Current-driven instabilities in forced current sheetsterpconnect.umd.edu/~sitnov/testpage_files/recpubl/2003...Current-driven instabilities in forced current sheets M. I. Sitnov,1

Current-driven instabilities in forced current sheets

M. I. Sitnov,1 A. T. Y. Lui,2 P. N. Guzdar,1 and P. H. Yoon3

Received 8 July 2003; revised 4 December 2003; accepted 31 December 2003; published 11 March 2004.

[1] The nonlocal kinetic linear stability analysis of the non-Harris thin current sheetequilibrium, namely the thin current sheet embedded in a thicker anisotropic plasma sheet[Sitnov et al., 2000a, 2000b], with respect to current-driven instabilities is performed usingthe finite element technique. In contrast to the Harris sheet, the new equilibrium becomespossible due to the plasma anisotropy outside the sheet caused by two warmcounterstreaming field-aligned beams and complex ion orbits that cannot be described insuch thin current sheets in terms of the conventional magnetic moment. It is found that incontrast to the case of the Harris sheet, the analogs of the drift-kink instability in thesecurrent sheets can have significant growth rates for the realistic ion-to-electron mass andtemperature ratios. The unstable modes share the properties with both the lower-hybridand drift-kink modes. In particular, the unstable modes resemble the lower-hybrid driftmodes as they are more highly structured across the sheet than the drift-kink instability(DKI) and assume both odd (DKI-like) and even parity solutions. On the other hand, incontrast to the lower-hybrid drift instability (LHDI) and like the DKI, the unstable modeshave much larger wavelength, electromagnetic component, and significantly perturb thecentral current region. The possible role of the current-driven instabilities in magneticreconnection and magnetic annihilation as well as the geophysical implications such as thecurrent disruption in the geomagnetotail during substorms are also discussed. INDEX

TERMS: 2772 Magnetospheric Physics: Plasma waves and instabilities; 2744 Magnetospheric Physics:

Magnetotail; 3230 Mathematical Geophysics: Numerical solutions; 2788 Magnetospheric Physics: Storms and

substorms; 7827 Space Plasma Physics: Kinetic and MHD theory; KEYWORDS: current-driven instability, thin

current sheet, reconnection onset, magnetic annihilation, current disruption

Citation: Sitnov, M. I., A. T. Y. Liu, P. N. Guzdar, and P. H. Yoon (2004), Current-driven instabilities in forced current sheets,

J. Geophys. Res., 109, A03205, doi:10.1029/2003JA010123.

1. Introduction

[2] Current sheets are the key structure elements of manymagnetized plasma formations. They maintain oppositelydirected magnetic fields created by the dynamo process.Their filamentation or disruption provide the reverse pro-cesses, which transform the magnetic field energy into theparticle energy and are known as magnetic reconnection andmagnetic annihilation. In collisional plasmas these processesare controlled by plasma resistivity. However, in hightemperature plasmas typical for fusion devices, Earth’smagnetosphere, and solar corona collisions are negligible.They cannot explain energy transformation processes and inparticular their characteristic time scales [e.g., Biskamp,2000]. The corresponding collisionless mechanisms involveexcitation of plasma waves, wave-particle interaction, andplasma turbulence. This is why the stability problem of the

current sheets in collisionless plasmas is crucial for deter-mining the onset conditions and time scales of magneticreconnection and magnetic annihilation. In particular thestability of the current sheet in the tail of Earth’s magneto-sphere determines the onset of magnetic reconnection andcurrent disruption phenomena, which are the main mecha-nisms of magnetospheric substorms [Baker et al., 1996; Luiet al., 1992; Lui, 1996]. Instabilities also play an importantrole in the magnetic reconnection experiments involvingweakly collisional plasmas [Carter et al., 2002; H. Ji et al.,Electromagnetic fluctuations during fast reconnection in alaboratory plasma, submitted to Physical Review Letters,2004, hereinafter referred to as Ji et al., submitted manu-script, 2004].[3] However, in spite of the long research efforts, the

main unstable modes of the current sheets responsible forthe onset of the explosive energy release remain poorlyunderstood. One of the most impressive examples is themechanism of the substorm onset in the Earth’s magnetotail,the main reservoir where the magnetic field energy isaccumulated and then suddenly released during substorms.Originally, this explosive release of energy was explainedby the onset of the reconnection with the X-line pattern dueto the collisionless tearing mode [Laval et al., 1966; Coppiet al., 1966]. However, as was shown later, the presence ofeven a very small component Bn of the magnetic field

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109, A03205, doi:10.1029/2003JA010123, 2004

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

2Applied Physics Laboratory, Johns Hopkins University, Laurel,Maryland, USA.

3Institute for Physical Science and Technology, University of Maryland,College Park, Maryland, USA.

