Reconnection-driven particle acceleration in relativistic shear flows
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Reconnection-driven particle acceleration in relativistic shear flows
Lorenzo Sironi, Michael E. Rowan, and Ramesh Narayan Department of Astronomy and Columbia Astrophysics Laboratory, Columbia University, New York, NY 10027, USA Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA (Received xx; Revised xx; Accepted xx)
Submitted to ApJLABSTRACTParticle energization in shear flows is invoked to explain non-thermal emission from the boundariesof relativistic astrophysical jets. Yet, the physics of particle injection, i.e., the mechanism that allowsthermal particles to participate in shear-driven acceleration, remains unknown. With particle-in-cellsimulations, we study the development of Kelvin-Helmholtz (KH) instabilities seeded by the velocityshear between a relativistic magnetically-dominated electron-positron jet and a weakly magnetizedelectron-ion ambient plasma. We show that, in their nonlinear stages, KH vortices generate kinetic-scale reconnection layers, which efficiently energize the jet particles, thus providing a first-principlesmechanism for particle injection into shear-driven acceleration. Our work lends support to spine-sheathmodels of jet emission — with a fast core/spine surrounded by a slower sheath — and can explain theorigin of radio-emitting electrons at the boundaries of relativistic jets. INTRODUCTIONShear flows are ubiquitous in space and astrophysicalplasmas. The free energy of the velocity shear is ofteninvoked to accelerate charged particles to non-thermalenergies (e.g., Rieger 2019) — via a mechanism akinto the Fermi process in converging flows (Fermi 1949).Shear-driven acceleration relies on particles scattering inbetween regions that move toward each other because ofthe velocity shear. This results in a secular energy gain,as long as the particle mean free path is sufficiently longto sample a significant velocity gradient. In fact, themajor unknown of shear-driven acceleration models isthe so-called “injection stage,” i.e., the mechanism(s) topromote thermal particles — that cannot participate inshear acceleration, due to their short mean free path —to non-thermal energies.Shear layers may be prone to the Kelvin-Helmholtzinstability (KHI), driven by the transfer of momentumacross the shear interface. The KHI has been thor-oughly studied with linear stability analysis (e.g., Blu- [email protected]@[email protected] men et al. 1975; Ferrari et al. 1978, 1980; Sharma &Chhajlani 1998; Komissarov 1999; Bodo et al. 2004; Os-manov et al. 2008; Prajapati & Chhajlani 2010; Sobac-chi & Lyubarsky 2018; Berlok & Pfrommer 2019) andfluid-type simulations, including relativistic effects andmagnetic fields (e.g., Keppens et al. 1999; Ryu et al.2000; Zhang et al. 2009; Hamlin & Newman 2013; Millaset al. 2017). In shear layers with flow-aligned magneticfields, the KHI opens a new possibility for dissipation: inaddition to feeding off the free energy from the velocityshear, KH vortices can wrap up the field lines onto them-selves, leading to dissipation via reconnection (Faganelloet al. 2008; Nakamura & Fujimoto 2008; Faganello et al.2010; Faganello et al. 2012; Henri et al. 2013; Nakamuraet al. 2013; Nakamura & Daughton 2014; Fadanelli et al.2018; see also Tolman et al. 2018, for reconnection inKHI-stable shear flows).In this Letter, we employ fully-kinetic particle-in-cell(PIC) simulations to demonstrate that particles are ef-ficiently accelerated at reconnecting current sheets thatare self-consistently generated by the nonlinear stages ofthe KHI (Faganello et al. 2008; Faganello et al. 2012).Our study is motivated by the limb-brightened appear-ance of relativistic jets, e.g., in Cygnus A (Boccardiet al. 2016) and M87 (Walker et al. 2018). Instabil-ities at relativistic jet boundaries are seen in general- a r X i v : . [ a s t r o - ph . H E ] S e p relativistic magnetohydrodynamic (MHD) simulations(Chatterjee et al. 2019), which however cannot probethe physics of particle acceleration. Our work providesa first-principles mechanism for particle injection intoshear-driven acceleration in relativistic magnetically-dominated jets. This lends support to spine-sheathmodels of jet emission (Sikora et al. 2016) and can ex-plain the origin of radio-emitting electrons at the bound-aries of relativistic jets. NUMERICAL METHOD AND SETUPWe perform ab initio
PIC simulations with TRISTAN-MP (Buneman 1993; Spitkovsky 2005). We conducttwo-dimensional (2D) simulations in the xy plane, re-taining all three components of particle velocities andelectromagnetic fields. In the initial state, the fluid bulkmotion is along y , and the gradient of the velocity isalong x . The domain has length L y along y (0 ≤ y/L y ≤ L x = 3 L y along x ( − . ≤ x/L y ≤ . | x | /L x (cid:46) .
