First order particle acceleration in magnetically-driven flows
aa r X i v : . [ a s t r o - ph . H E ] D ec Draft version October 15, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
FIRST ORDER PARTICLE ACCELERATION IN MAGNETICALLY-DRIVEN FLOWS
Andrey Beresnyak
Naval Research Laboratory, Washington, DC 20375
Hui Li
Los Alamos National Laboratory, Los Alamos, NM, 87545
Draft version October 15, 2018
ABSTRACTWe demonstrate that particles are regularly accelerated while experiencing curvature drift in flowsdriven by magnetic tension. Some examples of such flows include spontaneous turbulent reconnectionand decaying magnetohydrodynamic (MHD) turbulence, where magnetic field relaxes to a lower-energy configuration and transfers part of its energy to kinetic motions of the fluid. We show thatthis energy transfer which normally causes turbulent cascade and heating of the fluid, also results ina first-order acceleartion of non-thermal particles. Being very generic, this acceleration mechanism islikely to be responsible in production of non-thermal particle distribution in magnetically-dominantenvironments such as solar chromosphere, pulsar magnetosphere, jets from supermassive black holesand γ -ray bursts. Subject headings: magnetohydrodynamics—particle acceleration INTRODUCTIONMagnetically-dominated environments are fairly com-mon in Astrophysics and a sizable fraction of astrophys-ical objects, that we know, is made of extremely rarefiedplasma. These objects are only visible to us because theycontain a non-thermal particle population which domi-nates emission across most of the objects spectrum. Abasic idea that particles must be accelerated, therefore,has been around for some time. One of the main ele-ments of the explanation of how energy gets to particlesinvolves generic mechanisms of how energy may be dis-sipated in an almost inviscid and perfectly conductingmedium, i.e. how energy is transferred to small scaleswhere is is available to particles. Of these mechanismsthe most popular are three: a) discontinuities in the fluidmotion e.g. shocks, b) turbulence and c) dicontinuities inthe magnetic field, e.g. current sheets. Following the ideaof Fermi (1949) that collisionless particles can get energyby scattering in fluid motions, especially converging mo-tions, the diffusive shock acceleration mechanism (DSA)(Krymskii 1977; Bell 1978; Malkov & O’C Drury 2001)has become rather popular in explaining non-thermalemission, as well as energetic charged paricles, detectedat Earth, known as cosmic rays. The acceleration rate,however, is related to the scattering rate, which is criti-cally bound by the Bohm limit, which makes accelerationto higher energies slower and slower.The observations of some variable astrophysical ob-jects, however, suggested extremely fast accelerationtimescales, incompatible with diffusive shock accelera-tion. Some radio jets powered by supermassive blackholes exhibit ∼
10 min variations in TeV emissions,e.g., Aharonian et al. (2007); Aleksi´c et al. (2011). Suchfast time variabilities, along with other emission re-gion constraints, have been argued in Giannios (2013)as being evidence for mini-jets generated by reconnec-tion. Recent observations of gamma-ray flares fromCrab (Abdo et al. 2011) have suggested that the im- pulsive nature of the energy release and the associatedparticle acceleration might need an alternative explana-tion as well (Clausen-Brown & Lyutikov 2012). It hasalso been suggested that reconnection plays a crucialrole in producing high energy emissions from gamma-raybursts (Zhang & Yan 2011). As far as magnetically dom-inated enviroments are concerned, e.g., the solar coronaor the pulsar wind nebula, it is natural to expect themajor source of energy to be magnetic energy, which iswhy reconnection and associated phenomena has beena active field of study, see, e.g. Uzdensky (2015) fora review. It would be especially interesting to find ifthere a generic mechanism to transfer magnetic energyinto particles and to conceptually understand the na-ture of recent numerical results that demonstrated thatin both MHD fluid simulations (Kowal et al. 2012) andab-initio plasma simulations (Sironi & Spitkovsky 2014;Guo et al. 2014; Uzdensky & Rightley 2014) of reconnec-tion there is a regular acceleration of particlesParticle acceleration is often classified into “first or-der Fermi” mechanism where particles are gaining en-ergy regularly, e.g., by colliding with converging mag-netic mirrors and “second order Fermi” where particlescan both gain and lose energy (Fermi 1949). These twoare not mutually exclusive and represent two differentterms in the equation for the evolution of the distribu-tion function: the