Particle acceleration in kink-unstable jets
Jordy Davelaar, Alexander A. Philippov, Omer Bromberg, Chandra B. Singh
DDraft version May 27, 2020
Typeset using L A TEX twocolumn style in AASTeX63
Particle acceleration in kink-unstable jets
Jordy Davelaar ,
1, 2
Alexander A. Philippov ,
2, 3
Omer Bromberg , and Chandra B. Singh Department of Astrophysics/IMAPP, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA Moscow Institute of Physics and Technology, Dolgoprudny, Institutsky per. 9, Moscow region, 141700, Russia The Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel (Received March 6, 2020; Revised April 24, 2020; Accepted May 22, 2020)
Submitted to ApJLABSTRACTMagnetized jets in GRBs and AGNs are thought to be efficient accelerators of particles, however, theprocess responsible for the acceleration is still a matter of active debate. In this work, we study the kink-instability in non-rotating force-free jets using first-principle particle-in-cell simulations. We obtainsimilar overall evolution of the instability as found in MHD simulations. The instability first generateslarge scale current sheets, which at later times break up into small-scale turbulence. Reconnectionin these sheets proceeds in the strong guide field regime, which results in a formation of steep powerlaws in the particle spectra. Later evolution shows heating of the plasma, which is driven by small-amplitude turbulence induced by the kink instability. These two processes energize particles due to acombination of ideal and non-ideal electric fields.
Keywords: acceleration of particles — plasmas — instabilities — magnetic reconnection — turbulence INTRODUCTIONMagnetized relativistic jets are efficient particle accel-erators. They are observed in a broad variety of as-tronomical sources, e.g., X-ray binaries, Active GalacticNuclei (AGN), and gamma-ray bursts (GRBs), see e.g.Pudritz et al. (2012) for a review on jets. These sourcesare typically observed over the entire electromagneticspectrum from radio to γ -rays, and are considered asmain candidates for accelerating ultra-high-energy cos-mic rays. Their observed spectral energy distributionssuggest that a large fraction of the radiatively impor-tant electrons are non-thermal. However, the way thesejets accelerate electrons is still uncertain. An effectivemechanism for particle acceleration in highly magne-tized flows is the dissipation of magnetic energy via re-connection in thin current sheets (Zenitani & Hoshino2001; Cerutti et al. 2014; Sironi & Spitkovsky 2014; Guoet al. 2014). The reconnection is driven by the plas- Corresponding author: Jordy [email protected] moid instability (Loureiro et al. 2007), which continu-ously breaks current sheets into plasmoids separated byX-points. In the case of relativistic reconnection, strongelectric fields in the vicinity of X-points accelerate elec-trons up to γ max ≈ σ (Werner et al. 2016), where σ = B / (4 πm e nc ), B is the magnetic field strength, m e is the electron mass, and n is the plasma numberdensity. A secondary acceleration phase that happensinside the plasmoids pushes particles to higher energies(Petropoulou & Sironi 2018). The study of reconnectionis usually done with kinetic plasma simulations, whichmodel reconnection from first principles by using Harrissheets as initial conditions. However, it is still unknownif and where such sheets can form in realistic jets, andwhat the geometry of the reconnecting magnetic field is.Global magnetohydrodynamics (MHD) simulationsshow that near the launching site jets expand andquickly loose transverse causal contact, making themstable for current-driven instabilities (Tchekhovskoy &Bromberg 2016; Bromberg & Tchekhovskoy 2016). Asthe pressure of the confining medium becomes impor-tant, the flow is re-collimated and regains causal con-tact. As a result, the toroidal hoop stress becomes ef- a r X i v : . [ a s t r o - ph . H E ] M a y Davelaar et al. fective, and compresses the flow into forming a nozzle,which may become prone to internal kink-instability.In