25, and a weakly magnetized stationaryambient plasma at | x | /L x (cid:38) .
25 (which we call “wind,”since it should represent the wind of the accretion flow).The simulation is performed in the wind frame.The jet is composed of electron-positron pairs with co-moving density n (including both species) and a smallthermal spread, moving with four-velocity Γ β = 1 . y (we also report results for Γ β = 3 and 10).The jet carries an energetically-dominant magnetic field.The in-plane field strength B j ,y is parameterized by themagnetization σ j ,y = B ,y / (4 πn m e c ), where m e is theelectron mass and c the speed of light. We also initial-ize an out-of-plane field B j ,z ≡ B z = B j ,y tan θ and itsassociated motional electric field E j ,x = − β B j ,z . Ourreference runs employ σ j ,y = 6 . θ = 75 ◦ (corre-sponding to θ (cid:48) = 65 ◦ in the jet frame for Γ β = 1 . y ) and toroidal(along z ) fields in the jet frame (see, e.g., Alves et al.2014; Alves et al. 2015; Liang et al. 2013a,b; Nishikawaet al. 2014, 2016; Pausch et al. 2017 for PIC studies ofshear instabilities in unmagnetized plasmas).The wind is composed of an electron-ion plasma withdensity n w = 128 n (including both species). We typ-ically employ a mass ratio m i /m e = 25, but we havetested that the late-time particle spectrum is the samefor m i /m e = 5, 25, and 100, and also for the artificialcase of an electron-positron wind ( m i /m e = 1). As weshow below, particle energization primarily involves thejet particles, and so our results are insensitive to themass ratio in the wind. The wind is initialized with an out-of-plane field B w ,z = 0 . B j ,z . The wind plasmabeta β p ≈ [ B ,y + (Γ − B j ,z ) ] /B ,z ≈ (cid:29)
1, so thewind is particle-dominated.Our unit of length is the skin depth of jet par-ticles, c/ω p = (cid:112) m e c / (4 πe n ), which we resolvewith 11.3 cells ( e is the positron charge). The elec-tron skin depth and Debye length in the wind aremarginally resolved. Our reference runs have L y ≡ L =3840 cells ≈ c/ω p , but we also present larger runswith L y = 3 L ≈ c/ω p to demonstrate that weachieve asymptotically-converged results.The wind and jet properties are smoothly connectedwith spatial profiles varying as tanh[2 π ( x − x SL ) / ∆]in the vicinity of the shear layers at | x | = x SL ≈ . L x . Our emphasis is on wide shear layers (withthickness ∆ (cid:29) c/ω p ), in application to realistic jet/windboundaries. For ∆ (cid:29) c/ω p , the KHI growth shouldbe independent of kinetic physics, and in fact ourmeasured growth rates are in good agreement withMHD expectations (e.g., Bodo et al. 2004). To en-sure that we start from MHD-scale initial conditions,we choose ∆ to be larger than the largest kineticscale, i.e., the Larmor radius of wind ions, r L , i ≈ (Γ − B j ,z /B w ,z ) (cid:112) ( m i /m e )( n /n w ) c/ω p . We typicallyemploy ∆ = 64 c/ω p , but we report identical resultsobtained with ∆ = 192 c/ω p . The spatial profiles oftemperature, charge density and electric current den-sity in the shear layer follow from pressure equilibriumand Maxwell’s equations.We employ 4 particles per cell in the jet. For computa-tional convenience, in the wind we typically use particleswith a larger numerical weight (fixing the overall windmass density, this gives a lower macro-particle count),but we have carefully checked that our results are insen-sitive to this choice. RESULTSThe temporal evolution of the KHI is presented inFig. 1, where color indicates the out-of-plane field B z ,with in-plane field lines overlaid. The instability devel-ops in two stages: a mode with wavelength λ ≈ L y / λ ≈ L y grows at later times. The correspondinggrowth rates are in good agreement with MHD lineardispersion analysis. The vortices created by the non-linear stages of the KHI bend the in-plane field lines,creating anti-parallel configurations prone to reconnec-tion (panel (d)). The final stage (panel (e)) is character-ized by: ( i ) the persistence of a nearly-unperturbed jetcore (yellow) surrounded by a sheath of weaker B z (red),whose width is ≈ . L y ; ( ii ) the presence of magnetized“clouds” of jet material — on a variety of scales, from Figure 1.
