1. Yosuke Mizuno Institute of Astronomy National Tsing-Hua University
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 ~ 3M9 hour
For the TeV emission to escape pair creation Γem>50 is required (Begelman, Fabian & Rees 2008)
But PKS , 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)
PKS (Aharonian et al. 2007)
See also Mrk501, PKS
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. Applicability of Hydrodynamic Approximation
To apply hydrodynamic approximation, we need the condition:
Spatial scale >> mean free path
Time scale >> collision time
These are not necessarily satisfied in many astrophysical plasmas
E.g., solar corona, galactic halo, cluster of galaxies etc.
But in plasmas with magnetic field, the effective mean free path is given by the ion Larmor radius.
Hence if the size of phenomenon is much larger than the ion Larmor radius, hydrodynamic approximation can be used.

10. Applicability of MHD Approximation
MHD describe macroscopic behavior of plasmas if
Spatial scale >> ion Larmor radius
Time scale >> ion Larmor period
But MHD can not treat
Particle acceleration
Origin of resistivity
Electromagnetic waves

11. 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

12. 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)

13. Difference in Ideal & Resistive RMHD
Evolution of EM fields
ideal
resistive
unnecessary to solve Ampere’s law in ideal MHD
E field can be determined directly from Ohm’s law
with Ohm’s law
2 additional equations should be solved
Disadvantage of RRMHD
Courant condition
∇・E=4πq should be satisfied as well as ∇・B=0

14. Basic Equations for Ideal / Resistive RMHD
Ideal RMHD
Hyperbolic equations
Source term
Stiff term

15. 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

16. 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

17. Basics of Numerical RMHD Code
Non-conservative form (De Villier & Hawley (2003), Anninos et al.(2005))
U=U(P) - conserved variables,
P – primitive variables
F- numerical flux of U
where
Merit:
they solve the internal energy equation rather than energy equation. → advantage in regions where the internal energy small compared to total energy (such as supersonic flow)
Recover of primitive variables are fairly straightforward
Demerit:
It can not applied high resolution shock-capturing method and artificial viscosity must be used for handling discontinuities

18. 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)

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

20. Difficulty of Handling Shock Wave
Numerical oscillation (overshoot)
Diffuse shock surface
Time evolution of wave equation with discontinuity using Lax-Wendroff scheme (2nd order)
initial
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)

21. 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
Fi-1/2 Ui
Fi+1/2
Pi-1 Pi
Pi+1

22. Reconstruction Cell-centered variables (Pi)
→ right and left side of Cell-interface variables(PLi+1/2, PRi+1/2)
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)
MPWENO5 (Balsara & Shu 2000)
Logarithmic 3rd order limiter (Cada & Torrilhon 2009)
Piecewise linear interpolation
PLi+1/2
PRi+1/2
Pni-1
Pni
Pni+1

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

24. Approximate Riemann Solver
HLL Approximate Riemann solver: single state in Riemann fan
HLLC Approximate Riemann solver: two-state in Riemann fan (Mignone & Bodo 2006, Honkkila & Janhunen 2007)
HLLD Approximate Riemann solver: six-state in Riemann fan (Mignone et al. 2009)
Roe-type full wave decomposition Riemann solver (Anton et al. 2010)

25. Wave speed (for ideal RMHD)
To calculate numerical flux at each cell-boundary via Riemann solver, we need to know wave speed in each directions
lE=vi, entropy wave
Alfven waves
Magneto-acoustic waves are found from the quartic equation
Some simple estimation for fast magnetosonic wave
=> Leismann et al. (2005), no numerical iteration

26. Constrained Transport (for Ideal MHD)
Differential Equations
The evolution equation can keep divergence free magnetic field
If treat the induction equation as all other conservation laws, it can not maintain divergence free magnetic field
→ We need spatial treatment for magnetic field evolution
Constrained transport scheme
Evans & Hawley’s Constrained Transport (need staggered mesh)
Toth’s constrained transport (flux-CT) (Toth 2000)
Fixed Flux-CT, Upwind Flux-CT (Gardiner & Stone 2005, 2007)
Other method
Diffusive cleaning (GLM formulation) (better method for AMR or RRMHD)

27. Flux interpolated Constrained Transport
２D case
Toth (2000)
Use the “modified flux” f that is such a linear combination of normal fluxes at neighbouring interfaces that the “corner-centred” numerical representation of divB is kept invariant during integration.
k+1/2
k-1/2
j-1/2
j+1/2