term describing advection in energyspace and the term describing diffusion in energy space.Speaking of practical applications, the first order mech-anism usually dominates, if present. The outcome ofacceleration depends on the rate at which particle gainenergy, called acceleration rate r acc and the rate of par-ticle escape from the system, r esc . If escape is negligi-ble and r acc is constant with energy, the energy growsexponentially. Also, if r esc /r acc does not depend on en-ergy, the stationary solution for the particle distributionis a power-law, with the power law index determined by − − r esc /r acc , see, e.g. Drury et al. (1999). Various en-vironments, such as supernova shocks, were thought tosatisfy this condition and produce power-law distributedcosmic rays, which become consistent with observationsafter being modified by diffusion. Acceleration withinmany orders of magnitude in energy was regarded as aresult of a large-scale physical layout of the accelerationsite, e.g., the planar shock can be thought of as a setof large-scale converging mirrors. The very same pic-ture could also be applied to the large-scale reconnectionsite, where the two sides of the inflow effectively workas converging mirrors (de Gouveia dal Pino & Lazarian2005). In this paper we deviate from this mindset of theproblem that achive scale-free acceleration just becausethere is only a single scale – the scale of the system. In-stead we will try to find regular acceleration over largeenergy ranges in systems that do not necessarily pos-sess global regular structure – however they could still bescale-free in statistical sense, such as turbulent systems.Normally, turbulent environments are expected to be re-gions of second-order acceleration, see, e.g., Schlickeiser(2002); Cho & Lazarian (2006). In this paper we pointto the mechanism of regular or first order accelerationthat was overlooked in the literature. This mechanismis inherently related to a certain statistical measure ofenergy transfer in turbulence and, therefore, does notrely on a particular geometry and is very robust. As wewill show below, the direction of energy transfer frommagnetic field to kinetic motions and the sign of curva-ture drift acceleration are inherently related, so that insystems with the average positive energy transfer frommagnetic energy to kinetic motions there is an averagepositive curvature drift acceleration, while in the oppo-site case, there is an average curvature drift cooling.One of the commonly considered cases of magnetically-driven flows is magnetic reconnection. A significant ef-fort was put into understanding on non-ideal plasma ef-fects that could both cause reconnection and create non-zero parallel component of the electric field (Pritchett2006; Fu et al. 2006). While non-ideal effects are indeedrequired for the individual field lines to break and re-connect, their influence is limited to fairly small scales,typically below the ion skin depth d i . In this paper, weinstead focus on larger scales and ignore non-ideal effectsfor the following two reasons. Firstly, it has been arguedthat understanding of global energetics of large-scale re-connection, such as the amount of magnetic energy dis-sipated per unit time, does not necessarily require de-tailed knowledge on how individual field lines break andreconnect (Lazarian & Vishniac 1999; Eyink et al. 2011;Loureiro et al. 2007; Beresnyak 2013). These global en-ergetic parameters could be more important for acceler-ation to high energies than local non-ideal effects. Sec-ondly, in order to understand high energy particle accel-eration, one normally has to consider plasma dynamicson scales much larger than d i , that is on MHD scales.While modern simulations such as Sironi & Spitkovsky(2014); Guo et al. (2014); Uzdensky & Rightley (2014),are able to reach box sizes of several hundred d i , sometheory work is needed to disentangle contribution fromMHD field and the non-ideal field to acceleration. MHD FLOWS AND ENERGY TRANSFERWell-conductive plasmas can be described on largescales as inviscid and perfectly conducting fluid (ideal
Fig. 1.—
