Dissipation measures in weakly-collisional plasmas
O. Pezzi, H. Liang, J.L. Juno, P.A. Cassak, C.L. Vasconez, L. Sorriso-Valvo, D. Perrone, S. Servidio, V. Roytershteyn, J.M. TenBarge, W.H. Matthaeus
MMNRAS , 1–17 (2021) Preprint 5 January 2021 Compiled using MNRAS L A TEX style file v3.0
Dissipation measures in weakly-collisional plasmas
O. Pezzi , , ★ , H. Liang , J. L. Juno , P. A. Cassak , C. L. Vásconez , L. Sorriso-Valvo , , D. Perrone ,S. Servidio , V. Roytershteyn , J. M. TenBarge & W. H. Matthaeus Gran Sasso Science Institute, Viale F. Crispi 7, I-67100 L’Aquila, Italy INFN/Laboratori Nazionali del Gran Sasso, I-67100 Assergi (AQ), Italy Istituto per la Scienza e Tecnologia dei Plasmi, Consiglio Nazionale delle Ricerche, Via Amendola 122/D, 70126 Bari, Italy Center for Space Plasma and Aeronomic Research, University of Alabama in Huntsville, Huntsville, AL 35899, USA Department of Physics and Astronomy, University of Iowa, Iowa City IA 54224, USA Department of Physics and Astronomy and Center for KINETIC Plasma Physics, West Virginia University, WV, 26506, USA Departamento de Física, Escuela Politécnica Nacional, Ladrón de Guevara E11-253, 170525 Quito, Ecuador Swedish Institute of Space Physics, Ångström Laboratory, Lägerhyddsvägen 1, SE-751 21 Uppsala, Sweden ASI – Italian Space Agency, via del Politecnico snc, I-00133 Rome, Italy Dipartimento di Fisica, Università della Calabria, I-87036 Rende (CS), Italy Space Science Institute, Boulder, CO 80301, USA Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA Bartol Research Institute and Department of Physics and Astronomy, University of Delaware, Newark, Deleware 19716, USA
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
The physical foundations of the dissipation of energy and the associated heating in weakly collisional plasmas are poorlyunderstood. Here, we compare and contrast several measures that have been used to characterize energy dissipation and kinetic-scale conversion in plasmas by means of a suite of kinetic numerical simulations describing both magnetic reconnection anddecaying plasma turbulence. We adopt three different numerical codes that can also include inter-particle collisions: the fully-kinetic particle-in-cell vpic, the fully-kinetic continuum
Gkeyll , and the Eulerian Hybrid Vlasov-Maxwell (HVM) code. Wedifferentiate between i) four energy -based parameters, whose definition is related to energy transfer in a fluid description of aplasma, and ii) four distribution function -based parameters, requiring knowledge of the particle velocity distribution function.There is overall agreement between the dissipation measures obtained in the PIC and continuum reconnection simulations, withslight differences due to the presence/absence of secondary islands in the two simulations. There are also many qualitativesimilarities between the signatures in the reconnection simulations and the self-consistent current sheets that form in turbulence,although the latter exhibits significant variations compared to the reconnection results. All the parameters confirm that dissipationoccurs close to regions of intense magnetic stresses, thus exhibiting local correlation. The distribution function-based measuresshow a broader width compared to energy-based proxies, suggesting that energy transfer is co-localized at coherent structures,but can affect the particle distribution function in wider regions. The effect of inter-particle collisions on these parameters isfinally discussed.
Key words: turbulence – magnetic reconnection – (Sun:) solar wind
Understanding energy dissipation and heating in weakly collisionalplasmas is a key challenge in the study of space and astrophysicalplasmas, such as the solar corona, solar wind, the outer magne-tospheres of planets, and compact astrophysical objects. At vari-ance with neutral fluids and collisional plasmas (e.g. magnetoflu-ids), inter-particle collisions are typically weak in these systems andare often neglected. One example is during magnetic reconnection,where magnetic fields with a reversing component effectively breakand cross-connect at length scales at and below the gyroradius ofthe particles (kinetic scales), allowing the conversion of magnetic ★ Email: [email protected] energy to kinetic, thermal, and non-thermal energy (Yamada et al.2010). Another example is that weakly-collisional plasmas are oftenobserved to be in a strongly turbulent state (Federrath et al. 2010;Bruno & Carbone 2016; Narita 2018; Beresnyak 2019; Fraternaleet al. 2019). Hence, they are characterized by the cross-scale transferof fluctuation energy from large injection scales to smaller, kineticscales, where energy dissipation is expected to occur (Schekochihinet al. 2009; Howes 2015b; Matthaeus et al. 2015; Vaivads et al. 2016;Verscharen et al. 2019). These features have, at least, two profoundimplications.First, weakly-collisional plasmas can in principle explore thewhole phase space (i.e., both in spatial and velocity coordinates),as opposed to collisional systems for which the particle distribu-tion functions (VDFs) do not greatly differ from a Maxwellian. In- © a r X i v : . [ phy s i c s . p l a s m - ph ] J a n O. Pezzi et al. deed, VDFs in weakly-collisional plasmas frequently display non-equilibrium features, such as temperature anisotropy and agyrotropy,rings, beams of accelerated particles, etc. (Marsch 2006; Servidioet al. 2012, 2015; Lapenta et al. 2017; Perri et al. 2020). In turbu-lence, this interesting dynamics has been envisioned as a phase-spacecascade (Tatsuno et al. 2009; Kanekar et al. 2015; Schekochihinet al. 2016; Parker et al. 2016; Servidio et al. 2017; Plunk & Tat-suno 2011), where the plasma exhibits a power-law scaling –typicalof turbulence– in both physical and velocity space. In this perspec-tive, by expanding particle VDFs in velocity space using Hermitepolynomials, a power-law Hermite spectrum was recently observed,reflecting the presence of fine velocity-space structures, in both in-situ observations (Servidio et al. 2017) and numerical simulations(Pezzi et al. 2018; Cerri et al. 2018). This clearly illustrates thenecessity of using kinetic models to describe such plasmas. More-over, non-Maxwellian structures in particle VDFs are observed to besignificant in the vicinity of intense current sheets or other nearbycoherent structures such as vortices, where MHD-like dissipation isthought to occur (Osman et al. 2011; Servidio et al. 2012; Matthaeuset al. 2015; Parashar & Matthaeus 2016).Second, concepts about energy dissipation and conversion in neu-tral fluids and magnetofluids do not necessarily carry over to weakly-collisional systems. The rate and parametric dependence of dissi-pation and kinetic-scale energy conversion for such plasmas is ex-tremely challenging and is not thoroughly understood. One challengeis that various physical mechanisms and processes can cause dissi-pation and kinetic scale energy conversion in various physical cir-cumstances. In weakly collisional plasmas in which kinetic effectsplay a crucial role in the dynamics, such as during turbulence andreconnection, it may be difficult to assess quantitatively the relativeimportance of the various mechanisms in the overall dissipation bud-get, and likewise the net effect of a given dynamical process maybebe unclear due to reversibility. To address these issues different mech-anisms and concepts of dissipation have been proposed and, hence,several dissipation surrogates have been adopted (see, e.g., Vaivadset al. 2016; Matthaeus et al. 2020, for recent reviews). However, aunified and general picture of when and where different views workbetter is still lacking.Another important question is the following: do the sites identi-fied potential sites of dissipation correspond to regions where inter-particle collisions, although weak, dissipate energy in an irreversibleway? Addressing this question has encouraged the examination ofa different concept of dissipation associated with the growth of en-tropy due to collisions (TenBarge et al. 2013; Navarro et al. 2016;Pezzi et al. 2019c; Liang et al. 2019). Although collisions typicallyact on large characteristic times (Spitzer Jr. 1956; Vafin et al. 2019),their effects are enhanced where particle VDFs exhibit strong dis-tortions, since intense velocity-space gradients are dissipated veryquickly by collisions (Landau 1936; Rosenbluth et al. 1957; Balescu1960; Schekochihin et al. 2009; Pezzi et al. 2016; Pezzi 2017). Non-Maxwellian VDFs make intra-species collision operators non-zero,thus activating this dissipation channel.In the following, we classify eight different dissipation proxies intotwo general classes that we call “energy-based” and “VDF-based”.The energy-based definition describes dissipation as a transfer ofenergy within a fluid-like description. Often such transfer occursfrom an ordered component (e.g. magnetic or bulk flow fluctuations)into a random (e.g., internal) component. On the other hand, VDF-based surrogates directly quantify the presence of non-equilibriumfeatures in the particle VDF. The two classes of dissipation proxiesare correlated, since the distortion of the particle VDF is often theconsequence of a transfer of energy and vice versa. The dissipation proxies here adopted do not explicitly distinguish signatures asso-ciated with particular phenomena, e.g. Landau damping, cyclotrondamping, or stochastic processes (Chandran et al. 2010; Numata &Loureiro 2015; Li et al. 2016; Chen et al. 2019). Future studies willanalyze the connection between these dissipation measures –usefulto detect potential sites of inhomogeneous dissipation in a turbulentenvironment– and the underlying plasma processes, e.g. highlightedthrough the field-particle correlation (Klein & Howes 2016; Kleinet al. 2017; Chen et al. 2019; Klein et al. 2020).In this work, we investigate numerically the functionality of severaldissipation proxies belonging to the two classes introduced above.We focus on two different types of numerical simulations: mag-netic reconnection in a single current sheet and the developmentof a turbulent cascade at kinetic scales. We exploit three differentnumerical Boltzmann-Maxwell algorithms, of both Lagrangian andEulerian type, that can include inter-particle collisions. In partic-ular, we adopt the fully kinetic particle-in-cell vpic code (Bowerset al. 2008), and two Eulerian Vlasov-Maxwell codes. These lattercodes are the fully kinetic continuum Vlasov-Maxwell solver imple-mented in the
Gkeyll simulation framework (Juno et al. 2018) andthe Hybrid Vlasov-Maxwell (HVM) code with kinetic protons andfluid electrons (Valentini et al. 2007). We find that the dissipationmeasures well characterize significant features of both magnetic re-connection and turbulence, such as the reconnection diffusion regionand the intermittent current sheets surrounding turbulent vortices.The dissipation surrogates evaluated from the PIC and continuumreconnection simulations agree at a wide extent: slight differencesresult from the presence of secondary islands in the continuum case.Qualitative correspondences between the signatures in the reconnec-tion simulations and the self-consistent current sheets generated inturbulence is also found, despite larger variations observed in theturbulent case with respect to the magnetic reconnection one. Theparameters show a regional correlation : their local peaks take placein similar spatial regions, but they are not necessarily point-to-pointcorrelated (Yang et al. 2019; Matthaeus et al. 2020). When includ-ing inter-particle collisions, peaks of the dissipation proxies are ingeneral weaker than in the associated collisionless system, suggest-ing that the slow yet incessant effect of collisions locally reduce thetransfer of energy and the presence of non-Maxwellian features. Byconsidering the effect of both intra- and inter- species collisions, weconfirm that the former mainly dissipate non-Maxwellian features inthe particle VDF, although they may have an indirect effect also onenergy transfer through the pressure tensor isotropization (Del Sartoet al. 2016). On the other hand, the latter also significantly affectenergy-based parameters. Finally, by adopting a suite of differentalgorithms and numerical codes, the current work aims at provid-ing a further contribution to the “turbulence dissipation challenge”(Parashar et al. 2015), on which several recent efforts have been ded-icated (e.g, Pezzi et al. (2017); Perrone et al. (2018); González et al.(2019)).The paper is structured as follows. In Section 2, we define anddiscuss the dissipation measures investigated in the current work.In Section 3, numerical models and algorithms adopted for the cur-rent analysis are described. Sections 4 and 5 report numerical resultsobtained in the simulations of reconnection and the onset of kineticturbulence, respectively. In Section 6, we show one-dimensional pro-files of the dissipation proxies close to the reconnecting current sheetand a typical current sheet observed in the turbulence simulation.Finally, conclusions and discussions are presented in Section 7.