Copyright 2004 by the American Geophysical Union.0148-0227/04/2003JA010123$09.00

A03205 1 of 15

normal to the sheet plane, which magnetizes plasma elec-trons, results in strong stabilization of the tearing mode[Lembege and Pellat, 1982; Pellat et al., 1991]. Furtherstudies [Sitnov et al., 1998, 2002] clarified that the onset ofthe X-line reconnection via tearing modes becomes possiblewhen the tail current sheet is long enough to allow forkinetic response of the electron species resulting fromdifferent motions of trapped and passing particles. In fact,recent Geotail measurements showed that the formation ofthe near-Earth neutral line and plasmoids during substormsusually starts in the premidnight sector of the magnetotailbetween XGSM = �20RE and XGSM = �30RE [Nagai et al.,1998; Ieda et al., 1998]. Closer to the Earth, the tail currentsheet is tearing-stable and changes of the magnetic topologyare not favorable energetically. As a result, the perturbationsdue either to the immediate solar wind trigger or to quasi-static changes in the process of magnetospheric convectionshould result in the formation of MHD discontinuities ratherthan the change of magnetic topology [Syrovatskii, 1971;Kulsrud and Hahm, 1982; Schindler and Birn, 1993]. Themore detailed MHD [Birn and Schindler, 2002] and kinetic[Pritchett and Coroniti, 1994, 1995; Hesse et al., 1996;Lottermozer et al., 1998] modeling shows that these dis-continuities are represented by thin current sheets (TCS)with thickness comparable to the thermal ion gyroradiusbased on the field outside the sheet.[4] At the early stage of its formation the TCS may be

dominated by the electron current because the differencebetween the electron and ion response to the convectionfields leads to a negative charging of the central plasmasheet and the corresponding additional drift of both elec-trons and ions in the direction of the original electron drift inthe Harris-type sheet [Pritchett and Coroniti, 1994]. Later,owing to the penetration of the convection field inside thesheet, the unmagnetized ions can be accelerated and candominate the current [Burkhart et al., 1992; Pritchett andCoroniti, 1992; Holland and Chen, 1993; Kropotkin et al.,1997; Sitnov et al., 2000a, 2000b]. The current is generatedby the quasi-adiabatic motion of ions that form the counter-streaming flows outside the TCS [Speiser, 1965]. AfterBurkhart et al. [1992], these TCSs are called forced currentsheets. They may serve as the outflow regions of the X-linereconnection pattern [Hill, 1975; Lottermozer et al., 1998].Note however that the penetration of the convection electricfield inside the TCS is not necessarily related to theformation of the X-line. More correctly, these two processesshould be considered as different manifestations of the sameglobal process of the energy transformation in the tail-likesystems.[5] Concentration of the current density in the TCS forms

a free energy source for current-driven instabilities, whichdo not change the initial magnetic field topology as theypropagate in the dawn-dusk direction. The most extensivelystudied instability of this class is the lower-hybrid driftinstability (LHDI) [Krall and Liewer, 1971; Davidson et al.,1977; Huba et al., 1977, 1980]. However, the classicalLHDI is most unstable at the edges of the current sheet andit does not affect significantly the central region. Later, Luiet al. [1991, 1995] and Yoon and Lui [1996] found anotherclass of instabilities, cross-field current instabilities (CFCI),driven by different dynamics of unmagnetized ions andmagnetized electrons at the center of the current sheet. At

the same time, simulations [Zhu et al., 1992; Ozaki et al.,1996; Pritchett and Coroniti, 1996; Zhu and Winglee, 1996]revealed the drift-kink instability (DKI), which stronglydistorted the main equilibrium current like the CFCI modes.Lapenta and Brackbill [1997], Yoon et al. [1998], andBuchner and Kuska [1999] also found the drift-sausageinstability (DSI), which had the opposite parity compared tothe DKI (even profile of the dawn-dusk component of theelectromagnetic potential in the case of the Coulombgauge). However, Daughton [1998, 1999] performed thenonlocal kinetic linear stability analysis of DKI, DSI, andLHDI modes and reported no evidence of the DSI. Moreimportantly, this analysis and later particle simulations[Hesse and Birn, 2000; Pritchett and Coroniti, 2001]revealed a strong decrease of the DKI growth rate withthe increase of the mass ratio mi/me, making that mostpromising mode irrelevant for the case with realistic mi/me =1836. We should note that recently Yoon et al. [2002]performed a nonlocal stability analysis of the Harris currentsheet for current-driven instability based on the two-fluidformalism and found an entire class of unstable eigenmodesof either parity. The ground-state solutions of this classresemble the DKI and DSI modes, while the higher-ordersolutions increasingly behave as the LHDI as the orderincreases. The latter finding has been partly confirmed bythe kinetic stability analysis of Daughton [2003]. Heshowed in particular that in a very thin Harris TCS withthe thickness L < r0i (r0i is the thermal ion gyroradius in thefield B0 outside the sheet), the modes of both parities similarto the classical LHDI but having larger wavelength(ky

ffiffiffiffiffiffiffiffiffiffiffir0ir0ep �1) may still have the significant growth rates

and larger electromagnetic components, which are localizedin the central region of the sheet.[6] Meanwhile, Yoon and Lui [1996] and Yoon et al.