the context of astrophysical jets, the kink instabil-ity is generally divided into two types: internal kink ,which grows at the jet’s core and is not affecting the jetboundaries and, external kink , which grows on the jetboundaries and perturbs the entire jet body. Bromberg& Tchekhovskoy (2016) and Tchekhovskoy & Bromberg(2016) showed that internal kink mode that grows atre-collimation nozzles of collimated jets could lead toefficient magnetic energy dissipation, reducing the jet’smagnetization parameter, σ , which is high before theflow enters the nozzle, down to σ ≈ B φ /B z ≈ . z defines the direction along the jet’s axis, and φ corre-sponds to a toroidal direction with respect to the sameaxis). Coincidentally, similar conditions are expected incollimation nozzles of relativistic jets.Particle acceleration in the process of kink instabil-ity was studied using PIC simulations by Alves et al.(2018). They considered a pressure supported jet wherethe toroidal magnetic field component dominates andfound significant particle acceleration solely due to thegeneration of an ideal coherent electric field along thejet axis. Since their setup is pressure supported, force-balance implies, ∇ p = (cid:126)J/c × (cid:126)B , which effectively trans-lates to p ≈ B / π (hereafter, p is the plasma pressure,and (cid:126)J = c ∇ × (cid:126)B/ π is the plasma current density).Therefore, their setup considers an effective, ”hot”, mag-netization σ h = B / πw ≈
1, where w = ε + p is the gasrelativistic enthalpy, and ε is the plasma internal energy.AGN jets are, however, thought to be launched with σ h (cid:29) σ h (cid:29) B φ /B z ≈ .
0. We findno coherent axial electric field in our setups, and findthat particle acceleration occurs due to a combinationof reconnection and turbulence. NUMERICAL SETUPThe first setup we consider is a force-free non-rotatingjet originally investigated with MHD simulations byMizuno et al. (2009) and by Bromberg et al. (2019).The magnetic field profile consists of a strong verticalfield, B z , dominated core surrounded by a region domi-nated by a toroidal field component, B φ . The magneticfield profile is given by, B z = B [1 + ( r/r core ) ] ζ , (1) B φ = B z r core r (cid:115) [1 + ( r/r core ) ] ζ − − ζ ( r/r core ) ζ − , (2)where B is a scale factor that determines the value ofmagnetization parameter at the axis, r core sets the sizeof the kink unstable core, and r is the cylindrical radius.For r (cid:29) r core both field components asymptotically ap-proach zero. The free parameter ζ sets the behavior ofthe magnetic pitch, P = rB z /B φ . For ζ < r , for ζ = 1 the pitch is constant, andfor ζ > r . In this workwe consider two representative values of ζ , ζ = 0 . ζ = 1 .
44 (Decreasing Pitch,DP). The radial profile of the pitch is important for theglobal evolution of the instability. In the case where thepitch is increasing with the cylindrical radius, resonantsurfaces confine the instability to the kink unstable core(Rosenbluth et al. 1973), while in the case of a decreas-ing pitch profile the instability becomes disruptive.We also consider a force-free setup by Bodo et al.(2013), which has a non-monotonic pitch profile and astrong confining vertical magnetic field outside of thekink unstable core. We term this profile as embedded article acceleration in kink-unstable jets B φ = B Rr (cid:114)(cid:16) − e − ( r/r core)4 (cid:17) , (3) B z = B RP r (cid:114)(cid:16) − √ π ( r core /P ) erf [( r/r core ) ] (cid:17) , (4)where R is the cylindrical radius of the domain’s outerboundary, and the parameter P is the value of themagnetic pitch at the axis. We consider a value of P = 1 . r core . The magnetic field configuration qual-itatively differs from the IP and DP setups, since for r > r core the axial component of the magnetic field, B z ,asymptotes to a constant value. This vertical magneticfield leads to a strong confinement of the jet.We perform our simulations in the frame co-movingwith the jet, thus the plasma is initially at rest. Weuse the