampere’s law

Equation becomes stiff at high conductivity

ConstraintsApproaching Divergence cleaning method (Dedner et al. 2002, Komissarov 2007)

Introduce additional field & (for numerical noise)advect & decay in time

Stiff EquationProblem comes from difference between dynamical time scale and diffusive time scale => analytical solution

Ampere’s lawdiffusion (stiff) term

Komissarov (2007)

Analytical solution

Operator splitting method

Hyperbolic + source termSolve by HLL method

source term (stiff part)Solve (ordinary differential) eqaution

Flow Chart for Calculation (RRMHD)

Step1: integrate diffusion term in half-time step

Step2: integrate advection term in half-time step

Step3: integrate advection term in full-time step

Step4: integrate diffusion term in full-time step

Un=(En+1/2, Bn)

(En+1, Bn+1)=Un+1

Strang Splitting Method

General (Approximate) EoS• In the theory of relativistic perfect single gases, specific enthalpy is a function of temperature alone (Synge 1957)

temperature p/K2, K3: the order 2 and 3 of modified Bessel functions

• Constant -law EoS (ideal EoS) :• : constant specific heat ratio

• Taub’s fundamental inequality(Taub 1948)

→ 0, eq → 5/3, → ∞, eq → 4/3

• TM EoS (approximate Synge’s EoS) (Mignone et al. 2005)

Mignone & McKinney 2007

Solid: Synge EoSDotted: ideal + =5/3Dashed: ideal+ =4/3Dash-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=B ０ , vx=0, k: wave number, A: amplitude of wave

=p=1, B0=0.46188 => vA=0.25c, ideal EoS with =2

Using high conductivity=105

1D CP Alfven wave propagation test

Solid: exact solutionDotted: Nx=50Dashed: Nx=100Dash-dotted: Nx=200

Numerical results at t=4 (one Alfven crossing time)

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

• (L, pL, ByL) = (1, 1, 0.5), (R, pR, By

R)=(0.125, 0.1, -0.5)

• Ideal EoS with =2Orange solid: =0Green dash-two-dotted: =10Red dash-dotted: =102

Purple dashed: =103

Blue dotted: =105

Black solid: exact solution in ideal RMHD

Smooth change from a wave-like solution (=0) to ideal-MHD solution (=105)

1D Shock-Tube Test (Balsara 2)

• Aim: check the effect of choosing EoS in RRMHD

• Balsara Test 2

• Using ideal EoS (=5/3) & approximate TM EoS

• Changing conductivity from =0 to 103

• Mildly relativistic blast wave propagates with 1.3 < < 1.4

1D Shock-Tube Test (Balsara 2)

Purple dash-two-dotted: =0Green dash-dotted: =10Red dashed: =102

Blue dotted: =103

Black solid: exact solution in ideal RMHD

FRSR

CD

SS & FS

The solutions: Fast Rarefaction, Slow Rarefaction, Contact Discontinuity, Slow Shock and Fast Shock.

1D Shock-Tube Test (Balsara 2)

FRSR

CD

SS & FS

Purple dash-two-dotted: =0Green dash-dotted: =10Red dashed: =102

Blue dotted: =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:

• Uniform gas pressure p=1.0

• Density:=1.0 in the region vsh=0.5, =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 layervsh=0.5 => relative =2.29

p=0.5, t=1.0

A0=0.1,=0.1

Growth Rate of KHI

Purple dash-two-dotted: =0Green dash-dotted: =10Red dashed: =102

Blue dotted: =103

Black solid: =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

Amplitude of perturbation

Volume-averaged Poloidal field

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 modelDensity & gas pressure: Uniform density & gas

pressure outside current sheet, b=pb=0.1

Magnetic field:

Current:

Resistivity (anomalous resistivity in r<r):

b=1/b=10-3, 0=1.0, r=0.8 Electric field:

Global Structure of Relativistic MRPlasmoid

Slow shock

Strong current flow

t=100

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 rate

Reconnection outflow speed

Solid: ideal EoSDashed: TM EoS

time

Magnetic energy

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