A model of global magnetic reconnection rate in relativistic collisionless plasmas
Yi-Hsin Liu, Michael Hesse, Fan Guo, William Daughton, Hui Li
AA model of global magnetic reconnection rate in relativistic collisionless plasmas
Yi-Hsin Liu, Michael Hesse, Fan Guo, William Daughton, and Hui Li NASA-Goddard Space Flight Center, Greenbelt, MD 20771 Los Alamos National Laboratory, Los Alamos, NM 87545 (Dated: October 16, 2018)A model of global magnetic reconnection rate in relativistic collisionless plasmas is developed andvalidated by the fully kinetic simulation. Through considering the force balance at the upstreamand downstream of the diffusion region, we show that the global rate is bounded by a value ∼ . ∼ O (1) and the local inflow speed approaches the speed oflight in strongly magnetized plasmas. The derived model is general and can be applied to magneticreconnection under widely different circumstances. PACS numbers: 52.27.Ny, 52.35.Vd, 98.54.Cm, 98.70.Rz
Introduction–
Magnetic fields often serve as the majorenergy reservoirs in high energy astrophysical systems,such as pulsar wind nebulae [1–4], gamma-ray bursters[5–7] and jets from active galactic nuclei [8–10], whererelativistic cosmic rays and gamma rays of energies up toTeV are generated explosively [11, 12]. Among the pro-posed physics processes (e.g.,[13–15]) that could unleashthe magnetic energy, magnetic reconnection is consideredto be a promising mechanism. For comparison, collision-less shocks, regarded to be efficient for particle accel-eration in weakly magnetized plasmas, are inefficient indissipating energy and accelerating non-thermal particlesin magnetically dominated flows [13]. Hence the study ofmagnetic reconnection in these exotic systems continuesto be an interesting topic in high energy astrophysics.One of the most important issues in relativistic recon-nection studies is how fast magnetic energy can be dis-sipated in the reconnection layer, which determines thetime scale of the explosive energy release events. Anotherrelated problem is the mechanism of non-thermal parti-cle acceleration [16–25]. Proposed mechanisms includethe direct acceleration by the reconnection electric fieldat the diffusion region [25, 26], the Fermi mechanism atthe outflow regions that involves particles bouncing backand forth between reconnection outflows emanated fromdifferent x-lines [20, 27, 28], and many other ideas (e.g.,[29–31]). In collisionless plasmas, the energy gain of aparticle must come from the work done by the electricfield ∼ q (cid:82) E · v dt . Thus, determining the reconnectionelectric field in the relativistic limit is crucial to deter-mine the acceleration rate and efficiency.In such magnetically-dominated plasmas, the magneticenergy density is much larger than the rest mass energydensity and the Alfv´en speed approaches the speed oflight. Early theoretical work suggested that the magneticreconnection rate in the relativistic limit may increasecompared to the non-relativistic case due to the enhancedinflow arising from the Lorentz contraction of plasmapassing through the diffusion region [32, 33]. However,it was later pointed out that the thermal pressure withina pressure-balanced current sheet will constrain the out- flow to mildly relativistic conditions, where the Lorentzcontraction is negligible [34] and a relativistic inflow istherefore impossible.Recently, fully kinetic simulations by Liu et al. [35]showed that the local inflow speed approaches the speedof light, and the reconnection rate normalized to theimmediately upstream condition of the diffusion regioncan be enhanced to ∼ O (1) in strongly magnetized plas-mas. However, the global reconnection rate normalizedto the far upstream asymptotic value remains (cid:46) . ad hoc localized resistivity[39, 40], otherwise, the current sheet collapses to the longSweet-Parker layer [41, 42]. A mechanism for the local-ized diffusion region is therefore essential to model thereconnection rate. In this Letter, we derive the relationbetween the global rate and the degree of localizationthrough considering the force balance at the upstreamand downstream of the diffusion region. We then proposea mechanism that naturally leads to the localization insuch collisionless plasmas. Simulation setup–
