Scaled laboratory experiments explain the kink behaviour of the Crab Nebula jet

Abstract

The remarkable discovery by the Chandra X-ray observatory that the Crab nebula's jet periodically changes direction provides a challenge to our understanding of astrophysical jet dynamics. It has been suggested that this phenomenon may be the consequence of magnetic fields and magnetohydrodynamic instabilities, but experimental demonstration in a controlled laboratory environment has remained elusive. Here we report experiments that use high-power lasers to create a plasma jet that can be directly compared with the Crab jet through well-defined physical scaling laws. The jet generates its own embedded toroidal magnetic fields; as it moves, plasma instabilities result in multiple deflections of the propagation direction, mimicking the kink behaviour of the Crab jet. The experiment is modelled with three-dimensional numerical simulations that show exactly how the instability develops and results in changes of direction of the jet.

ARTICLE

Received 16 Feb 2016 | Accepted 31 Aug 2016 | Published 7 Oct 2016

C.K. Li1, P. Tzeferacos2, D. Lamb2, G. Gregori3, P.A. Norreys3, M.J. Rosenberg1, R.K. Follett4,5, D.H. Froula4,5,M. Koenig6,7, F.H. Seguin1, J.A. Frenje1, H.G. Rinderknecht1, H. Sio1, A.B. Zylstra1, R.D. Petrasso1, P.A. Amendt8, H.S. Park8, B.A. Remington8, D.D. Ryutov8, S.C. Wilks8, R. Betti4,5, A. Frank4,5, S.X. Hu4, T.C. Sangster4,P. Hartigan9, R.P. Drake10, C.C. Kuranz10, S.V. Lebedev11 & N.C. Woolsey12

The remarkable discovery by the Chandra X-ray observatory that the Crab nebulas jet periodically changes direction provides a challenge to our understanding of astrophysical jet dynamics. It has been suggested that this phenomenon may be the consequence of magnetic elds and magnetohydrodynamic instabilities, but experimental demonstration in a controlled laboratory environment has remained elusive. Here we report experiments that use high-power lasers to create a plasma jet that can be directly compared with the Crab jet through well-dened physical scaling laws. The jet generates its own embedded toroidal magnetic elds; as it moves, plasma instabilities result in multiple deections of the propagation direction, mimicking the kink behaviour of the Crab jet. The experiment is modelled with three-dimensional numerical simulations that show exactly how the instability develops and results in changes of direction of the jet.

1 Plasma Science and Fusion Center, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139 USA.

2 Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637, USA. 3 Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK. 4 Laboratory for Laser Energetics, University of Rochester, Rochester, New York 14627, USA.

5 Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA. 6 LULI-CNRS, Ecole Polytechnique, CEA: Universit Paris-Saclay; UPMC Univ Paris 06: Sorbonne Universits, F-91128 Palaiseau cedex, France. 7 Institute of Laser Engineering, Osaka University, Suita,Osaka 565-0871, Japan. 8 Lawrence Livermore National Laboratory, Livermore, California 94551, USA. 9 Department of Physics and Astronomy,Rice University 6100 S. Main, Houston, Texas 77521, USA. 10 Department of Atmospheric, Ocean and Space Science, University of Michigan, 2455 Hayward Street, Ann Arbor, Michigan 48103, USA. 11 The Blackett Laboratory, Imperial College London, London SW7 2BW, UK. 12 Department of Physics, University of York, York YO10 5D, UK. Correspondence and requests for materials should be addressed to C.K.L. (email: mailto:[email protected]

Web End [email protected] ).

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Scaled laboratory experiments explain the kink behaviour of the Crab Nebula jet

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13081

X-ray images from the Chandra X-ray Observatory1,2 show that the South-East jet in the Crab nebula changes direction every few years (Supplementary Fig. 1). This

fascinating phenomenon is also seen in jets associated with pulsar wind nebulae and other astrophysical objects35, and therefore is a fundamental feature of astrophysical jet evolution that needs to be understood613. The South-East Crab nebula jet is a highly collimated, mildly relativistic gas outow from a pole of the rapidly rotating Crab pulsar, conned by a toroidal magnetic eld (B) and accelerated outwards initially by means of magnetic elds (Poynting ux) drawing from the pulsar rotational energy6,815.