[1996] have shown that the current-driven instabilities maysurvive in the case of the realistic mass ratio due to the bulkflow velocity shear in the initial TCS equilibrium. Further-more, recent simulations [Hesse et al., 1998; Horiuchi andSato, 1999; Shinohara et al., 2001; Lapenta and Brackbill,2002; Daughton, 2002] revealed the formation of the TCSprofiles with the bulk flow velocity shear as a nonlineareffect of the LHDI with further excitation of the DKI andKelvin-Helmholtz instabilities (KHI) as a consequence ofthat shear flow effect [Yoon et al., 1996]. The problemhowever is that the LHDI should be strongly suppressed bythe normal magnetic field Bn. According to Pritchett andCoroniti [2001], in the case of the LHDI the Bn field resultsin an effective component of the wave vector parallel to thelocal magnetic field, which strongly stabilizes the LHDI[Gladd, 1976]. Indeed, Pritchett and Coroniti [2001] andPritchett [2002] found no signatures of the LHDI even formoderate mass ratio in the presence of a very weak normalcomponent of the equilibrium magnetic field. As a result,both linear and nonlinear stages of the TCS evolution revealno signatures of DKI, DSI, or KHI. Therefore it still remainsunclear whether the large-scale instabilities such as the DKIor KHI can actually grow in in the Harris TCS with Bn 6¼ 0for realistic mass and temperature ratios. Note here that theeffective shear of the bulk flow speed can be provided in thecurrent sheet by the background plasma, and it may be acause of the high DKI growth rate in the case of realisticmass ratio [Daughton, 1999; Karimabadi et al., 2003a,

A03205 SITNOV ET AL.: CURRENT-DRIVEN INSTABILITIES

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2003b]. However, the models with a uniform backgroundplasma are known to have an additional free energy sourcefor the ion-ion instability because the ion distribution istwo-humped in the velocity space in the direction of thecurrent. As a result, the assumption of a uniform back-ground density is sometimes considered as artificial, the oneto be eliminated [e.g., Shinohara et al., 2001].[7] Thus the absence of robustly growing modes in thin

Harris sheet equilibria is in contradiction to the growingobservational evidence of wave activity during substorms,earthward of the forming X-line. In addition to the originalAMPTE/CCE observations of the current disruption pertur-bations at radial distance r � 8RE [Takahashi et al., 1987;Lui et al., 1988, 1992; Lopez et al., 1989; Ohtani et al.,1995, 1998], Bauer et al. [1995] statistically investigatedlow-frequency waves in the region �19RE < XGSM < �9RE

using AMPTE/IRM data. They found in particular that themagnetic field fluctuations increase with increasing flowvelocity in the plasma sheet. With the use of recent Geotailobservations Fairfield et al. [1998, 1999], Sigsbee et al.[2002] and Shiokawa et al. [2002] reported strong waveactivity between �10 and �15RE during substorms. Also,recent Cluster measurements [Sergeev et al., 2003; Volwerket al., 2003] revealed modes corresponding to the DKI andpossibly even the DSI [Runov et al., 2003] at �19RE duringsubstorms. For the theoretical interpretation of these obser-vations it is necessary to explore a mechanism of theexcitation of the instabilities similar to the DKI and KHI,which do not involve the preliminary excitation of theLHDI.[8] In this paper we show that the modes similar to the

DKI and KHI, which have larger wavelength and electro-magnetic component than the lower-hybrid modes andsignificantly perturb the central current sheet region, mayhave large enough growth rates even for the realistic massratio in a non-Harris thin current sheet embedded in athicker anisotropic plasma sheet [Sitnov et al., 2000a,2000b], also known as the forced current sheet (hereinafterreferred to as FCS) [Burkhart et al., 1992]. The FCS hasbulk flow velocity shear, which is a natural consequence ofthe quasi-adiabatic ion motion and plasma anisotropy out-side the sheet. Moreover, this shear is not related to anyadditional background plasma, and therefore the system isfree from the ion-ion instabilities. According to the currentdisruption model of substorms [Lui, 1996], the currentdisruption starts when the near-Earth current sheet thinsenough to make ions unmagnetized with respect to the fieldBn and accelerated by the dawn-dusk convection electricfield which penetrates the sheet. This is exactly how theFCS is formed [e.g., Hill, 1975; Lottermoser et al., 1998;Nakamura et al., 1998]. The FCS represents therefore thenatural model for a considerable region of the tail currentsheet during active periods. It is shown to be unstable withrespect to current-driven instabilities with the frequencyaround the ion gyrofrequency outside the sheet for a widerange of wave numbers.[9] The FCS equilibrium is fully self-consistent for the

case of zero normal magnetic field just like the popularHarris equilibrium [Harris, 1962]. As shown by Sitnov et al.[2003a], these two types of current sheet equilibria aredifferent limiting cases of a more general class of one-dimensional (1-D) equilibrium models. In contrast to the

