Transverse electron-scale instability in relativistic shear flows
TTransverse electron-scale instability in relativistic shear flows
E. P. Alves, ∗ T. Grismayer, R. A. Fonseca,
1, 2 and L. O. Silva † GoLP/Instituto de Plasmas e Fus˜ao Nuclear, Instituto Superior T´ecnico,Universidade de Lisboa, 1049-001 Lisbon, Portugal DCTI/ISCTE - Instituto Universit´ario de Lisboa, 1649-026 Lisbon, Portugal (Dated: October 18, 2018)
Abstract
Electron-scale surface waves are shown to be unstable in the transverse plane of a shear flow inan initially unmagnetized plasma, unlike in the (magneto)hydrodynamics case. It is found thatthese unstable modes have a higher growth rate than the closely related electron-scale Kelvin-Helmholtz instability in relativistic shears. Multidimensional particle-in-cell simulations verify theanalytic results and further reveal the emergence of mushroom-like electron density structures inthe nonlinear phase of the instability, similar to those observed in the Rayleigh Taylor instabilitydespite the great disparity in scales and different underlying physics. Macroscopic ( (cid:29) c/ω pe ) fieldsare shown to be generated by these microscopic shear instabilities, which are relevant for particleacceleration, radiation emission and to seed MHD processes at long time-scales. PACS numbers: 52.38.Kd, 52.35.Tc, 52.35.Mw, 52.38.Dx, 52.65.Rr a r X i v : . [ phy s i c s . p l a s m - ph ] M a y fundamental question in plasma physics concerns the stability of a given plasma config-uration. Unstable plasma configurations are ubiquitous and constitute important dissipationsites via the operation of plasma instabilities, which typically convert plasma kinetic energyinto thermal and electric/magnetic field energy. Plasma instabilities can occur at micro-scopic (particle kinetic) and macroscopic (magnetohydrodynamic, MHD) scales, and aregenerally studied separately using simplified frameworks that focus on a particular scaleand neglect the other. This approach conceals the role that microscopic processes may haveon the macroscopic plasma dynamics, which in many scenarios cannot be disregarded. It isnow recognized, for instance, that collisionless plasma instabilities operating on the electronscale in unmagnetized plasmas, such as the Weibel [1] and streaming instabilities [2], play acrucial role in the formation of (macroscopic) collisionless shocks in astrophysical [3–7] andlaboratory conditions [8, 9]. These microscopic instabilities result from the bulk interpene-tration between plasmas and are believed to be intimately connected to important questionslike particle acceleration and radiation emission in astrophysical scenarios [10, 11].Plasma shear flow configurations can host both microscopic and macroscopic instabilitiessimultaneously, although the former have been largely overlooked. Shear flow settings havebeen traditionally studied using the MHD framework [12–14], where the Kelvin-Helmholtzinstability (KHI) is the only instability known to develop [15]. Only very recently havecollisionless unmagnetized plasma shear flows been addressed experimentally [16] and usingparticle-in-cell (PIC) simulations, revealing a rich variety of electron-scale processes, suchas the electron-scale KHI (ESKHI), dc magnetic field generation, and unstable transversedynamics [17–22]. The generated fields and modified particle distributions due to thesemicroscopic processes can strongly impact succeeding macroscopic dynamics of the shearflow. However, whilst both the ESKHI and the dc magnetic field formation mechanismhave been treated theoretically, an analytical description of the transverse shear instability,observed in kinetic simulations, is still lacking. An established analytical description iscrucial in order to assess its relevance in different physical scenarios.This Letter focuses on the shear surface instability that occurs in the plane perpendicularto that of the ESKHI. These new unstable modes explain the transverse dynamics andstructures observed in PIC simulations in [17, 19, 20, 22]. We label this effect the mushroominstability (MI) due to the mushroom-like structures that emerge in the electron density.We analyze the stability of electromagnetic perturbations in the transverse