A cartoon of several acceleration mechanisms in mag-netized environments. The upper panel depicts reconnection withinflow and outflow where particles can be accelerated regularly dueto the gradient drift and the large-scale electric field The accelera-tion term is dominated by v z and E z . The middle panel depicts ini-tial stages of spontaneous reconnection which has negligible inflow,therefore the gradient drift term averages out. In this case accel-eration is mostly driven by contracting field lines which drive fluidmotion and at the same time cause the curvature drift of particles.Note that the direction of drift is typically perpendicular to thefluid motion. The dominant acceleration term is associated with E x and the field curvature of B y component. The bottom paneldepicts the same mechanism in a more homogeneous and isotropicsetup of decaying MHD turbulence, which also has contracting fieldlines. In this situation all vector components contribute equally. MHD). The ideal MHD equations allows for exchangebetween thermal, kinetic and magnetic energies. TheLorentz force density, multiplied by the fluid velocity, u · [ j × B ] /c is the amount of energy transferred frommagnetic to kinetic energy. While macroscopic (i.e. ki-netic plus magnetic) energy is expected to be conservedin the ideal MHD, it is not the case in real systems whichhave non-zero dissipation coefficients. This is qualita-tively explained by the nonlinear turbulent cascade thatbrings macroscopic energy to smaller and smaller scalesuntil it dissipates into thermal energy. One of the im-portant examples of this is the spontaneous reconnec-tion where the thin current layer becomes turbulent andstarts dissipating magnetic energy at a constant rate.The small scales of these turbulent flows resemble “nor-mal” MHD turbulence which has equipartition betweenmagnetic and kinetic energies (Beresnyak 2013), it is alsotrue that kinetic and magnetic part of the cascade eachcontribute around half of the total cascade rate. There-fore, if we assume that the turbulent cascade is being fedwith magnetic energy, approximately half of the mag-netic energy has to be transferred into kinetic energybefore equipartition cascade sets in. It follows that the B to v energy transfer must be positive on average andcould be approximated by one half of the volumetric en-ergy dissipation rate ǫ , the main parameter of turbulence.The term u · [ j × B ] /c is the Eulerian expression for thework done by magnetic tension upon the fluid element.This term can be rewritten as the sum of − ( u · ∇ ) B / π ,advection of magnetic energy density by the fluid, and T bv = u · ( B · ∇ ) B / π, (1)the actual energy transfer between B and v . For thepurpose of future calculations we will separate the T bv inthe following way: T = 14 π u · ( B · ∇ ) B =14 π ( u · B )( b · ∇ ) B + B π u · ( B · ∇ ) b = X + D , (2)where we designated b = B /B , a unit magnetic vec-tor. The term X contains cross helicity density u · B .We argue that in those systems where h u · B i = 0, whichinclude many physically relevant cases, such as sponta-neous reconnection, the whole X term could average out(see also Fig. 2). The second term D contains magneticfield curvature ( B · ∇ ) b and will be important for sub-sequent calculation of curvature drift. CURVATURE DRIFTTo explore the implications of magnetic energy transferin non-thermal particle acceleration, it is instructive toconsider the particles motion in slowly-varying electricand magnetic field, which can be described in the so-called drift approximation. The leading drift terms areknown as electric, gradient and curvature drifts. Whileelectric drift, proportional to [ E × B ], can not produceacceleration, the other two drifts can. For example,imagine the configuration of the reconnection with theinflow, Fig 1 top panel. The gradient drift ∼ [ B × ∇ B ]is along − z , and so is the electric field in the ideal case E = − [ u × B ] /c . Their product will be positive and willresult in acceleration which is due to particles being com-pressed by the converging inflow. This mechanism doesnot work in the initial, most energetic stages of spon-taneous reconnection which has negligible inflow, Fig. 1middle panel (Beresnyak 2013). It is this initial stagethat has the highest volumetric dissipation rate, how-ever. Fig 1 illustrates why curvature drift acceleration isimportant in this configuration. It also turns out that inany magnetically-driven turbulent environment, such asdepicted on the bottom panel of Fig 1, curvature driftacceleration will accelerate particles on average.Let us look carefully at the term which is responsiblefor acceleration by curvature drift, d E dt = − E k B [ u × B ] · [ b × ( b · ∇ ) b ] , (3)see, e.g., Sivukhin (1965), where E k = v k p k / γmv k / E k /B ) u · ( B · ∇ ) b . It now becomes clear that thisterm is related to the transfer rate between magnetic andkinetic energies, in particular it is a fraction of D : d E dt = E k πB D . (4)The physical meaning of this equation is that, givenefficient particle scattering, so that E k = E /