Harris model, the FCS equilibrium becomes possible due tothe plasma anisotropy and characteristics of the ion orbits inthin current sheets that cannot be described in terms of theconventional magnetic moment (although the orbits remainadiabatic and fully integrable). This is why the stabilityanalysis of the FCS equilibrium for the case Bn = 0 is asvalid and self-consistent as a similar analysis of the Harrissheet [e.g., Daughton, 1998, 1999, 2003].[10] Like practically all other linear kinetic stability

analyses of the current-driven instabilities [Lapenta andBrackbill, 1997; Daughton, 1998, 1999, 2003; Silin et al.,2002], the present study is limited to the case of zero normalmagnetic field Bn = 0. It is believed nevertheless to clarifythe important aspects of the stability picture in the case of asmall nonzero Bn for the following reasons. First, like theHarris equilibrium, which locally keeps its original form inthe presence of a small finite normal component of themagnetic field Bn [e.g., Schindler, 1972], the FCS equilib-rium is not changed significantly in the latter case. Asargued below, in the case of a small finite Bn the FCSmodel describes the sheets where the magnetic tension isbalanced by the ion inertia of the quasi-adiabatic ions.However, this condition for small Bn can be reduced tothe form, which is independent of Bn. Similarly, the mainnew ingredient of the FCS model, namely the invariant ofthe quasi-adiabatic motion [e.g., Sonnerup, 1971] remainsapproximately constant on the time scales considered.[11] The question, whether one can neglect the influence

of the Bn component in the stability analysis of the current-driven instabilities, is still an open question. The 3-Dparticle simulations [Pritchett and Coroniti, 2001] suggestthat the influence of the Bn field is drastically different forthe LHDI-like and DKI-like modes. While the lower-hybriddrift modes are stabilized due to the aforementioned Gladd[1976] mechanism, this is not the case for the DKI modes,as they are much less structured along the normal to thesheet plane. We show below that though the unstable FCSmodes share some properties with the LHDI, they are muchless structured and thus may avoid the stabilization.[12] The structure of the paper is as follows. In section 2

we describe the analog of the Harris [1962] self-consistentmodel for FCS class of TCSs as well as some reduceddescriptions of the model useful for the subsequent numer-ical stability analysis. The basic system of the linearizedVlasov-Maxwell equations for the considered class ofcurrent-driven instabilities is derived in section 3. Thesolution of the linearized Vlasov equation resulting in theintegrodifferential system of equations for perturbed elec-trostatic and electromagnetic potentials is given in section 4.The results of the numerical analysis of this system usingthe finite element approach are presented in section 5. Theseresults are discussed and summarized in section 6.

2. Self-Consistent Model of the ForcedCurrent Sheet

[13] For many years the Harris model of current sheets[Harris, 1962] and its modification for the case of a nonzeronormal component Bn of the equilibrium magnetic field[Schindler, 1972] represented the only class of the self-consistent models used in applications to the magnetosphericcurrent sheets. In Harris-type models with finite Bn the


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plasma is isotropic and the magnetic field line tension isbalanced by the pressure gradient along the tail axis. Thereis, however, another way to balance the magnetic field linetension, namely, by the inertia of the counter-streamingion flows penetrating the current sheet [Hill, 1975] andproviding the current due to their quasi-adiabatic motion[Schindler, 1965; Sonnerup, 1971]. The correspondingforce balance condition [e.g., Burkhart et al., 1992] canbe obtained by integrating the force balance equation overthe current sheet thickness

B0Bn=4p ¼ mi

Zvxvzfi vð Þd3v; ð1Þ

where B0 is the x-component of the magnetic field outsidethe sheet in the GSM coordinate system. Integration overthe velocity space is made outside the sheet, and forsimplicity we neglect the effects of the stochastic scatteringinvestigated by Burkhart et al. [1992]. In spite of the factthat the finite value of the normal magnetic field Bn iscrucial for the force balance (equation (1)), as this fieldprovides both the magnetic tension and the maindynamical features of the Speiser ions balancing thattension, the parameter Bn formally disappears from thesubsequent approximate equilibrium theory. In particular,assuming the ion gyrotropy outside the sheet, one canconvert intergration in equation (1) to the pitch angle andgyrophase coordinates (vx, vy, vz) ! (f, v?, vk) with f (vx,vy, vz) ! (2p)�1 f (v?, vk), d

3v ! v?dv?dvkdf, vx = vkcos q0 � v? cos f sin q0, vz = vk sin q0 + v? cos f cos q0,and tan q0 = Bn/B0, and after integrating over thegyrophase f, equation (1) can be written as

B0Bn=4p ¼ mi sin q0 cos q0

Z 1

0

v?dv?

�Z 1

�1v2k � v2?=2� �

fi v?; vk� �

dvk: ð2Þ

In the limit Bn/B0 1 it takes the form

B20=4p ¼ 2mi

Z 1

0

v?dv?

Z 1

0

v2k � v2?=2� �

fi v?; vk� �

dvk; ð3Þ

where Bn is absent. It is also known as the so-calledmarginal firehose condition [Rich et al., 1972; Hill, 1975].In the case of the shifted Maxwellian distribution f � exp[�v?