MNRAS , 1–17 (2021) issipation measures in plasmas In this section, we introduce the framework and summarize the dis-sipation proxies adopted in the present work. We consider a weakly-collisional plasma, composed of protons ( 𝑝 ) and electrons ( 𝑒 ). TheBoltzmann-Maxwell equations, describing non-relativistic plasmas,in cgs units are: 𝜕 𝑓 𝛼 𝜕𝑡 + 𝒗 · 𝜕 𝑓 𝛼 𝜕 𝒓 + 𝑞 𝛼 𝑚 𝛼 (cid:16) 𝑬 + 𝒗 𝑐 × 𝑩 (cid:17) · 𝜕 𝑓 𝛼 𝜕 𝒗 = 𝜕 𝑓 𝛼 𝜕𝑡 (cid:12)(cid:12)(cid:12)(cid:12) coll (1) ∇ · 𝑬 = 𝜋𝜌 𝑐 (2) ∇ · 𝑩 = ∇ × 𝑬 = − 𝑐 𝜕 𝑩 𝜕𝑡 (4) ∇ × 𝑩 = 𝑐 𝜕 𝑬 𝜕𝑡 + 𝜋𝑐 𝒋 , (5)where 𝑓 𝛼 ( 𝒓 , 𝒗 , 𝑡 ) is the 𝛼 -species VDF ( 𝛼 = 𝑝, 𝑒 ); 𝑬 ( 𝒓 , 𝑡 ) and 𝑩 ( 𝒓 , 𝑡 ) are the electric and magnetic fields; and 𝑞 𝛼 , 𝑚 𝛼 , and 𝑐 are the 𝛼 -species charge, mass and the light speed. The chargeand electric current densities are respectively 𝜌 𝑐 = (cid:205) 𝛼 𝑞 𝛼 𝑛 𝛼 and 𝒋 = (cid:205) 𝛼 𝑞 𝛼 𝑛 𝛼 𝒖 𝛼 , where 𝑛 𝛼 = ∫ 𝑑 𝑣 𝑓 𝛼 is the 𝛼 -species numberdensity and 𝑛 𝛼 𝒖 𝛼 = ∫ 𝑑 𝑣 𝒗 𝑓 𝛼 is the 𝛼 -species number flux den-sity. In the following subsections, we omit the collisional operator 𝜕 𝑓 𝛼 / 𝜕𝑡 | coll for simplicity. The energy-based dissipation proxies can be introduced from theenergy equations: 𝜕 E 𝑓𝛼 𝜕𝑡 + ∇ · (cid:16) 𝒖 𝛼 E 𝑓𝛼 + 𝒖 𝛼 · 𝑷 𝛼 (cid:17) = ( 𝑷 𝛼 · ∇) · 𝒖 𝛼 + 𝑛 𝛼 𝑞 𝛼 𝒖 𝛼 · 𝑬 (6) 𝜕 E 𝑡ℎ𝛼 𝜕𝑡 + ∇ · (cid:16) 𝒖 𝛼 E 𝑡ℎ𝛼 + 𝒉 𝛼 (cid:17) = − ( 𝑷 𝛼 · ∇) · 𝒖 𝛼 (7) 𝜕 E 𝑚 𝜕𝑡 + 𝑐 𝜋 ∇ · ( 𝑬 × 𝑩 ) = − 𝒋 · 𝑬 (8)where E 𝑓𝛼 = 𝜌 𝛼 𝒖 𝛼 / 𝛼 -species, E 𝑡ℎ𝛼 = 𝑚 𝛼 ∫ 𝑑 𝑣 𝑓 𝛼 ( 𝒗 − 𝒖 𝛼 ) / 𝛼 -species and E 𝑚 = ( 𝑬 + 𝑩 )/ 𝜋 is theelectromagnetic energy density. In the above equations, 𝑷 𝛼 = 𝑚 𝛼 ∫ 𝑑 𝑣 ( 𝒗 − 𝒖 𝛼 )( 𝒗 − 𝒖 𝛼 ) 𝑓 𝛼 (9) 𝒉 𝛼 = ∫ 𝑑 𝑣 ( 𝒗 − 𝒖 𝛼 ) ( 𝒗 − 𝒖 𝛼 ) 𝑓 𝛼 (10)are the pressure tensor and vector heat flux, respectively.The left-hand sides of Eqs. (6–8) contain terms involving diver-gences of fluxes. These terms can be locally important and corre-spond physically to energy transport (Pezzi et al. 2019a). Assumingthere is no flux across the domain boundaries (e.g., with periodicboundary conditions), they have no net effect on the global energypartition of the system. Energy transfer between bulk flow and mag-netic energy is described by the 𝒋 · 𝑬 term, while conversion of energybetween bulk flow and thermal occurs through the pressure-strain in-teraction ( 𝑷 𝛼 · ∇) · 𝒖 𝛼 . Including intra-species collisional effects(e.g. proton-proton and electron-electron) would not introduce anextra term in Eqs. (6–8). However these collisions indirectly affectthese equations by thermalizing the particle VDF, thus reducing thenongyrotropic terms in the pressure tensor and, in turn, having an impact on the pressure-strain interaction term (Del Sarto et al. 2016).On the other hand, inter-species collisions (electron-proton) insert anexplicit inter-species energy transfer term in the energy equations.The rate of work per unit volume done by the electric field onparticles, 𝒋 · 𝑬 , dubbed the Zenitani measure when evaluated in theelectron fluid frame, has been widely adopted to describe dissipationin magnetic reconnection (Zenitani et al. 2011; Phan et al. 2018)and plasma turbulence (Wan et al. 2015). It is often evaluated in thereference frame co-moving with the considered species, i.e. 𝐷 𝛼 = 𝒋 (cid:48) · 𝑬 (cid:48) = 𝒋 · (cid:16) 𝑬 + 𝒖 𝛼 𝑐 × 𝑩 (cid:17) − 𝜌 𝑐 ( 𝒖 𝛼 · 𝑬 ) , (11)where 𝒋 (cid:48) and 𝑬 (cid:48) are the current density and the electric field inthe reference frame co-moving with the species 𝛼 , respectively. Asshown in Zenitani et al. (2011), 𝑛 𝑒 𝐷 𝑒 = 𝑛 𝑝 𝐷 𝑝 and, in a singly ion-ized quasi-neutral system such as the ones considered in the presentwork, 𝐷 𝑒 (cid:39) 𝐷 𝑝 . 𝐷 𝛼 contains both reversible and irreversible con-tributions since the electric field 𝑬 has contributions from both re-versible (e.g. wave-particle interactions etc.) and irreversible (e.g.collisional resistive) processes. We note that the probability distribu-tion function (PDF) of 𝐷 𝛼 and other dissipation measures are almostsymmetric between negative and positive values (See Fig. 4 of Wanet al. (2015)). Net dissipation, which is ultimately an integral overspace and time, arises from slight asymmetry (skewness) in the tailsof the PDF.More recently, the pressure-strain interaction ( 𝑷 𝛼 · ∇) · 𝒖 𝛼 hasbeen analyzed to understand dissipative mechanisms in weakly-collisional plasmas (Yang et al. 2017a,b; Chasapis et al. 2018a; Yanget al. 2019; Sitnov et al. 2018; Pezzi et al. 2019a; Matthaeus et al.2020). This term is commonly decomposed as − ( 𝑷 𝛼 · ∇) · 𝒖 𝛼 = − 𝑃 𝛼 𝜃 𝛼 − 𝚷 𝛼 : D 𝛼 , (12)where 𝑃 𝛼,𝑖 𝑗 = 𝑃 𝛼 𝛿 𝑖 𝑗 + Π 𝛼,𝑖 𝑗 ; 𝑃 𝛼 = 𝑃 𝛼,𝑖𝑖 / 𝜃 𝛼 = ∇ · 𝒖 𝛼 ; D 𝛼,𝑖 𝑗 = (cid:0) 𝜕 𝑗 𝑢 𝛼,𝑖 + 𝜕 𝑖 𝑢 𝛼, 𝑗 (cid:1) / − 𝜃 𝛼 𝛿 𝑖 𝑗 /
3; and 𝛿 𝑖 𝑗 and 𝜕 𝑖 denotethe Kronecker delta and a partial derivative with respect to the 𝑖 –th spatial coordinate, respectively. The first term on the right-handside of Eq. (12), called P- 𝜃 𝛼 , is associated with plasma expansionand compression. The last term, called Pi-D 𝛼 , is associated with thetrace-less (anisotropic and off-diagonal) parts of the pressure tensorand the symmetric part of the velocity strain and describes the rate ofwork per unit volume done by flow shear. The spatial integral of thePi-D 𝛼 term measures the thermal energy gain (Pezzi et al. 2019a).Here, we use the convention that Pi-D 𝛼 and P- 𝜃 𝛼 include the minussigns in Eq. (12), so that positive values tend to locally increase thethermal energy. The Pi-D 𝛼 term has been adopted to provide insightson the mechanisms that transfer energy towards smaller scales, whereit is dissipated (Yang et al. 2019). The motivation for this term be-ing associated with dissipation arises from the MHD framework, inwhich traceless pressure-tensor terms are related to viscous dissipa-tion (Braginskii 1965). Hence, it is interesting to explore whetherthis connection remains valid in a weakly-collisional system.Another dissipation proxy is associated with the cross-scale con-version of energy in its nonlinear transfer during the turbulent cas-cade (Frisch 1995). A proxy of the scale-dependent local energytransfer rate (LET) in a weakly-collisional plasma at proton inertialscales may be estimated through the combined velocity, magneticfield and current fluctuations as (Sorriso-Valvo et al. 2018b,a, 2019) 𝜖 𝑝,ℓ = (cid:16) | Δ 𝒖 𝑝 | + | Δ 𝒃 | (cid:17) Δ 𝑢 𝑝,ℓ ℓ − ( Δ 𝒖 𝑝 · Δ 𝒃 ) Δ 𝑏 ℓ ℓ − 𝑑 𝑝 | Δ 𝒃 | Δ 𝑗 ℓ ℓ + 𝑑 𝑝 ( Δ 𝒃 · Δ 𝒋 ) Δ 𝑏 ℓ ℓ (13)where 𝑑 𝑝 is the proton skin depth. The magnetic field is in ve- MNRAS000
3; and 𝛿 𝑖 𝑗 and 𝜕 𝑖 denotethe Kronecker delta and a partial derivative with respect to the 𝑖 –th spatial coordinate, respectively. The first term on the right-handside of Eq. (12), called P- 𝜃 𝛼 , is associated with plasma expansionand compression. The last term, called Pi-D 𝛼 , is associated with thetrace-less (anisotropic and off-diagonal) parts of the pressure tensorand the symmetric part of the velocity strain and describes the rate ofwork per unit volume done by flow shear. The spatial integral of thePi-D 𝛼 term measures the thermal energy gain (Pezzi et al. 2019a).Here, we use the convention that Pi-D 𝛼 and P- 𝜃 𝛼 include the minussigns in Eq. (12), so that positive values tend to locally increase thethermal energy. The Pi-D 𝛼 term has been adopted to provide insightson the mechanisms that transfer energy towards smaller scales, whereit is dissipated (Yang et al. 2019). The motivation for this term be-ing associated with dissipation arises from the MHD framework, inwhich traceless pressure-tensor terms are related to viscous dissipa-tion (Braginskii 1965). Hence, it is interesting to explore whetherthis connection remains valid in a weakly-collisional system.Another dissipation proxy is associated with the cross-scale con-version of energy in its nonlinear transfer during the turbulent cas-cade (Frisch 1995). A proxy of the scale-dependent local energytransfer rate (LET) in a weakly-collisional plasma at proton inertialscales may be estimated through the combined velocity, magneticfield and current fluctuations as (Sorriso-Valvo et al. 2018b,a, 2019) 𝜖 𝑝,ℓ = (cid:16) | Δ 𝒖 𝑝 | + | Δ 𝒃 | (cid:17) Δ 𝑢 𝑝,ℓ ℓ − ( Δ 𝒖 𝑝 · Δ 𝒃 ) Δ 𝑏 ℓ ℓ − 𝑑 𝑝 | Δ 𝒃 | Δ 𝑗 ℓ ℓ + 𝑑 𝑝 ( Δ 𝒃 · Δ 𝒋 ) Δ 𝑏 ℓ ℓ (13)where 𝑑 𝑝 is the proton skin depth. The magnetic field is in ve- MNRAS000 , 1–17 (2021)