Astrophysical jets can be studied in a controlled environment using appropriately scaled laboratory experiments that reproduce and study critical physical aspects; even though laboratory-generated supersonic plasma jets and astrophysical jets have very different scales, they can have similar dimensionless hydrodynamic and magnetic eld parameters (as will be shown below) and therefore can share common physical properties1618. These important similarities allow us to scale our laboratory results to the conditions in the Crab nebula, showing that the laboratory approach provides an incisive platform for studying various properties of astrophysical jets. To mimic the kink behaviour of the Crab jet, a laboratory experiment requires magnetic elds with the right properties: the elds must have a strong azimuthal (toroidal) component generated near the target where the jet is launched, and the elds must be embedded in (frozen-in), and advected with, the fast moving magnetized plasma ow.

The development and use of diagnostics that enable visualization and quantication of magnetic elds and magnetohydro-dynamic (MHD) instabilities is as important as the creation of the plasma jet itself. Most conventional plasma diagnostics, using X rays and optical photons, are sensitive to plasma density and temperature but not to electromagnetic elds1921. The recently

developed method of monoenergetic proton radiography22 (Methods section) is sensitive to electromagnetic elds and can provide spatial visualization and quantitative measurements.

Here we report experiments using scaled plasma jets, generated by high-power lasers, to reproduce and model the Crab jet (Methods section and Supplementary Fig. 2). Magnetic elds and current-driven MHD instabilities taking place in the jet, visualized and measured directly by monoenergetic proton radiography22, have been unambiguously identied as the mechanisms that cause such a unique jet kink behaviour. We show how the toroidal magnetic eld embedded in the supersonic jet triggers plasma instabilities and results in considerable deections throughout the jet propagation, mimicking the kinks in the Crab jet. We also demonstrate that these kinks are stabilized by high jet velocity, consistent with the observation that instabilities alter the jet orientation but do not disrupt the overall jet structure. Our laser experiments produce plasma jets characterized by higher plasma temperatures (4BkeV) and faster ow velocities (4B1,000 km s 1) that are at least one to two orders of magnitudes higher (faster) than those achievable by other experimental approaches1921. Our experiments also produce plasma jets that have magnetic Reynolds numbers large enough for the magnetic eld to be frozen into the plasma ow. Consequently, the plasma in the jet must follow the eld topology and its evolution, which is locally kinked but globally collimated along the propagation axis. We successfully model these laboratory experiments with a validated three-dimensional (3D) numerical simulation, which in conjunction with the experiments provide compelling evidence that we have an accurate model of the most important physics behind the observed kinking of the Crab nebula jet. These experiments not only advance our knowledge of the structure and dynamics of the Crab jet, but also open up opportunities for laboratory study of jets from a variety of other astrophysical objects.

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Figure 1 | Experiments and proton radiographs. (a) Schematic of a laser-beam-irradiated, cone-shaped target, and resulting plasma jet comprised of ions and electrons, also indicating the resulting toroidal magnetic eld directions (Methods). Side-on (proton ux into the paper) radiographic images show the proton uence distribution at t0 4.70 ns with 15-MeV protons and at t0 4.92 ns with 3.3-MeV protons, where t0 is the time when the lasers turned on.

The enlarged image shows a sequence of clumps and changes of jet direction. Cartoons in b illustrate the congurations of self-generated, spontaneous magnetic elds (B1 and B2) associated with the two plasma plumes. The resultant magnetic eld can be decomposed into poloidal (BP BRBz) and

toroidal (Bj) components. The eld structure is crucial for the excitation of the kinks. (c) Lineouts from the images in a along the axes of the plasma jets. The unit of the vertical axis is proton counts, which is proportional to the proton uence. (d) Schematic illustrations of the fastest growing MHD current-driven instabilities: mode m 0 (sausage, leading to jet propagation clumping) and m 1 (kink, leading to jet direction changing). Higher modes

(m41) are also expected to be excited, but will have smaller effects and are not illustrated here.

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ResultsLaser-driven scaled plasma jet. In our laboratory experiments, we form plasma jets through the collision of expanding, laser-driven plasma plumes (Fig. 1). The laser-foil interaction induces self-generated magnetic elds due to the Biermann battery effects23 (qB/qtprne rTe, where ne and Te are

electron density and temperature, respectively) that are predominantly toroidal with respect to each plume and to the jet that is formed from the collision of these plumes (Fig. 1b). The magnetic elds are embedded in and advance with the moving jet (qB/qtpr (vj B), where vj is jet velocity),

mimicking the fundamental scenario that magnetic elds are anchored in the rapidly spinning Crab pulsar and advected with the Crab jet.