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

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

30. 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

31. Time Evolution (ideal RMHD)
System of Conservation Equations
We use multistep TVD Runge-Kutta method for time advance of conservation equations (RK2: 2nd-order, RK3: 3rd-order in time)
RK2, RK3: first step
RK2: second step (a=2, b=1)
RK3: second and third step (a=4, b=3)

32. 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

33. Recovery step
The GRMHD code require a calculation of primitive variables from conservative variables.
The forward transformation (primitive → conserved) has a close-form solution, but the inverse transformation (conserved → primitive) requires the solution of a set of five nonlinear equations
Method
Noble’s 2D method (Noble et al. 2005)
Mignone & McKinney’s method (Mignone & McKinney 2007)

34. Noble’s 2D method
Conserved quantities(D,S,t,B) → primitive variables (r,p,v,B)
Solve two-algebraic equations for two independent variables W≡hg2 and v2 by using 2-variable Newton-Raphson iteration method
W and v2 →primitive variables r p, and v
Mignone & McKinney (2007): Implemented from Noble’s method for variable EoS

35. 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)

36. 2. 1. Numerical Tests ideal RMHD

37. 1D Relativistic MHD Shock-Tube
Exact solution: Giacomazzo & Rezzolla (2006)

38. 1D Relativistic MHD Shock-Tube
Balsara Test1 (Balsara 2001)
Mizuno et al. 2006
The results show good agreement of the exact solution calculated by Giacommazo & Rezzolla (2006).
Minmod slope-limiter and CENO reconstructions are more diffusive than the MC slope-limiter and PPM reconstructions.
Although MC slope limiter and PPM reconstructions can resolve the discontinuities sharply, some small oscillations are seen at the discontinuities. FR SR SS FR CD
Black: exact solution, Blue: MC-limiter, Light blue: minmod-limiter, Orange: CENO, red: PPM
400 computational zones

39. Advection of Magnetic Field Loop
No advection
Advection of a weak magnetic field loop in an uniform velocity field
2D: (vx, vy)=(0.6,0.3)
3D: (vx,vy,vz)=(0.3,0.3,0.6)
Periodic boundary in all direction
Run until return to initial position in advection case B2
Advection
Volume-averaged magnetic energy density (2D) 3D B2
Nx=512
256
128
Advection
No advection

40. Cylindrical Explosion
Cylindrical explosion in magnetized medium
(propagation of strong shock waves)
Pc=1.0, rc=0.01 (R<1.0)
Pe=5x10-4, re=10-4
Bx=0.1 (uniform)

41. Numerical Tests

42. 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= => vA=0.25c, ideal EoS with G=2
Using high conductivity s=105

43. 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

44. 1D Self-Similar Current Sheet Test
Assumption: Magnetic pressure << gas pressure
Magnetic field configuration: B=[0, By(x,t),0], By(x,t) changes sign within a thin current layer (thickness Dl)
The evolution of thin current layer is a slow diffusive expansion due to resistivity and described by diffusion equations
As the thickness of the layer becomes much larger than Dl, the expansion becomes self-similar:
c=t/x2, erf: error function
Test simulation:
Initial solution at t=1 with p=50, r=1, E=v=0, s=100 with G=2

45. 1D Self-Similar Current Sheet Test
dotted & dashed: analytical solution at t=1 & 10
Solid: numerical solution at t=10
Numerical Simulation shows good agreement with exact solutions with moderate conductivity regime

46. 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)

47. 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

48. 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.

49. 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 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

50. 2D Cylindrical Explosion
Cylindrical blast wave expanding into uniform magnetic field
Standard test for ideal RMHD code
No exact solution in multi-dimensional tests
Initial Condition
Density, pressure:
R<0.8, p=1, r=0.01
0.8<R<1.0 decrease exponentially to ambient gas
R>1.0 p=r=0.001
Magnetic field: uniform in x-direction with B=0.05
Using different conductivity s=0-105
Using ideal EoS with G=4/3 and TM EoS

51. Global Structure in Cylindrical Explosion
qualitatively similar to ideal RMHD results
No different between ideal EoS and TM EoS

52. 1D cut of Cylindrical Explosion
Purple dash-two-dotted: s=0
Green dash-dotted: s=10
Red dashed: s=102
Blue dotted: s=103
Black solid: s=105
At high conductivity (s > 103) no difference (recovers ideal MHD solution)
conductivity ↓ maximum gas and mag pressure ↓
No mag pressure increase for s=0

53. 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

54. Growth Rate of KHI Initial linear growth with almost same growth rate
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

55. 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

56. 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

57. 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

58. 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:

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

60. 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

61. 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