2D evolution of the out-of-plane field B z (colorscale) — in units of the initial jet field B z ≡ B j ,z — at (a) ω p t = 80, (b) ω p t = 3262, (c) ω p t = 4216, (d) ω p t = 5171,and (e) ω p t = 12569, with in-plane field lines overlaid. Themagnetized jet is initially at | x | (cid:46) c/ω p , surrounded bythe wind. plasma scales up to ≈ . L y — in pressure equilibriumwith the surrounding wind plasma.The evolution of the KHI is further analyzed in Fig. 2.The jet starts with bulk four-velocity Γ β y = Γ β = 1 . σ y ≈ σ j ,y = 6 . σ z = σ y tan θ ≈ . σ i ≡ B i / πn m e c is the mag-netization contributed by the field component B i ). As aresult of the KHI, the in-plane field lines are twisted andfolded, and a significant B x develops at the jet bound-aries, with peak magnetization σ x ≈ Figure 2.
Temporal evolution of y -averaged profiles, withcolors from blue to red referring to the same times as pan-els in Fig. 1. (a) Bulk four-velocity along y , in units of thespeed of light, where in each cell the fluid speed is com-puted by averaging over velocities of individual particles.(b) Local magnetization contributed by B x (solid lines), i.e., σ x = B x / (4 πn m e c ). Dotted lines represent the magneti-zation from in-plane fields, i.e., σ x + σ y , at the initial (blue)and final (red) times. (c) Electron internal energy densitynormalized to the initial rest-mass energy density of jet elec-trons. Dotted lines refer to jet electrons only, at the initial(blue) and final (red) times. in (b)). Since part of the resulting magnetic energy willbe dissipated by reconnection, the peak value of σ x canbe taken as a proxy for the characteristic magnetiza-tion of reconnecting current sheets. Since σ x (cid:38)
1, KHI-driven reconnection occurs in the relativistic regime.The end stage (red lines) shows a velocity profile char-acterized by a fast jet core (at | x | (cid:46) c/ω p ), mov-ing nearly at the initial four-velocity Γ β = 1 .
3, sur-rounded by wings (or, a sheath) of slower moving ma-terial (at 100 (cid:46) | x | (cid:46) c/ω p ), with Γ β y ≈ . β = 3and Γ β = 10). In the sheath near | x | ≈ c/ω p ,the in-plane magnetic energy density (dotted red in (b))is nearly in equipartition with the electron energy den-sity (solid red in (c)), or equivalently, the plasma beta β p ≈
1. This is a generic outcome of relativistic recon-nection (e.g., Sironi et al. 2016).The nonlinear development of the KHI leads to ef-ficient particle acceleration (the temporal evolution of
Figure 3.
Top: Temporal evolution of the electron spec-trum (at the times indicated in the legend, corresponding tothe panels in Fig. 1), for all the electrons (dashed), jet elec-trons only (solid), and jet positrons (dotted). Bottom: atthe final time ω p t = 12569, spatial dependence of the elec-tron spectrum (see legend), for all the electrons (dashed) andjet electrons only (solid). In both panels, the vertical blackdashed line is at the bulk energy Γ − the electron spectrum is in the top panel of Fig. 3).Wind electrons populate a non-relativistic Maxwellian(dashed lines), while the spectrum of jet electrons ini-tially peaks at their bulk energy Γ − ≈ . λ ≈ L y / λ ≈ L y mode). Inthe final stage (red lines), the spectrum extends up to acutoff energy γ e − ≈
30, as expected from reconnection-driven particle acceleration if the in-plane magnetization ≈
10 (Werner et al. 2016; Petropoulou & Sironi 2018),as inferred from Fig. 2(b). The electron spectrum ateven later times (not shown) is nearly identical to thered curve, i.e., the system has reached a a steady state.At all times, the spectrum of jet positrons (dotted) isnearly identical to the one of jet electrons (solid).
Figure 4.