xy plane of2 collisionless plasma shear flow with velocity profile (cid:126)v = v ( x ) (cid:126)e z . We assume a cold( v th (cid:28) v , where v th is the thermal velocity) unmagnetized plasma, and impose charge andcurrent neutrality, n e = n i = n , v e = v i (subscripts e and i refer to electron and ionquantities, respectively) to guarantee initial equilibrium. We use a two-fluid model wherethe relativistic equations of motion of the electron and ion fluid are coupled to Maxwell’sequations. We assume linear perturbations in the fluid quantities of the form f = f + δf with δf = ¯ δf ( x ) e iky − iωt ( ω ∈ C and k ∈ R are frequency and wavenumber, respectively)with all zeroth order quantities being zero except for n ( x ) and v ( x ). We find, for theperturbed current densities, δj x = − e (1 + m e /m i ) n δv ex , δj y = − e (1 + m e /m i ) n δv ey , δj z = − e (1 + m e /m i )( n δv ez + δnv ). Substituting in Maxwell’s equations, we can simplify theoriginal set of ten coupled equations in a reduced 2 × ¯¯ (cid:15).δ E = , where thecoefficient of the tensor (cid:15) ij are operators of the form (cid:15) ij = (cid:80) m =0 C ij,m ( ω, k y , v ( x ) , n ( x )) ∂ x m and δ E = ( δE y , δE z ). In order to obtain analytical results we use a step velocity shear anddensity profile of the form v ( x ) = v − + ( v + − v − ) H ( x ) and n ( x ) = n − + ( n + − n − ) H ( x )where H is the Heaviside function. Integrating ¯¯ (cid:15).δ E = for x (cid:54) = 0 and using the continuityof δE y and δE z across the shear interface, we find solutions corresponding to evanescentwaves: δE y,z ( x ) = ¯ δE y,z (0) e − k ±⊥ | x | where k ±⊥ = (cid:112) D ±⊥ /c , D ±⊥ = c k + ω pe ± γ ± − ω , ω pe ± = e n ± (1 + m e /m i ) /(cid:15) m e and γ ± = 1 / (cid:112) − ( v ± /c ) . By evaluating the derivative jump of theelectric fields across the shear interface we arrive at ¯¯I .δ E = , where δ E = ( ¯ δE y (0) , ¯ δE z (0))and I ij = a + ij k + ⊥ + a − ij k −⊥ ; a = ( ω − ω pe /γ ) D − ⊥ , a = − ( kv /ω )( ω pe /γ ) D − ⊥ , a = a and a = − − ( k c /ω − ω pe /γ )( v /c ) D − ⊥ . The dispersion relation is finally obtainedby solving det( ¯¯I ) = . In the special case, n + = n − = ¯ n and v + = − v − = ¯ v , the growthrate reads Γ ω pe = 1 √ (cid:32)(cid:115) k ¯ v ¯ γ ω pe + D (cid:12) − D (cid:12) (cid:33) / , (1)with Γ = Im( ω ) and D (cid:12) = 1 / ¯ γ + k c /ω pe .The fastest growing mode of this unstable branch ( ∂ k Γ = 0) is found at k → ∞ (it willbe shown later that finite thermal effects and/or smooth velocity shear profiles introduce acutoff an finite k ). This limit corresponds to the maximum growth rate of the instabilityΓ max for a given shear flow Lorentz factor, and it is given by Γ max /ω pe = ¯ v /c √ ¯ γ . In thelimit, n − (cid:29) n + /γ , γ + (cid:29) v − = 0, relevant for astrophysical jet/interstellar mediumshear interaction, the maximum growth rate yields Γ max /ω pe + (cid:39) / √ γ + .3he MI and ESKHI [17] growth rates are compared for different shear Lorentz factorsand velocities in Figure 1. It is clear that the ESKHI has higher growth rates than the MIfor subrelativistic settings. However, the MI growth rate decays with ¯ γ − / , slower than theESKHI, which decays with ¯ γ − / , as shown in Figure 1.b. Therefore, given that the noisesources for both instabilities are similar, the MI is the dominant electron-scale instability inrelativistic shear scenarios.To verify the theoretical model and better understand the underlying feedback cycle of theMI, we first analyze the evolution of a single unstable mode in an electron-proton ( e − p + )shear flow using the PIC code OSIRIS [23, 24]. We simulate a domain with dimensions20 × c/ω pe ) , resolved with 40 cells per c/ω pe , and use 36 particles per cell per species. Theshear flow initial condition is set by the velocity field ¯ v = +0 . c for L x / < x < L x /
4, and¯ v = − . c for x < L x / ∪ x > L x /