The kinetic simulation is performedusing a Particle-in-Cell code- VPIC [43], which solves thefully relativistic dynamics of particles and electromag-netic fields. The relativistic Harris sheet [24, 35, 36, 44–46] is employed as the initial condition. The initialmagnetic field B = B x tanh( z/λ )ˆ x corresponds to alayer of half-thickness λ . Each species has a distribu-tion f h ∝ sech ( z/λ )exp[ − γ d ( γ L mc + mV d u y ) /T (cid:48) ] inthe simulation frame, which is a component with a peakdensity n (cid:48) and temperature T (cid:48) boosted by a drift velocity ± V d in the y-direction for positrons and electrons, respec-tively. In addition, a non-drifting background component f b ∝ exp( − γ L mc /T b ) with a uniform density n b is in-cluded. Here u = γ L v is the the space-like components of4-velocity, γ L = 1 / [1 − ( v/c ) ] / is the Lorentz factor ofa particle, and γ d ≡ / [1 − ( V d /c ) ] / . The drift velocityis determined by Amp´ere’s law cB x / (4 πλ ) = 2 eγ d n (cid:48) V d .The temperature is determined by the pressure balance a r X i v : . [ a s t r o - ph . H E ] M a y t! pe R G R L V in,L /c B xL /B x FIG. 1: The evolution of measured global reconnection rate R G , local rate R L , local inflow speed V in,L /c and B xL /B x in a plasma of σ x = 89. The blue circle marks the deviationof R L from R G . The grey dashed line at value 0 . B x / (8 π ) = 2 n (cid:48) T (cid:48) . The resulting density in the simula-tion frame is n = γ d n (cid:48) . In this Letter, the primed quan-tities are measured in the fluid rest (proper) frame, whilethe unprimed quantities are measured in the simulationframe unless otherwise specified. Densities are normal-ized by the initial background density n b , time is nor-malized by the plasma frequency ω pe ≡ (4 πn b e /m e ) / ,velocities are normalized by the light speed c , and spatialscales are normalized by the inertial length d e ≡ c/ω pe .The domain size is L x × L z = 384 d e × d e and is re-solved by 3072 × λ = d e , n b = n (cid:48) , T b /m e c = 0 . ω pe / Ω ce = 0 .
05 where Ω ce ≡ eB x / ( m e c ) is a cy-clotron frequency. The upstream magnetization param-eter is σ x = B x / (4 πw ) with enthalpy w = 2 n (cid:48) b m e c +[Γ / (Γ − P (cid:48) . Here Γ is the ratio of specific heats and P (cid:48) ≡ n (cid:48) b T (cid:48) b the total thermal pressure. For Γ = 5 / σ x = 89 in this run. A localized perturbationwith amplitude B z = 0 . B x is used to induce a domi-nant x-line at the center of simulation domain. Simulation results–
In this Letter, we define the globalreconnection rate as R G ≡ cE y / ( B x V A ) and the lo-cal reconnection rate as R L ≡ cE y / ( B xL V AL ). Sub-scripts “0” and “L” indicate quantities far from, andimmediately upstream of, the diffusion region where thefrozen-in condition E + V e × B = 0 breaks ( | z | (cid:46) . d e [35]). E y is the reconnection electric field at the x-lineand the Alfv´en speed in the relativistic limit [49–52] is V A = c [ σ x / (1 + σ x )] . and V AL = c [ σ xL / (1 + σ xL )] . with σ xL (cid:39) ( B xL /B x ) σ x . The evolution of reconnec-tion rates are plotted in Fig. 1, along with the local elec-tron inflow speed, V in,L , and the ratio of magnetic fields zz zzx x V ez V ez | B x | B x (a)(b) FIG. 2: The morphology of relativistic magnetic reconnectionat t = 600 /ω pe . In (a), the V ez and a cut at x = 0; In (b),the | B x | and a cut of B x at x = 0. The white contour is thein-plane magnetic flux. To better illustrate the variation ofthe upstream field in (b), we have put an upper limit B x inthe color scale, which artificially reduces the | B x | around themagnetic islands at outflow exhausts. B xL /B x . Before a quasi-steady state is reached, boththe local and global rates increase as the simulation pro-gresses. The deviation of the local rate from the globalrate occurs at time t (cid:39) /ω pe and B xL /B x (cid:39) . R G reaches a plateau of value (cid:39) .
15 at t (cid:38) /ω pe while R L continues to grow and B xL /B x continues to drop.The local rate R L eventually reaches a plateau of value (cid:39) . B xL /B x reaches a plateau of value (cid:39) . t (cid:38) /ω pe . The local inflow speed basicallytraces the local rate because of the frozen-in condition E y (cid:39) V in,L B xL /c and V AL (cid:39) c in this case, which leads R L = V in,L /V AL (cid:39) V in,L /c . The values of these twoquantities can approach ∼ O (1) with a larger σ x , asreported before [35, 53, 54].To get a better idea of the spatial variation of the inflowvelocity and magnetic fields at the quasi-steady state,the V ez and B x at time t = 600 /ω pe are shown in Fig. 2with the in-plane magnetic flux overlaid. Immediatelyupstream of the intense thin current sheet, the | V ez | peaksat | z | (cid:39) d e with value (cid:39) .
