Current-Driven Kink Instability
in Magnetically Dominated
Relativistic Jets
Yosuke Mizuno
Institute for Theoretical Physics
Goethe University Frankfurt am Main
Mizuno, Lyubarsky, Nishikawa, & Hardee 2012, ApJ, 757, 16
DESY, Germany, November 27, 2014
Contents
•
•
•
•
Introduction: Relativistic Jets
Introduction: Current-Driven Kink Instability
3D (G)RMHD Simulation Code RAISHIN
3D RMHD Simulations of CD Kink Instability
(static plasma column case), effect of radial density
and magnetic pitch profile
• 3D RMHD Simulations of CD Kink Instability
(rotating relativistic jet case), jet flow & rotation
effect
• Summary
Relativistic Regime
• Kinetic energy >> rest-mass energy
– Fluid velocity ~ light speed (Lorentz factor gamma >> 1)
– Relativistic jets/ejecta/wind/blast waves (shocks) in AGNs, GRBs, Pulsars
• Thermal energy >> rest-mass energy
– Plasma temperature >> ion rest mass energy
– GRBs, magnetar flare?, Pulsar wind nebulae
• Magnetic energy >> rest-mass energy
– Magnetization parameter sigma (Poynting to kinetic energy ratio) >> 1
– Pulsars magnetosphere, Magnetars
• Gravitational energy >> rest-mass energy
– GMm/rmc2 = rg/r > 1
– Black hole, Neutron star
• Radiation energy >> rest-mass energy
– Supercritical accretion flow
Applicability of MHD Approximation
• Magnetohydrodynamics (MHD) describe
macroscopic behavior of plasmas if
– Spatial scale >> ion Larmor radius
– Time scale >> ion Larmor period
• But MHD can not treat
– Kinetic properties of plasma
–Particle acceleration
–Origin of resistivity
Relativistic Jets
• A tremendous, elongated and
collimated outflows of plasma
with relativistic speed (~ light
speed)
• Many sources for relativistic
jets (from stellar size to galaxy
size)
• Jets (Outflows) are common in
the universe
• Lunching from accreting
compact objects (Black Holes)
• Observed relativistic jet (at large
scale) is kinetic energy is
dominated = magnetic field is
weak
M87 jets (AGNs)
Fundamental Problems in
Relativistic jets
• How are the jets formed near the compact objects (BHs)?
(formation mechanism)
• How are the jets accelerated up to relativistic speed?
(acceleration mechanism)
• How the collimated jet structure make? (collimation
mechanism)
• How the jets remain stable over large distance? (Jet stability
mechanism)
• How to emit the radiation in the relativistic jets? (Radiation
mechanism and particle acceleration mechanism)
Simulation and Theory of Jet
Formation & Acceleration
• Relativistic jet is formed and
accelerated by macroscopic plasma
(MHD) process
• Collimated jet is formed near the
central BH and accelerates g >> 1
• But, it has problems
– Most of energy remains in
Poynting energy (magnetic energy)
– Acceleration needs take long time
(slow acceleration efficiency)
• Inconsistent with current observation
=> Rapid energy conversion
GRMHD
(dissipation)
simulations
(McKinney 06)
Magnetic
field lines
Jets
MHD process
(schematic picture)
Jet Energetics
Gravity, Rotational energy (BH or accretion disk)
Efficient conversion to EM energy
Poynting flux (magnetic energy)
Easy to get ~ equipartition,
hard to get full conversion
Jet kinetic energy
Magnetic field is a medium for a transmission not a source
Dissipation in the Relativistic Jet
Shocks


Time-dependent energy injection (internal shock)
Change of external medium spatial structure (recollimation shock)
Magnetic Reconnections
 Magnetic field reversal or deformation of ordered magnetic field
Turbulences

Leads from MHD instabilities in jets
MHD Instabilities



Kelvin-Helmholtz instability at jet shear boundary
Current-Driven Kink instability at jet interior
=> Turbulence in the jets and/or magnetic reconnection?