4, where L i is the size of the simulation box in the i thdirection. The system is initially charge and current neutral. Periodic boundary conditionsare imposed in every direction. In order to ensure the growth of a single mode, both e − p + temperatures are set to zero, and an initial harmonic perturbation δv x = ¯ δv x cos( k seed y ) inthe velocity field of the electrons, with ¯ δv x = 10 − c and k seed = 2 π/L y , is introduced to seedthe mode k seed of the instability.The evolution of the electron density, the out-of-plane current density J z , and the in-plane magnetic field are presented in Figure 2, which zooms in on the shear interface at x = L x /
4. The initial velocity perturbation δv x transports electrons across the velocityshear gradient, producing a current imbalance in δJ z . This current induces an in-planemagnetic field (namely δB y ) that, in turn, enhances the velocity perturbation δv x via the v × δB y force. The enhanced velocity perturbation then leads to further electron transportacross the velocity shear gradient in a feedback loop process, which underlies the growth ofthe instability in the linear stage. The surface wave character of the fields, predicted by thelinear theory, is also observed in Figure 2.b1.The MI eventually enters the nonlinear phase when the growing magnetic fields becomestrong enough to significantly displace the electrons and distort the shear interface. Thenonlinear distortion of the shear interface in the electron fluid (Figure 2.a2) leads to the for-mation of electron surface current filaments (Figure 2.b2). These surface current filamentseffectively translate in a strong dc (non-zero average along the y direction) out-of-plane cur-rent structure on either side of the shear interface. Note that the protons in the background4emain unperturbed on these time-scales due to their inertia, and also contribute to theformation of the dc current. The dc current structure induces a strong dc magnetic field in B y , as seen by the uniform field lines along the shear interface in Figures 2.b2-3. The dcmagnetic field, in turn, continues to drive the shear boundary distortion via the v × δB y force, effectively mixing the electrons across the shear, and further enhancing the dc currentstructure in an unstable loop. The development of the dc magnetic field has been previouslyshown to be associated to other electron-scale shear flow processes, like the nonlinear devel-opment of the cold ESKHI and electron thermal expansion effects [18]. Here we show thatthe MI is an additional mechanism capable of driving the dc magnetic field. The nonlineardistortion of the shear boundary in the electron fluid, driven by the dc field physics, ulti-mately gives rise to the formation of the mushroom-like electron density structures shownin Figure 2.a3. Interestingly, these electron density structures are very similar to those pro-duced by the Rayleigh-Taylor instability [25–27] despite the great disparity between scales(electron-kinetic versus hydro/MHD scales) and different underlying physics.The evolution of the ratio of total magnetic field energy to initial particle kinetic energy( (cid:15) B /(cid:15) p ), for the single mode k seed = 2 π/ ω pe c simulation, is presented in the inset of Figure3. The exponential growth associated with the linear development of the instability isobserved for 10 (cid:46) tω pe (cid:46)
30, matching the theoretical growth rate. The instability saturatesat tω pe (cid:39)
40, approximately when the size of the mushroom-like density structure is onthe order of 2 π/k seed (Figure 2.a3). Various simulations with different k seed values wereperformed, verifying the dispersion relation of Eq. 1 (Figure 3.a).The inclusion of thermal effects should introduce an instability cutoff at a finite wavenum-ber k cutoff ∼ π/λ D ( λ D is the Debye length), since the thermal pressure force F p ∼ k ( k B T ⊥ )( k B and T ⊥ are the Boltzmann constant and electron temperature perpendicular to v ), canovercome the electric/magnetic forces that drive the unstable arrangement of the electriccurrents at high k [28]. Yet such effects must be incorporated through kinetic theory, particu-larly when v th ∼ ¯ v , since the thermal expansion of electrons may mitigate the velocity sheargradient, impacting the development of the instability. Interestingly, however, the thermalexpansion is greatly reduced when considering ultra-relativistic ( γ (cid:29)