65, where B x drops to a value (cid:39)
3. Because of the thin current sheet, d e -scale secondarytearing modes [35] are generated repeatedly, which canbe seen in Fig. 2. Note that R G reaches the plateau inFig. 1 long before the generation of secondary tearingmodes. The enhancement of V ez /c closer to the diffusion (a)(b) B ·r B z ⇡ ( r· P ) zi,e X j mn j V j ·r U jz totaltotal Z B ·r B z ⇡ dz B ⇡ r B ⇡ Z ( r· P ) z dz Z i,e X j mn j V j ·r U jz dz z FIG. 3: In (a), the force balance in the z-direction along x = 0in Fig. 2; In (b), the pressure balance along x = 0. region is anti-correlated with the reduction of B x because E y (cid:39) V ez B x /c should be spatially uniform in a quasi-steady state under the 2D constraint, per Faraday’s law.To get a clue of how the B xL drops from B x , we ex-amine the force balance across the x-line at x = 0. Bycombining the momentum equations for electrons andpositrons [35, 55], the equation of force balance can bederived as e,p (cid:88) j mn j V j ·∇ U j + ∇ B π + ∇· P − B · ∇ B π = − e,p (cid:88) j mn j ∂∂t U j (1)Here the pressure tensor P ≡ (cid:80) e,pj (cid:82) d u vu f j − n j V j U j ,and subscripts “e” and “p” stand for electrons andpositrons respectively. U ≡ (1 /n ) (cid:82) d u u f is the firstmoment of the space-like components of 4-velocity, and V ≡ (1 /n ) (cid:82) d u v f as usual. On the left hand side ofEq. (1), the terms represent the inertial force, magneticpressure gradient force, plasma thermal gradient forceand magnetic tension, respectively. In the upstream re-gion the magnetic pressure is balanced by the tensionforce as shown in Fig. 3(a). The thermal pressure is neg-ligible because of the small plasma β ≡ P/ ( B / π ) (cid:39) . B x from the value far upstreamat | z | (cid:38) d e to the value immediately upstream of thediffusion region at | z | (cid:39) . d e as shown in the profile of B / π in Fig. 3(b). Simple model–
When the current sheet pinches locally,it implies a curved upstream magnetic field as illustrated in Fig. 4(a). The local magnetic field immediately up-stream of the diffusion region, B xL , becomes smaller than B x , so that the magnetic pressure gradient force bal-ances the magnetic tension. A larger degree of local-ization implies a larger curvature, and a smaller B xL ,as indicated by the “line-density” of the in-plane flux inboth Fig. 4(a), and the upstream region in Fig. 2. Hence,even though the local reconnection rate can be enhancedsignificantly due to the normalization, the global recon-nection rate may not increase much.To estimate this effect in the β (cid:28) ∇ B / π (cid:39) B · ∇ B / π , at point 1marked in Fig. 4(b): B x − B xL π ∆ z (cid:39) (cid:18) B x + B xL (cid:19) B z π ∆ x . (2)Note that ∇· B = 0 is also satisfied at point 1. The B x atpoint 1 is linearly interpolated from B x and B xL . Theupstream inertial force can be formally ordered out, andit is also negligible in Fig. 3.A curved upstream magnetic field naturally impliesan flaring angle, and that is measured by ∆ z/ ∆ x (cid:39) B z / [( B x + B xL ) / z/ ∆ x (cid:39) B zL /B xL . Weobtain the relation, B zL B xL (cid:39) (cid:115) − B xL /B x B xL /B x . (3)This expression suggests that a larger opening angle re-quires a further reduction of B xL /B x . In this sense, B xL /B x gauges the localization of sheet pinch. When B xL /B x →
0, the opening angle approaches 45 ◦ in thismodel.Combined with E y (cid:39) B zL V out,L /c , the reconnectionrates are R G (cid:39) (cid:18) B xL B x (cid:19) (cid:18) B zL B xL (cid:19) (cid:18) V out,L V A (cid:19) ; R L (cid:39) (cid:18) B zL B xL (cid:19) (cid:18) V out,L V AL (cid:19) (4)and the local inflow speed is V in,L (cid:39) R L V AL . (5)Using Eq. (3) and the outflow speed V out,L ∼ V AL , thepredicted R G , R L and V in,L /c as functions of B xL /B x are plotted in Fig. 4(d) as dashed-lines. If B xL /B x = 1,the opening angle is zero and reconnection is not ex-pected. In the limit of B xL /B x →
0, the reconnect-ing component vanishes and reconnection ceases (i.e., R G = 0).However, the geometrical constrain can reduce the out-flow speed from V AL when the opening angle approaches45 ◦ . This correction can be modeled through analyzingthe force-balance in the x-direction at point 2 of Fig. 4(c): (a)(b) B xL B x B z x z B z B B x B xL diffusion region (d) R G B zL B zL B xL /B x R L V in,L /cB xL B zL L (c) opening angle V out,L V out,L FIG. 4: The cartoons of magnetic field lines upstream of thediffusion region ( z >