Ultra-Fast TeV Flare in Blazars
• Ultra-Fast TeV flares are observed in
some Blazars (AGN jets).
• Variation timescale:
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 bulk jet speed is not so high (g ~ 510)
• Emitter: compact & extremely fast
• Proposed Model:
• Magnetic Reconnection inside jet
(Giannios et al. 2009)
Giannios et al.(2009)
Key Questions of Jet Stability
• When jets propagate outward, there are possibility to grow of two
major instabilities
– Kelvin-Helmholtz (KH) instability
• Important at the shearing boundary flowing jet and external medium
• In kinetic-flux dominated jet (>103 rs)
– Current-Driven (CD) instability
• Important in existence of twisted magnetic field
• Twisted magnetic field is expected jet formation simulation & MHD theory
• Kink mode (m=1) is most dangerous in such system
• In Poynting-flux dominated jet (<103 rs)
Questions:
• How do jets remain sufficiently stable?
• What are the Effects & Structure of instabilities in particular jet
configuration?
We try to answer the questions through 3D RMHD simulations
The Five Regions of AGN Jet Propagation
• Hot Spot/Lobe: ~109 rS (~100 kpc; or 20’)
–  Outer jet is not Poynting-Flux Dominated
• Kinetic-Flux-Dominated (KFD) Jet: ~103 – 109 rS
(0.1 – 105 pc; 1 mas – 20’)
• Transition Region: ~102.50.5 rS (< 0.1 pc; < 1 mas)
– Poynting-Flux Dominated (PFD)  KFD
• MHD Acceleration/Collimation Region: ~10 – 102.50.5 rS
(1 – < 100 mpc; 10 as – < 1 mas)
– The Jet “Nozzle”
• Jet Launching Region: The Accretion Flow; ~5 – 50 rS
(0.5 – 5 mpc; 5 – 50 as)
– Probably unresolved or slightly resolved
CD Kink Instability
• Well-known instability in
laboratory plasma (TOKAMAK),
astrophysical plasma (Sun, jet,
pulsar etc).
• In configurations with strong
toroidal magnetic fields, currentdriven (CD) kink mode (m=1) is
unstable.
• This instability excites large-scale
helical motions that can be
strongly distort or even disrupt
the system
• Distorted magnetic field structure
may trigger of magnetic
reconnection.
Schematic picture of CD kink instability
Kink instability in experimental plasma
lab (Moser & Bellan 2012)
Previous Work for CD Kink Instability
• For relativistic force-free configuration
– Linear mode analysis provides conditions for the instability
but say little about the impact instability has on the system
(Istomin & Pariev (1994, 1996), Begelman(1998), Lyubarskii(1999),
Tomimatsu et al.(2001), Narayan et al. (2009))
– Instability of potentially disruptive kink mode must be
followed into the non-linear regime
• Helical structures have been found in Newtonian
/relativistic simulations of magnetized jets formation
and propagation (e.g., Nakamura & Meier 2004; Moll et al. 2008;
McKinney & Blandford 2009; Mignone et al. 2010)
Purpose
• We investigate detail of non-linear behavior of
relativistic CD kink instability in relativistic jets
– Relativistic: not only moving systems with relativistic speed
but any with magnetic energy density comparable to or
greater than the plasma energy density.
– First task: static configuration
• In generally, rigidly moving flows considered in the proper frame
• The simplest ones for studying the basic properties of the kink
instability
– Second task: rotating relativistic jets
• Consider differentially rotating relativistic jets motivated from
analytical work of Poynting-flux dominated jets (Lyubarsky 2009)
4D General Relativistic MHD Equation
•
General relativistic equation of conservation laws and Maxwell equations:
∇n ( r U n ) = 0
∇n T n = 0
(conservation law of particle-number)
(conservation law of energy-momentum)
∂Fnl  nFl  lF n = 0
∇F
n
=-J
n
(Maxwell equations)
FnUn = 0
•
Ideal MHD condition:
•
metric： ds2=-a2 dt2+gij (dxi+b i dt)(dx j+b j dt)
•
Equation of state : p=(G-1) u
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.
u
U : velocity four vector. Ju : current density four vector.
n
∇ : covariant derivative. gn : 4-metric. a : lapse function, bi : shift vector, gij : 3-metric
n
n
n
 n
s n
lk
T : energy momentum tensor, T = pg + r h U U +F F s -gnF Flk/4.