2, the accel- eration rate is determined by the half of the local energytransfer rate 8 π D /B , not including the X term. A CASE STUDYWe can test the general ideas outlined above in twophysical cases that feature turbulent energy transfer frommagnetic to kinetic energies. Using spontaneous recon-nection and the decaying MHD turbulence simulated nu-merically, we can directly calculate the discussed termsand compare them. The spontaneous reconnection nu-merical experiment was started with thin planar currentsheet and small perturbations in u and B and was de-scribed in detail in Beresnyak (2013), while the decayingMHD turbulence was similar to our previous incompress-ible driven simulations in Beresnyak & Lazarian (2009),except that there was no driving and the initial condi-tions were set as a random magnetic field with wavenum-bers 1 < k < T , D , and X terms and presentedthem in Fig. 2.The spontaneous reconnection case had fairly stablereconnection rate, this also corresponded to the approx-imately constant D integrated over the volume. The X term didn’t seem to be sign-definite and contributed rel-atively little. Gradient drift acceleration was also negli-gible, possibly due to the absence of global compression.Keeping in mind that all the energy had to come frommagnetic energy, and given that the dissipation rate wasapproximately constant, it was no surprise that the aver-age D term evolved relatively little. It should be noted,however, that in the spontaneous reconnection experi-ment the width of the reconnection region was grow-ing approximately linear with time (Beresnyak 2013),so the D term magnitude, pertaining to the reconnec-tion region itself was much higher than that of an av-eraged D over the total volume. Given the reconnec-tion layer thickness l ( t ), the local D could be estimatedas (1 / v r ( B / π ) /l ( t ), where v r is a reconnection rate,which was v r ≈ . v A in Beresnyak (2013) and the 1 / η ≪ l ( t ). This wouldcorrespond to acceleration rate of (1 / v r /l ( t ) and canbe very high, because the l ( t ) could be as small as theSweet-Parker current sheet width or the ion skin depth.The decaying MHD turbulence experienced tworegimes: 1) the initial oscillation when excessive amountof magnetic energy was converted into kinetic energyand the bounce back and partial inverse conversion after-wards; 2) the self-similar decay stage of MHD turbulence.The first stage had the strongest conversion term whichwas dominated by D . All terms integrated over time weremostly accumulated within the first 1-2 Alfv´enic times. SCALE-FILTERED QUANTITIESIt is known from turbulence theory that the energytransfer rate T can be demonstrated to be local in scale,under relatively weak assumptions (Aluie & Eyink 2010;Beresnyak 2012). The scale-locality means that eachscale contributes to the transfer independently. We alsoknow empirically that most of the transfer between mag-netic and kinetic energies happens on relatively large ratescumulative cutoff wavevector reconnectionturbulence reconnection reconnection time -0.200.20.40.60.8 0 1 2 3 4 5 6 7 ratescumulative turbulence time Fig. 2.—
A case study of terms related to curvature drift acceleration and energy conversion in spontaneous reconnection and decayingMHD turbulence. Left panel: case A, volumetrically averaged energy conversion rate T and curvature acceleration rate D in MHDsimulation with turbulent current layer produced by spontaneous reconnection with setup described in Beresnyak (2013). Right panel:case B, the same for decaying turbulence generated by random initial field. On both panels we also show cumulatives R {T , D} dt . Thereconnection case (left panel) is characterized by approximately constant turbulent dissipation rate and it also show stable rate of energyconversion T , while D closely follow T . The decaying turbulence (middle panel) shows a burst of energy conversion rate within a fewdynamical times (Alfv´enic times). In both cases the gradient drift acceleration term (not shown) is relatively negligible. Bottom panelshows the scale-dependency of the D term, by plotting D l which is obtained by calculating D with a coarse-grained dynamical quantities v , B (Gaussian low pass filter in Fourier space with a cutoff wavenumber k = 2 π/l ). D l has a physical meaning of all energy transfer B to v accumulated down to scale l , for which reason it asymptotes to a constant at small l or large cutoff wavenumbers. The inset in theright panel shows kinetic and magnetic spectra in case A to demonstrate the range of scales within which magnetic energy is transferredto kinetic – down to k ≈
