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