Figure 1a shows proton radiographs of the plasma jets at different times, indicating a structure that is collimated throughout its propagation but has a sequence of clumps and changes of direction along its length. These features reect perturbations in the magnetic eld structure around the jet (Fig. 2), and they grow locally and expand at each axial position where the jet is unstable. The shape of the jet is serpentine due to the kink instability, and so can be viewed as comprised of ellipsoidal blobs that are typically viewed with the proton radiography from an angle yB45. Adopting this picture, we can apply the analysis of ref. 24 for ellipsoidal blobs. Interpreting the structures seen in the proton radiography images as caustics, we use the criterion for caustic formation24, which indicates that B40.8 MG for y 45. Shown

in Fig. 1d are cartoons illustrating the most feasible and fast growing MHD current-driven instabilities: Mode m 0 (sausage)

arises as the Bj tension is enhanced by radial contraction, responsible for the axis pinching when |Bj|4O2|BP|.

Mode m 1 (kink) arises when the strength and pressure of Bj

increase at the inside of the kinks and decrease outside. |Bj||BP| 1l(2prj) 14a, with a 1 is the KruskalShafranov

criterion for the kink instability25, where l is the modulation wavelength and rj the jet radius. In astrophysical jets, effects like expansion can tend to stabilize the jet10, resulting in a larger than but of order unity. This scenario is illustrated by experiment with a at target (Fig. 3) where the plasma jet is stabilized when magnetic eld is overwhelmed by the parallel components as the toroidal components around the jet are too weak to excite the MHD instability. The unstable modes have a growth rate g comparable to the inverse of the time required for phase velocity

of an Alfvn wave (vA B/O4pr) to cross the unstable jet

column26

g G

2prj l

vArj : 1

Using the measured values rjB0.5 mm and lB0.60.7 mm, and vAB1,000 km s 1 around the jet launching region, we nd gB3 109 s 1, which is consistent with the instability evolution

time implied by Fig. 1a.

Modelling of experiments with numerical simulation. To model the observations of the plasma jet, a 3D numerical simulation was performed with the radiation-MHD code FLASH27,28 (Methods section). The simulation was post-processed to provide a more complete physical picture of the jet behaviour, leading to the images in Fig. 4 showing the spatial variations of various quantities at tEt0 5.0 ns in the plane containing the jets axis.

Figure 4a shows that a modulated central spine (backbone) region with stronger eld strength is formed, and is surrounded by asymmetrically distributed, weaker elds around the jet core. When the eld is sufciently large and has nonuniform toroidal components Bj, current-driven MHD kink modes are excited with the susceptibility increasing with increasing |Bj/BP|

(Fig. 4b). Such a structure is conrmed by the corresponding distribution of b 8pnkT/B2 (the ratio of plasma thermal to

magnetic pressures) in Fig. 4c: in the jet core boB1, showing the ow is magnetically dominated, while in the surrounding plasma b41. This indicates that the jet is globally collimated due to inertial connement and magnetic tension, but locally kinked.

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Figure 2 | Two images connected with each radiograph. (a) Image of proton uence versus position, taken with 3-MeV DD protons at 4.92 ns from the onset of the laser drive on the subject cone target, showing a clear kink structure which indicates that the jet propagation was subjected to plasma instabilities. (b) Image displaying mean proton energy versus position shows a very uniform distribution, with no hint of the density structure of the jet. The latter would be expected if Coulomb scattering43

were important, indicating that the structures seen in the uence image are due to deections of protons by magnetic elds.

Jet

Figure 3 | Plasma jet and magnetic eld congurations generated by a laser-driven at target. (a) Cartoons of face-on and side-on views of a plasma jet and associated toroidal magnetic elds after reconnection due to collision of two plasma plumes from a laser-driven plastic foil. (b) Radial distribution of magnetic elds indicate the toroidal components around the jet are too weak to excite the MHD instability (overwhelmed by the parallel components, that is, Bj/Bpoo1). (c) Proton side-on radiographic image shows the jet is stable to MHD instabilities when toroidal components are weak (white arrow points the position of at foil target). It also suggests that in this type of experiment the toroidal elds generated by the plasma current are too weak to destabilize the jet propagation. The jet is predominately collimated by inertial connement due to the hydrodynamic compression produced by the collision of the two plumes.