30 with thick arrows depicting the energy transfer. scales which are comparable to the outer scale of the sys-tem, while below outer scale there is little average trans-fer due to an approximate equipartition between kineticand magnetic energies. For example, the reconnectingturbulent current layer has most of its T transfer withina factor of a few of the scale of the layer thickness, whilein the decaying turbulence problem it is within a factorof a few from the outer scale of the system. Let us des-ignate T l as a transfer calculated from quantities whichwere filtered by low-pass Gaussian filter with a cutoffwavenumber of 1 /l . Keeping in mind of locality we willconclude that only scales larger than l will contribute to T l . This means that T l will be constant for small l andwill start decreasing when l approaches the outer scale ofthe problem L . In general, we can not deduce the samefor D l and X l , but keeping in mind that X l contributesrelatively little in two cases that we considered, D l ≈ T l in those cases. Fig. 2 demonstrates this behavior on thebottom panel, where energy transfer is operating betweenwavenumbers 2 and 20 in the reconnection case (1/20 isapproximately the layer width at this point), and the D l mostly changes within the same range of scales. Thedecaying turbulence case has the transfer more localizedaround the outer scales.In terms of drift, the particle with Larmor radius r L will “feel” magnetic and electric fields on scales largerthan r L , while the scales smaller or equal to r L will con-tribute to particle scattering. It follows that the “effec-tive” D will be D r L , an implicit function of energy. Com-bining this with the result obtained above that D l goes toa constant for small l we conclude that the accelerationrate will also go to a constant for particles with r L smallerthan the system size. A similar, more hand-waving ar-gument, is that the term D l could be roughly approx-imated as B l v l /l , resembles turbulent energy transferrate, which is scale-independent. Interestingly, start-ing with scale independent energy transfer in turbulencewe arrived at the energy-independent acceleration rates.Given the generality of the arguments presented aboveit is not surprising that energy-independent rates wereindeed observed in simulations Guo et al. (2014). ACCELERATION IN SPONTANEOUSRECONNECTION The development of the thin current sheet instabilityresults in turbulence and reconnection in a sense of dissi-pated magnetic energy. This process will come throughtwo distinct regimes, the regime without significant out-flow for times smaller than
L/v A (Beresnyak 2013) andthe stationary reconnection with outflow for larger times(Lazarian & Vishniac 1999). Let us consider the firstregime which has larger dissipation rate per unit vol-ume, because the current layer thickness l ( t ) = v r t isrelatively small. We use v r for the reconnection rateand t as the time since the beginning of spontaneous re-connection. Let us assume that the current layer thick-ness is limited from below by the ion skin depth. Wewill have acceleration rate of 1 / (4 t ) for all times largerthan d i /v r but smaller than L/v A . The solution for en-ergy, therefore, will be E = E ( tv r /d i ) / where E is theinitial energy, e.g., the thermal energy. The particle’senergy will be E max = 0 . E ( L/d i ) / given the recon-nection rate v r = 0 . v A . This, however, is only themaximum energy of accelerated particles, as only a tinyfraction of particles were contained in the original thincurrent sheet and started accelerated from initial time d i /v r (Guo et al. 2014). With proper stochastic treat-ment and assuming that the escape rate r esc is negligiblecompared with acceleration rate in this no-outflow prob-lem, we expect to see the power law particle spectrumwith the − E max .The subsequent development of an outflow willdo three things: first it will enable the inflowand therefore the extra acceleration term associatedwith gradient drift or converging magnetic mirrors(de Gouveia dal Pino & Lazarian 2005). Secondly, itwill stabilize acceleration rate for the curvature drift ac-celeration at v A / L , as the current layer is no longer ex-panding. Thirdly, it will enable particle escape throughthe outflow. In this regime the spectrum will extendfrom E max to higher energies, up to Larmor radii of thelarge scale of the current sheet L . The spectral slopeof this extension will be determined by − − r esc /r acc ,where r acc should account for all acceleration and cool-ing mechanisms, such as gradient drift acceleration andoutflow cooling. The detailed analysis of this stage willbe the subject of a future work.The electron spectra observed in solar X-ray flares arefitted with the thermal component with temperature ofseveral keV and the steep power law component. Thisis consistent with our picture, as the rather shallow − E max , is likelyto thermalize. Also, the outflow phase will extend