Structure and dynamics of a new class of thin current …sitnov/testpage_files/recpubl/2005JA...Structure and dynamics of a new class of ... 2006), Structure and dynamics of a new

Structure and dynamics of a new class of thin

current sheets

M. I. Sitnov,1 M. Swisdak,2 P. N. Guzdar,1 and A. Runov3

Received 3 November 2005; revised 7 April 2006; accepted 21 April 2006; published 12 August 2006.

[1] New results on a steady state Vlasov theory of current sheets, which generalizes theHarris (1962) model by assuming anisotropic and nongyrotropic plasmas and usingthe invariant of particle motion in regions of strong gradients, are presented with the aimto explain multiprobe observations of thin current sheets in the geomagnetotail andlaboratory experiments, including the effects of current sheet embedding and bifurcation.The dynamics of these sheets is explored using a full particle code with more realisticmass ratio and anisotropy parameters than those used in our earlier works. The resultsrelevant to 2001 CLUSTER observations, with the sheet thickness appreciably exceedingthe thermal ion gyroradius, include ion distributions and pressure tensor components,which reveal the important role of nongyrotropic effects on the structure of these sheets.Their flapping motions are distinguished by north-south asymmetry of current profiles,quasi-rectangular shape of the flapping waves, and their small propagation speed,suggesting an explanation of their propagation toward the flanks of the tail sheet. Themain effect of the ion anisotropy on the sheets with thickness less than the thermal iongyroradius, relevant to 2003 CLUSTER observations and laboratory experiments, istheir charging, which may limit their minimum thickness, while their structure can bemodified by electron anisotropy. Other distinctive features of these sheets are three-peakedcurrent density profiles, found both in simulations and in the steady state theory, thenorth-south asymmetry of flapping sheets, and the shape of flapping waves, which isdrastically different from the case of thicker sheets.

Citation: Sitnov, M. I., M. Swisdak, P. N. Guzdar, and A. Runov (2006), Structure and dynamics of a new class of thin current sheets,

J. Geophys. Res., 111, A08204, doi:10.1029/2005JA011517.

1. Introduction

[2] For more than 4 decades, the kinetic studies of currentsheets in space and laboratory plasmas, including manyaspects of magnetic reconnection, have been based on theHarris equilibrium model [Harris, 1962]. It was used inparticular to fit data on the current sheet structure [Sanny etal., 1994; Yamada et al., 2000]. It was the dominatingequilibrium used in the kinetic stability analysis [Pritchett etal., 1991; Daughton, 1999] and as an initial condition inparticle simulations [Shinohara et al., 2001; Ricci et al.,2004]. The key element of the Harris theory was the choiceof particle distributions, expressed as the exponential func-tions of two invariants of motion, the total particle energyand the component of the canonical momentum along thecurrent direction. The resulting shifted Maxwellian distri-butions described isotropic ion and electron species. Theygave rise to a large family of isotropic equilibria for

various geometries [Schindler, 1972; Birn et al., 1975;Kan, 1973, 1979; Lembege and Pellat, 1982; Zwingmann,1983; Wang and Bhattacharjee, 1999; Manankova et al.,2000; Brittnacher and Whipple, 2002] and dominantcurrent-carrying species [Yoon and Lui, 2004].[3] Significant deviations from the Harris model were

also reported, with the most notable effects being observedfor thin current sheets, whose thickness is comparable to afew thermal ion gyroradii based on the field outside thesheet. In particular, McComas et al. [1986], Sergeev et al.[1993], and Sanny et al. [1994] found relatively thin currentsheets with unusually large current densities, embedded intomuch thicker plasma sheets. At the same time, Sergeev et al.[1993] reported several cases, in which the current density,estimated using the magnetic field difference measured bytwo spacecraft ISEE 1 and 2, had a minimum at the centerof the sheet. However, all those earlier results were based onone or at best two spacecraft observations. Therefore theycould not fully resolve whether the observed effects indeedarose from crossing the complicated spatial structures orthey just reflected changes in time of more conventionalplasma formations. As a result, they left significant freedomfor theoretical interpretations. For instance, Sergeev et al.[1993] explained their current splitting as an effect of theplasma anisotropy, while Hoshino et al. [1996] interpretedsimilar statistical results on the double-peaked current

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1Institute for Research in Electronics and Applied Physics, Universityof Maryland, College Park, Maryland, USA.

2Icarus Research Inc., Bethesda, Maryland, USA.3Space Research Institute, Austrian Academy of Sciences, Graz,

Austria.

Copyright 2006 by the American Geophysical Union.0148-0227/06/2005JA011517$09.00

A08204 1 of 19

http://dx.doi.org/10.1029/2005JA011517

sheets as either signatures of magnetic reconnection asso-ciated with the slow shocks or statistical effects arising fromaveraging in time of the flapping motions of Harris-typecurrent sheets.[4] The observational picture has become much more

transparent after the launch of the four-spacecraft CLUSTERmission, which allowed, for the first time, the unambiguousseparation of spatial and temporal variability. It also allowedfor high time resolution to reconstruct the current structureduring each separate traversal of the current sheet and toavoid statistical averaging over many flapping periods.Already, first studies of the 2001 tail period data with thespacecraft separation of 1500–2000 km [Nakamura et al.,2002; Runov et al., 2003; Sergeev et al., 2003] confirmedthat the current sheet profile may strongly differ from theHarris model, and in particular, may be bifurcated, that is,split into two sheets. Moreover, it was shown that at least insome cases the current bifurcation effect could not beexplained either by the classical picture of collisionlessreconnection or by flapping motions. For example, Sergeevet al. [2003] reported on a very small (less than a few nT)dawn-dusk component of the magnetic field, in contrast tothe expected magnitude of that component By � 0.4 B0 (B0

is the lobe magnetic field outside the sheet) in the vicinityof the X-line [Arzner and Scholer, 2001]. Also, consistentwith the Geotail observations [Hoshino et al., 1996; Asanoet al., 2004], they found current bifurcation in the absenceof any significant plasma flows typical for reconnection.The high time resolution of the magnetic field measure-ments (1 s) showed that the bifurcation was not anaveraged effect of current sheet oscillations.[5] These first results were complemented recently by a

statistical analysis [Asano et al., 2005], which shows thatthe overall occurence of bifurcated sheets among stable(nonflapping) and thin (with the average thickness of aboutfour ion gyroradii) current sheets is 17%. They have alsoshown that such thin sheets are even more frequentlyembedded within thicker sheets, and overall, the significantdeviations from the Harris model are rather the rule than theexception for such thin sheets.[6] It is interesting that the multiprobe measurements are

much easier to set up in the laboratory. For instance, up to60 probes, equivalent to 20 virtual satellites are available inthe PPPL Magnetic Reconnection Experiment (MRX)[Yamada et al., 2000], which allow one to reconstruct thecurrent and plasma density profiles with high resolution.The first MRX results revealed the amazing consistencybetween the Harris current density profile and those ob-served in the experiment [Yamada et al., 2000]. Yet, closerexamination of this experiment also reveals significantdeviations from the Harris model. An important character-istic of the Harris sheet is the absence of the bulk flowvelocity shear because current and plasma density profilesare similar. However, the MRX measurements indicate aclear difference between these profiles with the plasmasheet being significantly thicker [Carter, 2001]. Anotherpuzzling feature of the MRX current sheets is that theirthickness scales as the ion inertial length c/wpi [Yamada etal., 2000, Figure 3a], while the Harris sheet thickness L isnot supposed to scale with c/wpi because the ratio Lwpi/c is afree parameter in the theory (for details, see sections 2 and4.1). A distinctive feature of the MRX experiment, espe-

cially in the context of its comparison with CLUSTERobservations, is that the MRX current sheets are usuallythinner than the thermal ion gyroradius based on themagnetic field outside the sheet. Such thin current sheetscould not be resolved in the 2001 tail data, when theseparation between spacecraft appreciably exceeded oneion gyroradius (�400 km) [Asano et al., 2005; Runov etal., 2005a]. However, they have been detected in the 2003tail observations with 250 km CLUSTER tetrahedron scale[Nakamura et al., 2004].[7] Thus both space and laboratory observations demand