O. Pezzi et al. locity units 𝒃 ≡ 𝑩 / √︁ 𝜋𝜌 , where 𝜌 is the proton mass density. Δ 𝒈 ≡ 𝒈 ( 𝒓 + ℓ ) − 𝒈 ( 𝒓 ) is the increment of a field 𝒈 between two spatialpoints separated by a distance ℓ and is used to describe structuresor fluctuations of that size. The subscript ℓ indicates longitudinalincrement, i.e. the projection along the increment vector ℓ . In thiswork, LET values are associated with the starting point of the in-crement vector and, for all simulations, the LET parameter 𝜖 𝑝,ℓ iscomputed with ℓ = 𝑑 𝑝 and by averaging the increments taken in thepositive horizontal and vertical directions. Since the LET parameter 𝜖 𝑝,ℓ is introduced within a fluid framework, we do not compute itwith ℓ below proton scales, where electron kinetic effects becomeimportant. The definition of LET is motivated by the isotropic formof the Politano-Pouquet scaling law for the mixed third-order fluctua-tions in turbulent incompressible MHD plasmas (Politano & Pouquet1998; Sorriso-Valvo et al. 2002, 2007), in this case also includingthe Hall-MHD terms (Galtier 2008; Ferrand et al. 2019; Bandyopad-hyay et al. 2020; Vásconez et al. 2020). Under the assumption ofstationarity, homogeneity and large Reynolds’ number, the associ-ated scaling coefficient (cid:104) 𝜖 𝑝 (cid:105) ( (cid:104) ... (cid:105) indicating the ensemble average)is the constant mean energy transfer (or, in the stationary steadystate, the dissipation) rate of the turbulent cascade (Kolmogorov1941). In 3D fluid, fully developed turbulence, the sign of the av-eraged third-order moment has to be negative, corresponding to anensemble-averaged global net transfer towards the small scales (thenonlinear turbulent energy cascade). Other terms locally contribut-ing to the energy transfer vanish and are disregarded here. Theserepresent the different fluid contributions to the turbulent transfer ofenergy: kinetic and magnetic turbulent energy transported by veloc-ity fluctuations, coupled magnetic and velocity Alfvénic fluctuationstransported by magnetic structures, and two associated Hall terms. Inweakly-collisional plasmas, the LET may be unable to describe thecontribution of compressibility, as well as the role of possible non-thermal features. The latter might enter in the energy budget via thepressure-tensor contributions. However, it can provide informationon the local transfer of ordered energy towards small scales, whereit is made available for conversion through various possible mecha-nisms, including dissipation. In particular, according to the definitionused in this work, negative LET terms can be thought of as locallycontributing to the nonlinear energy transfer towards small scales,and positive LET terms describe a contribution to energy transfer tolarge scales. In non-turbulent systems as well as in systems displayinga large-scale structure (e.g. a single reconnecting current sheet), theinterpretation of the LET sign is more difficult, since the assumptionof homogeneity is not satisfied. However, for the present analysis,peaks of LET indicate the local concentration of cross-scale energytransfer that can be made available for dissipation, regardless of thesign. The first parameter belonging to the class of VDF-based dissipationmeasures is the pressure agyrotropy (Scudder & Daughton 2008).Agyrotropy is a measure of differences in the plasma temperatures inthe two perpendicular directions to a given axis (Scudder & Daughton2008; Aunai et al. 2013; Swisdak 2016). Standard axis orientationsfor computing agyrotropy are along the local magnetic field or themean magnetic field. Other coordinate systems, such as the minimumvariance frames of the particle VDF, have been also adopted (Ser-vidio et al. 2015; Pezzi et al. 2017). We consider here the agyrotropyparameter √ 𝑄 𝛼 proposed by Swisdak (2016), evaluated relative tothe local magnetic field. Writing the pressure tensor 𝑷 𝛼 in the coor-dinate system where the local magnetic field is parallel to the 𝑧 -axis as 𝑷 𝛼 = (cid:169)(cid:173)(cid:171) 𝑃 𝛼, ⊥ 𝑃 𝛼, 𝑃 𝛼, 𝑃 𝛼, 𝑃 𝛼, ⊥ 𝑃 𝛼, 𝑃 𝛼, 𝑃 𝛼, 𝑃 𝛼, (cid:107) (cid:170)(cid:174)(cid:172) , (14)the agyrotropy parameter is defined as 𝑄 𝛼 = 𝑃 𝛼, + 𝑃 𝛼, + 𝑃 𝛼, 𝑃 𝛼, ⊥ + 𝑃 𝛼, ⊥ 𝑃 𝛼, (cid:107) . (15) 𝑄 𝛼 can be computed in an arbitrary coordinate system, as explainedin App. A of Swisdak (2016).The variety of non-Maxwellian structures observed during mag-netic reconnection or in a turbulent plasma is much richer than justpressure agyrotropies. Another proxy, usually named 𝜖 but here called 𝜉 since 𝜖 is already used to indicate the LET proxy, was proposedby Greco et al. (2012). Here we propose a slightly different, non-dimensional definition (the original definition had dimensions of 𝑣 − / ): 𝜉 𝛼 ( 𝒓 , 𝑡 ) = 𝑣 / 𝑡ℎ,𝛼 ( 𝒓 , 𝑡 ) 𝑛 𝛼 ( 𝒓 , 𝑡 ) √︄∫ 𝑑 𝑣 [ 𝑓 𝛼 ( 𝒓 , 𝒗 , 𝑡 ) − 𝑔 𝛼 ( 𝒓 , 𝒗 , 𝑡 )] . (16)Here, 𝑔 𝛼 is the equivalent Maxwellian distribution function associ-ated with 𝑓 𝛼 , i.e. constructed using the local values of the density 𝑛 𝛼 , bulk speed 𝒖 𝛼 and total temperature 𝑇 𝛼 of the 𝛼 -species; while 𝑣 𝑡ℎ,𝛼 = √︁ 𝑘 𝐵 𝑇 𝛼 / 𝑚 𝛼 is the (local) thermal speed.Similar measures to identify non-Maxwellian VDFs were con-structed using kinetic entropy. The kinetic entropy density 𝑠 𝛼 is 𝑠 𝛼 ( 𝒓 , 𝑡 ) = − 𝑘 𝐵 ∫ 𝑑 𝑣 𝑓 𝛼 ( 𝒓 , 𝒗 , 𝑡 ) log 𝑓 𝛼 ( 𝒓 , 𝒗 , 𝑡 ) . (17)Note the total entropy 𝑆 𝛼 = ∫ 𝑠 𝛼 𝑑 𝑟 of a collisional system is non-decreasing from the Boltzmann H-theorem, but the entropy densitymay locally increase or decrease (Pezzi et al. 2019c). It is possibleto define a velocity-space entropy density 𝑠 vel 𝛼 retaining only thespatially local contribution to entropy from permutations of particlesin velocity space (Liang et al. 2019) as: 𝑠 vel 𝛼 ( 𝒓 , 𝑡 ) = 𝑠 𝛼 ( 𝒓 , 𝑡 ) + 𝑘 𝐵 𝑛 𝛼 ( 𝒓 , 𝑡 ) log (cid:18) 𝑛 𝛼 ( 𝒓 , 𝑡 ) Δ 𝑣 (cid:19) (18)where Δ 𝑣 is the volume of the cell in velocity space. Using thesedefinitions, two dimensionless non-Maxwellianity parameters havebeen introduced:¯ 𝑀 KP ,𝛼 ( 𝒓 , 𝑡 ) = 𝑠 M ,𝛼 ( 𝒓 , 𝑡 ) − 𝑠 𝛼 ( 𝒓 , 𝑡 )( / ) 𝑘 𝐵 𝑛 𝛼 ( 𝒓 , 𝑡 ) = 𝑠 velM ,𝛼 ( 𝒓 , 𝑡 ) − 𝑠 vel 𝛼 ( 𝒓 , 𝑡 )( / ) 𝑘 𝐵 𝑛 𝛼 ( 𝒓 , 𝑡 ) , (19)proposed by Kaufmann & Paterson (2009), and¯ 𝑀 𝛼 ( 𝒓 , 𝑡 ) = ¯ 𝑀 KP ,𝛼 ( 𝒓 , 𝑡 ) + log (cid:16) 𝜋𝑘 𝐵 𝑇 𝛼 𝑚 𝛼 ( Δ 𝑣 ) / (cid:17) , (20)proposed by Liang et al. (2020). Here, 𝑠 M ,𝛼 ( 𝒓 , 𝑡 ) is the entropy den-sity evaluated using the equivalent Maxwellian distribution function 𝑔 𝛼 ( 𝒓 , 𝒗 , 𝑡 ) associated with the VDF 𝑓 𝛼 ( 𝒓 , 𝒗 , 𝑡 ) , given by 𝑠 M ,𝛼 = 𝑘 𝐵 𝑛 𝛼 (cid:34) + log 2 𝜋𝑘 𝐵 𝑇 𝛼 𝑚 𝛼 𝑛 / 𝛼 (cid:35) . (21)and 𝑠 velM ,𝛼 ( 𝒓 , 𝑡 ) is computed according to Eq. (18).In a system with a fixed number of particles and total energy, theMaxwellian distribution has the maximum entropy. Hence, ¯ 𝑀 KP and MNRAS , 1–17 (2021) issipation measures in plasmas ¯ 𝑀 (along with 𝜉 ) are positive definite. These parameters measure allhigher-order VDF disturbances from the local Maxwellian beyondits second order moment. This retains information about dissipation,since high-order variations from the Maxwellian coincide with thepresence of fine velocity-space structures that, in turn, are dissipatedby collisional effects (Pezzi et al. 2016). The parameters introduced in the previous section are computedusing the results of kinetic numerical simulations performed withdifferent codes. All simulations in this work are 2.5 dimensional inspace (quantities depend on two dimensions, but vectors have threecomponents) and 3 dimensional in velocity space. For all codes,quantities are presented using a normalization based on an arbitrarymagnetic field strength 𝐵 and density 𝑛 . Spatial and temporal scalesare normalized to the proton inertial length 𝑑 𝑝 = 𝑐 / 𝜔 𝑝 𝑝 and the pro-ton cyclotron time Ω − 𝑐 𝑝 , respectively, where 𝜔 𝑝 𝑝 = √︃ 𝜋𝑛 𝑒 / 𝑚 𝑝 is the proton plasma frequency based on 𝑛 and Ω 𝑐 𝑝 = 𝑒𝐵 / 𝑚 𝑝 𝑐 is the proton cyclotron frequency based on 𝐵 . Thus, velocities arenormalized to the Alfvén velocity 𝑐 𝐴 = 𝑑 𝑝 Ω 𝑐 𝑝 ; electric fields arenormalized to 𝑐 𝐴 𝐵 / 𝑐 ; pressures and temperatures are normalizedto 𝐵 / 𝜋 and 𝑚 𝑝 𝑐 𝐴 / 𝑘 𝐵 , respectively; and entropy is normalized toBoltzmann’s constant 𝑘 𝐵 (see Liang et al. (2019) for a detailed dis-cussion of the units of the continuous Boltzmann entropy). Derivedunits of the dissipation measures are therefore as follows: 𝐷 𝛼 , Pi-D 𝛼 , and P- 𝜃 𝛼 are Ω 𝑐 𝑝 𝐵 / 𝜋 , 𝜖 𝑝 is 𝑐 𝐴 Ω 𝑐 𝑝 , while 𝑄 𝛼 , 𝜉 𝛼 , ¯ 𝑀 𝐾 𝑃,𝛼 ,and ¯ 𝑀 𝛼 are dimensionless. For the present analysis, we adopt the particle-in-cell vpic codeand two different Eulerian Vlasov-Maxwell codes: the fully-kinetic
Gkeyll code and the hybrid-kinetic HVM code.vpic utilizes a three-dimensional, relativistic, fully-kinetic explicitalgorithm (Bowers et al. 2008). vpic has been widely adopted for bothcollisionless and weakly collisional plasma simulations, includingsimulations of magnetic reconnection and plasma turbulence (e.g.,Daughton et al. (2009); Daughton et al. (2011); Karimabadi et al.(2013); Roytershteyn et al. (2013); Wan et al. (2015); Roytershteynet al. (2015)). The code includes several models of binary collisions,including the particle-pairing Coulomb collision algorithm of Tak-izuka & Abe (1977) capable of accurately reproducing the Landaucollisional integral over a wide range of parameters. The latter modelis used in this study.