0) in (a), and the geometry of recon-nection in (b). The dimension of the diffusion region in (c).The predictions with σ x = 89 in (d), the dashed lines use V out,L = V AL . The orange vertical line corresponds to that inFig. 1. n (cid:48) mU out /L + B zL / πL (cid:39) ( B zL / B xL / / πδ , wherethe inertial force becomes important. The outflow can berelativistic, U out ∼ γ out V out,L ∼ V out,L / (1 − V out,L /c ).Assuming the incompressibility of plasmas, then the as-pect ratio of the diffusion region δ/L ∼ B zL /B xL , andthe outflow speed becomes V out,L (cid:39) c (cid:115) (1 − B zL /B xL ) σ xL − B zL /B xL ) σ xL . (6)This expression suggests that when δ/L (cid:28) B zL /B xL (cid:28)
1) then V out,L ∼ V AL . When δ/L → ◦ ), the outflow tension is balanced by the mag-netic pressure and the outflow vanishes. Plugging Eqs.(3)and (6) back to Eqs. (4)-(5), we get the solid curvesin Fig. 4(d). This correction further constrains the re-connection rate when the opening angle is larger and B xL /B x is smaller.This model suggests that during the pinching of thecurrent sheet, a weak localization with B xL /B x (cid:46) . R G to ∼ .
2, then it varies slowly overa wide range of B xL /B x . The local rate R L and localinflow speed V in,L /c can reach ∼ O (1) under stronger lo-calization. The evolution of reconnection rates in Fig. 1can be qualitatively described by this model through de-creasing B xL /B x . The rates in the quasi-steady stateat time t = 600 /ω pe of Fig. 1 also compares well withthe prediction at B xL /B x (cid:39) .
22 with the predicted R G (cid:39) . R L (cid:39) .
69 and V in,L (cid:39) . c . Given the sim-plicity of this model, this agreement is quite remarkable.While the localization mechanism may vary in differ-ent systems, we point out a natural tendency that canlead to the B xL /B x reduction in such plasmas: A diffu-sion region sandwiched by a large B xL (cid:39) B x at d e -scale(i.e., where the frozen-in condition is broken) requiresthe current sheet plasma to have a huge thermal pres-sure to balance the magnetic pressure, and a high driftspeed to support the current. For instance, the initial d e -scale current sheet has T (cid:48) = 100 m e c , n (cid:39)
10 and γ d V d (cid:39)
10. However, the maximum possible reconnec-tion electric field may not be efficient enough in heatingand accelerating the cold non-drifting inflowing plasmabefore they exit the diffusion region [56], hence the B xL drops significantly until the d e -scale current sheet be-comes sustainable in the quasi-steady state. If this dropcontinues with a larger σ x , reconnection in the more ex-treme limit is prone to choke itself off in the quasi-steadystate. Discussion–
Knowing the magnitude of electric field isessential for estimating the acceleration of super-thermalparticles in highly magnetized astrophysical systems.This study suggests that the magnitude of the reconnec-tion electric field is bounded by ∼
30% of the reconnect-ing component of magnetic field, even in the large- σ x limit. While a weak localization of the diffusion regionis required, the global reconnection rate R G ∼ . − . B xL /B x , but the local rate and localinflow speed are. This explains the large difference be-tween the local and global reconnection rates observed inthe simulation.In this model, a larger σ x has little effect on theprofile of the global rate R G , but it could make the localinflow speed closer to the speed of light [35]. In addition,the effect of a guide field can be included by making therelevant Alfv´en speed V A = c [ σ x / (1 + σ x + σ g )] . with σ g ≡ ( B g /B x ) σ x accounting for the effect a guide field B g . This expression is basically the projection of thetotal Alfv´en speed in the outflow direction [35, 36, 52]. Aguide field also has little effect on R G , but it significantlyreduces the local inflow speed and the magnitude of thereconnection electric field through reducing the speed ofAlfv´enic outflows, as observed in Liu et al. [35]. Theprediction in the non-relativistic and low- β limit can beobtained by taking σ x (cid:28)
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