relativistic PIC code Tristan-MP (Spitkovsky2005). The simulations are performed in a Cartesianthree-dimensional computational box. The box lengthin z , L z , is set to fit two wavelengths of the fastest-growing kink mode λ max = 8 πP / P is thevalue of the pitch at the axis (Appl et al. 2000). Weinitialize our calculations with a cold uniform electron-positron plasma with temperature T = 10 − m e c /k B ,and a density of ten particles per cell giving a total of ∼ particles in the simulation box. We set both elec-trons and positrons to drift in opposite directions withvelocities (cid:126)v dr = ± (cid:126)J/ ne to generate the currents thatsupport the initial magnetic field profile. The simula-tions are run up to t = 300 r core /V A , where V A is theAlfv´en speed defined as V A = c (cid:112) σ / (1 + σ ), and σ isthe magnetization at the jet axis, σ = B / (4 πm e nc ).We set r core = 60 cells, and use grid sizes of: a) DP,3000 × × × x, y, z ) directions respectively. We studied thedependence of our results on the scale separation byvarying the ratio between the size of the kink unsta-ble core and the plasma skin depth, d e = c/ω p , where ω p = (cid:112) πe n/m e is the plasma frequency. We varied d e from 3 to 6 cells. The simulations presented in this Let-ter use a scale separation of r core /d e = 20, where d e = 3cells, which is sufficient to recover the overall MHD evo-lution (see Bromberg et al. (2019) and Appendix A). Inthe z-direction, we apply periodic boundary conditions,while at the boundary in the x-y plane we have an ab-sorbing layer for both fields and particles (Cerutti et al.2015). For all three setups, we present simulations forthree values of the magnetization parameter at the axis, σ = 10 , ,
40, which correspond to β = 8 πnT /B =2( k B T /m e c )(1 /σ ) = [20 , , × − . Larmor gyra- tion period 2 π/ √ σ ω p is resolved with at least a fewtime steps for all simulation setups. RESULTSOur PIC simulations show the same global behav-ior found in our MHD simulations (Bromberg et al.2019). The sufficiently large separation between fluidand kinetic scales allows us to obtain similar growth-rates in the linear stage, and a comparable amount ofelectromagnetic energy dissipation as in the MHD sim-ulations (between 15-20% of the initial electromagneticenergy in all three setups, see appendix A). Initially,the most unstable mode is a kink mode with a longitu-dinal wavenumber l = 2, and an azimuthal wave number m = − . It gives rise to a global helical current sheetat the edge of the kink unstable core. Later on the l = 2 mode transforms into an l = 1 mode. Eventually,the global current sheet breaks up generating small-scalecurrent sheets and turbulence that mediate further dis-sipation of the magnetic energy. A similar behavior wasobserved in our MHD simulations.In all three setups, we observe particle energizationdue to an electric field that is parallel (non-ideal) orperpendicular (ideal) to the local magnetic field direc-tion. As the instability becomes non-linear, we observea strong burst of particle energization due to a non-ideal electric field, which takes place in current sheetsat the jet’s periphery (see Fig. 1 for a 3D visualizationof a down-sampled distribution of simulation particlescolor-coded by their Lorentz factors). These sheets havestrong guide fields, which are comparable to the recon-necting field at the periphery and can become up to ∼ α , of the DF, f ( γ ) ∝ γ − α . Inour work we find α ≈ −
5, which is in agreement withtheir results for comparable strengths of the reconnect-ing and guide magnetic field components. At this stage,we find the maximum energy of accelerated particles to The longitudinal wavenumber is defined as l = k z L z / π , where k z is the component of the wave vector in the z direction. The az-imuthal wavenumber, m , defines the type of mode, where modeswith | m | = 1 are known as kink modes. For a more detaileddiscussion on the properties and behavior of the unstable mode,see Bromberg et al. 2019 Davelaar et al.
Figure 1.
