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
Pulsar Wind Nebulae
Pulsar
magneto
sphere
Termination Shock
Pulsar
wind
e+ , e - , (ions?)
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)
Pulsar Wind
Nebulae (obs.)
Vela
(Pavlov et al. 2001)
3C58
(Slane et al. 2004)
G54.1+0.3
(Lu et al. 2002)
G320.4-1.2
(Gaensler et al. 2002)
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 0.001-0.01 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)
Dependence on s to shock downstream
Kennel & Coroniti 1984
structure
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)
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
Constraining s in PWNe
Smaller s, jet
does not formed
s=0.03
s=0.003
Larger s, PWNe
elongates
s>0.01 required for
Jet formation
(a factor of 10 larger
than within 1D spherical
MHD models)
s=0.01
(Del Zanna et al. 2004)
Obliquely rotating Pulsar magnetosphere
Spitkovsky (2006)
• 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
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
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 currentdriven 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
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 KurskalShafranov (KS) criterion
Schematic picture of CD kink instability
– Unstable wavelengths:
l > |Bp/Bf |2pR
• However, rotation and shear motion could
significant affect the instability criterion
3D RMHD simulation of CD kink instability in
helical force-free field (Mizuno et al. 2009)
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
3D GRMHD code RAISHIN
Mizuno et al. 2006a, astro-ph/0609004
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)
Numerical Schemes
* 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 & 3rdorder)
* Recovery step: Noble 2 variable method, Mignore-McKinney 1
variable method
* Equation of states: constant G-law EoS, variable EoS for ideal gas
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)
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
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)
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
N: total number of modes, fk: random phase,
ak:x,y,: random direction
• 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
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
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
Time Evolution of Volume Averaged
Quantities
Ep=rhg2-p, Em=B2/2, Et=Ep+Em
Plasma energy
Total energy
• Initial slow evolution in linear
magnetic energy 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
• 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
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
Radial Profile
Case A
Radial field
Axial field
Toroidal field
Gas pressure
Radial profile of toroidal- and
axial- averaged quantities for
case A
• 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)
• For different initial perturbation profiles, evolutionary timescale is different but physical
behavior is similar (not shown here)
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
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
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)