Gkeyll is a highly extensible code framework which containssolvers for a number of systems of equations of relevance to plasmaphysics, including multi-moment multi-fluid (Wang et al. 2015), con-tinuum gyrokinetics (Shi et al. 2019; Mandell et al. 2020), and con-tinuum Vlasov-Maxwell (Juno et al. 2018; Hakim & Juno 2020).
Gkeyll ’s Vlasov-Maxwell solver utilizes the discontinuous Galerkinfinite element method for phase-space discretization and a strong-stability preserving Runge-Kutta method for the integration in time.The conservative, discontinuous Galerkin implementation of the non-linear Dougherty operator (Dougherty 1964) is adopted to includeintra-species collisions (Hakim et al. 2020) (see Juno (2020) forfurther details).HVM integrates the Vlasov-Maxwell system within the hybridframework, assuming quasi-neutrality and neglecting the displace-ment current density (Mangeney et al. 2002; Valentini et al. 2007). The proton Vlasov equation is discretized on a phase-space grid andintegrated numerically, while electrons are assumed to be a masslessisothermal fluid. A generalized Ohm’s law for evaluating the electricfield in Faraday’s law is coupled to the Vlasov equation. Proton-proton collisions have been recently included through the nonlinearDougherty operator (Pezzi et al. 2015; Pezzi et al. 2019b,c).
We discuss the two classes of numerical simulations in the presentwork. The first class employs both collisionless and weakly-collisional vpic and
Gkeyll simulations of a single current sheetthat undergoes symmetric anti-parallel magnetic reconnection. Inthe collisionless case, we find that vpic and a separate PIC code P3D(Zeiler et al. 2002), adopted in Liang et al. (2020), provide consis-tent and qualitatively similar results. Although the vpic and
Gkeyll simulations are very similar in their choice of parameters, there aresmall differences we make note of in the subsequent discussion. Wefirst describe the vpic simulations.The vpic reconnection simulations use a domain size of 𝐿 𝑥 × 𝐿 𝑧 = ×
25, with periodic boundary conditions in 𝑥 and perfectlyconducting boundaries on 𝑧 . A single-current-sheet initial conditionis used, with magnetic field given by 𝐵 𝑥 ( 𝑧 ) = tanh [( 𝑧 − 𝐿 𝑧 / )/ 𝑤 ] ,where 𝑤 = . 𝑇 𝑒 = /
12 and 𝑇 𝑝 = /
12 for electrons and protons, respectively; bothtemperatures are initially uniform over the whole domain. The densityis set to balance plasma pressure in the fluid sense, with 𝑛 ( 𝑧 ) = sech [( 𝑧 − 𝐿 𝑧 / )/ 𝑤 ] + 𝑛 𝑏 , where 𝑛 𝑏 = . 𝛽 for this simulation is 𝑛 𝑏 𝑘 𝐵 ( 𝑇 𝑒 + 𝑇 𝑝 )/( 𝐵 / 𝜋 ) = .
2. The proton-to-electron mass ratio is 𝑚 𝑝 / 𝑚 𝑒 =
25 and the speed of light 𝑐 =
15. These choices enforcethat the plasma is non-relativistic (the thermal and Alfvén speedsare much less than the light speed), which is appropriate for thenon-relativistic treatment of kinetic entropy. We employ a time stepof Δ 𝑡 ≈ . × − . The smallest electron Debye length for thissimulation (based on the maximum density of 1 + 𝑛 𝑏 ) is 𝜆 𝐷𝑒 = . Δ 𝑥 = Δ 𝑧 = . ≈ . 𝜆 𝐷𝑒 ( 𝑁 𝑥 = 𝑁 𝑧 = for a density equal to one. We simulate threedifferent electron-ion collision frequencies: 𝜈 = 𝜈 = . Ω 𝑐𝑒 = . Ω 𝑐 𝑝 and 𝜈 = . Ω 𝑐𝑒 = . Ω 𝑐 𝑝 . All types of collisions(electron-electron, electron-ion, and ion-ion) are taken into account.For each type of collision, the variance of the scattering angle inthe Takizuka-Abe algorithm is chosen to yield correct ratio of therespective collision frequencies (see Takizuka & Abe (1977) for moredetails).In addition to the parameters for the PIC simulation, the kineticentropy diagnostic requires other parameters, discussed in detail inAppendix B of Liang et al. (2019). For the PIC simulation, we usea velocity space grid scale of Δ 𝑣 ≈ . 𝑣 𝑡ℎ,𝑒 for electrons and Δ 𝑣 ≈ . 𝑣 𝑡ℎ,𝑖 for ions. We use a velocity range for binning theparticles from − 𝑐 to 𝑐 in each dimension for electrons and from − . 𝑐 to 0 . 𝑐 for ions.The Gkeyll reconnection simulation also uses a single currentsheet initial condition, in a domain of size, 𝐿 𝑥 × 𝐿 𝑧 = 𝜋 × 𝜋 ,which compared to the PIC simulation is a similar size in 𝑥 butabout half the size in 𝑧 . The boundary conditions are periodic in 𝑥 , a reflecting wall boundary condition in 𝑧 . The initial magneticfields and plasma parameters are the same as the vpic simulations: 𝑤 = . 𝑚 𝑝 / 𝑚 𝑒 = , 𝑇 𝑝 = / , 𝑇 𝑒 = / , and 𝑛 𝑏 = .
2. Like-wise, there is only one Maxwellian component in the current sheet,
MNRAS000
MNRAS000 , 1–17 (2021)
O. Pezzi et al. but 𝑐 =
50. Since the continuum Vlasov method in
Gkeyll avoidsaliasing errors associated with under-resolving the Debye length inPIC methods while still conserving energy, we choose a coarser con-figuration space grid resolution to save on the computational cost ofa continuum method while still resolving the reconnection dynam-ics. Our grid resolution is Δ 𝑥 = Δ 𝑧 ≈ . ≈ 𝜆 𝐷𝑒 ( 𝑁 𝑥 = 𝑁 𝑧 =
32) with piecewise quadratic Serendipity polynomials withina grid cell (Arnold & Awanou 2011). The velocity space range isfrom − 𝑣 𝑡ℎ,𝛼 to 6 𝑣 𝑡ℎ,𝛼 with a velocity space grid of Δ 𝑣 𝛼 = 𝑣 𝑡ℎ,𝛼 along with piecewise quadratic Serendipity polynomials in velocityspace. Zero-flux boundary conditions are employed in velocity spaceto ensure energy conservation. We choose a constant collisionalityof 𝜈 𝑒𝑒 = .
01 for electron-electron collisions and 𝜈 𝑝 𝑝 = .