1) shear flows, inci-dentally the regime where the MI is most significant, allowing the use (to some extent) of thecold fluid treatment outlined above. When considering a plasma of temperature T R (definedin its rest frame) and drifting relativistically in the z direction with mean velocity v = β c ,5ost of the particles have a large Lorentz factor and thus (cid:104) β (cid:105) = (cid:82) d (cid:126)pf ( (cid:126)p ) β (cid:39)
1, where f ( (cid:126)p ) is the Juettner distribution [29, 30]. The velocity dispersion in the direction of the driftis (cid:104) ( β z − β ) (cid:105) = (cid:104) β z (cid:105) − β (cid:39)
0. Since (cid:104) β (cid:105) = (cid:104) β ⊥ + β z (cid:105) , we find that (cid:104) β ⊥ (cid:105) (cid:39) − β = 1 /γ (exact for ξ = k B T R /m e c (cid:29) γ . The averageparticle Lorentz factor increases with T R as (cid:104) γ (cid:105) ∼ γ (1 + (1 + µ ( ξ )) ξ ) [30], where µ ( ξ ) = 3(3 /
2) for ξ (cid:29) ξ (cid:28) max /ω pe ∼ / (cid:112) (cid:104) γ (cid:105) (for the symmetric shear scenario) due to the enhanced average relativistic particle mass.Hence, thermal expansion effects remain negligible as long as c (cid:112) (cid:104) β ⊥ (cid:105) / Γ max (cid:28) c/ω pe (cid:112) (cid:104) γ (cid:105) [18], which implies γ (cid:29) e − e + shearconfiguration consisting of a hot relativistic jet, with γ + = 50 and ξ + = 0 . , ,
5, and acold stationary background, with γ − = 1 and ξ − = 10 − ; we consider n − = n + = n . Wesimulate these configurations in a domain of 200 ×
100 ( c/ω pe ) resolved with a 4000 × ξ + (Figure 3.b), and is consistent withthe temperature-enhanced average relativistic particle mass predicted analytically. The evo-lution of the electron density around one of the shear interfaces of the system is presentedin Figure 4 for the case ξ + = 1. At early times, k (cid:46) k cutoff structures are observed in theelectron density in Figure 4.a1. These small-scale structures, which are essentially surfacecurrent filaments, magnetically interact with each other and merge (Figure 4.a2-3), forminglarger-scale structures in a similar manner to the current filament merging dynamics asso-ciated with the current filamentation instability [31]. The evolution of the self-generatedmagnetic field structure is also illustrated in Figure 4. In addition, saturation of the self-generated magnetic field ( B sat ) is achieved when Γ max ∼ ω B = eB sat /m (cid:104) γ + (cid:105) , based onmagnetic trapping considerations [32].We have also considered the effects of a smooth shear in the development of the MI. Wehave performed 2D PIC simulations with velocity profiles of the form v ( x ) = ¯ v tanh( x/L v ),where L v is the shear gradient length, which reveal the persistent onset of the MI at gradientlengths up to L v = c/ω pi (cid:29) c/ω pe (where ω pi is the ion plasma frequency). The MI can thusbe of relevance to the physics on the ion length/time scale. It is found that the introductionof the finite shear length L v also leads to a fastest growing mode at a finite k max (cid:39) π/L v .The growth rate Γ max of the MI decreases with increasing L v , but remains higher than theESKHI for the same L v in relativistic scenarios.6nterestingly, due to its electromagnetic nature, the MI is found to operate in the absenceof contact between flows, i.e., in the case of a finite gap L g separating the shearing flows(Figure 5), highlighting the different nature of the MI compared to the (bulk) two-stream andcurrent filamentation instabilities. This setting is closely connected to the work explored in[33] on the development of optical instabilities in (subrelativistic) nanoplasmonic scenarios,consisting of shearing metallic slabs separated by a nanometer-scale gap; the developmentof such instabilities results in an effective non-contact friction force between slabs [34, 35].The role of the MI, however, was overlooked since only subrelativistic configurations wereconsidered; the transverse MI mode will be predominant in the relativistic regime. The newMI modes are found by taking into account the new boundary conditions imposed by thegap. The new eigenmodes of the system have a more complex spatial structure; the surfacewaves peak at the flow boundaries, evanesce with k ⊥ | n = n e and k ⊥ | n =0 in the plasmaand vacuum regions, respectively, and couple in the gap. Therefore, the interaction betweenflows across the gap is strong as long as L g k ⊥ | n =0 ∼