From left to right: decreasing pitch (DP), increasing pitch (IP), and embedded pitch (EP) cases. In the top row,thick green lines show magnetic field lines. Subsampled distribution of energetic particles is visualized as dots colorcoded bytheir Lorentz factors. Plots are computed at t = 60 , , r core /V A correspondingly, the onset times of the acceleration episodein each configuration (see bottom panel). The middle row shows distribution functions (DFs) for all three setups, each set oftwo plots shows DFs at the end of the simulation on the left for all three σ = 10 , ,
40 values, and the time evolution of thespectrum of the σ = 40 run on the right. Panel b also includes Maxwellians fitted to the DFs, panel e, and h show powerlaws fitted to the DFs. The bottom row shows statistics of the acceleration events as a function of simulation time and particleenergy. For a given particle at a particular energy, we classify the acceleration episode based on if parallel or perpendicularelectric field dominates particle energization. N (cid:107) and N ⊥ are the numbers of parallel and perpendicular acceleration events,respectively. Initial particle distribution is a Maxwellian with a low temperature, 10 − m e c /k B , and all the spectra correspondto energized particles with γ > scale as γ max ≈ χr core /r L0 , where r L0 = m e c /eB is anominal cold relativistic gyroradius, and χ ≈ / .In all our setups, we find that the self-excitedturbulence is small-amplitude, e.g. the mean field This conclusion is based on our simulations with differentstrengths of the jet’s magnetic field. Increasing the jet’s sizeis numerically expensive in our current setups, as the jet signif-icantly expands laterally during the simulation time. We willconduct a systematic study of the dependence of γ max on thejet’s size in the future work. is stronger compared to the fluctuating compo-nent. We evaluate the amplitude of turbulenceas ξ = | ( B ( (cid:126)x ) − (cid:104) B ( (cid:126)x ) (cid:105)| / (cid:104) B ( (cid:126)x ) (cid:105) , where (cid:104) B ( (cid:126)x ) (cid:105) = (cid:82) B ( (cid:126)x (cid:48) ) e −| (cid:126)x − (cid:126)x (cid:48) | / σ d (cid:126)x (cid:48) is the magnetic field strengthaveraged with a Gaussian kernel, and σ std = r core / σ std ∈ [ r core / , r core /
2] and found no qualitative differences inour conclusion based on this analysis. The value of ξ itself varies spatially. We quantify the amplitude of tur-bulent motions by measuring the range of ξ inside thekink unstable core. In all three setups we find ξ ≤ . article acceleration in kink-unstable jets Figure 2.
Formation of strong current layers in the onset of the non-linear stage of the kink instability. From left to rightcolumns: DP, IP, and EP cases. First row: slices of the axial component of the current, j z , in the x-y plane. Second row: slicesof the toroidal component of the current, j φ , in the x-z plane. Black/white lines show the in-plane components of the magneticfield. Insets show the distribution of E · B as color and highlight the E · B (cid:54) = 0 regions where in-plane magnetic field componentsshow anti-parallel orientation. A subsample of particles with γ > E · B values theyexperience. Their locations clearly correlate with strong current layers. The E · B colorbar is assigned to both the insets andthe particle colorcoding. The small-amplitude turbulence leads to heating of theplasma, which forms a secondary Maxwellian in the DF(see panels b and e in Fig. 1). The temperature of thisMaxwellian scales with the initial magnetization param-eter, namely, k B T /m e c ∝ σ . Particle energization atthis stage is dominated by the perpendicular componentof the electric field. To quantify the importance of bothparallel and perpendicular electric fields during the evo-lution of the instability we trace every tenth particlein our simulations with γ >
2. We classify individualacceleration events based on if the parallel or the per-pendicular electric field component dominates the ac-celeration by looking at the absolute values of energygained by each process. The statistics of accelerationepisodes are shown in Fig. 1, bottom row. In all threesetups, a large fraction of the particles undergo parallelacceleration immediately after the instability becomesnon-linear, while in the IP and DP case the perpen-dicular acceleration dominates at larger energies. Wefind that the number of acceleration events due to theparallel electric field increases at higher values of themagnetization parameter.In Fig. 3 we show an example of two particle trajecto-ries in the IP case