a significant generalization of the Harris theory. Althoughthe change of boundary conditions may produce a largevariety of modifications of the original Harris solution,including tangential discontinuities, islands, and other struc-tures, the simple analysis, given in the next section, showsthat it is not enough to explain the current bifurcation.Hence changes of the equilibrium distribution functionsmust be considered. Already in the late seventies, Cowley[1978] showed that rather modest values of the plasmaanisotropy should provide drastic changes in the currentsheet structure. In particular, the pressure anisotropy withPk > P? (Pk and P? are the components of the pressuretensor parallel and perpendicular to the local magnetic field)results in the formation of a thin current sheet embeddedwithin a much thicker sheet, while in the case P? > Pk aminimum of the current density at the center of the sheetappears. However, Cowley’s model was based on a fluidtheory of gyrotropic plasmas and the solution of the systemof the plasma momentum and Maxwell’s equations. It wasnot a steady state Vlasov theory. Moreover, it was soonrealized that even modest deviations from isotropy couldnot survive because of anisotropy-driven mirror andfirehose instabilities [Notzel et al., 1985; Hill and Voigt,1992]. In particular, according to Hill and Voigt [1992],deviations of the parameter P?/Pk from unity should notexceed one percent, provided that the plasma pressuretensor is gyrotropic. Against that background, findingproton distributions with P?/Pk = 1.04–1.34 in the tailprior all current disruption events studied by Lui et al.[1992] using the AMPTE CCE spacecraft, has stronglysuggested the significance of nongyrotropic effects in thincurrent sheets.[8] Significant nongyrotropic effects may indeed appear

in the current sheets when their thickness becomes compa-rable to the thermal ion gyroradius based on the fieldoutside the sheet. Ion orbits in such thin sheets differ fromthe conventional Larmor circle and become more similar tothe figure of eight [Speiser, 1965]. Their rotational symme-try relative to the local magnetic field vector is completelylost. However, the problem of generalizing the Harris modelto single out the contribution of figure-of-eight orbitsremained unattainable for many years. It has been solvedeventually as a part of the modeling activity stimulated byCLUSTER discoveries. One of the solutions, proposed bySitnov et al. [2003, hereinafter referred to as SGS], was toextend the set of integrals of motion used in the Harrismodel by including the so-called sheet invariant, an analogof the magnetic moment introduced for the systems withstrong magnetic field gradients, and to assume some plasmaanisotropy outside the sheet, where plasmas become gyro-tropic. The SGS results turned out, in fact, a kinetic and

A08204 SITNOV ET AL.: NON-HARRIS CURRENT SHEETS

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nongyrotropic analog of Cowley’s theory, as they showedthat the ion anisotropy outside the sheet with Tki > T?i (Tkiand T?i are ion temperatures parallel and perpendicular tothe local magnetic field) led to the current sheet embedding,while the opposite anisotropy led to its bifurcation. The newself-consistent kinetic theory of non-Harris current sheetshas been confirmed by particle simulations [Sitnov et al.,2004a], which have also revealed the remarkable structuralstability of the current bifurcation effect notwithstanding anumber of various instabilities and the resulting flappingmotions, that is large-amplitude north-south oscillations ofthe tail current sheet propagating along the dawn-duskdirection [Sergeev et al., 1998].[9] The properties of the flapping waves detected by

CLUSTER observations [Runov et al., 2003, 2005a;Sergeev et al., 2003, 2004] turned out to be anotherchallenge for theoreticians. According to the linear theoryof the kinetic kink instability of the Harris current sheet [Zhuand Winglee, 1996] proposed by Daughton [1999], theunstable kink-type waves should propagate with a phasevelocity close to the ion drift speed and in the same directionas the bulk ion flow. In contrast to these theoretical pre-dictions, Sergeev et al. [2004] and Runov et al. [2005a]found that the propagation speed of flapping waves is verysmall, much less than the ion bulk-flow speed inferred fromthe sheet thickness using the Harris model. Moreover, thestatistical analysis performed by Sergeev et al. [2004] andRunov et al. [2005a] revealed that the flapping motionspropagate flankward, that is, dawnward in the dawn sectorof the tail and duskward in the dusk sector.[10] The presently available SGS theory of non-Harris

sheets remains incomplete in a number of key points, whichlimit its comparison with observations as well as other thincurrent sheet models. First, only the lowest-order momentsof distribution functions (current and plasma density) havebeen investigated so far, whereas the ion pressure tensorcomponents, which reflect the core distinctive features ofthe model, namely anisotropy and nongyrotropy, are notavailable. Second, particle simulations using the newmodel were done with artificially large ion anisotropy andelectron-to-ion mass ratio. It remains unclear how theseartificial values of the key parameters affect the new featuresof the flapping motions previously reported. Also, the previ-ous runs were not long enough to reveal important nonlineareffects. Third, the original SGS theory and simulations wereaimed at the explanation of 2001 CLUSTER data, and as aresult, the features of non-Harris sheets with the thicknessless than the ion gyroradius were not addressed at all.[11] In this paper we report on the further development of

the theory of non-Harris current sheets. In section 2, webriefly overview the SGS equilibrium model and considerits applicability, taking into account the influence of thecomponents of the magnetic field normal to the sheet andparallel to the equilibrium current. In section 3, we describethe distinctive features of non-Harris sheets with the thick-ness exceeding a few thermal ion gyroradii, which are mostrelevant to the 2001 period of CLUSTER observations. Weprovide, in particular, the ion distribution functions forembedded and bifurcated current sheets as well as theirmoments, compare them with CLUSTER observations, andexplore the effects of plasma anisotropy and nongyrotropyon the current bifurcation. We also explore the flapping

motions of bifurcated current sheets, using an explicitparticle code with realistic ion anisotropy and smaller(compared to earlier runs) electron-to-ion mass ratio. Inter-esting nonlinear effects are also revealed using the extensionof one of the previously studied simulation cases. Insection 4 we consider the equilibrium structure and dynam-ics of non-Harris sheets with the thickness less than an iongyroradius, relevant to the 2003 CLUSTER observations ofthe magnetotail and laboratory experiments. The currentsheet properties in this case are shown to differ drasticallyfrom those of the thicker sheets considered in section 3. Theresults of our work are summarized in section 5.

2. Generalization of the Harris Model

[12] We start with a brief description of the Harris model[Harris, 1962], in order to highlight the properties thatneed to be modified, as well as to outline some possiblealternatives to our basic approach. One of the key ele-ments of this model is the set of ion and electrondistributions taken as simple exponential functions oftwo integrals of motion, the total particle energy Wa =mav

2/2 + qaf (f is the electrostatic potential) and acomponent of the canonical momentum along the currentdirection Pya = mavy + (qa/c)Ay

f0a / exp � Wa � vDaPya� �

=Ta� �

; a ¼ e; i ð1Þ

where Ay is the dawn-dusk component of the electromag-netic potential (here and below we use the GSM coordinatesystem). The parameters vDa, Ta, ma, qa denote the bulk-flow speed, temperature, mass, and charge for the species a,respectively. As a function of invariants of motion, thedistribution (1) automatically satisfies the Vlasov equation.Moreover, its exponential form suggests a simple solutionof the Poisson’s equation, f = 0, if the condition vDi/vDe =�Ti/Te holds. Then the only relevant Ampere’s equation canbe reduced to a Grad-Shafranov-type equation

d2Ay

dz2¼ � 8pn0 Ti þ Teð Þ

B0Lexp

2Ay

B0L

� �ð2Þ

with the bulk speed vDa = 2cTa/qaB0L, where L is a scaleparameter to be determined. With the boundary conditionsdAy/dzjz=0 = 0 and Ayjz!±1 = �B0Ljzj, this yields thesolution of this nonlinear eigenvalue problem in the form ofthe vector potential profile (eigenfunction) Ay = �B0Llog[cosh(z/L)] and the condition (eigenvalue) b0 = 8pn0Ti/B0

2 =Ti/(Ti + Te). As a consequence, the magnetic field profilehas the well-known hyperbolic-tangent shape Bx = B0 tanh(z/L), while the plasma density profile takes the form n =n0cosh

�2(z/L) similar to the current density profile j =ne(vDi � vDe). The parameter L determines the character-istic thickness of the sheet, and it can be written in theform L = (vTa/vDa)r0a = (vTi/vDi)

ffiffiffiffiffib0

p(c/wpi), where r0a

and vTa are the gyroradius in the field B0 and thermalspeed of the species a.[13] Thus the Harris model describes current sheets that

have similar profiles for the current and plasma densitieswith a single peak at Bx = 0 and a single scale, therebyeliminating the possibility of both current sheet embeddingand bifurcation. Also, the current is dominated by ions


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(jvDij > jvDej), provided that Ti > Te, a situation typical formagnetospheric plasmas. Finally, the thickness L can bearbitrary in units of the ion inertial length c/wpi, becauseL/(c/wpi) = (vTi/vDi)

ffiffiffiffiffib0

pis a free parameter in the theory.

