1. Relaxation of Pulsar Wind Nebula via Current-Driven Kink Instability
Yosuke Mizuno (水野 陽介)
Institute of Astronomy
National Tsing-Hua University
Collaborators
Y. Lyubarsky (Ben-Gurion Univ), K.-I. Nishikawa (NSSTC/UAH), P. E. Hardee (Univ. of Alabama, Tuscaloosa)
Mizuno et al., 2011, ApJ, 728, 90

2. Pulsar Wind Nebulae Pulsar magnetosphere Pulsar wind
Termination Shock
Pulsar wind
Pulsar wind nebula
electromagnetic fields
Synchrotron & IC radiation
Pulsar wind nebulae (PWNe) are considered as relativistically hot bubbles continuously pumped by e+-e- plasma and magnetic field emanating from pulsar
Pulsar loses rotation energy by generating highly magnetized ultra-relativistic wind
Pulsar wind terminates at a strong reverse shock (termination shock) and shocked plasma inflates a bubble with in external medium
From shocked plasma Synchrotron and Inverse-Compton radiation are observed from radio to gamma-ray band (e.g., Gaensler & Slane 2006)

3. Pulsar Wind Nebulae (obs.)
Vela
(Pavlov et al. 2001)
3C58
(Slane et al. 2004) G
(Lu et al. 2002) G
(Gaensler et al. 2002)

4. Simple Spherical Model of PWNe
Close to pulsar, energy is carried mostly by electromagnetic fields as Poynting flux
Common belief: at termination shock, wind must already be very weakly magnetized
Magnetization parameter s (ratio of Poynting to kinetic energy flux) needs to be as small as at termination shock (e.g., Rees & Gunn 1974, Kennel & Coroniti 1984)
Such low value of s is puzzling because it is not easy to invent a realistic energy conversion mechanism to reduce s to required level (s problem) (reviews by Arons 2007; Kirk et al. 2009)

5. Dependence on s to shock downstream structure
Kennel & Coroniti 1984
Postshock speed
At shock downstream
c/3
s>>1: effectively weak (magnetic energy dominated)
s<<1: significant fraction of total energy in upstream converted to thermal energy in downstream
s>>1: almost constant with relativistic speed
s<<1: velocity just after shock becomes c/3 limit, then decreasing
From radio observation of Crab nebula, expanding velocity is 2000km/s at 2pc (s~0.003)

6. Axisymmetric RMHD Simulations of PWNe
Del Zanna et al.( 2004)
Flow magnitude
Synchrotron emission map
Extensive axisymmetric RMHD simulations of PWNe show that the morphology of PWNe including jet-torus structure with s~0.01(e.g., Komissarov & Lyubarsky 2003, 2004, Del Zanna et al. 2004, 2006)
If magnetization were larger, then the nebula would be elongated by magnetic pinch effect beyond observational limits

7. Termination Shock structure
Axisymmetric RMHD simulations of PWNe
Komissarov & Lyubarsky 2003, 2004; Del Zanna et al. 2004, 2006; Bogovalov et al. 2005
Del Zanna et al. 2004
Fsin2()
sin2()
Bsin()G()
Flow magnitude
A: ultrarelativistic Pulsar wind
B: subsonic equatorial outflow
C: supersonic equatorial funnel
D: bright arch
a: termination shock front
b: rim shock
c: FMS surface

8. Constraining  in PWNe =0.03 =0.003 >0.01 required for
Smaller s, jet does not formed
=0.003
=0.03
Larger s, PWNe elongates
>0.01 required for
Jet formation
(a factor of 10 larger
than within 1D spherical MHD models)
=0.01
(Del Zanna et al. 2004)

9. Dependence on Field Structure
=0.03
b=100
b=10
B()
(Del Zanna et al. 2004)

10. Synchrotron Emission maps
X-rays
optical
=0.025, b=10
(Weisskopf et al 00)
(Hester et al 95)
=0.1, b=1
Emax is evolved with the flow
f(E)E-, E<Emax (Del Zanna et al. 2006)
(Pavlov et al 01)

11. Obliquely rotating Pulsar magnetosphere
In pulsar wind, most of energy transferred by waves, which an obliquely rotating magnetosphere excites near the light cylinder
In equatorial belt of wind, the sign of magnetic field alternates with pulsar period, forming stripes of opposite magnetic polarity (striped wind; Michel 1971, Bogovalov 1999)
Theoretical Modeling of pulsar wind suggest that most of wind energy is transported in equatorial belt (Bogovalov 1999; Spitkovsky 2006)
In the equatorial belt, magnetic dissipation of the striped wind would be a main energy conversion mechanism
Spitkovsky (2006)