2/vTi2 � (vk � vD)

2/vTi2], where vTi and vD are the

thermal velocity of ions and the bulk speed of theircounter-streaming flows, respectively, this balance equa-tion takes the form [Burkhart et al., 1992]

vA

vD¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

exp �d2� �

dffiffiffip

p1þ erf dð Þ½ �

s; ð4Þ

where d = vD/vTi is the measure of the ion anisotropy outsidethe sheet and erf is the ‘‘error function’’ [Abramowitz andStegun, 1972]. In the limit of strong anisotropy d� 1, whichcan also be considered as the limit of cold ions, equation (4)further transforms into the relation vA = vD, known as theWalen relation [Walen, 1944].[14] The self-consistent kinetic theory of the FCS equi-

libria [Sitnov et al., 2000a, 2000b] has been built on the

earlier contributions in this direction [Eastwood, 1972,1974; Rich et al., 1972; Hill, 1975; Francfort and Pellat,1976; Chen et al., 1990; Pritchett and Coroniti, 1992;Burkhart et al., 1992; Holland and Chen, 1993; Ashour-Abdalla et al., 1994; Kropotkin et al., 1997]. It is based on anew set of integrals of motion, namely, the total particleenergy W = miv

2/2 + ef (f is the electrostatic potential),which is also used in the Harris-type models, and the sheetinvariant Iz of the quasi-adiabatic motion [Schindler, 1965;Sonnerup, 1971; Whipple et al., 1990]

Iz ¼1

2p

Imvzdz; ð5Þ

which replaces the canonical momentum Py = mvy + (q/c) Ayused in the Harris model. Then the ion distribution,representing two counterstreaming Maxwellian beams out-side the sheet, can be described in the form

f0i I < w2 þ j� �

¼ n0

p3=2v3Ti

�exp �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw2 þ j� I

p� d2=3

� �2þI

�d2=3

� 1þ erf dð Þ ;

ð6Þ

Rwhere w = d2/3v/vD, j = 2ed4/3f0/mivD2, I = d�2/3Izw0/mvTi

2.It must be complemented by the distribution of ions that aretrapped inside the sheet. Its specific form

f0i I > w2 þ j� �

¼ n0

p3=2v3Ti

exp � d4=3 þ w2 þ jh i

d2=3n o

1þ erf dð Þ ð7Þ

satisfies in particular the condition of continuous transitionfrom trapped to transient distribution, which is necessary forself-consistency [Sitnov et al., 2000a]. The distinctivefeatures of the ion distribution (equations (6) and (7)) canbe grasped from Figure 1. It shows in particular how thedistribution of two field-aligned counterstreaming ionbeams outside the sheet transforms into a bean-shapeddistribution inside the sheet, which is asymmetric along they direction. It is this asymmetry in phase space that becomesthe mechanism of the current generation in forced currentsheets.[15] The electron distribution function can be taken as a

Maxwellian distribution over the particle energy with zerobulk flow speed [Sitnov et al., 2000b]

f0e ¼n0

p3=2v3Teexp � v2

v2Teþ ef0

Te

� �: ð8Þ

After substituting the distributions in equations (6)–(8) intothe system of Maxwell equations, it can be reduced to twoequations, namely, the generalized (nonlocal) Grad-Shafra-nov equation and the quasi-neutrality condition. Theirsolution gives us the self-consistent profiles of the magneticfield B(z) and the electrostatic potential f0(z). Moreover, intwo limiting cases of weak (d 1) and strong (d � 1) ion


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anisotropy outside the sheet the magnetic field profile canbe presented in the form

B zð Þ ¼ B0b z=Lð Þ; ð9Þ

where in the case of weak anisotropy the thickness L is ofthe order of the thermal ion gyroradius in the field outsidethe sheet L � r0i, while in the case of strong anisotropy L �d�1/3r0i, and the specific profiles b(z) are universal in thatsame sense as the profile tanh (z/L) in the case of the Harris[1962] equilibrium (for details see [Sitnov et al., 2000a,2000b]).[16] The use of the FCS equilibrium model in the stability

analysis requires an additional interpolation procedure tospeed up the process of the orbit integration. This is done inAppendix A using the scaling relation (equation (9)) and thesimilar relation for the electrostatic potential found in thework of Sitnov et al. [2000b] for the region of stronganisotropy, which is considered in detail in the followingstability analysis. In particular, the magnetic field is ap-proximated by the formula

b zð Þ ¼ tanhn mz1=n� �

; ð10Þ

while for the electrostatic potential we use the approximation

f0 zð Þ ¼ Tee�1sj0 exp �j1 zs�1=2

� �j2h i

; ð11Þ

where m,n, andji are constants ands=1� (4/3j0) log (10/d).The results of the interpolation (equations (10) and (11)) areshown in Figures 2 and 3. Figure 2 shows in particular thedifference between the universal profile (equation (9)),found as the solution of the Grad-Shafranov equations in thelimit d � 1, and its approximation (equation (10)) with theoptimized parameters m and n. Similar automodel profiles ofthe electrostatic potentials and their approximations based on

equation (11) are given in Figure 3 for different values of theparameter d.[17] The disappearance of the normal component of

the magnetic field from the force balance condition(equation (2)) suggests that the FCS equilibrium will existin the case of zero normal magnetic field just like thepopular Harris equilibrium. This suggestion was recentlyconfirmed by the theory of 1-D current sheets [Sitnov et al.,2003a], in which these two types of current sheet equilibriaappear as different limiting cases of a more general class of1-D equilibrium models. In contrast to the Harris model, theFCS equilibrium becomes possible due to the plasmaanisotropy outside the sheet and characteristics of the ion

Figure 1. Ion distributions given by equations (6) and (7) outside the sheet (R z0B(z0) dz0 = B0r0id

1/3, leftpanel) and at its center (z = 0, right panel) for d = 3. The distributions are color coded on the plane (vx/vD,vy/vD) and normalized by their maximum values.