[14] Generalizations of the Harris model, with the oper-ator d2/dz2 in (2) being replaced by its two-dimensional(2-D) analog r2 = d2/dx2 + d2/dz2, yield more sophisti-cated equilibria, including tail-like structures, magneticislands, and discontinuities [Schindler, 1972; Kan, 1979;Schindler and Birn, 1993; Manankova et al., 2000]. Atfirst sight, some of these equilibria, and in particular, thosewith magnetic islands, might provide the current bifurca-tion. However, as long as the structure of the right-handside of (2) is kept unchanged, any current bifurcationremains impossible. The reason is that bifurcation impliesthat the current density j as a function of the magneticfield Bx has the maxima dj/dBx = 0 for jBxj = Bbif 6¼ 0 (wedo not consider here the rather peculiar case when thismaximum current is less or equal to zero, making theprofiles j(Bx) and j(z) qualitatively different). Then, thederivative (dj/dAy) = (dj/dBx) (dBx/dAy) can be nonzero,consistent with (2), only if (dBx/dAy) ! 1 at jBxj = Bbif.This yields, however, an infinite current density at thepoint of the bifurcated current maximum dBx/dz = �(dBx/dAy)Bbif. We conclude that no boundary perturbation canprovide bifurcation for Harris-type systems, which requirestherefore either dynamical effects or changes of plasmadistributions leading to a modification of the right-handside of (2) as a function of Ay.[15] One approach to generalize the Harris model is

suggested by the equation (1), in which two invariants ofmotion Wa and Pya appearing in the distribution functiondo so in the form of a linear combination. The equilibriabased on more general combinations of these invariantswere considered in a number of works [Channell, 1976;Schindler and Birn, 2002; Mottez, 2003; Birn et al., 2004;Genot et al., 2005; Camporeale and Lapenta, 2005]. Theywere shown to reproduce important structure changes suchas the current sheet embedding [Schindler and Birn, 2002]and bifurcation [Birn et al., 2004; Genot et al., 2005;Camporeale and Lapenta, 2005], although that was typ-ically achieved at the expense of rather uneven distribu-tions, raising stability issues [Camporeale and Lapenta,2005]. However, probably the biggest problem in thatapproach is the choice of the specific combination of theinvariants Wa and Pya, different from the Harris case, andits physical motivation. In some cases [Mottez, 2003;Genot et al., 2005], it is proposed that the necessarycombination be derived from the given spatial structureof the current sheet, even though this problem is usuallyill-posed.[16] Another way of generalization was suggested by the

unusual properties of the particle dynamics near the mag-netic field reversal, when its gyroradius becomes compara-ble to either the current sheet thickness or the curvatureradius of the magnetic field (if there is a component Bz ofthis field normal to the sheet plane). Then the particle orbitsmay strongly differ from Larmor circles and resemble morefigures of eight. In general, the particle dynamics, includingfigure-of-eight orbits, is very complicated, since it does notobey the conventional guiding center theory and may beeven chaotic [Chen, 1992, and references therein]. How-

ever, when the current sheet is thin enough, and inparticular, when its thickness obeys the condition

L � r0a B0=Bzð Þ2; ð3Þ

the particle dynamics becomes approximately adiabatic or‘‘quasi-adiabatic’’ [Buchner and Zelenyi, 1989], similar tothe case of a weakly curved magnetic field. In this case anadditional integral of motion, the so-called quasi-adiabaticor sheet invariant can be introduced [Schindler, 1965;Sonnerup, 1971; Francfort and Pellat, 1976; Whipple et al.,1986; Buchner and Zelenyi, 1989]

I að Þz ¼ 1

2p

IPzadz; ð4Þ

where Pza is the z-component of the canonical momentum.The condition (3) is readily satisfied for ions in the regionsof high current density in the geomagnetotail, suggestingthat these regions are produced by quasi-adiabatic ions[Kaufmann et al., 2001]. It is even more relevant for oxygenions, whose quasi-adiabatic dynamics appreciably modifiescurrent and plasma densities at substorm onset [Kistler etal., 2005].[17] The models utilizing the features of the quasi-

adiabatic ion motion [Eastwood, 1972, 1974; Burkhart etal., 1992; Holland and Chen, 1993] and in particular, usingthe sheet invariant Iz

(i) in place of the canonical momentumPyi in the distribution function f0i = f(Wi, Iz

(i)) [Francfort andPellat, 1976; Kropotkin et al., 1997; Sitnov et al., 2000],introduce a natural scale of current sheets, related to thethermal ion gyroradius r0i. Following Burkhart et al. [1992],we will call them forced-current sheet or FCS models.Owing to the features of quasi-adiabatic orbits [see, e.g.,Zelenyi et al., 2003, Figure 3] the FCS models look quitepromising in describing both embedding [Sitnov et al., 2000]and bifurcation [Harold and Chen, 1996] effects. Moreover,since the major structure features of such models aredetermined by the universal form (4) of the invariant Iz

(i),they must weakly depend on the details of the distributionfunction. Yet, the FCS models have a severe limitation, asthey have in fact no limit of isotropic plasmas. Reducinganisotropy results in either a catastrophe of the model atsome point [Burkhart et al., 1992; Zelenyi et al., 2002] or inan unlimited growth of the current sheet thickness or theplasma density [see Sitnov et al., 2003, p. 13,038]. It mayalso require introducing multicomponent plasmas [Hollandand Chen, 1993; Harold and Chen, 1996]. These complica-tions are not surprising, because the FCSs originated as themodels of magnetic merging [Alfven, 1968; Hill, 1975],where the magnetic tension is balanced by the ion inertiarather than by the pressure gradient as is the case in isotropicHarris-type equilibria [Schindler, 1972].[18] The problem of a missing link to isotropic equilibria

has been resolved in the SGS model of thin current sheets[Sitnov et al., 2003], which has become a generalization ofthe Harris and FCS models. This model is based on a set ofAmpere’s equation and the quasi-neutrality condition, wherethe electron and ion distributions are taken in the form

f0a / exp � 2Wa � w0aIað Þ

z

2Tjjaþ vDaPya

Tjja� w0aI

að Þz

2T?a

� �ð5Þ


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with w0a = eB0/mac (e = jqaj). They are constructed fromthe appropriate integrals of motion, namely Wa, Pya, and theinvariant Iz

(a) given by (4). Note that we adopt the invariantIz(a) for both ions and electrons, although the condition (3)for electrons can hardly be satisfied in the geomagnetotail[e.g., Runov et al., 2005a], and their dynamics should betreated more correctly as adiabatic as discussed in moredetail in [Sitnov et al., 2003]. However, the assumption (5)is more consistent with the following 2-D particle simula-tions, which usually require Bz = 0 to ensure the forcebalance along the x-direction. Besides, this difference isessential only for studying the effects of the electron speciesanisotropy.[19] In the limit Tka = T?a = Ta the distribution (5) takes

the classical isotropic form (1) [Harris, 1962; Schindler,1972]. In another limiting case vDa ! 0 with Tke = T?e, theelectron distribution becomes a pure Maxwellian, while theion distribution becomes similar to the counterstreaming iondistribution of the FCS model [see Sitnov et al., 2000,equation (8)]. The SGS theory addresses a more generalcase vDa 6¼ 0 and Tka 6¼ T?a.[20] Let us consider the magnetic field in the form B =

(Bx(z), By, Bz), where By and Bz are constants, which aremuch less than the asymptotic value of the x-componentjBx(±1)j = B0. Then the invariant (4) can be rewritten in theform

I að Þz vy; vz; z

� �¼ 2ma

p

Z z1

z0

dz0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiW 2

1 vy; vz; z; z0� �

�W 22 vy; z; z0� �q

ð6Þ

where W1 =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2y þ v2z þ 2qa=mað Þ f zð Þ � f z0ð Þ½ �

q, W2 = vy +

(qa/mac)R z0

zBx(z

00)dz00, and the limits of integration in (6)

are given by the equation W1(vy, vz, z, z0,1) ± W2(vy, z, z0,1) =0, with z0 < z < z1 and the additional restriction that z0 = 0 ifits formal solution becomes negative.[21] Note that the finite magnetic field components By

and Bz does not appear in the equation (6), whichdetermines the invariant Iz

(a), and as a consequence thewhole equilibrium theory does not depend on these com-ponents. This can be understood if we consider the particleHamiltonian