002 forproton-proton collisions for the Dougherty collision operator. Recon-nection is initiated using a magnetic perturbation with a spectrumof random wave modes in the first 20 modes of the system withr.m.s. amplitude 𝛿𝐵 / 𝐵 = × − . These random perturbationsbreak the symmetry of the continuum kinetic initial condition andallow for the study of the standard 𝑚 = 𝐿 𝑥 = 𝐿 𝑦 = 𝐿 = 𝜋 ×
20. Periodic boundary condi-tions are imposed for the spatial domain. Velocity space is discretizedwith 71 grid-points in the range 𝑣 𝑗 = (cid:2) − 𝑣 𝑡ℎ, 𝑝 , 𝑣 𝑡ℎ, 𝑝 (cid:3) ( 𝑗 = 𝑥, 𝑦, 𝑧 ) , with the boundary condition 𝑓 ( 𝑣 𝑗 > 𝑣 𝑡ℎ, 𝑝 ) =
0. The ini-tial equilibrium is characterized by spatial homogeneity, Maxwellianproton VDFs, and a background uniform out-of-plane magnetic field 𝑩 = 𝒆 𝑧 with 𝛽 𝑝 =
2. This equilibrium is perturbed at 𝑡 = 𝜹𝑩 and bulk speed 𝜹𝒖 = ± 𝜹𝒃 fluctuations( 𝜹𝒃 in Alfvén speed units). Energy is injected at large scales, i.e. 𝑘 ∈ [ , ] 𝑘 ( 𝑘 = 𝜋 / 𝐿 ), with a flat energy spectrum and randomphases. The r.m.s. amplitude of the fluctuations is 𝛿𝐵 / 𝐵 = / 𝑡 =
0. Electron inertia effects are neglected in Ohm’s law, while elec-tron temperature is set equal to the initial ion temperature. A smallresistivity ( 𝜂 (cid:39) − ) is introduced to suppress numerical instabili-ties and does not play a significant role in the plasma dynamics. Theadopted numerical resolution captures two decades of perpendicularwavenumbers: one above and one below the proton skin depth 𝑑 𝑝 .We consider two simulations, characterized by a different proton-proton collisional frequency 𝜈 , namely collisionless ( 𝜈 =
0) andweakly-collisional ( 𝜈 = − ) (see Pezzi et al. (2019c) for furtherdetails). Figure 1 displays the set of implemented proxies for protons in thevpic simulations. The left column collects the results for the col-lisionless simulation, i.e. with 𝜈 =
0. We initially focus on theseresults, and discuss the effects of collisions in Section 4.3. The dataare taken at time 𝑡 (cid:39) Ω − 𝑐 𝑝 , after the peak of the reconnectionrate. To compute energy-based parameters involving spatial deriva-tives, Gaussian smoothing is used to filter the noise (e.g. Birdsall &Langdon (2004)).The Zenitani parameter 𝐷 𝑝 shows signatures consistent with pre-vious studies (Zenitani et al. 2011; Swisdak 2016), being peakedwith a positive value in the diffusion region. In the exhausts, thereis oscillatory behavior especially on small scales in the primary is-land. This is likely due to time-domain structures (Mozer et al. 2015) such as electron holes, which form Debye-scale bi-directional electricfields. Such structures are at small scales and produce local energyconversion between the particles and fields.The LET parameter 𝜖 𝑝,ℓ shows a large-scale pattern peaked insidethe magnetic island that is only weakly modulated in the horizontaldirection, as well as a weaker signal approximately coincident withthe electron diffusion region (EDR). As pointed out in Section 2.1,extracting information about LET is challenging in this reconnectionsimulation, where the background field is inhomogeneous and thefields are not in a steady state of fully developed turbulence. The signsof LET are opposite on either side of the magnetic reversal becauseLET is calculated with an increment with a component in the positive 𝑧 direction. The contribution of various terms to the local 𝜖 𝑝,ℓ (notshown) reveals that the Hall terms (in particular the current-helicityterm, i.e., the last term in Eq. 16) dominate the nonlinear energytransfer. This is related to the presence of Hall-scale electric currentsand to the magnetic configuration of the reconnection region, whichthus are the main drivers of the nonlinear interactions. Furthermore,the prevalent positive sign observed for the MHD cross-helicity term(not shown) suggests that nonlinear interactions are inhibited by thestrong presence of coupled velocity-magnetic field (Alfvénic) fluc-tuations, which reduce the effective transfer of energy and possiblythe onset of turbulence.The Pi-D 𝑝 plot shows that the pressure-strain term is positiveyet small near the X point on a length scale in the inflow directionbeyond the EDR. Protons undergo meandering orbits in this region,producing non-gyrotropic VDFs, while the reconnection inflow andoutflow are associated with bulk velocity shear: this produces a non-zero Pi-D 𝑝 . Inside the islands, a bipolar (positive/negative) signal isfound. The strong negative region suggests energy is locally beingconverted from thermal energy to bulk kinetic energy, perhaps ina region where counterstreaming beams including reflected ions atthe dipolarization front where the denser current sheet populationis being pushed downstream by the reconnected magnetic field areconverted into bulk flow. In contrast, P- 𝜃 𝑝 , which has a peak valueabout a factor of 2 larger than Pi-D 𝑝 , tends to be quite structured andpositive in most of the island, is negative in the EDR, and is small inthe ion diffusion region (IDR). These results make sense physically:the plasma in the island is undergoing compression due to the bulkflows, so that P- 𝜃 𝑝 is positive. In the diffusion region, when upstreammagnetic flux tubes enter the region of weaker magnetic field, theyexpand, leading to negative 𝜃 𝑝 = ∇ · 𝒖 𝑝 .Moving to VDF-based parameters, the proton agyrotropy √︁ 𝑄 𝑝 parameter indicates a proton gyro-scale region of non-gyrotropy sur-rounding the diffusion region and the whole island at this stage of theevolution, owing to the complicated distribution functions that ap-pear where protons undergo meandering orbits. Local √︁ 𝑄 𝑝 maximaoccur in the inner shell of the magnetic island, where both the LETand Pi-D 𝑝 are locally peaked. Turning to the non-Maxwellianity pa-rameters, the 𝜉 𝑝 parameter similarly shows structure at proton scalesin both the diffusion region and the islands. The structure of ¯ 𝑀 KP , p and ¯ 𝑀 𝑝 are qualitatively quite similar to 𝜉 𝑝 , as expected. In eachcase, the protons are most strongly non-Maxwellian in the EDR,with non-zero values also in the IDR and in the island. The maindifference between the measures is that the entropy-based diagnos-tics appear more localized than the quadratic measure, suggestingthat these proxies require relatively more strongly non-Maxwellianstructures to attain appreciable values (owing to the natural log intheir definition) relative to the 𝜉 𝑝 measure.Summarizing the collisionless simulation results for protons, alleight dissipation measures in question show structure in and aroundthe current sheet and magnetic island. For the parameters in this MNRAS , 1–17 (2021) issipation measures in plasmas Figure 1. (Color online) Comparison of various dissipation proxies for protons, from the vpic simulations with 𝜈 = 𝜈 = .
25 (center)and 𝜈 = .
25 (right). The proxies are computed at 𝑡 =
22 for the collisionless run and 𝑡 =
26 for the weakly-collisional runs. From top to bottom: theZenitani parameter 𝐷 𝑝 ; the LET parameter 𝜖 𝑝,ℓ with ℓ (cid:39) 𝑑 𝑝 ; the pressure-strain interaction Pi-D 𝑝 ; the pressure dilatation P- 𝜃 𝑝 ; the agyrotropy √ 𝑄 𝑝 ; thenon-Maxwellian indicators 𝜉 𝑝 ; ¯ 𝑀 KP ,𝑝 ; and ¯ 𝑀 𝑝 . Solid lines indicate separatrices. MNRAS000
26 for the weakly-collisional runs. From top to bottom: theZenitani parameter 𝐷 𝑝 ; the LET parameter 𝜖 𝑝,ℓ with ℓ (cid:39) 𝑑 𝑝 ; the pressure-strain interaction Pi-D 𝑝 ; the pressure dilatation P- 𝜃 𝑝 ; the agyrotropy √ 𝑄 𝑝 ; thenon-Maxwellian indicators 𝜉 𝑝 ; ¯ 𝑀 KP ,𝑝 ; and ¯ 𝑀 𝑝 . Solid lines indicate separatrices. MNRAS000 , 1–17 (2021) O. Pezzi et al. simulation, the quantities that are most strongly peaked in the EDRare 𝐷 𝑝 , 𝜉 𝑝 , ¯ 𝑀 𝐾 𝑃, 𝑝 , and ¯ 𝑀 𝑝 . The strongest measure for the IDR is √ 𝑄 𝑝 , with Pi-D 𝑝 , 𝜉 𝑝 , ¯ 𝑀 𝐾 𝑃, 𝑝 , and ¯ 𝑀 𝑝 also displaying structure. Inthe island, 𝜖 𝑝 , Pi-D 𝑝 and P- 𝜃 𝑝 are significant, with 𝐷 𝑝 revealingsignificant electron-scale variations. In analogy with protons, electron dissipation proxies are displayed inFig. 2. One exception is that the LET parameter is not computed forelectrons, as discussed in Section 2.1. As expected, 𝐷 𝑒 (cid:39) 𝐷 𝑝 due toquasi-neutrality. Again, we first focus on the collisionless case in theleft column.The other energy-based parameters, the electron pressure-straininteraction terms Pi-D 𝑒 and P- 𝜃 𝑒 , have highly structured patternsat much smaller scales than their proton counterparts, as expected.In the current sheet, Pi-D 𝑒 and P- 𝜃 𝑒 are both confined to the EDR,with almost no signal in the IDR. Pi-D 𝑒 is positively peaked inthe magnetic island close to the X-point. There are strong bands ofPi-D 𝑒 near the upstream edges of the EDR, where velocity sheardue to electron meandering orbits is significant. In the island, bothPi-D 𝑒 and P- 𝜃 𝑒 have strong variations in the small-scale structuresdiscussed in the previous subsection. The intense electric fields areexpanding and compressing the electron fluid as seen in P- 𝜃 𝑒 , andthese fluctuations in the local velocity shear give a non-zero Pi-D 𝑒 .There is also coherent structure of Pi-D 𝑒 as the electrons leave theEDR and develop a velocity shear as they move around the pre-existing magnetic island.For the VDF-based proxies, all four proxies √ 𝑄 𝑒 , 𝜉 𝑒 , ¯ 𝑀 KP , e and¯ 𝑀 𝑒 are peaked in the EDR where the strong signature of 𝐷 𝑒 ispresent. For √ 𝑄 𝑒 , it is peaked at the upstream edges of the EDRwhere the meandering orbits meet the upstream electrons, and isrelatively smaller near the magnetic field reversal where distributionshave a characteristic wedge shape (Ng et al. 2011). Interestingly,the agyrotropy √ 𝑄 𝑒 is only non-zero in the EDR, while the non-Maxwellianities 𝜉 𝑒 , ¯ 𝑀 KP , e and ¯ 𝑀 𝑒 are non-zero in both the EDRand IDR. The reason for this is that the electrons upstream in theIDR are trapped (Egedal et al. 2005), and it has been shown that theyproduce gyrotropic distributions elongated in the parallel direction(Egedal et al. 2008, 2013). Consequently, the agyrotropy in the IDRis zero, while the non-Maxwellianity is non-zero, as is seen in thesimulation results.All four VDF-based proxies also show strong signatures close tothe separatrices, where complicated distributions at the boundariesbetween upstream plasma and the magnetic island occur. This is akey distinction between these proxies and the energy-based proxies,which are not peaked near the separatrices. This signature suggeststhat the energy conversion in the island and exhaust is not takingplace near the edges of the islands, but more towards the core asthe bent field lines straighten. The VDF-based proxies also displaynon-zero signals at the small-scale structures in the exhaust. We now turn to the effect of inter- and intra-species collisions onthe dissipation measures for both protons and electrons. To put thenumerical collisionality in perspective, we compare it to two knowncritical collision frequencies for reconnection. Collisionless (Hall)reconnection transitions to collisional (Sweet-Parker) reconnectionat a critical resistivity 𝜂 𝑐 (Cassak et al. 2005). The initial currentsheet thickness 𝑤 = . 𝑑 𝑝 , so it is expected that collisionless reconnection will occur for smallenough resistivity. From Fig. 3 of Cassak et al. (2005), the criticalresistivity is 𝜂 𝑐 𝑐 / 𝜋𝑐 𝐴 𝑑 𝑝 (cid:39) .
2, which in normalized units forthis study is 𝜂 𝑐 (cid:39)
1. Then, the critical collision frequency is 𝜈 𝑐 = 𝜂 𝑐 𝑛 𝑒 𝑒 / 𝑚 𝑒 (cid:39)
5. Therefore, for 𝜈 = .
25 and 1.25, as is used here inthe vpic simulations, reconnection is expected to remain Hall-like.The time scale for magnetic diffusion in the electron current sheetis 4 𝜋𝑑 𝑒 / 𝜂𝑐 = / 𝜈 , so the diffusion time scales are 4 and 0.8 for 𝜈 = 𝑑 𝑒 /( . 𝑐 𝐴𝑒 ) (cid:39)
2, where 𝑐 𝐴𝑒 is theelectron Alfvén speed. Consequently, collisions are expected to havea noticeable effect in the EDR in the vpic simulations for 𝜈 = . 𝜈 = .
25. A second critical collisionfrequency is that at which collisions affect electron trapping upstreamof the EDR, which is approximately 𝜈 = . 𝜈 = .
25 andsignificantly affected for 𝜈 = .
25. In contrast, the collisionality forthe
Gkeyll simulation is very low, below both thresholds, so theevolution is essentially collisionless.Figure 1 displays the proton dissipation proxies for the 𝜈 = . 𝜈 = .
25 (right) vpic simulations. Data are from 𝑡 (cid:39) 𝜈 = .
25 and 1 .
25, when the magnetic energy of thesesimulations is nearly the same as the collisionless case at 𝑡 =
22. The 𝜈 = .
25 case is just after the peak in reconnection rate, as for the 𝜈 = 𝜈 = .
25 case is just before the peak in reconnectionrate, which explains why the island is somewhat smaller for this case.The energy-based parameters should be affected by the presenceof inter-species collisions due to the exchange of energy betweenspecies. For 𝜈 = .