that exhibit acceleration due to eithera parallel or a perpendicular electric field. In the case ofparallel acceleration (particle 1), the energization hap-pens in the current sheet at the edge of the kink unstablecore, where E · B (cid:54) = 0. In the perpendicular case (par- ticle 2), the particle is initially accelerated by a parallelelectric field and then ends up in the turbulent core,where it undergoes further acceleration to higher ener-gies mediated by the perpendicular electric field. Theseparticle trajectories are representative for all three se-tups, although the relative contribution of parallel andperpendicular episodes differs, as can be seen in Fig. 1.The DP simulation shows a strong acceleration eventaround t = 60 r core /V A , as is shown in Fig. 1c. At thistime the l = 2 mode forms a helical current sheet at theedge of the kink unstable core, see Fig. 1a. The sheetis produced by the relative shear between the magneticfield inside the jet’s core and at the periphery and is sup-ported by strong currents (see Fig. 2a). These currentlayers contain most of the energized particles and corre-late with locations where E · B (cid:54) = 0. In these layers, someof the magnetic field components exhibit anti-parallelorientations, see inset in Fig. 2d where B z is the re-connecting field component. This shows that non-idealelectric fields in current sheets are the driving mecha-nism of the energization. The statistics of accelerationevents in the DP case is shown in Fig. 1c, where theburst of acceleration events at t = 60 r core /V A coincideswith the increasing number of non-thermal particles inthe DF (see Fig. 1b, right panel). Clearly, a majority ofthe particles is initially accelerated via parallel electricfields. At later times a second acceleration stage due toa perpendicular electric field in turbulence pushes the Davelaar et al.
Figure 3.
Trajectories of two accelerated particles in the IPcase. Top panel shows E · B in the x-z plane, overplotted withtrajectories of a particle (1) that undergoes mainly parallelacceleration, and a particle (2) that undergoes perpendicularacceleration. Lower panel shows the time-integrated work ofthe electric field, E · v , along the trajectory of these particles,the contribution of parallel and perpendicular components tothe integrated E · v , and particle Lorentz factors as a functionof time. The dashed lines correspond to particle 1, and solidlines correspond to particle 2. Particle 1 is predominantlyaccelerated by a parallel electric field in the current layer atthe edge of the kink unstable core, while particle 2 experi-ences strong acceleration by perpendicular electric fields inthe jet’s core. particles to higher γ values. For all three values of σ ,the DF shows the growth of a secondary Maxwellianwith a temperature that scales linearly with σ , as isexpected from the energy conservation argument . Themeasured amplitude of the turbulence for σ = 40 is ofthe order of ξ ≤ . t = 110 r core /V A . At this time, the l = 1 mode devel-ops a current sheet at the jet’s periphery. Again, thelocation of particle acceleration correlates with currentsheets where E · B (cid:54) = 0, as can be seen in Fig. 2. Thestatistics of acceleration events in Fig. 1f clearly shows The Larmor radius of particles with γ = σ in the jet’s core is r L = σ r L0 = √ σ d e , which corresponds to 0 . r core for σ =40. The size of the kink unstable core, however, grows to ∼ . λ max ∼ r core in the non-linear stage, so further particleacceleration is in principle possible. The plasma skin depth, d e = (cid:112) m e c (cid:104) γ (cid:105) / πe n , also increases as a result of the heating (seealso Appendix A for the discussion of the scale separation in theDP case). that at this time, the majority of particles are acceler-ated due to parallel electric fields. The resulting spectrain Fig. 1 shows a power law with α ≈ . σ = 40,and a secondary Maxwellian that slowly grows over time.We measure the amplitude of turbulence in the core tobe of the order of ξ ≤ .
05, which is smaller compared tothe DP simulation. This can explain the slower growthof the secondary Maxwellian in the spectra.For the EP case, at t = 50 r core /V A the particle ac-celeration starts when the l = 2 mode grows. Again,current sheets coincide with locations of E · B (cid:54) = 0,where particles are accelerated due to parallel electricfields. The resulting DF shows a clear power-law withindex α ≈ σ = 40, and a modest steepening ofthe spectrum for lower values of σ . The turbulencein the EP setup has a small-amplitude, of the order of ξ ≤ .