Trajectory of a representative high-energy elec-tron. (a) Time evolution of the electron Lorentz factor (blacksolid) and of the parallel work W (cid:107) = − e (cid:82) t E (cid:107) v (cid:107) dt (cid:48) /m e c (dashed red), where E (cid:107) = E · ˆ b and v (cid:107) = v · ˆ b (here, E isthe electric field, v the electron velocity and ˆ b = B /B themagnetic field unit vector). (b) and (c): 2D structure of theout-of-plane current J z (in units of J = en c ) and of themean internal energy per electron υ e (in units of m e c ), atthe time ω p t = 4750 of particle injection (vertical dashedorange in panel (a)). The electron position at this time isindicated by the circle. (d) 2D structure of the bulk four-velocity along y , in units of the speed of light, at ω p t ≈ ω p t = 6080 (blue dashed inpanel (a)) to ω p t = 7250 (green dashed in (a)). Most of the electron and positron acceleration is lo-calized at the jet boundaries, with nearly identical out-comes from the left and right side (bottom panel inFig. 3, cyan and yellow lines). The core of the jet(green line) retains a narrow spectrum centered at thebulk energy Γ − ≈ .
7. The high-energy particlesat | x | (cid:38) c/ω p (blue and red lines) were initially inthe jet, and now they reside in the magnetized cloudsembedded in the wind (Fig. 1(e)).The trajectory of a representative high-energy elec-tron is displayed in Fig. 4. The top panel shows thatthe first stage of particle acceleration ( ω p t ≈ Figure 5.
Dependence of the electron spectrum on phys-ical parameters, for all the electrons (dashed) and jet elec-trons only (solid). Unless otherwise noted, we employ nu-merical and physical parameters as described in Sect. 2 forour reference run. (a) We vary the simulation-frame angle θ = arctan( B j ,z /B j ,y ) as indicated in the legend, keeping theinitial B ,y + B ,z fixed. Spectra refer to ω p t = 12410. (b) Wevary the layer width ∆ (in units of c/ω p ) and the box size L y (see legend). Spectra refer to t ≈ L y /c , corresponding to ω p t = 10500 for L y = L and to ω p t = 31500 for L y = 3 L .(c) We vary the ion mass m i (in units of m e , see legend).Yellow, cyan, and red spectra refer to our reference box at ω p t = 11137, while green and blue spectra refer to a boxwith wider ∆ = 192 c/ω p and larger size L y = 3 L , and theyare measured at ω p t = 33411 (so, all spectra are measuredat t ≈ L y /c ). (d) We vary the jet bulk four-velocity (seelegend). Spectra refer to simulations with m i /m e = 1 and∆ = 16 c/ω p at ω p t = 13523. marked by the vertical orange line) is powered by E (cid:107) = E · ˆ b (dashed red in (a)). This is indeed expected forreconnection-powered acceleration with a strong non-alternating component (Ball et al. 2019; Comisso &Sironi 2019). During this injection stage, the electronis located within a reconnecting current sheet (panel(b)), where efficient particle acceleration/heating occurs(panel (c)). At later times, while E (cid:107) no longer resultsin acceleration, the electron energy still steadily grows— a similar two-stage acceleration process has been re-ported for magnetically-dominated plasma turbulence(Comisso & Sironi 2018, 2019). In this time range (be-tween the vertical dashed blue and green lines in (a)),the electron gains energy while moving back and forthacross the shear layer (panel (d)), as expected in shear-driven acceleration. At this stage, the electron orbitcovers a sizeable fraction of the shear layer width, andso it can experience a significant velocity gradient. To assess the generality of reconnection-powered injec-tion in KHI-unstable shear layers, in Fig. 5 we presentthe dependence of the spectrum of all electrons (dashed)and jet electrons (solid) on several physical parameters.Spectra are shown at sufficiently late times that the sys-tem is nearly in steady state. When varying the lab-frame angle θ = arctan( B j ,z /B j ,y ) at fixed B ,y + B ,z (panel (a)), we find that reconnection-driven particleacceleration is most efficient at intermediate angles,60 ◦ (cid:46) θ (cid:46) ◦ . At smaller angles ( θ = 40 ◦ ), the shearlayer is KHI-stable. In the absence of in-plane fields( θ = 90 ◦ ), reconnection cannot operate, and we reportonly marginal evidence for accelerated particles, withcutoff energy much smaller than in our reference run(see also Cerutti & Giacinti 2020). The high-energyspectral cutoff does not significantly vary for angles60 ◦ (cid:46) θ (cid:46) ◦ . This is a consequence of the fact that thetypical magnetization of KHI-generated current sheets(as tracked by the peak σ x ) is nearly the same, for θ inthis range. In turn, this is due to a combination of twoopposite effects: at larger θ , the initial σ j ,y is smaller, yetthe KHI is more effective in folding the field lines (pre-cisely because of the weaker tension of in-plane fields),which results, overall, in comparable magnetizations ofthe self-generated current sheets. Given that black-holejets start with poloidal fields (here, along y ), while theyare dominated by toroidal fields (here, along z ) at largedistances, our results in Fig. 5(a) may help put con-straints on the distance where KHI-driven reconnectionand ensuing particle acceleration is most effective.We have also tested the dependence of our steady-state spectra on the shear layer width ∆ and the boxsize L y , demonstrating that our results hold in the MHDlimit L y (cid:29) c/ω p and ∆ (cid:29) c/ω p (panel (b)). Electronspectra also show only a weak dependence on the ion-to-electron mass ratio in the wind (panel (c)); this isnot surprising, given that particle acceleration mostlyinvolves electrons and positrons in the jet.Finally, panel (d) illustrates the dependence on theinitial jet velocity: with increasing Γ β , a separate pop-ulation emerges at high energies ( γ e (cid:38) γ e (cid:46) γ e (cid:38) CONCLUSIONSBy means of large-scale 2D PIC simulations, we havestudied the physics of particle acceleration in KHI-unstable shear layers, for the conditions expected at theboundary of relativistic magnetically-dominated jets.We start from shear layers much wider than kineticplasma scales. We find that the nonlinear evolutionof KH vortices leads to reconnection of the jet mag-netic field, which results in efficient acceleration of jetelectrons and positrons. The highest energy particles re-sulting from reconnection are further energized by shear-driven acceleration, i.e., reconnection can mediate parti-cle injection into shear acceleration. Our work lends sup-port to spine-sheath models of jet emission and can ex-plain the origin of radio-emitting electrons at the bound-aries of relativistic jets (see Ripperda et al. 2020 for analternative explanation).We defer an investigation of 3D effects to future work,though we note that simulations of both relativistic re-connection (e.g. Sironi & Spitkovsky 2014; Guo et al.2014; Werner & Uzdensky 2017; Sironi & Beloborodov2020) and magnetically-dominated plasma turbulence (e.g., Comisso & Sironi 2018, 2019) yield similar resultsbetween 2D and 3D, so we expect our conclusions to beapplicable to the 3D case. We also leave to future work amore detailed characterization of the properties of shear-accelerated particles (e.g., acceleration efficiency, power-law slope, scattering and acceleration rates).ACKNOWLEDGMENTSThis work is supported in part by NASA via theTCAN award grant NNX14AB47G and by the blackhole initiative at Harvard University, which is supportedby a grant from the Templeton Foundation. LS acknowl-edges support from the Sloan Fellowship, the CottrellScholar Award, NASA ATP NNX17AG21G and NSFPHY-1903412. The simulations have been performed atColumbia (Habanero and Terremoto), and with NASA(Pleiades) resources.APPENDIX
Figure 6.
2D structure of the out-of-plane field B z , in unitsof B z ≡ B j , z , from a simulation with θ = 60 ◦ , whose late-time spectrum is shown by the green line in Fig. 5(a). Bothpanels refer to ω p t = 3740, when the λ = L y / KHI-DRIVEN RECONNECTION PLASMOIDSIn the main body of the paper we have demonstratedthat the nonlinear development of the KHI naturallyproduces reconnection current sheets, which are con-ducive to efficient particle acceleration. Long recon-nection layers are known to be prone to the tearingmode instability (e.g., Uzdensky et al. 2010; Huang &Bhattacharjee 2012; Loureiro et al. 2012), which breaksthe current sheet into a chain of plasmoids/flux ropes.In Fig. 6, we show that this indeed occurs for KHI-generated reconnection layers: the KH vortex in theupper half of the left panel displays two reconnectionplasmoids (marked by the arrows in the zoom-in viewon the right panel). We argue that reconnection plas-moids, together with nonlinear structures generated bythe KHI, can provide the scattering required for efficientshear-driven acceleration.REFERENCES
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