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Figure 4 | Images of various physical properties in the 3D numerical simulation of the jet. (Images correspond to t t0 5 ns, the detailed evolution of

the simulation can be accessed online in Supplementary Movies 1,2,3 and 4). (a) The amplitudes of the self-generated magnetic elds that are advected with the jet ow show a collimated ow with a wiggling central spine. Outside the jet surface, the bulk ow has asymmetrically distributed magnetic elds. The white arrow indicates where the jet is thermally launched (zB2 mm). (b) Image showing the ratio of toroidal (Bt |Bj|) to poloidal (Bp |Bz BR|)

eld components. The image shows the locations where jet kinks take place and grow are correlated with the regions where Bj is stronger and asymmetrically distributed. (c) The corresponding distribution of the ratio b of plasma pressure to magnetic pressure. The jet core and surrounding region (bulk ow) have bo1. The instability occurs in the region where advection of the magnetic eld is dominant and bo1. (d) The simulated plasma density distribution shows clumps and kinks corresponding to the eld topology. (The units for x, y, and z axes are cm.)

The jet is not in force-free equilibrium: the gradient of thermal pressure is not in local balance with the sum of magnetic-pressure gradient and magnetic tension (hoop stress Bj2/4pr); that is,

@ @r

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The clumps and kinks shown in Fig. 4ac are similar to those we observe in the experiments (Fig. 1a). The distribution of the simulated plasma density depicted in Fig. 4d shows a clear kink structure that is correlated with the eld topology in Fig. 4a.

Validation of numerical simulations. The FLASH simulation was used to predict the physical properties of the jet (Methods section). These were compared quantitatively with the experimental measurements, including proton radiography (Fig. 1) and Thomson scattering29 (Supplementary Fig. 3). Figure 5a shows the measured jet positions and velocities which match the simulations well, providing compelling evidence for the validity of the numerical simulation. The velocity at the front of

the jet after it has been traveling for several nanoseconds is estimated to be vjB1,500 km s 1, indicating supersonic jet propagation with an internal Mach number MB3. Such a high jet velocity has two important effects on jet propagation. First, high Mach numbers suppress the Kelvin-Helmholtz instability, lessening the entrainment of the surrounding plasma in the jet plasma. Second, the high jet velocity helps to move the frozen-in non-uniform elds, leading to smoothing of asymmetric magnetic eld line distributions, stabilizing the jet. Further validation is provided in Fig. 5b,c, where plasma densities and temperatures measured using Thomson scattering are plotted against the time-resolved jet positions, respectively (Supplementary Fig. 3), and agree well with the numerical simulation. Again, this consistency greatly increases our condence that the simulation has captured the most important physics in the experiments.

DiscussionThe magnetization parameter ( B2/8prvj2, the ratio of the jet

magnetic to ram pressures) shown in Fig. 6a is sZ1 near the region where the jet was launched, and B10 210 3 near the

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Figure 5 | Comparison between measurements and numerical simulation. (a) Measured jet velocities (solid circles by protons (Fig. 1c) and open circles by Thomson-scattering (Supplementary Fig. 3), with measurement uncertainties (error bars) discussed therein, respectively), plotted as a function of position in the jet ow compare well with simulated values (blue line). The error bars of proton measured jet velocities indicate

DvB80120 km s 1, including measurement uncertainties and consequences of proton Coulomb scatterings. The increase in the simulated jet velocity as the ow propagates outwards, is a consequence of the gradient in the thermal plasma pressure, and leads to the decrease in the simulated jet density shown in (b) (green line). The measured plasma densities inferred from the Thomson-scattering data, which are shown as open red triangles, agree reasonably with those of the simulation. (c) The plasma temperatures T inferred from Thomson-scattering measurements (assuming TeBTi, in this relevant region, see Supplementary Fig. 3)29,44, which are shown as open red diamonds, compare reasonably well with those of the simulation (black line).

p Bcrab; 3

where the subscripts lab and crab refer to the laboratory and Crab nebula jets, respectively. As shown in Table 1, excellent MHD scaling is obtained with aB1.6 10 20, bB1.7 1025 and

cB1.1 1019.