01, which could explain the lack of a secondaryMaxwellian in the spectra. This correlates with a strongdominance of parallel acceleration events in the particleenergization history, which takes place over the entiresimulation duration in the EP case, as shown in Fig. 1i.Thin current sheets are known to be unstable to atearing instability, and subsequent plasmoid instabilityof secondary sheets (Loureiro et al. 2007). While lim-ited scale separation of our global simulations preventsus from observing the plasmoid instability, we do observethe initial tearing of current sheets generated by the rel-ative shear of the magnetic field at the jet’s boundary.An example of the IP case is presented in Fig. 4, wheredifferent quantities show plasmoid-like structures in dif-ferent parts of the current sheet at the jet’s boundary.We plan to study kink-unstable configurations presentedin this work with relativistic resistive MHD simulationswith adaptive mesh refinement (Ripperda et al. 2017), inorder to better resolve plasmoid chains in these currentsheets. DISCUSSION AND CONCLUSIONReconnection and turbulence in collisionless plasmawere studied so far in idealized periodic boxes. Ourstudy shows how they can be self-consistently excitedand energize particles in the process of kink instabilityin highly magnetized jets. We find that acceleration incurrent sheets dominates at low particle energies; andhappens due to non-ideal electric fields that lead to theformation of steep power laws in the DF, due to strongguide fields at the reconnection sites. The presence ofacceleration due to non-ideal electric fields is in contrastwith the study of Alves et al. (2018). This difference islikely caused by the fact that their pressure-supportedjet configuration corresponds to the case σ h ≈
1. As article acceleration in kink-unstable jets Figure 4.
Formation of plasmoids in the IP setup. The first row presents the y component of the current in x-z and x-y planes.The second row shows the z component of the electric field. In all panels, insets zoom into plasmoid-like structures. In all panelsdistances are measured in units of the fastest-growing kink mode λ max = 8 πP /
3, where P is the value of the pitch on the axis. we discuss above, we also find no coherent axial electricfield in our highly magnetized, force-free setups.While we observe plasmoid formation, our limitedscale separation does not allow the formation of a fullplasmoid chain, and a study of the Fermi-like process ofparticle acceleration in plasmoids (Petropoulou & Sironi2018). Future large-scale local simulations of reconnec-tion with a strong guide field are needed to investigatethis potentially important mechanism of particle accel-eration (Drake et al. 2006). We further find that en-ergization due to scatterings on small-amplitude turbu-lent fluctuations leads mostly to plasma heating. Thisis in contrast to local simulations of particle energiza-tion in high-amplitude turbulence (Zhdankin et al. 2013,2017; Comisso & Sironi 2018), which showed formationof prominent power laws. Motivated by our results inthe DP case, where particle energization in turbulenceerases the initial reconnection spectra, for the cases oflarge-amplitude turbulence we anticipate the power lawsto extend up to energies corresponding to the confine-ment condition, γ max ∼ r core /r L0 (Zhdankin et al. 2017).Future work should incorporate realistic jet struc-tures, including rotation and velocity shear, and developan understanding of how to extrapolate the results ofsimulations with limited scale separation, such as ours,to parameters of astrophysical systems. Similarly to this work, these studies will identify the geometry of currentsheets and quantify the amplitude of the excited turbu-lence and thus, allow to quantify particle accelerationand emission of energetic photons from kink-unstablejets in GRB and AGN from first principles.ACKNOWLEDGMENTSThe authors thank A. Bhattacharjee, L. Comisso, H.Hakobyan, B. Ripperda, L. Sironi, A. Spitkovsky and A.Tchekhovskoy for insightful comments over the courseof this project. J.D. is funded by the ERC SynergyGrant 610058, Goddi et al. (2017). The authors thankthe anonymous referee for insightful comments. O.B.and C.S. were funded by an ISF grant 1657/18 and byan ISF (I-CORE) grant 1829/12. O.B. and S.P. werealso supported by a BSF grant 2018312. S.P. acknowl-edges support by the National Science Foundation underGrant No. AST-1910248. The Flatiron Institute is sup-ported by the Simons Foundation. Software:
Tristan-MP (Spitkovsky 2005), python (Oliphant 2007; Millman & Aivazis 2011), scipy (Joneset al. 2001), numpy (van der Walt et al. 2011), matplotlib (Hunter 2007),
VisIt (Childs et al. 2005).APPENDIX
Davelaar et al. t [r core /V A ] E E M / E E M , m a x t [r core /V A ] t [r core /V A ] E / E m a x DP IP PICMHD EP Figure 5.