12. Obliquely rotating Pulsar magnetosphere
(Kirk & Lyubarsky 01)
In pulsar wind, most of energy transferred by waves, which an obliquely rotating magnetosphere excites near the light cylinder
In equatorial belt of wind, the sign of magnetic field alternates with pulsar period, forming stripes of opposite magnetic polarity (striped wind; Michel 1971, Bogovalov 1999)

13. Obliquely rotating Pulsar magnetosphere (cont.)
Spitkovsky (2006)
Theoretical Modeling of pulsar wind suggest that most of wind energy is transported in equatorial belt (Bogovalov 1999; Spitkovsky 2006)
In the equatorial belt, magnetic dissipation of the striped wind is main energy conversion mechanism

14. Dissipation of Alternating Fields
For simple wave decay, due to relativistic time dilation, complete dissipation could occur only on a scale comparable to or larger than radius of termination shock (Lyubarsky & Kirk 2001; Kirk & Skjaeraasen 2003)
But, alternating fields can annihilate at termination shock by strong deceleration of wind via magnetic reconnection (Petri & Lyuabrsky 2007)
After waves decay via magnetic reconnection: s < 1 (~0.1)
At quantitative level, s problem is partially solved if Poynting flux is converted into plasma energy via dissipation of oscillating part of field
1D RPIC simulation with σ = 45, Γ = 20 (dissipation occurs)
Petri & Lyubarsky 2007

15. Dissipation of Alternating Fields in Far Zone of Wind
Waves decay
wind accelerates
dissipation rate ↓
proper time
proper wavelength
The wave dissipation scale is about or larger
than the termination shock radius
(Lyubarsky & Kirk 2001; Kirk & Skjæraasen 2003)
The flow sharply decelerates at the shock
dissipation of alternating fields at the termination shock

16. Magnetic Reconnection at Termination Shock of striped pulsar wind
(l: wavelength of striped wind,
g1: Lorentz factor at upstream)
Full dissipation at
Initial condition
dissipation with σ = 45, g1 = 20
1D RPIC Simulation (Pétri & Lyubarsky, A&A, 2007)

17. Energy conversion at high latitudes
At high latitudes, magnetic field does not change sign (no reconnection occurs)
Fast magnetosonic waves may transport significant amount of energy
These waves can decay relatively easily (Lyubarsky 2003) but can release only a fraction of the Poynting flux into plasma (because at high latitudes, most of energy is carried by mean magnetic field)
Even though this fraction is still not known, this fraction is less than ½ because angular distribution of Poynting flux in pulsar wind is maximum at rotational equator, where mean field is zero

18. Another Possibility: CD Kink Instability in PWNe
At quantitative level, s problem is partially solved if Poynting flux is converted into plasma energy via dissipation of oscillating part of field (Petri & Lyubarsky 2007)
But, from residual magnetic field, s still cannot be as small as required (0.1~1).
Question still remains how the residual mean field s could become extremely small (0.001~0.01): need another mechanism
Begelman (1998) proposed that problem can be solved if current-driven kink instability destroys concentric field structure in pulsar wind nebula
As first step, we perform 3D evolution of simple cylindrical model of PWNe (Begelman & Li 1992) with growing CD kink instability using 3D RMHD simulation code

19. CD Kink Instability
Well-known instability in laboratory plasma (TOKAMAK), astrophysical plasma (Sun, jet, pulsar etc).
In configurations with strong toroidal magnetic fields, current-driven (CD) kink mode (m=1) is unstable.
This instability excites large-scale helical motions that can be strongly distort or even disrupt the system
For static cylindrical force-free equilibria, well known Kurskal-Shafranov (KS) criterion
Unstable wavelengths:
l > |Bp/Bf |2pR
However, rotation and shear motion could significant affect the instability criterion
Schematic picture of CD kink instability
3D RMHD simulation of CD kink instability in helical force-free field (Mizuno et al. 2009)

20. Purpose of Study
Begelman (1998) proposed that s problem can be solved if current-driven kink instability destroys concentric field structure in pulsar wind nebula
As first step, we perform 3D evolution of simple cylindrical model of PWNe (Begelman & Li 1992) with growing CD kink instability using 3D RMHD simulation code RAISHIN