Ha ¼ P2xa

2maþ

Pya � qa=cð ÞAy x; zð Þ� �2

2ma

þ Pza � qa=cð ÞAz zð Þ½ �2

2maþ qaf zð Þ ð7Þ

with the electromagnetic potential taken in the form

A ¼ 0;�Z z

Bxdz0 þ xBz;�xBy

� �ð8Þ

Since equation (6) is obtained by the substitution of Pza into(4) using (7) and (8), where, by definition, slow parameters,such as the coordinate x, must be taken constant, both By

and Bz disappear from the theory in the consideredapproximation. They may affect however the definitionsof anisotropy parameters, which are used below. Besides,the finite Bz field may result in violation of the invariant Iz

conservation. However, it is rather small as long as thecondition (1) is satisfied (for details, see Buchner andZelenyi [1989], Chen [1992], and Kropotkin et al. [1997,p. 22, 104]).[22] The basic equations of the SGS steady state Vlasov

theory can be found elsewhere [Sitnov et al., 2003, 2004a].Given the input parameters, including the electron-ion massand temperature ratios m = me/mi and t = T?e/T?i, theanisotropy ratio ha = T?a/Tka, and the dimensionless speedwDa = vDa/vT?a, the model yields the self-consistent profilesof the magnetic field Bx and the electrostatic potential f,which can then be substituted in the distributions (5) tofind all their relevant moments. The additional outputparameter of the model is the effective beta parameter b0 =8pn0T?i/B0

2. In the following we will use the dimensionlessvariables for the magnetic field Bx = bB0, the electro-magnetic potential Ay = �aB0r?0i (with r?0i = vT?i/w0i),the electrostatic potential f = T?ij/e, the coordinate z =r0iz, the particle velocities vy,z = v?Tawy,z, and thecorresponding sheet invariants Iz

(a) = I(a)mavT?ar?0a.[23] Ampere’s equation and the quasi-neutrality condition

can be complemented by the additional condition

wDe ¼ � t1=2m1=2hi=he �

wDi ð9Þ

which is similar to the condition vDi/vDe = �Ti/Te in theHarris model and provides the consistency of the equili-brium in the case of zero background population and zeroelectrostatic field outside the sheet. However, in thepresence of even a very small background populationcondition (9) becomes redundant thus allowing a greatervariety of electron-to-ion drift speed ratios. If present, thebackground population with a density nb, temperatures Tbaand zero bulk flow speed is specified by the dimensionlessparameters tba = Tba/T?i and eb = nb/n0.

3. Modeling Tail Current Sheet Probed byCLUSTER in 2001

3.1. New Equilibrium Features

[24] Just like in case of the equation (2) of the Harristheory, solving the basic set of equilibrium equations is anonlinear eigenvalue problem for the parameter b0 and thefunctions b(z) and j(z). The equations can be solved byiterations starting from the corresponding Harris solutionand updating at each step both the eigenfunctions b, j, andthe eigenvalue parameter b0, using the boundary conditionb(1) = 1. The details of this type of iteration procedure,including convergence control, were discussed earlier byKropotkin et al. [1997]. The left panels of Figures 1 and 2show two particular solutions for the parameters wDi =0.125, m = 1/1836, t = 1/4, and different ion anisotropyvalues hi = 1.2 (Tki < T?i) and hi = 0.8 (Tki > T?i) outside thesheet. These figures are similar to Figures 2 and 4 in thework of Sitnov et al. [2003] for the cases of embedded andbifurcated current sheets and differ from them by theirabscissa axes, which now reflect the magnetic field strengthBx = bB0. They also show an additional parameter, the bulkflow speed for each of the species. The new format providesa better way of comparison with CLUSTER observations.(In particular, plotting the profiles as functions of Bx


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eliminates ambiguities in estimating the spatial units, suchas the thermal ion gyroradius r0i, as well as the spatialprofiles themselves.) Figure 1 provides such a comparisonwith the bifurcated current sheet observed on 29 August2001 [Runov et al., 2004] (right panels show their Figure 5,which has been adapted to fit the specific set of plots shownin the left panels; in contrast to the left panel, only the iondrift speed is given in the bottom right panel). One cannotice similar profiles of the current density, plasma density(it has a plateau between the current density maxima, whichcan be seen in the real space when compared to the Harriscase [Sitnov et al., 2003]), as well as the characteristic shear

of the ion bulk-flow speed with a minimum at the center ofthe sheet [Runov et al., 2004]. Similar characteristic featuresof the embedded current sheets are shown in Figure 2. Theyare compared with the results of the recent statisticalanalysis performed by Runov et al. [2005b]. The current andplasma density profiles shown in the right panels ofFigure 2 are obtained by averaging 10 so-called center-peaked current sheets (for more detail on the current sheetclassification, see Runov et al. [2005b]). Consistent with thetheory, observations clearly show the current densitypedestal of a wide current sheet containing inside a well-distinguished thinner current sheet. A seeming discrepancy

Figure 2. Comparison of theory and observations for embedded current sheets. Left panels show twomodel solutions, one with hi = 0.8, he = 1, m = 1/1836, t = 1/4, and wDi = 0.125 drawn as in Figure 1, andthe other with hi = 0.95 and wDi = 0.07 (dotted lines) normalized to coincide with the first solution at b =0. Similar observational profiles on the right are inferred from the results of Runov et al. [2005b], wherethe field Bx is normalized to its lobe value Bl = B0 and data for Bx > 0 (stars) are combined with the caseBx < 0 (squares; Bx ! �Bx). Bottom right panel gives the ratio Vy = Jy/Np.

Figure 1. Comparison of theory and observations for bifurcated current sheets. Shown on the left arecurrent density J, plasma density n and bulk-flow speed Vy (in dimensionless units specified at the endof section 2) versus magnetic field b = Bx/B0 for the model with hi = 1.2, he = 1, m = 1/1836, t = 1/4,and wDi = 0.125; dashed and dash-dotted lines show electron and ion contributions. Right panels showsimilar CLUSTER observations [Runov et al., 2004].


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between theory and observations, which show the profilesJy(b) and Vy(b) with narrower embedded parts, can beexplained by the fact that both in our theory and inobservations the thickness of embedded sheets in real spaceis only a few ion gyroradii, while the thickness of thesurrounding plasma sheets in observations is an order ofmagnitude larger, which is not the case for the first group ofmodel curves shown in Figure 2 (according to Figure 2 inthe work of Sitnov et al. [2003], this plasma sheet is onlyabout two times as thick as the current sheet). To improvethe consistency with observations, we show in Figure 2another equilibrium solution (left panels, dotted lines) witha thicker surrounding sheet (wDi = 0.07) and smaller ionanisotropy (hi = 0.95), which turns out to be indeed moreconsistent with observations.[25] The ion distributions corresponding to bifurcated and

embedded current sheets are shown in Figures 3 and 4. Theplots are made using formula (5) and the self-consistentprofiles of the magnetic field and electrostatic potential.Compared to strongly non-Maxwellian distributions of FCSmodels (see, in particular, Burkhart et al. [1992] and Sitnovet al. [2004b]) and similar nonspherical distributionsobtained by Camporeale and Lapenta [2005] using themodel of bifurcated sheets by Birn et al. [2004], thesedistributions are quite spherical and only slightly differ fromshifted Maxwellian distributions of the Harris model. The

most notable non-Harris feature is found for bifurcatedsheets near the current maximum in the form of themushroom-like structure (Figure 3f). It resembles one ofthe electron distributions reported by Camporeale andLapenta [2005] (Figure 3, lower panel), where, however,they were observed outside the current sheet.[26] The distributions for embedded sheets look even

more spherical, although after some coarse-graining (20 �20 grid, not shown) the analog of Figure 4a reveals the lima-bean signatures characteristic for the similar cuts of forcedcurrent sheets [Burkhart et al., 1992, Figure 3h; Sitnov etal., 2004b, Figure 1]. Now this feature is only a rather smallcorrection to the core Harris part of the distribution, whichwas an important missing element of the FCS modelspreventing transitions to the limit of isotropic plasmas asdiscussed in section 2.