Figure 2. Universal magnetic field profile b(z) with z = z/L(equation (9)) for the case of strong anisotropy d � 1 (solidline) and its interpolation (equation (10)) with the parametersm = 0.517 and n = 0.897 (dashed line). The inset shows thedifference between the profiles.


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orbits in thin current sheets that cannot be described interms of the conventional magnetic moment, althoughthe orbits remain adiabatic and fully integrable in the caseBn = 0. This is why the stability analysis of the FCSequilibrium for the latter case is as valid and self-consistentas a similar analysis of the Harris sheet. The FCS equilib-rium is an exact equilibrium in the case of zero normalmagnetic field in particular because the invariant (equation(5)) is exact in the latter case (for details see, for instance,Kropotkin et al. [1997] and references therein). In the case ofnonzero Bn the invariant is only approximate. It is conservednevertheless much better than many other parameters,including the conventional magnetic moment [e.g., Whippleet al., 1990]. Nonadiabatic changes of Iz occur during thetransition from one type of the fast motion, namely theconventional Larmor rotation with the particle orbit notcrossing the z = 0 plane, to a figure-eight orbit crossingthe z = 0 plane. Changes of Iz during such transitions can bedescribed as a weak diffusion and they were studied in manypapers [Chen and Palmadesso, 1986; Buchner and Zelenyi,1989; Chen, 1992; Kropotkin et al., 1997]. Here it should benoted that the magnitude of nonadiabatic changes of Izduring the transitions between crossing and noncrossingorbits decreases with the decrease of the parameter Bn/B0.Also the characteristic period between these nonadiabaticjumps is of the order of the ion gyroperiod Wn in the field Bn.This is why the above changes of Iz can be neglected in thestability analysis as long as the characteristic frequenciesand growth rates are larger than Wn.

3. Basic Equations for the Nonlocal KineticStability Analysis

[18] The analysis of small perturbations in the FCS isbased on the linearized Vlasov equation, which can bewritten in the form

bL 0ð Þf1a ¼ � qa

maE1 þ

1

cv� B1ð Þ

� �rvf0a; ð12Þ

where

bL 0ð Þ ¼@

@tþ vrþ qa

maE0 þ

1

cv� B 0ð Þ� ��

rv

� �ð13Þ

and an arbitrary (tearing, kink, etc.) perturbation isdescribed by the potentials A1 and f1 with B1 = r � A1

and E1 = � rf1 � (1/c) (@/@t)A1.[19] We simplify the subsequent stability analysis by

neglecting the normal component of the magnetic fieldBn. Then the component of the particle velocity vx becomesan integral of motion. We also consider the particular caseof the current-driven instabilities with A1 = A(z) exp (gt +iky). Then the stability problem can be reduced to that forthe two-component vector-potential A1 = (0, Ay, Az), inwhich the components are additionally connected by theCoulomb gauge

@zA1z þ ikA1y ¼ 0: ð14Þ

Using this simplification, we exclude one of the Maxwellequations for the potential A1y and write other equations inthe form

r2A1z ¼ � 4pc

Xa

qa

Zvzgad

3v ð15Þ

r2f1 ¼ f1c zð Þ � 4pXa

qa

Zg1ad

3v; ð16Þ

where g1a = f1a � f1(qa/ma) (@f0a/@u) and

c zð Þ ¼ �4pXa

q2ama

Z@f0a@u

d3v ð17Þ

is the fixed function of the TCS profile. Taking into accountthe specific set of variables of the FCS distribution functionf0a = f0a (u + (e/mi)f, Iz), where u = v

2/2, and the specific formof the perturbed vector potential, the linearized Vlasovequation (13) for the perturbed distribution function g1a canbe rewritten as

bL 0ð Þg1a ¼ qa

ma

@f0a@u

gvy

cA1y þ

vz

cA1z � f1

� �þ qa

ma

@f0a@Iz

@Iz@vy

ikf1 þg

cA1y

hþ vz

c@zA1y �

vz

cikA1z

iþ qa

ma

@f0a@Iz

@Iz@vz

@zf1 þg

cA1z

hþ vy

cikA1z �

vy

c@zA1y

i: ð18Þ

4. Orbit Integration

[20] The solution of equation (18) can be presented in theform of the integrals over the unperturbed orbits