25, collisions affect the small-scale structures thatwere present in the collisionless case, especially in 𝐷 𝑝 and P- 𝜃 𝑝 .However, as expected, the large-scale structure of these parametersis not greatly altered for this collisionality. For 𝜈 = .
25, however,collisions significantly alter the dissipation proxies. The signals inthe magnetic islands are severely weakened, as are the signals in theEDR and IDR. Despite being weaker, the large-scale structure of themeasures is largely unchanged.In a similar way, VDF-based parameters are qualitatively unaf-fected by collisional effects for 𝜈 = .
25, with only weak quantitativedifferences. On the other hand, for 𝜈 = .
25, VDF-based parame-ters are strongly quantitatively affected. As collisions drive distribu-tions toward Maxwellianity, especially those with fine velocity-spacestructures, i.e. the non-Maxwellianity measures 𝜉 𝑝 , ¯ 𝑀 KP , p , and ¯ 𝑀 𝑝 ,are strongly decreased. The agyrotropy √ 𝑄 𝑝 is also reduced by strongcollisions, but not as drastically as the non-Maxwellianity measures,as it is less sensitive to sharp peaks in velocity space. This result againconfirms that collisional dissipation acts on different characteristictimescales depending on the scale of the velocity-space distortion inthe particle VDF: finer velocity space structures produce shorter dis-sipation timescales (Landau 1936; Rosenbluth et al. 1957; Balescu1960; Pezzi et al. 2016).We analyze now the effect of collisions on electron dissipationproxies for the vpic simulations, which bears many similarities tothe effect on proton dissipation proxies. The small-scale structures inthe island largely disappear even for the weaker collisionality of 𝜈 = .
25. However, at variance with the protons, P- 𝜃 𝑒 shows persistentsmall-scale structure even at large collisionality 𝜈 = .
25. The non-Maxwellianity proxies for electrons are very small for 𝜈 = .
25. Thisis consistent with collisions being dynamically important on the timescale of the electron transit through the EDR; they Maxwellianizealmost fully. In contrast, the EDR remains clearly visible in thenon-Maxwellianity measures for 𝜈 = .
25. The agyrotropy is non-zero at the edges of the EDR for all three simulations, suggesting
MNRAS , 1–17 (2021) issipation measures in plasmas Figure 2.
Same as figure 1, but for electrons instead of protons. that the meandering orbits are sufficient to produce this signal evenfor the highest collisionality considered here. The trapped electronsupstream of the EDR are weaker for 𝜈 = .
25 and nearly non-existentfor 𝜈 = .
25, consistent with the predictions from Le et al. (2015).We finally describe the weakly collisional continuum
Gkeyll sim-ulation. The
Gkeyll simulation includes only the effects of intra-species collisions.Figure 3 shows the proton dissipation proxies, plotted at a slightlyearlier time 𝑡 (cid:39)
18, but this time is after the peak of the reconnectionrate which takes place at an earlier time with respect to the vpic caseowing to the smaller system size. The plots reveal that the X-line is not located exactly in the center of the domain and there is a significantleft-right asymmetry. This is because the random perturbation to theinitial condition seeding reconnection breaks the symmetry in the 𝑥 direction.Three of the energy-based measures, 𝐷 𝑝 , Pi-D 𝑝 , and P- 𝜃 𝑝 , showbroad agreement with the vpic results in Figure 1. The Gkeyll simulation has a more structured Pi-D 𝑝 near the X-line due to asecondary island near 𝑧 = 𝑧 and 𝑥 − 𝑥 (cid:39) [ , ] , which developsshortly after the peak of the reconnection rate. The secondary islandhas a bipolar structure in Pi-D 𝑝 analogous to the bipolar structuredownstream in a dipolarization front in the vpic simulation, visible MNRAS000
18, but this time is after the peak of the reconnectionrate which takes place at an earlier time with respect to the vpic caseowing to the smaller system size. The plots reveal that the X-line is not located exactly in the center of the domain and there is a significantleft-right asymmetry. This is because the random perturbation to theinitial condition seeding reconnection breaks the symmetry in the 𝑥 direction.Three of the energy-based measures, 𝐷 𝑝 , Pi-D 𝑝 , and P- 𝜃 𝑝 , showbroad agreement with the vpic results in Figure 1. The Gkeyll simulation has a more structured Pi-D 𝑝 near the X-line due to asecondary island near 𝑧 = 𝑧 and 𝑥 − 𝑥 (cid:39) [ , ] , which developsshortly after the peak of the reconnection rate. The secondary islandhas a bipolar structure in Pi-D 𝑝 analogous to the bipolar structuredownstream in a dipolarization front in the vpic simulation, visible MNRAS000 , 1–17 (2021) O. Pezzi et al.
Figure 3.
Same dissipation proxies as plotted in Figure 1, but for the protonsin the
Gkeyll reconnection simulation. at 𝑧 = 𝑧 and 𝑥 − 𝑥 (cid:39) [− , − ] in Figure 1. The strongest differencebetween the two data sets is observed in the LET diagnostic 𝜖 𝑝 ,which displays broader features in the Gkeyll simulation. This islikely due to the coarser configuration space resolution.For the VDF-based diagnostics, the
Gkeyll simulation gives qual-itatively and quantitatively similar results in the exhaust, especiallythe inner shell of the magnetic island. However, small differencesarise in these diagnostics due to the secondary island in the
Gkeyll simulation. Indeed, this structure generates intense deviations fromthe thermal equilibrium due to the mixing and rapid rotation of pro-tons trying to align with the local magnetic field. Inside the protonscale island, we observe strong deformation of the proton VDF whichmanifests as an intense agyrotropy √︁ Q 𝑝 and non-Maxwellianity 𝜉 𝑝 ,¯M 𝐾 𝑃, 𝑝 and ¯M 𝑝 .Turning to electron dissipation proxies, the Gkeyll simulationresults display a good agreement with the vpic data. In fact, theoverall structure in energy-based diagnostics such as 𝐷 𝑒 and Pi-D 𝑒 and distribution function-based diagnostics such as √︁ Q 𝑒 and 𝜉 𝑒 ,agrees better for electrons than for protons. For example, 𝐷 𝑒 andPi-D 𝑒 are positive in the electron diffusion region, and we observeenhancement of all the distribution function-based diagnostics nearthe separatrices. This better agreement can be linked to the secondaryisland not having as dramatic an impact on the electron dynamics inthe magnetic island.The comparison of this wide array of diagnostics from these twodifferent codes in different regimes, from collisionless vpic to weaklycollisional Gkeyll to collisional vpic, reveals the diversity of infor-mation content each diagnostic contains. In many cases, we observelittle qualitative difference between the energy-based diagnosticsfrom different simulations while VDF-based diagnostics are moresensitive both qualitatively and quantitatively to the strength of col-lisions and subtle differences in the underlying kinetic evolution of
Figure 4.
Same dissipation proxies as plotted in Figure 2, but for the electronsin the
Gkeyll reconnection simulation. the reconnection process, such as the secondary island which formsin the
Gkeyll simulation.
We here describe the structure of dissipation proxies in plasma turbu-lence at kinetic scales. Since the HVM code using the hybrid modelneglects electrons, we only treat proton parameters. In these simu-lations, energy injected at large scale generates a cascade towardssmaller scales. The time corresponding to the most intense turbulentactivity is 𝑡 = 𝑡 ∗ =
30, in which a turbulent state characterized by anintermittent pattern of current sheets which delimit magnetic islandsand vortices (Servidio et al. 2015; Wan et al. 2015) is reached. Weshow simulation results at 𝑡 = 𝑡 ∗ in Fig. 5, with the output of thecollisionless (left) and collisional (right) simulation. Panels from (a)to (h) display 𝐷 𝑝 , 𝜖 𝑝,ℓ with ℓ (cid:39) 𝑑 𝑝 , Pi-D 𝑝 , P- 𝜃 𝑝 , √︁ 𝑄 𝑝 , 𝜖 𝑝 , ¯ 𝑀 KP , p ,and ¯ 𝑀 𝑝 , respectively.We begin by analyzing the energy-based parameters. 𝐷 𝑝 , LET 𝜖 𝑝,ℓ , and the Pi-D 𝑝 term are all peaked close to the most intensecurrent sheets. This confirms that current sheets are the sites withthe most intense local energy conversion and dissipation. 𝐷 𝑝 has apreferred sign, being positive in most of the regions of highest energyconversion. This implies there is a net conversion of energy from theelectromagnetic fields to the protons. Similarly, the predominantlynegative LET supports the standard picture of a direct global energycascade towards small scales. The regions of larger energy transferare generally located near the current sheets, and some complexityin the fine local details of the transfer can be observed. A detailedanalysis of each term of the right hand side of Eq. (13) reveals thatthe total energy (kinetic plus magnetic) available to be transported bythe longitudinal component Δ 𝑢 𝑝,ℓ is the main contributor to the LET MNRAS , 1–17 (2021) issipation measures in plasmas Figure 5. (Color online) Various dissipation surrogates evaluated at the time of maximum turbulent activity in HVM simulations, 𝑡 = 𝑡 ∗ . The two columns atleft refer to the collisionless case, and the two at the right are for the weakly-collisional case. Panels from (a) to (h) display 𝐷 𝑝 , 𝜖 𝑝,ℓ with ℓ (cid:39) 𝑑 𝑝 , Pi-D 𝑝 , P- 𝜃 𝑝 , √︁ 𝑄 𝑝 , 𝜉 𝑝 , ¯ 𝑀 KP , p , and ¯ 𝑀 𝑝 , respectively. parameter in this HVM simulation. In this case, it is also confirmed(not shown) that the global (Yaglom-Hall) law shows a linear scalingin the interval roughly corresponding to the MHD-turbulence range(2 𝑑 𝑝 (cid:46) ℓ (cid:46) 𝑑 𝑝 ), as also recently reported in similar simulationsetups by Sorriso-Valvo et al. (2018a) and Vásconez et al. (2020).As displayed in previous HVM (Pezzi et al. 2019a) and in PICsimulations (Yang et al. 2019), Pi-D 𝑝 is highly structured, havingboth positive and negative regions close to intense current sheets.Conversely, P- 𝜃 𝑝 is larger than its Pi-D 𝑝 counterpart and has signif-icant contributions both at the current sheets and in magnetic islandssince it is related to large-scale plasma compression (red) and rarefac-tion (blue). The most intense regions of P − 𝜃 𝑝 are at current sheets,reflecting the rapid collision or separation of large-scale magneticislands.The four VDF-based proxies bear many similarities. They arehighly-structured, with local peaks close to current sheets. As withthe reconnection simulations, there are also differences between these measures. It is more common to get appreciable values of √ 𝑄 𝑝 thanother VDF-based measures, especially ¯ 𝑀 𝐾 𝑃, 𝑝 and ¯ 𝑀 𝑝 . This indi-cates that the agyrotropy provides an overall picture of the presenceof large-scale kinetic effects in the VDFs (namely the 2 nd order VDFmoment), while fine-scale structures in the VDF are larger contribu-tors to non-Maxwellianity measures.The collisionless and weakly-collisional simulations do not revealsignificant differences in the energy-based parameters for the col-lision frequency in use. The inclusion of proton-proton collisionaleffects does not affect the statistical characteristics of turbulence atthe proton scale. This can be explained since, at variance with inter-species collisional effects, intra-species collisions do not generate aresistivity-like term which directly affects the electric field and hencefluid quantities. On the other hand, VDF-based parameters are dissi-pated by collisions, signature of the collisional thermalization. Sincethese parameters are sensitive to the presence of out-of-equilibriumstructure in the proton VDF, they are affected by intra-species colli- MNRAS000