3.2. Current Bifurcation: AnisotropyVersus Nongyrotropy

[27] The rather spherical shape of the ion distributionssuggests the importance of studying their moments and inparticular the pressure tensor components that may revealthe effects of anisotropy and nongyrotropy. In his seminalpaper, Cowley [1978] argued that small plasma anisotropy issufficient to provide strong changes of the current sheetstructure. Later the effects of embedding and bifurcation

Figure 3. Ion distribution cuts for the bifurcated current sheet shown in Figure 1.


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arising from plasma anisotropy and nongyrotropy weredemonstrated by Harold and Chen [1996]. Recently, Birnet al. [2004] proposed a model of the current bifurcationwith the plasma anisotropy being confined to the interiorof the thin current sheet, where the ion species becomesnongyrotropic. To explore the relative importance ofplasma anisotropy and nongyrotropy as the mechanismsof the current structure changes in the present theory, weexplored the profiles of the magnetic field, current, andplasma densities as well as ion pressure tensor componentsfor the original SGS model and its simple modification,where the linear dependence of the exponent in the iondistribution function (5) on the quasi-adiabatic invariantlog f0i � Iz

(i) is slightly modified: log f0i � F(Iz(i)) with

F(I) = I + s(1 � gI) exp (�dI2). This function, an exampleof which for the parameters s = 1, g = 1.8 and d = 0.125is given by the inset in Figure 6, makes a part of thedistribution with small values of the adiabatic invariant Iz

(i)

and hi > 1 stretched along the field line outside the sheetas shown in Figures 4d and 4e, while the rest of thedistribution remains similar to Figures 3g and 3h. The newdistribution may better reflect the features of the currentsheet formation and the influence of the Bz component. Inthe case of the finite Bz the ion orbits with small values ofIz(i) represent passing ions coming from the regions withstrong and weakly curved magnetic field where theirmotion is (locally) adiabatic. The influence of the convec-tion electric field on those orbits provides the drift towardthe neutral plane. Then their perpendicular temperature T?imust decrease by virtue of the magnetic moment conserva-tion, thus forming the negative slope of the function F(I). Incontrast, as argued by Sitnov et al. [2003], the trappedparticles with figure-of-eight orbits and large values of Iz

(i)

are heated nonadiabatically across the magnetic field. Thesetheoretical considerations are consistent with particlesimulations [Swisdak et al., 2005].

[28] We performed the comparison of the original SGSmodel and its modification described above for the param-eters m = 1/16, t = 1/4, hi = 1.1, he = 1.0 and wDi = 0.11. Asshown in Figure 5, for the original model this set providesthe clear current bifurcation with only 10% plasma anisot-ropy outside the sheet. At the same time, the modification ofthe distribution function, as shown by the inset in Figure 6,further reduces anisotropy outside the sheet down to 3%,keeping the basic structure elements, the magnetic field,current, and plasma density profiles (left panels in Figures 5and 6) practically unchanged. However, inside the sheet theion species is not only anisotropic but also appreciablynongyrotropic, as all three diagonal components of thepressure tensor are different (in particular, jPzz � Pxxj �jPyy � Pzzj).[29] We conclude that the effect of plasma nongyrotropy

on the current sheet is equally, if not more, important thanthat of the plasma anisotropy. It is interesting to note herethat in the new model with the reduced anisotropy outsidethe sheet, the profile of the local perpendicular temperatureT?(z) (Figure 6, top right panel) is very similar to thecorresponding profile provided by CLUSTER observations[Runov et al., 2004, Figure 5].

3.3. Flapping Motions

[30] The characteristic thickness of the current sheetsprobed by CLUSTER in 2001 tail period was determinedby two factors. The first one was the typical CLUSTERtetrahedron size �2000 km �5r0i. However, even group offour spacecraft is usually not enough to scan the wholecurrent sheet profile, and the researchers used the additionaleffect of the current sheet flapping motions [Sergeev et al.,1993, 1998] to overcome that problem [Runov et al., 2003,2005a; Sergeev et al., 2003, 2004]. Luckily, it turns out thatthe occurrence rate of the current sheet thickness Lfl ofthose flapping sheets peaks around the same value as the

Figure 4. Ion distribution cuts for the embedded current sheet shown in Figure 2.


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CLUSTER tetrahedron size: Lfl �5–10 r0i [Runov et al.,2005a]. Modeling the flapping motions of such not-too-thinsheets is rather difficult, as it requires large simulation boxesto obey the condition kL < 1 typical for all large-scaleinstabilities (here k is the wave number). At the same time,reducing the current sheet thickness L down to the scales lessthan r0i may prevent modeling some non-Harris features. Inparticular, the current bifurcation in the SGS model disap-pears for L < �2 r0i notwithstanding a significant ionanisotropy, while the bifurcated sheets, available in themodel [Birn et al., 2004] in that thickness region, are stronglystructurally unstable, and they collapse to a Harris-like sheetwithin one gyroperiod [Camporeale and Lapenta, 2005].[31] A compromise solution of that problem was proposed

by Sitnov et al. [2004a]. They considered the current sheet,which was thin enough (L � 3–5 r0i) and yet split intotwo currents owing to rather strong ion anisotropy (hi = 2).

Other parameters of the initial equilibrium taken for thisrun (hereafter Run 1) are the following: m = 1/16, t = 1/4,wDi = 0.3, and he = 1. The first results of these simulationsconfirmed the remarkable structural stability of ionanisotropy-based bifurcated current sheets and also revealeda number of unusual features of their flapping motions.However, it remains unclear to what extent these newfeatures are affected by excessively large anisotropy param-eter and electron-to-ion mass ratio. In particular, the largestion pressure anisotropy detected in the tail current sheet[Lui et al., 1992] corresponds to the parameter hi = 1.34,while the electron anisotropy is less by more than an orderof magnitude (A. T. Y. Lui, private communication, 2004),consistent with the limitations imposed by mirror and fire-hose instabilities (see section 1 for more details as well asKaufmann et al. [2002, 2005] for the long-term averagedanisotropy data). At the same time, the linear kinetic

Figure 6. Profiles shown in Figure 5 for the nonlinear exponent log f0i � F(Iz(i)) with F(I) = I + s(1 �

gI) exp (�dI2) and the parameters s = 1, g = 1.8 and d = 0.125 (see the inset).

Figure 5. Magnetic field, plasma, and current density profiles (left panels), as well as the localanisotropy profile T?/Tk and pressure components of the ion species for the bifurcated sheet withthe parameters m = 1/16, t = 1/4, hi = 1.1, he = 1.0, and wDi = 0.11. Inset shows the exponent log f0i �F(Iz

(i)) = Iz(i).


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stability analysis of the Harris sheet [Daughton, 1999]predicts significant changes of the dispersion and stabilityproperties of flapping motions, and in particular, the in-crease of their phase velocity with the decrease of the massratio down to its more realistic values. On the other hand, asimilar stability analysis of forced current sheets [Sitnov etal., 2004b] shows drastic changes of the flapping motions innon-Harris sheets. To resolve these issues, we performedanother run (Run 2) with the parameters hi = 1.4, m = 1/64,t = 1/4, wDi = 1/4, and he = 1, that is, with the realisticvalue of the plasma anisotropy and four times less electron-ion mass ratio, compared to Run 1.[32] The 2-D particle simulations are performed based on

the explicit code P3D [Zeiler et al., 2002], which isparallelized using MPI routines with 3-D domain decom-position and retains the full dynamics for both ions andelectrons. The original particle-loading procedure has beenmodified to allow loading the non-Maxwellian distributions(5) using a 3-D rejection method [Press et al., 1999] in thespace (z, vy, vz) with the initial number of particles per cellNi = 600 and the average number of accepted particles percell Na = 12 (Run 1) and 15 (Run 2). The simulation boxsize is (ly/d, lz/d) = (51.2, 25.6), where d = c/wpi is the ioninertial length based on the density n0, with Ny � Nz =(2048, 1024) grids. The speed of light is given by c/vA = 15,where vA is the Alfven speed based on the maximum plasmadensity, corresponding to the following value of anotherkey simulation parameter wpe/w0e = 1.875 in this run (andwpe/w0e = 3.75 in case of Run 1; here wpe and w0e are theelectron plasma and gyrofrequencies, respectively). Thetime step dt = 0.005w0i

�1 (Run 1) and 0.0025w0i�1 (Run 2),

with two substeps for fields, resolves both w0e�1 and wpe

�1,where wpe is the electron plasma frequency. The boundaryconditions are periodic in the y-direction. In the z-directionthe simulation box is bounded by conducting walls, whereparticles are specularly reflected. This reflection may

change the quasi-adiabatic invariant (4), which enters ourequilibrium theory. However, since the plasma density inour basic equilibrium falls down exponentially when jzj !1, the effect of reflection on the initial equilibrium cansafely be neglected.[33] The results of Run 2 are shown in Figures 7 and 8.