21. 4D General Relativistic MHD Equation
General relativistic equation of conservation laws and Maxwell equations:　　　
　　　　　　　∇n ( r U n ) = 0　　　 (conservation law of particle-number)
　　　　　　　∇n T mn = (conservation law of energy-momentum)
　　　　　　　∂mFnl + ∂nFlm + ∂lF mn = 0
　　　　　　　∇mF mn = - J n
Ideal MHD condition: FnmUn = 0
metric：　ds2=-a2 dt2+gij (dxi+b i dt)(dx j+b j dt)
Equation of state : p=(G-1) u
(Maxwell equations)
r : rest-mass density. p : proper gas pressure. u: internal energy. c: speed of light.
h : specific enthalpy, h =1 + u + p / r.
G: specific heat ratio.
Umu : velocity four vector. Jmu : current density four vector.
∇mn : covariant derivative. gmn : 4-metric. a : lapse function, bi : shift vector, gij : 3-metric
Tmn : energy momentum tensor, Tmn = pgmn + r h Um Un+FmsFns -gmnFlkFlk/4.
Fmn : field-strength tensor,

22. Conservative Form of GRMHD Equations (3+1 Form)
(Particle number conservation)
(Momentum conservation)
(Energy conservation)
(Induction equation)
U (conserved variables)
Fi (numerical flux)
S (source term)
√-g : determinant of 4-metric
√g : determinant of 3-metric
Detail of derivation of GRMHD equations
Anton et al. (2005) etc.

23. 3D GRMHD code RAISHIN
Mizuno et al. 2006a, astro-ph/
Mizuno et al. 2011, ApJ
RAISHIN dode utilizes conservative, high-resolution shock capturing schemes (Godunov-type scheme) to solve the 3D General Relativistic MHD equations (metric is static)
* Reconstruction: PLM (Minmod & MC slope-limiter), CENO, PPM, WENO, MP, MPWENO, WENO-Z, WENO-M, Lim03
* Riemann solver: HLL, HLLC, HLLD approximate Riemann solver
* Constrained Transport: Flux CT, Fixed Flux-CT, Upwind Flux-CT
* Time evolution: Multi-step TVD Runge-Kutta method (2nd & 3rd-order)
* Recovery step: Noble 2 variable method, Mignore-McKinney 1 variable method
* Equation of states: constant G-law EoS, variable EoS for ideal gas
Numerical Schemes

24. Ability of RAISHIN code
Multi-dimension (1D, 2D, 3D)
Special and General relativity (static metric)
Different coordinates (RMHD: Cartesian, Cylindrical, Spherical and GRMHD: Boyer-Lindquist of non-rotating or rotating BH)
Different spatial reconstruction algorithms (10)
Different approximate Riemann solver (3)
Different constrained transport schemes (3)
Different time advance algorithms (2)
Different recovery schemes (2)
Using constant G-law and variable Equation of State (Synge-type)
Parallelized by OpenMP (shared memory) and MPI (distributed memory)

25. 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, etc
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

26. Cylindrical Model of PWNe
This model (Begelman & Li 1992): quasi-static cylindrical configuration with purely toroidal magnetic field
The plasma within cylinder is relativistically hot and hoop stress is balanced by thermal pressure
Cylinder is confined on outside by non-magnetized plasma
Linear analysis shows that such configuration is unstable with respect to CD kink instability (Begelman 1998)

27. Initial Condition for Simulations
Radial profile
Toroidal field
pressure
We solve 3D RMHD equations in Cartesian coordinates
We consider hydrostatic hot plasma column containing a pure toroidal magnetic field with radius R and height Lz (magnetic hoop stress is balanced by gas pressure)
At R>1, hot plasma column is surrounded by a hot static unmagnetized medium with constant gas pressure
p0=105 r0c2 (relativistically hot, rc2 << pg), G=4/3 (adiabatic index)
Put small radial velocity perturbation
Computational domain: Cartesian box of size 6R x 6R x Lz (Lz=1R) with grid resolution of N/R,L=60
Boundary: periodic in axis direction, reflecting boundary in x, y direction
N: total number of modes, fk: random phase, ak:x,y,: random direction