30, in which a turbulent state characterized by anintermittent pattern of current sheets which delimit magnetic islandsand vortices (Servidio et al. 2015; Wan et al. 2015) is reached. Weshow simulation results at 𝑡 = 𝑡 ∗ in Fig. 5, with the output of thecollisionless (left) and collisional (right) simulation. Panels from (a)to (h) display 𝐷 𝑝 , 𝜖 𝑝,ℓ with ℓ (cid:39) 𝑑 𝑝 , Pi-D 𝑝 , P- 𝜃 𝑝 , √︁ 𝑄 𝑝 , 𝜖 𝑝 , ¯ 𝑀 KP , p ,and ¯ 𝑀 𝑝 , respectively.We begin by analyzing the energy-based parameters. 𝐷 𝑝 , LET 𝜖 𝑝,ℓ , and the Pi-D 𝑝 term are all peaked close to the most intensecurrent sheets. This confirms that current sheets are the sites withthe most intense local energy conversion and dissipation. 𝐷 𝑝 has apreferred sign, being positive in most of the regions of highest energyconversion. This implies there is a net conversion of energy from theelectromagnetic fields to the protons. Similarly, the predominantlynegative LET supports the standard picture of a direct global energycascade towards small scales. The regions of larger energy transferare generally located near the current sheets, and some complexityin the fine local details of the transfer can be observed. A detailedanalysis of each term of the right hand side of Eq. (13) reveals thatthe total energy (kinetic plus magnetic) available to be transported bythe longitudinal component Δ 𝑢 𝑝,ℓ is the main contributor to the LET MNRAS , 1–17 (2021) issipation measures in plasmas Figure 5. (Color online) Various dissipation surrogates evaluated at the time of maximum turbulent activity in HVM simulations, 𝑡 = 𝑡 ∗ . The two columns atleft refer to the collisionless case, and the two at the right are for the weakly-collisional case. Panels from (a) to (h) display 𝐷 𝑝 , 𝜖 𝑝,ℓ with ℓ (cid:39) 𝑑 𝑝 , Pi-D 𝑝 , P- 𝜃 𝑝 , √︁ 𝑄 𝑝 , 𝜉 𝑝 , ¯ 𝑀 KP , p , and ¯ 𝑀 𝑝 , respectively. parameter in this HVM simulation. In this case, it is also confirmed(not shown) that the global (Yaglom-Hall) law shows a linear scalingin the interval roughly corresponding to the MHD-turbulence range(2 𝑑 𝑝 (cid:46) ℓ (cid:46) 𝑑 𝑝 ), as also recently reported in similar simulationsetups by Sorriso-Valvo et al. (2018a) and Vásconez et al. (2020).As displayed in previous HVM (Pezzi et al. 2019a) and in PICsimulations (Yang et al. 2019), Pi-D 𝑝 is highly structured, havingboth positive and negative regions close to intense current sheets.Conversely, P- 𝜃 𝑝 is larger than its Pi-D 𝑝 counterpart and has signif-icant contributions both at the current sheets and in magnetic islandssince it is related to large-scale plasma compression (red) and rarefac-tion (blue). The most intense regions of P − 𝜃 𝑝 are at current sheets,reflecting the rapid collision or separation of large-scale magneticislands.The four VDF-based proxies bear many similarities. They arehighly-structured, with local peaks close to current sheets. As withthe reconnection simulations, there are also differences between these measures. It is more common to get appreciable values of √ 𝑄 𝑝 thanother VDF-based measures, especially ¯ 𝑀 𝐾 𝑃, 𝑝 and ¯ 𝑀 𝑝 . This indi-cates that the agyrotropy provides an overall picture of the presenceof large-scale kinetic effects in the VDFs (namely the 2 nd order VDFmoment), while fine-scale structures in the VDF are larger contribu-tors to non-Maxwellianity measures.The collisionless and weakly-collisional simulations do not revealsignificant differences in the energy-based parameters for the col-lision frequency in use. The inclusion of proton-proton collisionaleffects does not affect the statistical characteristics of turbulence atthe proton scale. This can be explained since, at variance with inter-species collisional effects, intra-species collisions do not generate aresistivity-like term which directly affects the electric field and hencefluid quantities. On the other hand, VDF-based parameters are dissi-pated by collisions, signature of the collisional thermalization. Sincethese parameters are sensitive to the presence of out-of-equilibriumstructure in the proton VDF, they are affected by intra-species colli- MNRAS000 , 1–17 (2021) O. Pezzi et al.
Figure 6. (Color online) One-dimensional cuts vertically through the x-line showing the various dissipation surrogates in the reconnecting current sheet for thevpic collisionless reconnection simulation. The top row refer to proton proxies, displaying energy-based parameters (left, with 𝜖 𝑝,ℓ scaled by a factor of 10 tomake it more visible) and VDF-based parameters (right). In the right panel, the Zenitani measure 𝐷 𝑝 is showed as a reference. The bottom row is analogous forthe electron proxies. sional effects. The effect of collisions is less visible in √︁ 𝑄 𝑝 since col-lisions preferentially dissipate fine velocity-space structures (Pezziet al. 2016), which contribute less to √︁ 𝑄 𝑝 . The effect of collisionsis visible in 𝜉 𝑝 and in the entropy density-based non-Maxwellianityproxies. The maximum values of these three parameters are smallerby about 10% in the weakly-collisional simulation than the collision-less simulation. Similarly, average values are reduced by about 20%for 𝜉 𝑝 and 30% for ¯ 𝑀 KP , p and ¯ 𝑀 𝑝 when including collisions. The ef-fect of collisions becomes more significant at later times (not shownhere, see Fig. 4 of Pezzi et al. (2019c)). Indeed, at the final time of thesimulation, maxima of 𝜉 𝑝 and of the entropy density-based proxiesare respectively about 20% and 30% smaller when collisional ef-fects are considered, while their average values are reduced by 40%and 60%, respectively. The different dissipation (in terms of bothmaximum and averaged values) of 𝜉 𝑝 and the entropy density-basedproxies suggests that the latter react more to collisional effects than 𝜉 𝑝 since entropy-based proxies are dissipated more efficiently viacollisions. We conclude by presenting one-dimensional cuts of the dissipationsurrogates adopted in this work across a typical current structure.We include both the reconnecting current sheet and a typical cur-rent sheet observed in the turbulent HVM simulation. We focus onthe collisionless simulation, since our aim is mainly discussing howthese parameters look when crossing a particular structure. More-over, for the magnetic reconnection simulations, we show cuts foronly the vpic runs, since
Gkeyll and vpic provide similar results. These plots help reveal comparative proxy structure near to the cur-rent sheets, and may also be useful for future comparison with in-situ spacecraft observations, e.g., for the Pi-D measure (Chasapis et al.2018b). For applications of these results to observations or labora-tory experiments, it is important to emphasize that the results areundoubtedly sensitive to the plasma parameters and to the reduced2D physical-space dimensionality (Howes 2015a; Li et al. 2016).Note also that, since some of these diagnostics explicitly requirethe computation of spatial derivatives (e.g. Pi-D and the Zenitanimeasure), multi-spacecraft observations are necessary.Figure 6 shows the dissipation proxies along a cut in the recon-nection simulation through the X-point in 𝑧 . The vertical dotted anddashed lines mark the upstream edges of the EDR and IDR, respec-tively. These edges are defined by the location at which the electronand ion out-of-plane currents are 20% of their maximum (Shay et al.2001). The upper panels show the results for the protons, and thelower panels are for electrons, with energy-based measures in the leftplots and VDF-based measures on the right. For the energy-basedmeasures, 𝐷 𝑒 (cid:39) 𝐷 𝑝 shows a clear peak in the EDR. The LET param-eter 𝜖 𝑝,ℓ , scaled by a factor of 10 to make it easier to see, displays thenegative-positive double-peaked structure in the EDR seen in Figure1, and is negligible elsewhere. The sign of 𝜖 𝑝,ℓ is the same sign as 𝐵 𝑥 for this current sheet. The electron and ion Pi-D and P- 𝜃 showmoderately intensified signals within the EDR with tails extendinginto the IDR. Compared to the Zenitani measure, both P- 𝜃 and Pi-Ddisplay rapidly-fluctuating patterns but only Pi-D is positive definitewithin the IDR, i.e. the same sign of 𝐷 𝑝 .For the VDF-based measures, the agyrotropies √ 𝑄 𝑒 and √︁ 𝑄 𝑝 reveal a double peak structure the diffusion region of each speciescorrelated with the size of its diffusion region. As discussed in Sec-tion 4.2, this shape is caused by the meandering motions of the MNRAS , 1–17 (2021) issipation measures in plasmas Figure 7. (Color online) Same as figure 6, but at 𝑥 − 𝑥 = 𝑑 𝑝 . particles traversing their diffusion regions at their gyroscale andis therefore a characteristic shape of √ 𝑄 in the diffusion regionof anti-parallel reconnection. For both electrons and ions, the non-Maxwellianity proxies 𝜉 , ¯ 𝑀 𝐾 𝑃 , and ¯ 𝑀 , show intensified signals overthe IDR with relatively strong peaks in the EDR. As discussed in Sec-tion 4, the electron non-Maxwellianities not only show the peaks inthe EDR but also broad intensified signals over the IDR. As analysedin Liang et al. (2020), the broad intensified signals are due to thegyrotropic distributions created by trapped electrons (Egedal et al.2005). This non-zero signal is coincident with negligible signal inthe electron energy-based proxies 𝐷 𝑒 , Pi-D 𝑒 and P- 𝜃 𝑒 .Figure 7 similarly displays the dissipation proxies along a verticalcut through the reconnection simulation, except at 𝑥 − 𝑥 =
5, i.e.through the reconnection exhaust. The format is the same as Figure 6.For this cut, the exhaust is between the separatrices at 𝑧 − 𝑧 (cid:39) ± . 𝐷 𝑒 = 𝐷 𝑝 , 𝜖 𝑝 , Pi-D and P- 𝜃 show intensified, yet noisy, signals. The signalsinclude positive and negative values due to complex flows in thisregion, although for this cut the Pi-D 𝑝 measure has a sizeable neg-ative value. The VDF-based proxies for both electrons and ions areintensified in the exhaust, as well. The electron VDF-based proxieshave peaks near the separatrices, due to the counter-streaming elec-tron flows (Hesse et al. 2018; Liang et al. 2020). The VDF-basedproxies are slightly broader than the exhaust, which results from thefinite Larmor radius (FLR) effects near the separatrices as shown bythe non-gyrotropy √︁ 𝑄 𝑝 . Although FLR effects are not seen much inthe energy-based diagnostics, they are picked up by the VDF-baseddiagnostics 𝜉 𝑝 , ¯ 𝑀 𝐾 𝑃, 𝑝 and ¯ 𝑀 𝑝 .Finally, we show in Fig. 8 the dissipation proxies for the collision-less HVM turbulence simulation along a 1D cut close to the currentsheet near ( 𝑥, 𝑦 ) = ( , ) . Similar results are obtained in the col-lisional simulation. All the energy-based parameters (top panel ofFig. 8) are peaked within the same region, i.e. 𝑦 (cid:39)
36. The VDF-based proxies (bottom panel of Fig. 8) also reveal clear maxima near the current structure. However, the maxima are somewhat broader inwidth ( ∼ −
5) than the energy-based parameters ( ∼ √︁ 𝑄 𝑝 shows large-scale fluctuations quitefar from the current structure. The broader distribution observed forthe VDF-based parameters suggests that energy transfer is localizedclose to the coherent structures, but can affect the particle distributionfunction in a larger region around these structures.The 1D cuts of the dissipation proxies for both the reconnectionand turbulence simulations support the idea that peaks of dissipationmeasures in a reconnecting current sheet or a turbulent environmentcharacteristically occur in coincident spatial regions, but not nec-essarily at the same exact spatial position. This is consistent withthe notion of regional correlations, suggested by Yang et al. (2019);Matthaeus et al. (2020). The selected structure has more of a resem-blance to the 1D cut of the reconnection simulation at the X-pointthan at the reconnection exhaust. However, there are qualitative andquantitative differences. Energy-based parameters are dominated bythe Zenitani measure in the reconnection simulation, while a positiveP- 𝜃 is the strongest in the turbulence simulation. LET oscillates in thereconnection simulation, but is strongly negative in the turbulenceone. On the other hand, qualitative agreement between the two struc-tures is found for the VDF-based surrogates, although some featuresin the reconnection simulation (e.g. double peaks in the agyrotropy)are not present in the turbulence simulation perhaps because of thedifferent numerical resolution. Furthermore, for the cut through thecurrent sheet in turbulence, the parameters depend on time due to theturbulent interaction of larger scale magnetic eddies, so wide vari-ation in parameters across different current sheets and at differenttimes is expected. MNRAS000
5) than the energy-based parameters ( ∼ √︁ 𝑄 𝑝 shows large-scale fluctuations quitefar from the current structure. The broader distribution observed forthe VDF-based parameters suggests that energy transfer is localizedclose to the coherent structures, but can affect the particle distributionfunction in a larger region around these structures.The 1D cuts of the dissipation proxies for both the reconnectionand turbulence simulations support the idea that peaks of dissipationmeasures in a reconnecting current sheet or a turbulent environmentcharacteristically occur in coincident spatial regions, but not nec-essarily at the same exact spatial position. This is consistent withthe notion of regional correlations, suggested by Yang et al. (2019);Matthaeus et al. (2020). The selected structure has more of a resem-blance to the 1D cut of the reconnection simulation at the X-pointthan at the reconnection exhaust. However, there are qualitative andquantitative differences. Energy-based parameters are dominated bythe Zenitani measure in the reconnection simulation, while a positiveP- 𝜃 is the strongest in the turbulence simulation. LET oscillates in thereconnection simulation, but is strongly negative in the turbulenceone. On the other hand, qualitative agreement between the two struc-tures is found for the VDF-based surrogates, although some featuresin the reconnection simulation (e.g. double peaks in the agyrotropy)are not present in the turbulence simulation perhaps because of thedifferent numerical resolution. Furthermore, for the cut through thecurrent sheet in turbulence, the parameters depend on time due to theturbulent interaction of larger scale magnetic eddies, so wide vari-ation in parameters across different current sheets and at differenttimes is expected. MNRAS000 , 1–17 (2021) O. Pezzi et al.