Figure 7 summarizes them in the form of the fast Fouriertransform (FFT) analysis of instabilities, while Figure8 gives a snapshot of the current sheet at the end of therun. Notwithstanding a significant difference in the simula-tion parameters compared to Run 1, the new run revealsstrong resemblance with the part of Run 1 reported bySitnov et al. [2004a]. In particular, the initial bifurcatedequilibrium remains basically stable (Figure 8b) and theLHDI develops only at the outer edges of the split currentlayers (Figures 7b and 8b). Recall that this structuralstability drastically differs from the collapse of a bifurcatedsheet based on the model [Birn et al., 2004] reported byCamporeale and Lapenta [2005]. The most obvious reasonfor this difference is the thickness of current sheets consid-ered by Camporeale and Lapenta [2005] and in our model.In our relatively thick sheet (L� 4 r0i), the initial bifurcationis provided by massive ions, making the bifurcation effectquite robust. In contrast, in much thinner sheets considered inthe work of Camporeale and Lapenta [2005] (with L � r0i),the major role for causing current bifurcation can be playedby electrons (as discussed in section 4.1, in our model ionscannot provide bifurcation at these scales at all). Thereforeone can expect their equilibrium to be far more sensitive tovarious instabilities with electron scales, which may quicklydestroy the sheet. Another explanation, which is still to bechecked, is the fundamental difference between these twoequilibrium models. Like forced current sheets, our bifur-cated sheet is maintained by quasi-adiabatic dynamics ofions in strongly curved magnetic fields. These dynamical

Figure 7. Run 2: FFT analysis of the field Bx for the LHDI (top panels) and the large-scale flappinginstability (bottom panels) in terms of (a,d) the FFT spectra, (b,e) eigenmode profiles across the sheet forthe modes with the given wave number m, and the time history of these modes. Bm is the amplitude of theFFT transform and hBm

2 i is the corresponding intensity averaged over all z. w = w0i; r = r0i.


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features may indeed be far more fundamental and robustand more difficult to destroy.[34] The spectrum of LHD waves (Figure 7a) is similar to

Run 1 [Sitnov et al., 2004a, Figure 4a], although it becomesbroader with the peak mode number m = 41 (correspondingto kr0et

�1/2 = 0.5), in contrast to m = 32 in Run 1. The LHDIgrowth rate (Figure 7c) decreases by a factor of two (basedon the comparison of its values averaged over the intervalw0it = 1–10), whereas the saturation amplitude of the LHDwaves is comparable to that in Run 1. Note that the growthrate scaling is consistent with the LHDI theory [Huba et al.,1980], which predicts it to be a fraction of the lower hybridfrequency. According to Figure 7c, the peak LHDI ampli-tude remains large for a long time (w0it > 30). A similareffect for the Harris sheet with the thickness L � r0i andsmall mass ratio (m = 1/400) was reported by Shinohara etal. [2001], who explained it by the interaction between theLHDI and large-scale flapping motions. However, we havenot detected any significant enhancement or even persis-tence of the LHD wave activity associated with the growthof large-scale flapping waves.[35] Although Run 2 is limited to only the initial stage of

the large-scale flapping motion instability (seen as a wavystructure near the neutral plane in Figure 8a), the FFTanalysis of this instability, presented in the lower panelsof Figure 7, allows the determination of a number of keyparameters of the flapping waves. In particular, the averagegrowth rate of the large-scale flapping waves g � 0.03 w0i

in the interval w0it = 50–90 is less than that in Run 1 by thefactor 1.5 (g � 0.046 w0i estimated as an average over theinterval w0it = 70–90, which is different from the maximumvalue g � 0.1 w0i reported by Sitnov et al. [2004a]). Thisreduction is less than one would expect for the drift kinkmode [Daughton, 1999, Figure 14], and it is rather consis-tent with the kinetic theory of the drift-kink instability in thepresence of a background plasma [Daughton, 1999], alsoknown as the ion-ion mode instability, suggesting a weakerdecrease of the growth rate in the presence of the bulk-flowvelocity shear. However, in agreement with Run 1, theproperties of the flapping motions differ drastically fromthose of drift-kink waves. In particular, the propagationspeed of flapping motions (it can be inferred, for instance,from the comparison of the profiles Bx(y, z = 0) for differentmoments t) is estimated as vfl � 0.06 vTi, that is four timesless than the ion drift speed vDi = wDivTi. Their frequencywfl = (2pm/ly) vfl � 0.03 w0i is much less than the classicalkink-mode frequency in this wavelength region (�w0i),although it is more consistent with observations [Sergeevet al., 2003], being closer to them even compared to theresults of Run 1. The resulting flapping motions have amixed parity (Figure 7e) and represent the correlated kink-type motions of two separate current layers. The analysis ofRun 2 suggests that the reduction of the ion anisotropy has arather minor effect on the current sheet dynamics becausethe development of the LHDI sharpens split current layersand decreases their thickness to the approximately same

Figure 8. Run 2: Color contours of (a) the magnetic field Bx, (b) current density Jy, (c) plasma density n,and (d) electric field Ey at the moment w0i t = 150. Here d = c/wpi = b0

�1/2 r0i = 1.21 r0i is the ion inertiallength based on the density n0.


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value (a few ion gyroradii) as in Run 1, independent of thedistance between the layers.[36] Thus the new simulations largely confirm the earlier

results of Run 1. The extension of this run can be usedtherefore to explore the nonlinear effects of flappingmotions, thereby strongly reducing the computationexpenses. In the work of Sitnov et al. [2004a], Run 1 waslimited to w0it = 100. Its extension up to w0it = 150 revealsseveral new effects that are particularly notable in view ofrecent 2001 CLUSTER observations. The main findings aresummarized in Figures 9 and 10. The magnetic field plot(Figure 9a) shows that the flapping motions become ratherlocalized and resemble solitary waves, consistent with therecent CLUSTER observations [Runov et al., 2005a].Moreover, Figure 9b suggests that the flapping motionsmay be associated with strong distortion of the equilibriumbifurcated structure making it asymmetric along the north-south direction. This is explicitly shown in Figure 9d,giving two cuts of the Jy component of the current densityin places marked as S1 and S2 in Figure 9b (the cuts areobtained by averaging the corresponding data Jy(y, z) overthe interval of two ion inertial lengths along the Y-directioncentered at the marked locations). These asymmetric currentdensity profiles are particularly interesting as they resemblethe asymmetric current profiles obtained using CLUSTERdata by Runov et al. [2005a] and shown in their Figure 2c.

Figure 9. Extension of Run 1: Color contours of (a) the magnetic field, (b) current density componentsJy, and (c) Jz as well as (d) one-dimensional (1-D) profiles of Jy (z) marked as S1 and S2 in Figure 9b at themoment w0it = 150. d = 1.3 r0i.

Figure 10. Extension of Run 1: 2-D circulation flow plotof the current density vector J(y, z) = (Jy, Jz) for bifurcatedsheet at the moment w0it = 150.


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[37] Another interesting feature of the current densitycomponent Jy shown in Figure 9b is its patchy structure.For example, the lower current layer in Figure 9b effectivelydisappears at y = 20d, whereas the upper current layerbecomes enhanced at this value of the Y-coordinate. Thecause of this patchiness can be grasped from Figure 9c,which gives the distribution of the Jz component. It reveals,in particular, two pairs of localized vertical currents con-necting the top and bottom parts of the bifurcated currentsystem. The complete 2-D picture of these current diver-sions and mergings of the bifurcated current layers is givenby the vector plot of the current density in Figure 10. Itclearly shows that the patches in Figure 9b appear when asignificant part of one of the initially split currents divertsand joins another major (top or bottom) current layer. Theselocalized current undulations, different for each of thebifurcated currents, seem to be the only plausible explana-tion of the very unusual quasi-rectangular shape of theflapping waves reported by Runov et al. [2005a]. It remainsunclear, however, if the current bifurcation necessary forthese rectangular structures is the feature of the initialequilibrium as in our simulations or it appears as a newquasi-steady state of the current sheet resulting from itsinstabilities. Note here that Asano et al. [2005] found thebifurcation effect only for 17% of the stable current sheets. Atthe same time, the possibility of the formation of new quasi-steady states of the current sheet was demonstrated in theearlier stage of the same run [Sitnov et al., 2004a] and is nowconfirmed by Run 2. The new bifurcated sheet with enhancedoff-center current density peaks was created at the nonlinearstage of the lower-hybrid drift instability, presumably be-cause of the electron anisotropy generated by that instability.

