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The CD Kink Instability in Magnetically Dominated Relativistic Jets *. Ken-Ichi Nishikawa 1 , Y. Mizuno 1 , Y. Lyubarsky 2 , P.E.Hardee 3 , 1 NSSTC/ CSPAR/ University of Alabama in Huntsville, USA, 2 Ben-Gurion University, Israel, 3 University of Alabama, Tuscaloosa, USA.

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The CD Kink Instability in Magnetically Dominated Relativistic Jets*

Ken-Ichi Nishikawa1 , Y. Mizuno1, Y. Lyubarsky2, P.E.Hardee3

, 1NSSTC/ CSPAR/ University of Alabama in Huntsville, USA, 2Ben-Gurion University, Israel, 3 University of Alabama, Tuscaloosa, USA

The relativistic jets associated with blazar emission from radio through TeV gamma-rays are thought to be accelerated and collimated by strong helically twisted magnetic fields with footpoints threading the black hole ergosphere and/or the surrounding accretion disk. The resulting magnetically dominated jet is current-driven (CD) unstable. In a resistive system instability may lead to magnetic reconnection, particle acceleration to the high energies required by the observed TeV emission, and also to the observed kinetically dominated jets far from the central engine. We have investigated the temporal development of current-driven kink instability in magnetically dominated relativistic jets via 3D RMHD simulations. In this investigation a static force-free equilibrium helical magnetic configuration is considered in order to study the influence of the initial configuration on the linear and nonlinear evolution of the instability. We find that the initial configuration is strongly distorted but not disrupted by the CD kink instability. The linear growth and nonlinear evolution of the CD kink instability depends moderately on the radial density profile and strongly on the magnetic pitch profile. Kink amplitude growth in the nonlinear regime for decreasing magnetic pitch leads to a slender helically twisted column wrapped by magnetic field. On the other hand, kink amplitude growth in the nonlinear regime nearly ceases for increasing magnetic pitch. We also present preliminary results showing the effect of velocity shear on the spatial and temporal development of the CD kink instability.

2. Instability of Relativistic Jets

  • Astrophysical Jets

Radio Observations of M87

Beam: ~ 0.4 x 0.2 mas, 0.3 mas ~ 0.024 pc ~ 42Rs

  • Two major instabilities:
    • Kelvin-Helmholtz (KH) instability
      • At the velocity shear surface between jet and external medium
    • Current-Driven (CD) instability
      • In the twisted magnetic field of magnetically dominate flows
  • Relativistic jets: outflow of highly collimated plasma
    • Microquasars, Active Galactic Nuclei, Gamma-Ray Bursts, Jet velocities ~c.
    • Generic systems: Compact object (White Dwarf, Neutron Star, Black Hole)+ Accretion Disk
  • Key Issues for Relativistic Jets
    • Acceleration & Collimation
    • Propagation & Stability
  • Modeling of Jet Production
    • Magnetohydrodynamics & Relativity (SR+GR)
  • Modeling of Jet Emission
    • Particle Acceleration & Radiation Mechanism

@ 15o jet viewing angle

5 mas ~ 1.55 pc ~ 2700 Rs

5 mas  0.4 pc ~700 Rs

M87: jet launching and collimation region

  • KH instability can lead to jet twisting, twisted filaments, limb brightening, shocks, turbulence, particle acceleration
  • CD instability can lead to jet twisting, twisted filaments, magnetic reconnection, particle acceleration

Black: constant density

Red: decreasing density

Solid: constant pitch

dotted: increasing pitch

Dashed: decreasing pitch

3. Motivation

(Mizuno et al., 2009)

4. Initial Conditions

(Acciari et al., 2009, Science, 325, 444)

Initial radial profile

  • For relativistic force-free configurations
    • Linear analysis provides conditions for instability but says little about
    • the impact on the system(Istomin & Pariev (1994, 1996), Begelman(1998),
    • Lyubarskii(1999), Tomimatsu et al.(2001), Narayan et al. (2009))
    • Instability of the potentially disruptive kink mode must be followed into
    • the non-linear regime
  • Helical structures have been found in simulations of strongly magnetized jets
  • (e.g., Nakamura & Meier 2004; Moll et al. 2008; McKinney & Blandford 2009)
  • We study the non-linear relativistic CD kink instability

Force-free helical magnetic field: CD kink unstable

a = characteristic radius of plasma column

Magnetic pitch (P=RBz/Bf): increasing, constant, decreasing

Density profile: constant or decreasing (r=r0 B2)

Numerical box: -16a < x, y < 16a, 0 < z < 16a (Cartesian coordinates:160 x 160 x 80 zones)

Boundary: periodic in axial (z) direction

Velocity perturbation:m=1(-1) andn=1(-1) modes

5. Results: Static Plasma Column

Time evolution

(volume-averaged kinetic energy transverse to the z-axis)

Density Isosurface & white magnetic field lines

Increasing pitch

Constant pitch

Decreasing pitch

Constant density

Decreasing density

tA: Alfven crossing time

Dotted: increasing pitch

Solid: constant pitch

Dashed: decreasing pitch

  • Increasing pitch: Amplitude growth ceases at late times.
  • Constant pitch:Amplitude growth slows at late times.
  • Decreasing pitch:Amplitude growth continues throughout simulation.
  • Initial exponential growth (linear phase) and subsequent non-linear evolution
  • Density Decline: more rapid growth & decline (less radial Alfven velocity decline)
  • Pitch increase: slower growth
  • Pitch decrease: more rapid growth

Consistent with non-relativistic linear analysis In Appl et al. (2000)

6. Results: Sub-Alfvenic Jet

Spatial Properties:

(Mizuno et al. 2010 in prep)

Temporal Properties:

(Mizuno et al. 2010, ApJ, submitted )

Kink Propagation: maximum density

position in x-y plane at z = 6a

  • Initial Conditions
  • Sub-Alfvenic jet (vj=0.2c) with force-free B field (KH stable)
  • Radial profile: decreasing density (r=r0 B2) with constant pitch
  • Jet velocity shear radius: Rj=a/2, a, 2a, 4a
  • Numerical box: -8a < x, y < 8a, 0 < z < 12a (160 x 160 x 120)
  • Boundary: periodic in axial (z) direction
  • Velocity perturbation: m=1(-1) andn=1(-1) modes
  • Initial Conditions
  • Sub-Alfvenic jet (vj=0.2c, Rj=1.0)with helical force-free magnetic
  • field established across computational domain
  • Radial profile: Decreasing density with constant magnetic pitch
  • Jet spine precessed to break symmetry
  • Numerical Box: 6Rj x 6Rj x 20 Rj (Cartesian: 180 x 180 x 400 zones)

Red: Rj=a/2, Orange: Rj=a, Green: Rj=2a,

Blue: Rj=4a, Black: no jet


Density Isosurfaces & velocity vectors


Density isosurfaces (color) with white magnetic field lines.



  • Precessional perturbation at inlet induces growth of the CD kink.
  • Helical structure propagates with continuous spatial kink growth.




  • Small jet radius: small kink propagation speed,
  • flow through kink.
  • Large jet radius: fast kink propagation speed,
  • kink embedded in flow.
  • Non-linear behavior most altered for Rj = a & 2a

For more detail, please see

Mizuno et al. 2009, ApJ, 700, 684

Mizuno et al. 2010, ApJ, submitted

*COSPAR 2010, Bremen, Germany, July 18-25, 2010