g1a ¼ qa

ma

@f0a@u

g

Z t

�1

1

cv0yAy z0ð Þ þ v0zAz z

0ð Þh i

� f z0ð Þ�

� exp gt0 þ ikyy0� �dt0 þ qa

ma

@f0a@Iz

Z t

�1

@I 0z@v0y

ikf z0ð Þ þ g

cAy z0ð Þ

hþ vz

c@zAy z0ð Þ� vz

cikAz z

0ð Þiexp gt0 þ ikyy

0� �dt0

þ qa

ma

@f0a@Iz

Z t

�1

@I 0z@v0z

@zf z0ð Þ þ g

cAz z

0ð Þ þ vy

cikAz z

0ð Þh

� vy

c@zAy z0ð Þ

i� exp gt0 þ ikyy

0� �dt0; ð19Þ

Figure 3. Profiles of the electrostatic potential in the TCSmodel [Sitnov et al., 2000b] (solid lines) and theirinterpolation (equation (11)) (dashed lines) for differentvalues of the parameter d.


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where f1 = f(z) exp (gt + iky), A1y,z = Ay,z(z) exp (gt + iky),y0 (t = 0) = y, z0 (t = 0) = z, v0y (t = 0) = vy, and v0z (t = 0) =vz. As a result, the equations (15)–(16) take the form

r2Az þ4pc

Xa

q2ama

Zvz g

@f0a@ueS1 þ @f0a

@IzeS2� �

d3v

�¼ 0 ð20Þ

r2f� fc zð Þ þ 4pXa

q2ama

Zg@f0a@ueS1 þ @f0a

@IzeS2� �

d3v

�¼0;

ð21Þ

where

eS1 ¼ Z 0

�1

1


0ð Þh i

� f z0ð Þ�

� exp gtþ ik y0 � yð Þ½ �dt ð22Þ

eS2 ¼ Z 0

�1

@I 0z@v0y

ikf z0ð Þ þ g

cAy z0ð Þ þ vz

c@zAy z0ð Þ � vz

cikAz z

0ð Þh i

� exp gtþ ik y0 � yð Þ½ �dtþZ 0

�1

@I 0z@v0z

@zf z0ð Þ þ g

cAz z

0ð Þh

þ vy

cikAz z

0ð Þ � vy

c@zAy z0ð Þ

i� exp gtþ ik y0 � yð Þ½ �dt ð23Þ

and t = t0 � t.[21] As shown by Daughton [1999], the expressions of

the type of eS1,2 can be reduced to the integration over thesingle period of the motion across the sheet (that may beeither the Larmor circle or the figure-eight orbit)

eS1 ¼ 1

1� exp �gtp þ ikDy

� � Z 0

�tp

1


0ð Þh i

� f z0ð Þ�

� exp gtþ ik y0p � y� �h i

dt ð24Þ

eS2 ¼ 1


� � Z 0

�tp

@I 0z@v0y

ikf z0ð Þ þ g

cAy z0ð Þ

hþ vz

c@zAy z0ð Þ � vz

cikAz z

0ð Þi� exp gtþ ik y0p � y

� �h idt

þ 1


� � Z 0

�tp

@I 0z@v0z

@zf z0ð Þ þ g

cAz z

0ð Þh

þ vy

cikAz z

0ð Þ� vy

c@zAy z0ð Þ

i� exp gtþ ik y0p � y

� �h idt; ð25Þ

where the displacement in the y-direction has the formy0(t) = nDy + y0p (t), Dy is the net drift in the y-directionduring the gyroperiod tp, and y0p (t) is the periodic function.[22] It is convenient for the purpose of the subsequent

numerical solution to present the basic set of equations (14),(20), (21), (24), and (25) in the dimensionless block form

L11 Ay

� �þ L12 Azð Þ þ L13 ef� �

¼ 0 ð26Þ

L21 Ay

� �þ L22 Azð Þ þ L23 ef� �

¼ 0 ð27Þ

L31 Ay

� �þ L32 Azð Þ þ L33 ef� �

¼ 0; ð28Þ

where the electrostatic potential is renormalized as f = efvTi/c.The dimensionless parameters used to describe the blocksLij include eg = g/w0i, h = kz, ey = y/r0a, ez = z/r0a, evy,z =vy,z/vTa , and et = tw0a, where w0a = eB0/mac. We alsoused the quasi-neutrality approximation r2f = 0 tosimplify the Poisson’s equation (21). Then the blockscan be written in the form

L11 ¼ iAy ð29Þ

L12 ¼@Az

@hð30Þ

L13 ¼ 0 ð31Þ

L21 ¼� 2eg w2pi

k2c2exp

ef0

Te

� �Z evzef0ebSe z;ev; ev0yAy z0ð Þn o� �

devydevzþ 2eg w2

pi

k2c2

Z evz @ef0i@ew bSi z;ev; ev0yAy z0ð Þ

n o� �d3ev

þw2pi

k2c2

Z evz @ef0i@ei � bSi�z;ev;� @ei0

@ev0y egAy z0ð Þ þ evz e@zAy z0ð Þh i

� @ei0@ev0zevy e@zAy

�z0� �

d3ev ð32Þ

L22 ¼d2Az

dh2� Az � 2eg w2

pi

k2c2exp

ef0

Te

� �Z evzef0ebSe z;ev; ev0zAz z0ð Þ

� �� devydevz þ 2eg w2

pi

k2c2

Z evz @ef0i@ew bSi z;ev; ev0zAz z

0ð Þ� ��

d3evþ

w2pi

k2c2

Z evz @ef0i@ei � bSi

z;ev;(� ikr0i

@ei0@ev0y evzAz z

0ð Þ

þ @ei0@ev0z"egAz z

0ð Þþikr0ievyAz

�z0�#)!