A similar effect of the current bifurcation induced by theanisotropic heating of electrons in the process of the lower-hybrid drift instability in the Harris sheet has been recentlyreported by Daughton et al. [2004]. Earlier, Greco et al.[2002] showed a turbulence-induced bifurcation effect evenin the presence of the finite Bz component, although theirstudies were not self-consistent, as they used a prescribedfluctuating magnetic field localized at the neutral plane.[38] One of the most interesting features of flapping

motions based on the considered equilibria compared tosmall-amplitude kink waves in Harris sheets [Daughton,1999] is their unusually small propagation speed. Indeed,the propagation speed of flapping waves vfl in our simu-lations is a small fraction of the Harris bulk-flow speed vDiand an even smaller fraction of the ion thermal speed vTi.Such a small propagation speed (vfl � 60–100 km/s for5 keV protons) is fully consistent nevertheless with theaverage propagation speed of flapping motions �57–145 km/s [Sergeev et al., 2004] and their period around3 min. It also explains the unusual propagation direction offlapping motions, which move duskward on the duskside ofthe tail and dawnward on its dawnside [Sergeev et al., 2004;Runov et al., 2005a]. Apart from the small speed of flappingmotions, this explanation invokes the significant convectionvelocity vc in the magnetotail, which is directed toward theflanks and is comparable in the absolute value with theion drift speed in the Harris model so that the averageY-component of the net ion bulk velocity on the dawnside ofthe tail is close to zero [Angelopoulos et al., 1993; Hori etal., 2000; Kaufmann et al., 2001]. The convection processshifts the system of reference as compared to the one used insimulations. Therefore the propagation velocity of flapping

Figure 11. Effects of ion and electron anisotropy and the background plasmas for super-thin currentsheets. Left panels show magnetic field, electrostatic potential, plasma, and current density profiles withthe parameters m = 1/64, t = 2/3, hi = 1.2, he = 1.0, wDi = 2 and b0 = 1.21. Dashed lines show thecorresponding Harris profiles. The potential is shown in the presence of a background plasma with theparameters eb = 0.01, tbi = 1, and tbe = 2/3 (solid line) and without the background (dash-dotted line).Electron and ion currents are marked by dash triple-dotted and dash-dotted lines, respectively. Centralpanels show the effect of the background in case when the parameter wDe given by the formula (9) isadditionally increased by the factor 2. Right panels show similar profiles for m = 1/64, t = 2/3, hi = 1, he =1.4, wDi = 2, b0 = 0.48 and zero background density.


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motions in the magnetotail system of reference vfl(m) can be

approximately described by the expression vfl(m) = vfl + jvcj

sgn (y). Since jvcj �vDi � vfl, one can expect vfl(m) > 0 on the

duskside of the tail and vfl(m) < 0 on its dawnside.

4. Current Sheet Modeling Relevant to 2003CLUSTER Period and MRX Experiment

4.1. Equlibrium Theory Results

[39] The properties of the non-Harris sheets in the presentmodel change drastically when their thickness becomes lessthan �2 r0i, corresponding to wDi > �0.5 (we will termthem ‘‘super-thin current sheets’’). At such small scales thecurrent sheet structure cannot depend on the features of thequasi-adiabatic ion orbits. At the same time, significantstructure changes can still be provided by the electronanisotropy. Furthermore, the difference between the motionsof ions and electrons can give rise to significant electrostaticeffects. Indeed, the left panels in Figure 11 show that thesame ion anisotropy hi = 1.2, which provided the currentbifurcation of thicker sheets (Figure 1), now results in onlya slight change of the magnetic field and plasma densityprofiles, as compared to the Harris model. At the same time,the second left panel in Figure 11 reveals a very strongbuildup of the electrostatic potential difference across thesheet. It becomes an order of magnitude larger as compared

Figure 12. The eigenvalue parameter b0 and the currentsheet thickness in units of the ion inertial length as functionsof the parameter vDi/vT?i for different ion anisotropyparameters hi.

Figure 13. Run 3: Color contours of (a) the magnetic field, (b) current density component Jy, and(c) electric field component Ey at the moment w0it = 2.4, as well as (d) 1-D profiles of Jy (z) obtainedby averaging the 2-D picture shown in the panel (b) and similar data at the moment w0it = 0 over theY-coordinate.


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to the case wDi = 0.125 [see Sitnov et al., 2003, Figure 4]and approaches the ion thermal potential T?i/e, consistentwith the results of another non-Harris sheet model [Birn etal., 2004]. As a consequence, the eigenvalue parameter b0(Figure 12, top panel) strongly differs from its Harris valueb0(H) = 1/(1 + t). Moreover, as one can see from Figure 12,this difference builds up rapidly with the increase of theparameter wDi (faster than any power function of thisparameter). An interesting consequence of this buildup isthat the current sheet thickness in the units of the parameter c/wpi does not decrease anymore with the growth of wDi, butinstead begins to grow again after reaching some minimumvalue Lmin (Figure 12, bottom panel). It is also interestingthat this minimum value rather weakly depends on the ionanisotropy and is of the order of unity within a significantrange of anisotropy values. This effect may be an explanationof the puzzling non-Harris scaling L � 0.35 c/wpi found inthe MRX experiment [Yamada et al., 2000].[40] The negative charging, which replaces the current

bifurcation for super-thin ion anisotropy-based currentsheets, can be explained by the excess of the ion figure-of-eight orbits and the corresponding ion redistribution offthe neutral plane as compared to the Harris case. Thepotential well created by such a negatively charged currentsheet can be much wider than the current sheet itself (see thedash-dotted line in the second left panel in Figure 11).However, the presence of even a very small amount of

background plasma shields the electrostatic field outside thesheet (solid line in the second left panel in Figure 11),although it does not affect other current sheet profiles. It isinteresting to note here that though the effect of negativecharging is expected to dominate 2003 CLUSTER obser-vations, having the necessary spatial resolution, it was firstdiscovered in super-thin (�100 km) current sheets using2001 CLUSTER data [Wygant et al., 2005] and the effect offlapping motions.[41] The effect described above resembles the negative

charging of thin current sheets found in the full-particle andhybrid simulations of the plasma sheet convection [Pritchettand Coroniti, 1995; Hesse et al., 1996], and just like inthose simulations, it is accompanied by some enhancementof the electron current. Indeed, the electron and ion currentprofiles shown in the left bottom panel of Figure 11 almostcoincide in spite of their expected ratio following from thetemperature ratio t = 2/3 and the Harris model. Moreover,in the present model the Harris-type condition (9) can becompletely waived in the presence of a very small back-ground population, which allows an arbitrary ratio betweenion and electron currents (for Harris current sheets, such aresult was recently reported by Yoon and Lui [2004]). Thiseffect is shown in the central panels of Figure 11, where theelectron current is further enhanced due to the additionalincrease of the parameter wDe given by the formula (9) bythe factor 2.[42] Having much smaller-sized orbits compared to ions,

the electrons are anticipated to provide significant structurechanges even in super-thin current sheets. This is confirmedby the plots in the right panels of Figure 11, which show theeffect of the current bifurcation in the case of the electronanisotropy parameter hi = 1.4. As shown by Daughton et al.[2004], this kind of bifurcation may appear as a result ofanisotropic heating of the electron species in the process ofthe lower-hybrid drift instability.