thisdistribution as a power-law to higher energies. DISCUSSIONWe demonstrated that magnetic configurations that re-lax to the lower states of magnetic energy will also reg-ularly accelerate particles, on timescales which are, typ-ically, Alfv´enic, but can be much shorter, e.g., in the be-ginning of spontaneous reconnection. This mechanism ofacceleration of collisionless nonthermal particles by MHDelectric field should not be confused with the accelerationof the bulk of the plasma by magnetic tension. Indeed,for partiles with low energies the drift terms could beneglected, i.e. expression (3) will trivially turn to zero.In the bulk fluid acceleration the energy gained by eachparticle does not depend on its initial energy, while fordift acceleration scenario it is proportional to the particleenergy.Some recent observations, e.g., (Aharonian et al. 2007;Abdo et al. 2011; Aleksi´c et al. 2011) suggested high en-ergy emission variability could be as fast as variability atlower energies, which is at odds with DSA, which predictsacceleration timescale proportional to diffusion coeffi-cient which is typically a positive power of energy. Thishas been pointed out as a moivation for reconnection sce-nario (Clausen-Brown & Lyutikov 2012; Giannios 2013;Zhang & Yan 2011).Particle acceleration during reconnection is a topic un-der intense study, but the mechanism discussed in thispaper is distinctly different from the direct accelerationby the reconnecting electric field at the X-line. In fact, wecompletely ignore non-ideal effects which produce local E k B . Also, our mechanism is not tied to a special X-point, but instead volumetric. An interesting first-orderacceleration mechanism in ideal MHD turbulence relatedto imbalanced turbulence and convergent field lines hasbeen suggested recently by Schlickeiser (2009).The regular acceleration due to converging field lineshave been suggested in de Gouveia dal Pino & Lazarian(2005), later it was pointed by Drury (2012) that anoutflow cooling should also be included. In this paperwe do not rely on simple transport equation, such asParker’s, therefore we relax the above approach’s require-ment that particles need to be almost isotropic. The acceleration in turbulent reconnection has been furthernumerically studied in Kowal et al. (2012). Plasma PICsimulations has also been increasingly used to understandparticle acceleration. The emphasis was mostly on thenon-ideal effects near X-line regions and interaction withmagnetic islands (Zenitani & Hoshino 2001; Pritchett2006; Fu et al. 2006; Drake et al. 2006; Oka et al. 2010;Dahlin et al. 2014). The change of energy due to curva-ture drift in a single collision of a particle with magneticisland was estimated analytically in Drake et al. (2006).An important question that was left out in that studywas whether this term will result in acceleration or cool-ing, on average. Without understanding this, it was notclear whether this process results in acceleration or de-celeration or the second-order effect. In this paper weunambiguously decide this by relating the answer to acertain well-known statistical measure in turbulence –the direction of transfer between magnetic and kineticenergies. We also showed that the curvature drift accel-eration does not require particles to be trapped in con-tracting magnetic islands, so their energy is not limitedby this requirement, meaning that the energy cutoff isnot related to the island size, instead it is related to thesystem size, see also Uzdensky & Rightley (2014).PIC simulations are limited in the range of scalesand energies they cover. Recent simulations inSironi & Spitkovsky (2014); Guo et al. (2014) demon-strated acceleration up to 100 MeV in electron energies,which is below maximum energy in most astrophysicalsources. Theory, therefore, is necessary to supplementconjectures based on the observed PIC distribution tails,explaining the underlying physical mechanism and mak-ing predictions for astrophysical systems which often fea-ture gigantic scale separation between plasma scales andthe size of the system. The feedback from simulations,nevertheless, was very useful, in particular the recentsimulations (Guo et al. 2014) that reached MHD scalesand confirmed the prediction that the curvature drift ac-celeration will dominate compared to the non-ideal elec-tric field acceleration. ACKNOWLEDGEMENTSWe are grateful to Fan Guo for sharing his resultson a preliminary stage. The work is supported by theLANL/LDRD program and the DoE/Office of FusionEnergy Sciences through CMSO. Computing resourcesat LANL are provided through the Institutional Comput-ing Program. We also acknowledge support from XSEDEcomputational grant TG-AST110057.