Current-Driven Kink Instability in Magnetically Dominated

Relativistic Jets

Yosuke MizunoInstitute 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 >> 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 (dissipation)

Jets

Magnetic field lines

GRMHD simulations(McKinney 06)

MHD process (schematic picture)

Jet Energetics

Gravity, Rotational energy (BH or accretion disk)

Poynting flux (magnetic energy)

Jet kinetic energy

Efficient conversion to EM energy

Easy to get ~ equipartition, hard to get full conversion

Magnetic field is a medium for a transmission not a source

Dissipation in the Relativistic JetShocksTime-dependent energy injection (internal shock)Change of external medium spatial structure (recollimation shock)

Magnetic ReconnectionsMagnetic field reversal or deformation of ordered magnetic field

Turbulences Leads from MHD instabilities in jets

MHD InstabilitiesKelvin-Helmholtz instability at jet shear boundaryCurrent-Driven Kink instability at jet interior=> Turbulence in the jets and/or magnetic reconnection?

Ultra-Fast TeV Flare in BlazarsPKS2155-304 (Aharonian et al. 2007)See also Mrk501, PKS1222+21• 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,

Fabian & Rees 2008)• But bulk jet speed is not so high ( ~ 5-10)• 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: ~102.50.5 rS (< 0.1 pc; < 1 mas)– Poynting-Flux Dominated (PFD) KFD

• MHD Acceleration/Collimation Region: ~10 – 102.50.5 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, 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

• Distorted magnetic field structure may trigger of magnetic reconnection.

Kink instability in experimental plasma lab (Moser & Bellan 2012)

Schematic picture of CD kink instability

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: 　　　

　　　　　　　∇ ( U ) = 0 　　　 (conservation law of

particle-number)

　　　　　　　∇ T = 0 (conservation law of energy-momentum)

　　　　　　　∂ F∂F∂F = 0

　　　　　　　∇ F = - J

• Ideal MHD condition: FU= 0

• metric ： ds2=-2 dt2+gij (dxi+i dt)(dx j+j dt)

• Equation of state : p=(-1) u: rest-mass density. p : proper gas pressure. u: internal energy. c: speed of light.h : specific enthalpy, h =1 + u + p /.: specific heat ratio.U : velocity four vector. J : current density four vector.∇ : covariant derivative. g : 4-metric. lapse function,ishift vector, gij : 3-metric

T : energy momentum tensor, T = pg +h UU+FF -gF

F/4.F : field-strength tensor,

(Maxwell equations)

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√ : determinant of 3-metricRelativistic density

momentum density

Total energy density

RAISHIN Code (3DGRMHD)• 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 -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)

Mizuno et al. 2006a, 2011c, & progress

Initial Condition for Static Plasma 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/B): constant, increase, decrease

• Density profile: constant or decrease (=0 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

Mizuno et al. (2009)

Force-Free Helical Magnetic FieldForce-free equilibrium:

Choose poloidal magnetic field:

Find toroidal magnetic field:

B0: magnetic amplitudea: characteristic radiusa=1/8L in this work: pitch profile parameter

Magnetic pitch (P= RBz/B) :

< 1 pitch increase⇒

=1 constant helical pitch ⇒ (same as previous non-relativistic work)

>1 helical pitch decrease⇒

Measured in Laboratory frame

Initial Radial Profile

Black: constant densityRed: decreasing density

Solid: constant pitchdotted: increase pitchDashed: decrease pitch

Magnetic pitch (P= RBz/B)

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: densityWhite line: magnetic field lines

Dependence on pitch profileIncrease pitch Decrease pitchConstant 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.

tA: Alfven crossing time

Time evolution

• 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)

Constant density Decrease density

tA: Alfven crossing time

Solid: constant pitchDotted: increase pitchDashed: decrease pitch

Volume-averaged kinetic energy transverse to the z-axis

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• Consider: Differential rotation relativistic jet with force-free

helical magnetic field• Solving RMHD equations in 3D Cartesian coordinates

• Magnetic pitch (P=RBz/B): constant (in no-rotation case)

• Angular velocity (=0,1,2,4,6)

• Density profile: decrease (=0 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

Mizuno et al. (2012)

Force-Free Helical Magnetic Field and Velocity

Force-free equilibrium:

Choose poloidal magnetic field:

Find toroidal magnetic field:

B0: magnetic amplitudeR0: characteristic radiusR0=1/4L in this work: pitch profile parameter: differential rotation parameter=1, =1 in this work

Magnetic pitch (P= RBz/B) :

Measured in comoving frame

Choose Angular velocity:

Jet Velocity(Drift velocity):

Initial Radial Profilesolid: 0=0dotted: 0=1dashed: 0=2dash-dotted: 0=4dash-two-dotted: 0=6

Toroidal field

Poloidal filed

Angular velocity

Axial jet velocity

Jet rotation velocity

Alfven velocity

Density

Magnetic pitch

Sound 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)

0=1

Color density contour with magnetic field lines

Dependence on Jet Rotation Velocity: growth rate

solid: 0=0dotted: 0=1dashed: 0=2dash-dotted: 0=4dash-two-dotted: 0=6

• 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, max~ 0.1vA/R0

• Growth rate of kink instability does not depend on jet rotation velocity

Alfven crossing time

Volume-averaged Kinetic energy of jet radial motion

Volume-averaged magnetic energy

Dependence on Jet Rotation Velocity:

3D Structure0=2 case: very similar to 0=1 case, excited n=1 mode• 0=4 & 6 cases: n=1 & n=2 modes start to grow near the axis region• Because pitch decrease with increasing 0

• 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 InteractionIn order to investigate the multiple mode interaction, perform longer simulation box cases with 0=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

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 (~3L) to excite instability

Mizuno et al. 2014, ApJ, 784, 167

3D Helical Structure

Density + B-field

Velocity +B-field

jet

• 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.

Density + B-field

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

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

Mizuno 2013, ApJS

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)

High resolution VLBI

observations

• In radio VLBI observation, angular resolution ∝/D, : wavelength, D: baseline

• Resolution ~ 50 as at ~1mm (300GHz) & D=4000 km

• In EHT project (with other sub-mm array), D > 9000 km => 20as at 345 GHz

• BH Cam project (ERC Synergy grant) supports EHT project from EU side (theoretical research and development of sub-mm array)

SMA

ALMA

GLT

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

rS11 as 8 as 0.3 as

Size of shadow 52 as 40 as 1.5 as

Submm-VLBI (Hawaii-ALMA-Gleenland)

BH shadow image (with jet) Moscibrodzka et al.(2014)

1mas = 200Rg 0.1mas = 20Rg

90 deg

60 deg

30 deg

intensity

• Based on 3D GRMHD Simulations of jet formation (HARM3D)

• Radiation is calculated by (general relativistic) radiation transfer code in post-process (ray-tracing method)