Fn : field-strength tensor,
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
Relativistic density
momentum density
Total energy density
RAISHIN Code (3DGRMHD)
Mizuno et al. 2006a, 2011c, & progress
• RAISHIN utilizes conservative, high-resolution shock capturing
schemes (Godunov-type scheme) to solve the 3D ideal GRMHD
equations (metric is static)
Ability of RAISHIN code
• Multi-dimension (1D, 2D, 3D)
• Special & General relativity (static metric)
• Different coordinates (RMHD: Cartesian, Cylindrical, Spherical and GRMHD:
Boyer-Lindquist coord. of non-rotating or rotating BH, Kerr-Schild coord. is
underdeveloped)
• Different schemes of numerical accuracy for numerical model (spatial
reconstruction, approximate Riemann solver, constrained transport schemes, time
advance, & inversion)
• Using constant G-law and approximate Equation of State (Synge-type)
• Parallel computing (based on OpenMP, MPI)
Free to download from my HP (www.th.physik.uni-frankfurt.de/~mizuno)
Initial Condition for Static Plasma
Mizuno et al. (2009)
Column of CDI
• Solving ideal RMHD equations using RAISHIN in
Cartesian coordinates
• Static Cylindrical Plasma column with force-free
equilibrium helical magnetic field
• Magnetic pitch (P=RBz/Bf): constant, increase, decrease
• Density profile: constant or decrease (r=r0 B2)
• Numerical box: -2L < x, y < 2L, 0 < z < 2L (Cartesian
coordinates:160 x 160 x 80 zones)
• Boundary: periodic in axial (z) direction
• Small velocity perturbation with m=1(-1) and n=1(-1)
modes
Force-Free Helical Magnetic Field
Force-free equilibrium:
Choose poloidal magnetic field:
Find toroidal magnetic field:
Measured in Laboratory
frame
B0: magnetic amplitude
a: characteristic radius
a=1/8L in this work
a: pitch profile parameter
Magnetic pitch (P= RBz/Bf) :
a < 1 ⇒ pitch increase
a=1 ⇒ constant helical pitch (same as previous non-relativistic work)
a >1 ⇒ helical pitch decrease
Initial Radial Profile
Magnetic pitch
(P= RBz/Bf)
Black: constant density
Red: decreasing density
Solid: constant pitch
dotted: increase pitch
Dashed: decrease pitch
density
Sound velocity
Alfven velocity
3D Structure
(Decrease density with Constant pitch )
•Displacement of the initial
force-free helical magnetic field
leads to a helically twisted
magnetic filament around the
density isosurface by CD kink
instability
• Slowly continuing outwards
radial motion is confined to a
lower density sheath around the
high density core
Color: density
White line: magnetic field lines
Dependence on pitch profile
Increase pitch
Constant pitch
tA: Alfven crossing time
Decrease pitch
Constant pitch: Amplitude growth slows at later time.
Increase pitch: 3D density structure looks similar to constant pitch case.
However, amplitude growth ceases at later time.
Decrease pitch: slender helical density wrapped by B-field developed.
Amplitude growth continues throughout simulation.
Time evolution
Volume-averaged kinetic energy transverse to the z-axis
Constant density
Decrease density
tA: Alfven crossing time
Solid: constant pitch
Dotted: increase pitch
Dashed: decrease pitch
• Initial exponential linear growth phase and subsequent non-linear evolution
• Density Decline: more rapid initial growth and decline (by more gradual radial decline
in the Alfven velocity) .
