Relaxation of Pulsar Wind Nebula via Current-Driven Kink Instability

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 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) G54.1+0.3 (Lu et al. 2002) G320.4-1.2 (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 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)

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) Synchrotron emission map Flow magnitude • 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. Constraining  in PWNe Smaller s, jet does not formed =0.03 =0.003 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)

8. Obliquely rotating Pulsar magnetosphere • In pulsar wind, most of energy transferred by waves, which an obliquely rotating magnetosphereexcites 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)

9. Dissipation of Alternating Fields 1D RPIC simulation with σ = 45, Γ = 20 (dissipation occurs) • 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 Petri & Lyubarsky 2007

10. 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

11. 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)

12. 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

13. 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) * 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

14. 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)

15. 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

16. 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)

17. Initial Condition for Simulations Radial profile pressure Toroidal field • 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=105r0c2 (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

18. 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

19. 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

20. 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 magnetic energy Plasma 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

21. 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

22. 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) Gas pressure Axial field • For different initial perturbation profiles, evolutionary timescale is different but physical behavior is similar (not shown here)

23. 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

24. 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

25. 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 sdecreases 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)