28. Cylindrical Model of PWNe
Based on cylindrical model of PWNe (Begelman & Li 1992), radial gas pressure and toroidal magnetic field profiles in hot plasma column are given by
Where x=r/R, h is found for any x from equation
In this solution, magnetic hoop stress is balanced by gas pressure
At x>1, hot plasma column is surrounded by hot static unmagnetized medium with constant gas pressure
where h0 is solution of eqs at x=1

29. Results (2D gas prssure)
Case A: perturbation
N=2, fk=0, n=1 mode in x-direction, n=2 mode in y-direction
Gas pressure
Initial small velocity perturbation excites CD kink instability n=1 mode in x-direction and n=2 mode in y-direction
radial velocity increases with time in linear growth phase
At about t=6R/c, CD kink instability shifts to nonlinear phase
In nonlinear phase, two modes interact and lead to turbulence in hot plasma column
Gas pressure within column, which was initially high to balance magnetic hoop stress, decreases because hoop stress weakens

30. Results (2D magnetic field)
Case A: perturbation
N=2, fk=0, n=1 mode in x-direction, n=2 mode in y-direction
As a result of CD kink instability, magnetic loops come apart and release magnetic stress

31. Time Evolution of Volume Averaged Quantities
Ep=rhg2-p, Em=B2/2, Et=Ep+Em
Initial slow evolution in linear growth phase lasts up to t=6R/c, and is followed by a more rapid evolution in nonlinear growth phase
In nonlinear phase, rapid decrease of magnetic energy ceases about t=11R/c
While magnetic energy declines, plasma energy increases because growth of CD kink instability leads to radial velocity increases which contributes kinetic energy
Plasma energy
magnetic energy
Total energy
At about t=11R/c, increase in plasma energy nearly ceases and hot plasma column is almost relaxed
Multiple-mode (dashed lines) lead to more gradual interaction, slower development of turbulent structure, and later relaxation of hot plasma column

32. Time Evolution of s
Volume-averaged magnetization parameter s in hot plasma column (R<1)
s=B2/rh (for hot plasma definition)
Initially, volume-averaged magnetization s =0.3 in hot plasma column
In linear growth phase, s gradually decreases
After transition to nonlinear phase, s rapidly decreases because the magnetic field strongly dissipates by the turbulent motion
When CD kink instability saturates, s~0.01

33. Radial Profile
Case A
Radial profile of toroidal- and axial- averaged quantities for case A
Radial field
Toroidal field
In linear phase, Br & Bz grow, while Bf & pg decline gradually beginning from near the axis
In nonlinear phase, Bf & pg decrease rapidly, and Br & Bz increase throughout hot plasma column
At end of nonlinear phase (t~11R/c), all magnetic field components become comparable and field totally chaotic
In saturation phase, magnetized column begins slow radial expansion (relaxation)
Axial field
Gas pressure
For different initial perturbation profiles, evolutionary timescale is different but physical behavior is similar (not shown here)

34. Discussion: Elongation of PWNe
Our simulation confirm scenario envisaged by Begelman (1998)
Toroidal magnetic loops come apart, hoop stress declines, and pressure difference across the nebula is washed out in nonlinear phase of CD kink instability
For this reason, elongation of PWNe cannot be correctly estimated by axisymmetric models
Because axisymmetric models retain a concentric toroidal magnetic field geometry
To understand the morphology of PWNe correctly, we should perform 3D RMHD simulations

35. Discussion: Radiation
Radiation from Crab nebula is highly polarized along axis of nebula (e.g., Michel et al. 1991, Fesen et al. 1992)
It is indicated that the existence of ordered toroidal magnetic field in PWNe
From our simulation results, we see that even though instability eventually destroys toroidal magnetic field structure, magnetic field becomes completely chaotic only at the end of nonlinear stage of development
Therefore toroidal magnetic field should dominate in central part of nebula that are filled by newly injected plasma

36. Summery
We have investigated development of CD kink instability of a hydrostatic hot plasma column containing toroidal magnetic field as a model of PWNe
CD kink instability is excited by a small initial velocity perturbation and turbulent structure develops inside the hot plasma column
At end of nonlinear phase, hot plasma column relaxes with a slow radial expansion
Magnetization s decreases from initial valule 0.3 to 0.01
For different initial perturbation profiles, timescale is a bit different but physical behavior is same
Therefore relaxation of a hot plasma column is independent of initial perturbation profile
Our simulation confirm the scenario envisaged by Begelman (1998)

37. Crab Nebula