• Pitch increase: slower growth
• Pitch decrease: more rapid growth
• Consistent with non relativistic linear analysis in Appl et al. (2000)
CD Kink Instability in Rotating
Relativistic Jets
• Here: we investigate the influence of jet rotation and bulk
motion on the stability and nonlinear behavior of CD kink
instability.
• We consider differentially rotating relativistic jets motivated from
analytical work of Poynting-flux dominated jets (Lyubarsky 2009).
• The jet structure relaxes to a locally equilibrium configuration if the
jet is narrow enough (the Alfven crossing time is less than the proper
propagation time). So cylindrical equilibrium configuration is
acceptable.
Initial Condition
Mizuno et al. (2012)
• Consider: Differential rotation relativistic jet with forcefree helical magnetic field
• Solving RMHD equations in 3D Cartesian coordinates
• Magnetic pitch (P=RBz/Bf): constant (in no-rotation case)
• Angular velocity (W0=0,1,2,4,6)
• Density profile: decrease (r=r0 B2)
• Numerical box: -3L < x, y < 3L, 0 < z < 3L (Cartesian
coordinates: 240 x 240 x 120 zones)
• Boundary: periodic in axial (z) direction
• Small velocity perturbation with m=1 and n=0.5 ~ 4
modes
Force-Free Helical Magnetic Field and Velocity
Force-free equilibrium:
Choose poloidal magnetic field:
Choose Angular velocity:
Find toroidal magnetic field:
Magnetic pitch (P= RBz/Bf) :
Jet Velocity
(Drift velocity):
Measured in
comoving frame
B0: magnetic amplitude
R0: characteristic radius
R0=1/4L in this work
a: pitch profile parameter
b: differential rotation
parameter
a=1, b=1 in this work
Initial Radial Profile
Angular
velocity
Toroidal
field
Axial jet
velocity
solid: W0=0
dotted: W0=1
dashed: W0=2
dash-dotted: W0=4
dash-two-dotted: W0=6
Poloidal
filed
Jet rotation
velocity
Magnetic
pitch
Sound
velocity
Density
Alfven
velocity
Time Evolution
of 3D Structure
• Displacement of the initial
force-free helical field leads to a
helically twisted magnetic
filament around the density
isosurface with n=1 mode by CD
kink instability
• From transition to non-linear
stage, helical twisted structure is
propagates in flow direction with
continuous increase of kink
amplitude.
• The propagation speed of kink
~0.1c (similar to initial
maximum axial drift velocity)
W0=1
Color density contour
with magnetic field lines
Dependence on Jet Rotation
Velocity: growth rate
Volume-averaged Kinetic energy of
jet radial motion
Volume-averaged
magnetic energy
solid: W0=0
dotted: W0=1
dashed: W0=2
dash-dotted: W0=4
dash-two-dotted: W0=6
Alfven crossing time
• First bump at t < 20 in Ekin is initial relaxation of system
• Initial exponential linear growth phase from t ~ 40 to t ~120 (dozen of Alfven
crossing time) in all cases
• Agree with general estimate of growth rate, Gmax~ 0.1vA/R0
• Growth rate of kink instability does not depend on jet rotation velocity
Dependence on Jet
Rotation Velocity:
3D Structure
 W0=2 case: very similar to W0=1
case, excited n=1 mode
• W0=4 & 6 cases: n=1 & n=2
modes start to grow near the axis
region
• Because pitch decrease with
increasing W0
• In nonlinear phase, n=1 mode
wavelength only excited in far
from the axis
• Propagation speed of kink is
increase with increase of angular
velocity
Multiple Mode Interaction
 In order to investigate the
multiple mode interaction, perform
longer simulation box cases with
W0=2 & 4
• n=1 & n=2 modes grow near the
axis region, in nonlinear phase,
growth of the CD kink instability
produces a complicated radially
expanding structure as a result of
the coupling of multiple
wavelengths
• Cylindrical jet structure is almost
disrupted in long-term evolution.