1. Simulations show that the mostunstable mode is found at k max ∼ π/L g and that the instability growth rate decreases asthe gap is broadened, Γ gapmax ∼ Γ max exp( − L g ω pe /c √ L v (cid:29) c/ω pe ), and, quite surprisingly, (iv) even in the ab-sence of contact between flows. Relativistic shear flow conditions may be reproduced in thelaboratory by propagating a globally neutral relativistic e − e + beam [36] in a hollow plasmachannel [37, 38], allowing experimental access to the MI and shear flow dynamics on theelectron scale. This will be explored and presented elsewhere. In the astrophysical context,7he unexpected ability of these microscopic shear instabilities (MI and ESKHI) to generatestrong (equipartition) macroscopic ( (cid:29) c/ω pe , in the case of e − p + shears) fields from initiallyunmagnetised conditions can directly impact particle acceleration and radiation emissionprocesses, and can seed the operation of macroscopic (MHD) processes at later times (e.g.magnetic dynamo).E. P. Alves and T. Grismayer contributed equally to this work. This work was partiallysupported by the European Research Council (ERC − − AdG Grant 267841) and FCT(Portugal) grants SFRH/BD/75558/2010 and IF/01780/2013. We acknowledge PRACE forawarding access to SuperMUC based in Germany at Leibniz research center. Simulationswere performed at the IST cluster (Lisbon, Portugal), and SuperMUC (Germany). ∗ [email protected] † [email protected][1] E. Weibel, Physical Review Letters , 83 (1959).[2] A. Bret, M. C. Firpo, and C. Deutsch, Physical Review E , 046401 (2004).[3] L. O. Silva, R. A. Fonseca, J. W. Tonge, J. M. Dawson, W. B. Mori, and M. V. Medvedev,Astrophysical Journal , L121 (2003).[4] J. T. Frederiksen, C. B. Hededal, T. Haugbolle, and A. Nordlund, Astrophysical Journal ,L13 (2004).[5] K. I. Nishikawa, P. Hardee, G. Richardson, R. Preece, H. Sol, and G. J. Fishman, Astrophys-ical Journal , 927 (2005).[6] A. Spitkovsky, Astrophysical Journal Letters , L5 (2008).[7] S. F. Martins, R. A. Fonseca, L. O. Silva, and W. B. Mori, The Astrophysical Journal ,L189 (2009).[8] F. Fi´uza, R. A. Fonseca, J. Tonge, W. B. Mori, and L. O. Silva, Physical Review Letters ,235004 (2012).[9] A. Stockem, F. Fi´uza, A. Bret, R. A. Fonseca, and L. O. Silva, Scientific Reports (2014).[10] M. V. Medvedev and A. Loeb, Astrophysical Journal , 697 (1999).[11] A. Gruzinov and E. Waxman, Astrophysical Journal , 852 (1999).[12] A. Frank, T. W. Jones, D. S. Ryu, and J. B. Gaalaas, Astrophysical Journal , 777 (1996).
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500 10 20 30 40t [ ω pe-1 ] ε B ( k = k s ee d ) / ε p -12 -8 -4 Γ t h e o r y ¯ v /c √ ¯ γ -8 -6 -4 -2 [ ω pe-1 ] ε B / ε p a b ξ + = 5 ξ + = 0 . ξ + = 1 FIG. 3. a) MI growth rate versus wavenumber for an e − p + shear flow with ¯ v + = − ¯ v − = ¯ v = 0 . c (solid line); red dots represent results of 2D PIC simulations. The inset shows the evolution of (cid:15) B /(cid:15) p for the single mode simulation with k seed = 2 π/ ω pe /c . b) Finite temperature effects onthe growth rate of the MI triggered by a hot relativistic e − e + jet with γ + = 50, shearing with acold stationary plasma with γ − = 1 and n + = n − . Dashed lines represent the theoretical slopeΓ max / ω pe + = 1 / (cid:112) (cid:104) γ + (cid:105) for each case. = a1x [c/ ω pe ] 100-10406080200 y [ c / ω p e ]
100 013 e - d e n s it y [ n ] x [c/ ω pe ]0-20-40 4020 p + p a2 x [c/ ω pe ] 100-10 013 e - d e n s it y [ n ] a3 22 FIG. 4. Electron density evolution in relativistic e − e + shear, with γ + = 50 and γ − = 1, at (a1) t = 80 /ω pe , (a2) t = 160 /ω pe and (a3) t = 400 /ω pe . Small-scale current filaments are excited atearly times, merging into large-scale structures as the instability develops. Vector field representsself-generated magnetic field structure. [c/ ω pe ]20100 30 5040406080200 y [ c / ω p e ] e - d e n s it y [ n ] − p + p a2 012 e - d e n s it y [ n ] x [c/ ω pe ]20100 30 5040 FIG. 5. MI development in the case of a finite gap between flows with L gap = 5 c/ω pe and¯ v = (cid:112) / c . Frames a) and b) reveal the electron density at times t = 0 /ω pe and t = 725 /ω pe ,respectively.,respectively.