In summary, our scaled laboratory experiments and validated numerical simulation reveal that the change in direction observed in the Crab jet can be attributed to magnetic elds and the associated MHD kink instabilities. This work not only advances our knowledge of such jet structure and dynamics, but also opens up tremendous opportunities in the laboratory to explore jets from a variety of other astrophysical objects, including active galactic nuclei, young stellar objects, X-ray binary systems and pulsar wind nebulae.

Methods

Experiments. In our experiment, performed at the OMEGA Laser Facility31 and illustrated schematically in Supplementary Fig. 2, the plasma jet was generated by the interaction of laser beams with a special target. The target was constructed with two 50-mm-thick, 3 3 mm plastic (CH) foils separated by 60. Each individual foil

was driven by two laser beams (0.351 mm in wavelength) at an angle B28 to the foil normal, with total energy B1,000 J in a 1-ns, square-top laser pulse with full spatial and temporal smoothing. The laser spot has a diameter of B850 mm determined by phase plate SG4 (dened as 95% energy deposition), resulting in a laser intensity of order of B2 1014 W cm 2. Laser ablation generated a plasma

p

vcrab; tlab a

b

c

q tcrab;

Blab c

jet head. These values compare very well to those of the Crab nebula, where observations and simulations indicate that sZ1 close to the pulsar pole where the jet is launched, and sB10 210 3 near the termination shock where the jet becomes subsonic13. The morphological similarities between the

Crab jet (Supplementary Fig. 1) and the laboratory jet can be clearly seen in the simulated current density map (Fig. 6b). The latter reveals kinks, knots, and large-scale radial deections that are reminiscent of the structures and dynamics observed in the Crab pulsar outow. This picture of a current-carrying jet is in

agreement with existing numerical efforts on modelling the Crab jet10,13 and the morphology mimics the jet structures observed in Chandra X-ray imaging1.

These similarities provide rigorous justication of the relevance of the plasma jet to the Crab nebula jet, preserving the facts that the energy ux is predominantly carried by the Poynting ux close to the pulsar pole and by the particles close to the termination shock. The consistency between the experiments and the simulation provides compelling evidence that strong toroidal magnetic elds and the associated MHD kink instabilities are the cause of the observed jet structure, and that the simulation has captured the basic physics behind kink behaviour in jets. Furthermore, this comparison conrms the hypothesis that the observed directional change of the Crab jet can be caused by strong toroidal magnetic elds and associated MHD kink instabilities.

Other evidence of the relevance of our experiments to the jet in the Crab nebula is provided by several important dimensionless parameters. Both jets have a Lorentz factor of the order of unity (G 1 for the laboratory plasma jet and GE1.09 for the Crab

Nebula jet30). Similarity in the MHD equations requires that the dissipative processes be negligible for both systems. This requirement is met if the viscosity, thermal conduction, and magnetic diffusion terms can be neglected in the momentum, energy, and generalized Ohms law equations. Equivalently, a number of corresponding dimensionless numbers, such as the Reynolds number Re( Lvj/n, the ratio of inertial forces to

viscous force, where L is jet scale size and n is the kinematic viscosity), the Pclet number Pe( Lvj/k, the ratio of heat

convection to conduction, where k is the thermal diffusivity), and the magnetic Reynolds number RMe( Lvj/Dm, the ratio of

ow velocity to diffusion velocity, where Dm is the magnetic diffusivity) must be large in both systems. Table 1 shows that all of these numbers are large, demonstrating that these important conditions are met. Table 1 also lists the other physical parameters and dimensionless numbers that are relevant to this laboratory jet and to the jet in the Crab nebula. To scale the laboratory results to the environment of the Crab nebula, the MHD equations need to be invariant under the transformations given below for the two systems17,18:

rlab arcrab; rlab brcrab; Plab cPcrab; vlab c

b

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a b

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Figure 6 | Spatial distributions of the magnetization parameter r and the current density J. (a) Simulated spatial distribution of the magnetization parameter s at t t0 5 ns. (The detailed evolution of the simulation can be accessed online in Supplementary Movie 5). Near the region where the jet was

launched (zB2 mm), both the core of the jet and the bulk ow have sZ1, while near the jet head sB10 210 3. (b) Simulated spatial distribution of the current density (J) at t t0 5 ns. The image clearly shows the kinked morphology of the jet. (The units for x, y, and z axes are cm.)