4.2. Dynamics of Super-Thin Current Sheets

[43] Being closer in structure to the Harris sheets, thesuper-thin sheets with the anisotropic ion species displaystrong resemblance with them in the dynamics too. Theresults of the relevant particle simulations (Run 3) with theparameters hi = 1.2, m = 1/64, t = 2/3, wDi = 2, he = 1, andwpe/w0e = 1.875 (except m and wpe/w0e their specific valuesare taken to be close to the parameters of the MRXexperiment [Yamada et al., 2000]) in the box (ly/d,lz/d) =(9.6, 9.6) with Ny � Nz = (384, 384) grids are shown inFigures 13, 15, and 16. Figure 13 shows the development ofthe lower-hybrid drift instability, for which the electricfield penetrates into the current sheet (Figure 13c) and isnot localized at its edges as is the case for a relativelythick (L > r0i) Harris sheet [Huba et al., 1980]. Also,according to Figures 13b and 13d, the LHDI significantlymodifies the current sheet profile by enhancing the currentdensity at the center of the sheet and at its edges. Theseeffects for the Harris sheet were discussed by Lapenta andBrackbill [2002], Lapenta et al. [2003], Scholer et al. [2003],Daughton et al. [2004], Lui [2004], and Ricci et al. [2004].However, in contrast to most of those Harris sheet-basedresults, which emphasized either the current sheet thinningor its bifurcation, Figure 13d clearly shows the formation ofthree separate current peaks. This can be explained by the

Figure 14. Magnetic field, electrostatic potential, plasma,and current density profiles for the triple-peaked super-thincurrent sheet with the parameters m = 1/64, t = 2/3, hi = 1,he = 1.35, wDi = 2. b0 = 0.58 and the modified electrondistribution based on the same function F(I) as is used inFigure 6 for ions. Background parameters are same as in leftpanels of Figure 11. The electron drift speed parameter wDi

obeys the equation (9).


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faster redistribution of the electrons under the influence ofboth the LHDI and the equilibrium electrostatic potential.Such three-peaked current sheet profiles were indeed foundby Nakamura et al. [2004] in the 2003 tail CLUSTERobservations. They can also be reproduced within theframework of our equilibrium model. As demonstrated inFigure 14, this can be achieved by using the same modifi-cation of the distribution function (now for electrons), whichwas used to reduce the anisotropy outside the relatively thickbifurcated sheet (Figure 6) log f0e � F(Iz

(e)) with F(I) = I + s(1 � gI) exp (�dI2) and the same parameters s = 1, g = 1.8and d = 0.125. The latter three-peaked current sheet isdifferent from the one created by the LHDI, which involvesalso the enhancement of the electron current at the centerof the sheet [see, e.g., Scholer et al., 2003, Figure 4].Nevertheless, it convincingly shows the potential of thepresent equilibrium model and it can be used for compar-ison with observations on the same grounds as oursimulation results.[44] Further evolution of the super-thin sheets is quite

similar to that of the Harris sheets [Daughton, 2002; Lapentaand Brackbill, 2002; Lapenta et al., 2003; Karimabadi etal., 2003; Scholer et al., 2003]. According to Figure 15, it isdominated by the kink mode. This type of the current sheetflapping is well known in the stability theory and simulationsof the classical Harris sheets. Comparison of Figure 15 andparticularly Figure 16 with the corresponding figures fromsection 3 (Figures 9 and 10) shows that the flappingmotions of super-thin non-Harris sheets are drastically

Figure 15. Run 3: Color contours of (a) the magnetic field, (b) current density components Jy, and (c) Jzas well as (d) 1-D profiles of Jy (z) marked as S1 and S2 in Figure 15b at the moment w0it = 4.8.

Figure 16. Run 3: 2-D circulation flow plot of the currentdensity vector J(y, z) = (Jy, Jz) for super-thin current sheet atthe moment w0it = 4.8.


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different from solitary flapping waves of relatively thickbifurcated current sheets, as well as from quasi-rectangularwaves observed by CLUSTER [Runov et al., 2005a].However, this kinking motion still has an interesting fea-ture, common with the bifurcated sheet flapping motions,but distinct from the classical kink waves in Harris sheets[e.g., Lapenta and Brackbill, 2002; Shinohara et al., 2001].As one can see from the comparison of our Figures 15a(red-yellow spots) and 15d with Figure 7 in the work ofLapenta and Brackbill [2002], in contrast to the classicalkink motions, the flapping motions of super-thin sheetsreveal the north-south current asymmetry even though theyare not bifurcated.

5. Conclusion

[45] Important energy transformation processes in mag-netized plasmas, including the onset of magnetic reconnec-tion, occur in current sheets that are critically thin. Recentmultiprobe observations showed that the structure anddynamics of thin sheets with thicknesses comparable tothe thermal ion gyroradius may become very unusual andinconsistent with the dominating theoretical picture basedon the Harris equilibrium [Harris, 1962]. In this paper weprovide a new theoretical picture of thin current sheetstructure and dynamics based on a generalization of theHarris model, which assumes anisotropic and nongyrotropicplasmas and uses the invariant of particle motion in regionsof strong gradients to describe the effects of anisotropy andnongyrotropy.[46] The most interesting results in the equilibrium theory

relevant to 2001 CLUSTER observations are the pressuretensor components of bifurcated current sheets and thefinding that the bifurcation can be obtained from our newequilibrium model with extremely small (�3%) plasmaanisotropy outside the sheet. This also suggests the impor-tant role played by nongyrotropic effects in modifying thestructure of thin current sheets. The ion distributions arequite close to isotropic ones. Yet, changing the resolution invelocity space reveals some characteristic features, such asthe mushroom-like structures in bifurcated sheets near thecurrent maxima.[47] At scales less than the thermal ion gyroradius, typical

for the 2003 CLUSTER and MRX observations, the role ofanisotropy on current sheet equilibria is found to be com-pletely different. The ions now do not cause any bifurcationregardless of their anisotropy, and it may require rather high(�40%) anisotropy of the electron species. At the sametime, small anisotropy of the ion species is found to create alarge potential difference across the sheet and thus limit itsminimum thickness.[48] The dynamics of thin non-Harris current sheets are

also quite distinct from the Harris case. Flapping motions ofCLUSTER 2001-type bifurcated current sheets differ fromconventional drift-kink waves due to the north-south asym-metry of the current profiles and the solitary structure offlapping waves, with the resulting flapping geometry beingsimilar to quasi-rectangular waves revealed in observations.Their frequency is one and a half orders of magnitude lessthan the ion gyrofrequency in the field outside the sheet,while their phase speed is several times less than the driftspeed of ions. Taking into account the absolute values and

directions of the convective plasma motions in the tail, thissmall speed allows explanation of the unexpected propaga-tion of flapping motions found in CLUSTER observationsand directed toward the flanks of the tail current sheet.[49] The relevance of these results to 2001 CLUSTER

observations is confirmed by Run 2 with the realistic ionanisotropy and lower electron-ion mass ratio. The bifurca-tion effect is shown to persist in spite of the development ofthe LHDI and large-scale flapping instabilities. The impactof the ion anisotropy reduction on the dynamics of bifur-cated sheets is small. The fourfold decrease of the mass ratiodecreases the growth rates of the LHDI and the large-scaleflapping instability by the factors 2 and 1.5, respectively,consistent with the earlier theoretical estimates. At the sametime, the frequency and propagation speed of flappingwaves are consistent with previous simulations as well aswith observations, although they are much less than thecorresponding parameters of the drift-kink mode in thinHarris sheets.[50] Particle simulations of the non-Harris sheets thinner

than the ion gyroradius complement recent studies ofsimilar Harris equilibria, where non-Harris featuresappeared as nonlinear dynamical effects. In particular, aslight pancake ion anisotropy enhances the nonlinear effectsof the lower-hybrid drift instability and results in theformation of three-peaked current sheets, which are foundin 2003 CLUSTER observations and successfully repro-duced in the new equilibrium theory. Flapping motionsfollowing the LHDI stage also reveal the north-southasymmetry of the current sheet profile, although they haveno rectangular wave signatures.[51] Thus the new class of the current sheet equilibria,

generalizing the Harris model, proved to be very useful inreproducing both structure and dynamical features of thincurrent sheets that were recently revealed in space andlaboratory as well as in predicting interesting new effects.

[52] Acknowledgments. The authors gratefully acknowledge usefuldiscussions with J. Drake, B. Rogers, V. Sergeev, R. Nakamura, Y. Asano,W. Baumjohann, L. Kistler, A. Lui, P. Pritchett, N. Tsyganenko, H. Ji,M. Yamada, and R. Kulsrud. This work was supported by NASA grantNAG513047 and NSF/DOE grant ATM0317253. Computations werecarried out at the National Energy Research Scientific Computing Centerat the Lawrence Berkeley National Laboratory.[53] Amitava Bhattacharjee thanks the reviewers for their assistance in

evaluating this paper.

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��P. N. Guzdar and M. I. Sitnov, Institute for Research in Electronics and

Applied Physics, University of Maryland, College Park, MD 20742, USA.([email protected])A. Runov, Space Research Institute, Austrian Academy of Sciences,

A-8042 Graz, Austria.M. Swisdak, Icarus Research Inc., P. O. Box 30780, Bethesda, MD

20824-0780, USA.


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