• The coupling of multiple unstable
wavelength is crucial to
determining whether the jet is
eventually disrupted.
CD Kink Instability in Sub-Alfvenic Jets:
Spatial Properties
Mizuno et al. 2014, ApJ, 784, 167
Purpose
• In previous study, we follow temporal properties (a few axial
wavelengths) of CD kink instability in relativistic jets using periodic
box.
• In this study, we investigate spatial properties of CD kink instability
in relativistic jets using non-periodic box.
Initial Condition
• Cylindrical (top-hat) non-rotating jet established across the
computational domain with a helical force-free magnetic field (mostly
sub-Alfvenic speed)
•Vj=0.2c, Rj=1.0
• Radial profile: Decreasing density with constant magnetic pitch
(a=1/4L)
• Jet spine precessed to break the symmetry (l~3L) to excite instability
3D Helical Structure
jet
Density
+ B-field
Density
+ B-field
Velocity
+B-field
• Precession perturbation from jet inlet produces the growth of CD
kink instability with helical density distortion.
• Helical kink structure is advected with the flow with continuous
growth of kink amplitude in non-linear phase.
• Helical density and magnetic field structure appear disrupted far from
the jet inlet.
Summery
• In CD kink instability of static plasma column, the initial
configuration is strongly distorted but not disrupted.
• In rotating relativistic jet case, developed helical kink structure
propagates along jet axis with continuous growth of kink
amplitude.
• The growth rate of CD kink instability depends on magnetic
pitch & radial density profiles and does not depend on the jet
rotation.
• The coupling of multiple unstable wavelength is crucial to
determining whether the jet is eventually disrupted in nonlinear
stage.
• The strongly deformed magnetic field via CD kink instability
may trigger of magnetic reconnection in the jet (need Resistive
Relativistic MHD simulation).
Relativistic Magnetic
Reconnection using
RRMHD Code
Mizuno 2013, ApJS
Assumption
• Consider Pestchek-type magnetic
reconnection
Initial condition
• Harris-type model（anti-parallel
magnetic field）
• Anomalous resistivity for
triggering magnetic reconnection （
r<0.8）
Results
• B-filed：typical X-type topology
• Density：Plasmoid
• Reconnection outflow: ~0.8c
Frontier of research
• Numerical simulations:
– Supercomputer becomes larger (Peta-scale computing), new technique
(GPU parallel computing)
– GRMHD simulations are possible
– Additional physics: Resistivity, radiation, microphysics, …
– Effects of time-dependence, non-asymmetry (3D)
• GR radiation transfer calculation is a key tool to connect MHD
simulations and observations (Need correct radiation process
including particle acceleration)
• Observations: mm & submm-VLBI trying to observe BH
shadow & jet launching region (EHT, BH Cam projects)
Source
SgrA*
M87
Cen A
BH Mass (Msun)
0.045 x 108
66 x 108
0.45 x 108
Distance
0.008 pc
16.7 Mpc
3.4 Mpc
High resolution
VLBI
observations
Submm-VLBI (Hawaii-ALMAGleenland)
rS
11 as
8 as
0.3 as
Size of shadow
52 as
40 as
1.5 as
• In radio VLBI observation, angular resolution
∝l/D,
l: wavelength, D: baseline
• Resolution ~ 50 as at l ~1mm (300GHz) &
D=4000 km
• In EHT project (with other sub-mm array), D >
9000 km => 20 as at 345 GHz
• BH Cam project (ERC Synergy grant) supports
EHT project from EU side (theoretical research
and development of sub-mm array)
GLT
SMA
ALMA
BH shadow image (with jet) Moscibrodzka et al.(2014)
1mas = 200Rg
0.1mas = 20Rg
• Based on 3D GRMHD
Simulations of jet
formation (HARM3D)
90 deg
• Radiation is calculated
by (general relativistic)
radiation transfer code
in post-process (raytracing method)
60 deg
30 deg
intensity