Table 1 | Physical parameters and similarity scaling between the laboratory jet and the Crab nebula jet.

Parameters and scales Plasma jet in OMEGA experiment* Scaled to the Crab nebulaw The kinked jet in the Crab nebulaw Temperature Te B300 eV B1130 eV

Ionization state Z B3.5 B1 Number density ne B5 1019 cm 3 B10 2 cm 3

Pressure P B4 105 bar B4 10 14 bar

Jet radius rj B5 10 2 cm B1 pc

Jet velocity vj B400 km s 1 o3 105 km s 1 B1.2 105 km s 1

Time scale t B10 9 s B1.5 years Bfew years Magnetic eld B B2 MG B0.6 mG B1 mG Thermal plasma beta b B0.11 oo1

Magnetization parameter s B16 Z1 Mach number M B3 441

Reynolds number Re B2 103 B2 1017

Pclet number Pe B15 B4 1015

Magnetic Reynolds number ReM B3 103 B1 1022

Biermann number Bi B6 B6 108

Radiation number P B3 105 B1 1018

*Near the region of jet launching. wNear the region of the pulsar pole.

The bold entries show the physical quantities from the two systems that can be directly compared through the scalings in equations (3), manifesting how the laboratory experiment parameters scale to match those of the Crab nebula jet.

plume on each foil, and the collision of these plumes forms a high Mach number plasma jet that propagates into the OMEGA chamber32. During laser illumination and heating, BMegagauss B elds (predominantly toroidal) are generated around each expanding, hemispherical plasma plume because of the Biermann battery effect23 due to non-collinear electron density and temperature gradients (rne rTe). The collision of the plasma plumes with B elds

of opposing sign eventually results in magnetic reconnection, leading to the formation a new magnetic topology with strong toroidal elds around the plasma jet32.

Proton radiography. Monoenergetic proton radiography22 has been developed on the OMEGA laser facility and utilized for backlighting of laser-produced plasma jets. From the Lorentz force (FL q(E v B)), deections due to magnetic elds

can be estimated as:

n

q A a

a

AmpVp

Z B dl 4

where a( 1 cm) and A( 28 cm) are distances from backlighter to the subject

target and to the detector in this experiment, respectively; mp is the proton mass and Vp is the proton velocity; q is the proton electric charge, n is the proton

deection distance and dl is the differential pathlength along the proton trajectory. This technology22 consists of a monoenergetic proton backlighter source and a matched imaging detector.

The backlighter is formed by an exploding-pusher implosion with aD3He- (deuterium-helium-3) lled, glass-shell capsule22 driven by 1630 of the60 OMEGA laser beams31. The capsule has a typical diameter B420 mm and shell thickness B2 mm, lled with 18 atm of equimolar D3He gas. The laser delivered B10 kJ in a 1 ns square pulse. Supplementary Table 1 summarizes the characteristics of the typical backlighter used in these experiments. The timing of the backlighter implosions was adjusted to provide radiographic images at different times relative to when the lasers turned on. The detection system33 consists of a layered assembly of metallic foils and solid-state nuclear track detector CR-39 on which backlighting protons are recorded at 100% efciency. The CR-39 has a chemical composition of C12H18O7. When a charged particle passes through

CR-39, it leaves a trail of damage along its track in the form of broken molecular chains and free radicals. The amount of local damage along the track is related to the local rate at which energy is lost by the particle. In particular, since dE/dx is different for protons at different energies, protons with different energies result in different track diameters. In this experiment, the CR-39 is etched for 23 h in a 6N solution of NaOH, which reveals the tracks with diameters on the order ofB10 mm. An automated microscope system scans and records information about

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the protons tracks, including their location on the piece of CR-39. Custom software is used to determine track properties and to transform that information into an image of proton uence incident on the CR-39.