Comparison of the linear growth rates of the instability and electromagnetic energy dissipation in PIC and MHDsimulations. From left to right: DP, IP, and EP case. Panels in the top row show the evolution of electric energy as a function oftime, which highlights a stage of exponential growth. Panels in the bottom row show the dissipation of electromagnetic energy.In all panels, red lines represent PIC simulations, and blue lines correspond to MHD simulations.A.
COMPARISON WITH MHDIn order to ensure that our simulations probe the large-scale behavior correctly, we compare the growth rates ofthe kink instability and electromagnetic dissipation rates of our PIC simulation with MHD simulations of the sameconfigurations from Bromberg et al. (2019). The simulation box sizes are identical, and we choose σ = 10, theseparation between the size of the kink-unstable core and the plasma skin depth in the case of PIC r core /d e =20, forthis comparison. To compute dissipation rates in both PIC and MHD simulations, we correct for the electromagneticenergy that leaves through the box boundary A (edge of the absorbing boundary for PIC, and the edge of the boxwith standard outflow boundary condition in the case of MHD).The growth rates of the electric energy are shown in the top panels of Fig. 5. In the PIC simulations, the onsetof the instability is slightly delayed with respect to MHD. We, therefore, shifted the PIC curves so that they overlapwith the MHD curves to ease the comparison of the rates by eye. The linear growth shows very similar rates in PICand MHD. In the PIC simulations, the instability initially kicks in on kinetic scales at the jet’s boundary, which is notobserved in the MHD simulations. This behavior is significantly more prominent in simulations with r core /d e = 10,which highlights the importance of using large scale separation in PIC simulations. The small scale plasma instabilitiescause some discrepancies between the linear growth rates at the very early times. Also, the initial amplitude of theelectric field is higher in the PIC runs because of the particle noise. However, when the kink instability grows and thejet expands at t ≥ r core /V A , the growth rate in PIC becomes indistinguishable from the one observed in MHD (seeBromberg et al. (2019) for MHD simulations). At this stage, the growth rates are observed to be nearly identical inPIC and MHD for all three setups. The magnetic field dissipation is shown in the bottom row of Fig. 5. In the DP,IP, and EP cases, the evolution and dissipation rates up to t = 200 r core /V A are very similar. This comparison showsexcellent agreement between the large scale behavior of the kink instability in the PIC simulations presented hereand the MHD simulations from Bromberg et al. (2019). In the DP case the MHD simulation continues to dissipate,while PIC saturates at around E EM /E EM , t=0 / ≈ .
8. The discrepancy is likely due to the fact that the separationbetween the jet scale and the skin depth scale shrinks because of the plasma heating during the turbulent stage of theinstability, which is most prominent in the DP case. For our box size, L z = 2 λ max , running MHD simulations furtherdoes not lead to the larger amount of the dissipation for the cases of IP and EP. However, the final state is not fully article acceleration in kink-unstable jets ∼
30% and ∼
40% for theIP and EP cases, correspondingly (see Fig. 6 in Bromberg et al. 2019). A significantly higher, up to 40%, amount ofdissipation is observed in MHD simulations of the DP case with the same box size as chosen in this Letter. As wementioned above, the decreased scale separation is a likely reason for this discrepancy.REFERENCES40% for theIP and EP cases, correspondingly (see Fig. 6 in Bromberg et al. 2019). A significantly higher, up to 40%, amount ofdissipation is observed in MHD simulations of the DP case with the same box size as chosen in this Letter. As wementioned above, the decreased scale separation is a likely reason for this discrepancy.REFERENCES