Figure 8. (Color online) Same as figure 6, but for the HVM collisionlessturbulence simulation. The cut is taken at 𝑥 = . 𝑑 𝑝 . As a result of the weak collisionality of many space and astrophysicalplasmas, several physical processes can locally contribute to the dis-sipation of energy and the heating of the plasma. During magneticreconnection, magnetic energy is efficiently converted to directedplasma flows, thermal energy, and energetic particles (Burch et al.2016; Torbert et al. 2018). During turbulence in a magnetized plasma,the cascade provides the global amount of energy needed for fluc-tuations to be dissipated at small scales (Marino et al. 2008). How-ever, which dissipative mechanisms are dominant and under whichconditions is still not sufficiently understood (Vaivads et al. 2016).Therefore, a multiplicity of dissipation surrogates have been adoptedin the literature to identify potential sites of dissipation. Owing tothe strongly nonlinear dynamics and the importance of physics atkinetic scales in such systems, numerical simulations of the Vlasov-Maxwell system (or, including collisions, the Boltzmann-Maxwellsystem) are the decisive tool to address the long-standing issue ofplasma heating and energy dissipation in magnetized plasmas, e.g.Parashar et al. (2009). For example, the “turbulence dissipation chal-lenge” (Parashar et al. 2015) motivated numerical work to comparedifferent algorithms on the solution of similar problems to assess thenature of the dissipation in magnetized turbulence (e.g. Pezzi et al.(2017); Perrone et al. (2018); González et al. (2019)).In the same spirit, we have conducted a survey of a number of dis-sipation surrogates with three different kinetic plasma codes — thefully-kinetic particle-in-cell vpic, the fully-kinetic Eulerian Vlasov-Maxwell
Gkeyll , and the Eulerian Hybrid Vlasov-Maxwell codes— to perform numerical simulations of two important physical phe-nomena in plasma physics. The first class investigates reconnection inan isolated current sheet, and the second concerns plasma turbulenceat kinetic scales. We have calculated and compared eight distinctdissipation proxies, delineated in Section 2. For the sake of clarity,we have categorized them in terms of (i) energy-based parameters, whose definition describes energy transfer and conversion; and (ii)VDF-based parameters, that are directly related to kinetic signaturesin the particle VDF. Energy-based parameters considered here are thepower density by electromagnetic fields on charged particles (Zeni-tani et al. 2011), the pressure-strain interaction (Yang et al. 2017a),and a local proxy of the turbulent energy transfer (Sorriso-Valvo et al.2018a). The VDF-based parameters are the local pressure agyrotropy(Swisdak 2016) and three measures describing how different a localdistribution function is from being Maxwellian (Greco et al. 2012;Kaufmann & Paterson 2009; Liang et al. 2019).Our findings are that each of the studied measures is non-zeroin key settings in reconnection and turbulence, including the recon-nection diffusion region and magnetic islands, and the intermittentmagnetic shear regions bordering magnetic eddies in a turbulent sys-tem. The region that each proxy is strongest highlights a potentiallydifferent aspect of the physics taking place, as is described in detailin Sections 4 and 5. It is intended that the discussion therein willcontribute to the assessment of dissipation and energy conversion insatellite and plasma laboratory experiment measurements. We hereremark on the importance of the VDF-based diagnostics, which re-veal further details about the underlying dynamics of the plasmacompared to the energy-based diagnostics. Indeed, the energy-baseddiagnostics give similar results for similar simulations in the mag-netic reconnection setup (i.e. vpic and
Gkeyll runs), as we expectsince these simulations which show little difference in many of themeasures of the plasma response to magnetic reconnection. However,the additional level of detail provided by VDF-based diagnostics al-lows us to more carefully ascertain the kinetic response of the plasmafrom small differences which arise naturally between two differentsimulation codes, and the added robustness of these diagnostics islikely to be especially useful when analyzing more general simu-lations and observations of real plasma systems such as the solarwind. In contrast, the energy-based diagnostics are directly related tothe turbulence cascade process that connects the energy containingscales to the kinetic range of scales where the velocity space structurebecomes increasingly important. Therefore both type of diagnosticsare helpful to form a complete picture of the dynamics leading todissipation.The spatial locations where each proxy has a local maximum canbe different between the different proxies. For example, in the mag-netic reconnection simulations, the Zenitani measure is peaked atthe X-point while the pressure-strain interaction terms are peakedinside the magnetic islands. This confirms a suggestion that bothenergy-based and VDF-based parameters display a regional correla-tion: structures identified by these parameters often occur in similarregions, although not necessarily exhibiting a point-to-point corre-lation. This underscores a key conclusion of this work: that no onesingle measure is universally the best measure for dissipation. Rather,employing a number of proxies is likely best for assessing the dissi-pation and energy conversion in a plasma system.We also consider the role of collisions by including their dy-namical effect ab-initio in the numerical simulations for both re-connection and turbulence. When including only weak intra-species(proton-proton) collisions in the turbulence simulations, the VDF-based proxies decreased while the energy-based proxies were largelyunchanged. This behavior is expected since such collisions driveparticle VDFs towards Maxwellian distributions, but they do notproduce a net energy transfer between different species althoughthey indirectly modify the energy equations by isotropizing the pres-sure tensor (Del Sarto et al. 2016). On the other hand, both energy-and VDF-based proxies are influenced by inter- and intra-species(electron-proton) collisions in simulations of magnetic reconnec-
MNRAS , 1–17 (2021) issipation measures in plasmas tion. In particular, small-scale structures are dissipated by collisions,and the distributions in the diffusion region are less non-Maxwellianand thermalize more rapidly than in the collisionless case.Interestingly, we find the peaks of dissipation proxies in the systemswith collisions are weaker than in the collisionless system. Indeed, allfour of the VDF-based dissipation proxies are local in position space,and therefore provide only a local measure of the complexity of thedistribution function. Meanwhile, the energy-based measures alsobecome weaker in collisional systems, especially visible in the re-connection simulations. This suggests that the rate of energy transferdescribed by the terms in question is locally greater in the collision-less systems than in the collisional case, although the total dissipation(i.e. integrated) in the system may be larger with collisions. Further, italso clearly illustrates that the energy-based measures contain infor-mation about both collisional and collisionless processes, implyingthat they are not yet capable of distinguishing reversible from irre-versible processes.The present study is not intended to be the final word on thistopic, as there are many avenues for future work. The diagnosticsconsidered here do not discriminate the underlying process whichmay lead to energy dissipation or conversion, e.g. Landau damping,cyclotron damping, phase-mixing, and stochastic heating (Chandranet al. 2010; Li et al. 2016). In this perspective, a different approach,based on the field-particle correlation , has been recently adopted toidentify the presence of particular signatures in the particle VDF(Klein & Howes 2016; Klein et al. 2017; Chen et al. 2019; Kleinet al. 2020). This method has the advantage of diagnosing energiza-tion directly in velocity space while the others here adopted involvean integration over the full velocity space. The field-particle corre-lation is also local in physical space and does not require spatialgradients, which would require multiple spacecraft for in situ obser-vations. Incorporating the field-particle correlation gives a visual wayto identify energy transfer between particles and fields, and would beinteresting to compare with the other proxies. Indeed, the diagnosticsconsidered in the present work are extremely useful to characterizepotential sites of intermittent dissipation, e.g. in structures close tointense current sheets. On the other hand, methods such as the field-particle correlation identifies basic plasma processes, e.g. Landaudamping. Connecting these two points of view would also contributeto addressing the fundamental question whether dissipation in plas-mas is uniform or intermittent (Vaivads et al. 2016) and deserves adedicated, future study. Moreover, kinetic plasma turbulence excitesin a very complex way an entire ensemble of genuinely kinetic de-grees of freedom, i.e. those related to velocity-space structures in theparticle VDF (Servidio et al. 2017; Pezzi et al. 2018), thus drivingfree energy towards finer and finer scales in velocity space whereit is dissipated through inter-particle collisions (Pezzi et al. 2019c).The relation of the dissipation surrogates considered here with theenstrophy phase-space cascade will be the subject of a future work.Finally, there are many necessary extensions to the current study.The present simulations varied collisionality, but for a particular setof field and plasma initial conditions, so the parametric dependenceof the conclusions attained herein should be the subject of futurework. Both the reconnection and turbulence simulations were 2D,and therefore 3D effects are not captured. The reconnection sim-ulations studied here employed a small system size in which theprotons do not fully couple back to the large scale systems, so it isimportant to revisit the proton dissipation measures in larger sys-tem sizes. The effect of proton-electron collisions on energy-basedparameters in turbulence should be addressed in future work. The tur-bulence simulations considered here addressed decaying turbulence,and comparisons to driven turbulence would be interesting. ACKNOWLEDGEMENTS
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