3D numerical simulation. The 3D Cartesian radiation-MHD simulation of the experiment was performed using FLASH27,28,34, a publicly available, multi-physics, nite-volume, shock-capturing code27. The simulation takes advantage of the full range of HEDP capabilities of the code, so as to accurately model the physical processes in play. The MHD equations are evolved using a directionally unsplit staggered mesh solver35, extended to three temperatures36, adding also the Biermann battery effect23,37,38. We include non-ideal effects such as explicit Spitzer resistivity, implicit thermal conduction and heat exchange, as well as multi-group radiation diffusion with multi-material tabulated opacities and equations of state. The laser energy deposition is accurately modelled using a 3D optical ray trace laser package39.

The computational domain spans 0.5 cm in X and Y, and 1 cm in Z (Supplementary Fig. 4), and is discretized on B3.3 107 zones (B20 mm cell size).

The reconstruction is carried out with a Piecewise Parabolic Method40, employing a minmod limiter. The Godunov uxes are recovered with an HLLC (Harten, Lax and van Leer-Contact)41 Riemann solver. Outow boundary conditions are imposed on all sides. The experimental target is modelled as two 3 3 mm

polystyrene foils at a density of 1.04 g cm 3 and room temperature with an angle of 60 between them. A 3o laser beam (comprised of 1.6 104 rays) with a 1 ns

square pulse prole and 1 kJ of energy illuminates each of the two foils. The incidence angle and the SG8 phase plates (with very similar characteristics to the SG4 plates used in the experiment) determine the spot size and shape for each beam. The beams point at the center of the foils, albeit one of the beams is offset by B100 mm in the Z direction towards the targets base to introduce an asymmetry that excites the m 1 (kink) mode. Conversely, we introduce a small-amplitude,

time-dependent, sinusoidal perturbation13,42 on the transverse velocity components in the interaction region where the jet is formed, so as to excite the m 0 (sausage) mode. The amplitude of the perturbation is 1% of the ow speed,

with a period 0.1 t 0.1 ns, where t is the systems timescale. The evolution of

the system is followed for 5 ns.

Data availability. The authors declare that the data supporting the ndings of this study are available within the article and its Supplementary Information les, and are available from the authors on request. The FLASH code is publicly available through the webpage of Flash Center, University of Chicago (ash.uchicago.edu).

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Acknowledgements

We thank the OMEGA operations and target fabrication crews for their assistance in carrying out these experiments and R. Frankel, and E. Doeg for their help in processing of CR-39 data used in this work. The experiments were supported in part by US DOE (Grant No. DE-FG03-09NA29553, No.DE-SC0007168), LLE (No.414090-G), NLUF (No.DE-NA0000877), FSC (No.415023-G) and LLNL (No. B580243). Numerical simulations were supported in part by the US DOE NNSA ASC under Field Work Proposal No. 57789 to the Argonne National Laboratory, and by NIH through resources provided by the Computation Institute and the Biological Sciences Division of the University of Chicago and Argonne National Laboratory (Grant 1S10OD018495-01). Partial support from the European Research Council under the European Communitys Seventh Framework Programme (FP7/2007-2013)/ERC grant agreements no. 256973 is acknowledged. The software used in this work was developed in part by the DOE NNSA ASC- and DOE Ofce of Science ASCR-supported Flash Center for Computational Science at the University of Chicago. This research used resources from the Directors Discretionary Program of the Argonne Leadership Computing Facility, supported by DOE Ofce of Science User Facility (DE-AC02-06CH11357).

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ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13081

Author contributions

C.K.L. conceived and led the experiments, and analyzed the data. P.T. and D.L. performed the FLASH numerical simulations, and contributed to data interpretation. M.J.R. contributed to execution and discussion of experiments. R.K.F. and D.H.F. supported the 4o Thomson scattering measurements. F.H.S. and R.D.P. contributed to the development of proton radiography and the discussion of experiments. M.K., J.A.F., H.G.R., H.S., A.B.Z., P.A.A., H.S.P., B.A.R., D.D.R., S.C.W., R.B., A.F., S.X.H., T.C.S., P.H., C.C.K., R.P.D., G.G., P.A.N., S.V.L. and N.C.W. contributed to support the experiments and technical discussions. C.K.L., P.T., G.G. and D.L. wrote the paper.

Additional information

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How to cite this article: Li, C. K. et al. Scaled laboratory experiments explain the kink behaviour of the Crab Nebula jet. Nat. Commun. 7, 13081 doi: 10.1038/ncomms13081 (2016).

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