d3ev ð33Þ

L23 ¼ 2eg w2pi

k2c2exp

ef0

Te

� �Z evzef0ebSe z;ev; vTi

vTeef z0ð Þ

� � �devydevz

� 2eg w2pi

k2c2

Z evz @ef0i@ew bSi z;ev; ef z0ð Þ

n o� �d3ev

þw2pi

k2c2

Z evz @ef0i@ei bSi z;ev; ikr0i

@ei0@ev0y ef z0ð Þ

( þ @ei0@ev0z e@zef z0ð Þ

)!d3ev

ð34Þ

L31 ¼� eg vTi

vTeexp

ef0

Te

� �Z ef0ebSe z;ev; ev0yAy z0ð Þn o� �

devydevzþ egZ @ef0i

@ew bSi z;ev; ev0yAy z0ð Þn o� �

d3evþ 1

2

Z@ef0i@ei � bSi

z;ev;( @ei0

@ev0y egAy z0ð Þ þ evz e@zAy z0ð Þh i

� @ei0@ev0zevy e@zAy z0ð Þ

)!d3ev ð35Þ


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L32 ¼� eg vTi

vTeexp

ef0

Te

� �Z ef0ebSe z;ev; ev0zAz z0ð Þ

� �� devydevz

þ eg Z @ef0i@ew bSi z;ev; ev0zAz z

0ð Þ� ��

d3evþ 1

2

Z@ef0i@ei � bSi z;ev; �ikr0i

@ei0@ev0y evz

( Az z

0ð Þ þ @ei0@ev0z"egAz z

0ð Þ

þ ikr0ievyAz z0ð Þ#)!

d3ev ð36Þ

L33 ¼� ef Ti

Teexp

ef0

Te

� �þ ef Z @ef0i

@ew d3evþ eg vTi

vTe

� �2

expef0

Te

� �Z ef0ebSe z;ev; ef z0ð Þn o� �

devydevz� eg Z @ef0i

@ew bSi z;ev; ef z0ð Þn o� �

d3evþ 1

2

Z@ef0i@ei bSi z;ev; ikr0i

@ei0@ev0y ef z0ð Þ þ @ei0

@ev0z e@zef z0ð Þ( ) !

d3ev:ð37Þ

In these equations the operator bSa is given by the formula

bSa z;ev; �f gð Þ ¼ 1

1� exp �egw0itp þ ikDy

� �� Z 0

�w0a tp

� z0; v0ð Þ

� exp g=w0að Þetþ ik y0p � y� �h i

det; ð38Þ

while the dimensionless distributions are as follows:

ef0e ¼ p�1 exp �ev2y � ev2z� �ð39Þ

ef0i ei < ev2 þ ej� �¼ 1

p3=2

exp �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiewþ ej0 �eip

� d� �2�eih i

1þ erf dð Þ ð40Þ

ef0i ei > ev2 þ ej� �¼ 1

p3=2

exp �d2 � ew� ej0

� �1þ erf dð Þ ; ð41Þ

where d = e�1, ew = ev2, ei = e�2/3I, ej0 = je�2/3 = ef0/Ti.[23] Further details of the orbit integration can be found,

for instance, in the work of Daughton [1999]. They differfrom the Harris CS case by the different form of the magneticfield profile b(z) equation (10) and the presence of theelectrostatic potential f0(z) (equation (11)). The calculationof the integral bSa starts from finding the turning points z0,1for Larmor and figure-eight orbits (note that the presence ofthe potential f0 does not affect the orbit classification).Given the turning points, one can find the correspondingperiod tp. Then bSa can be calculated with the orbits param-eters z0, v0 and y0p being updated at each time step using thestandard Runge-Kutta procedure [e.g., Press et al., 1999].

5. Finite Element Analysis

[24] To solve the set of equations (26)–(38), we use thefinite element method [Chen and Lee, 1985; Burkhart and

Chen, 1989; Brittnacher et al., 1995, 1998; Daughton,1998, 1999, 2003; Sitnov et al., 2002]. In this method theexpansion of potentials Ay,z and ef in a series of basisfunctions �i, i = 1, . . ., N, transforms the nonlocal eigen-value problem (equations (26)–(28)) into an algebraic one

MijCj ¼ 0; ð42Þ

where the elements of the matrix Mij are the following innerproducts:

�iLab �j

� � !; a; b ¼ 1; 2; 3 ð43Þ

and the potentials entering the blocks Lab are expanded asfollows:

Ay zð Þ ¼XNn¼1

Cn�n zð Þ; Az zð Þ ¼XNn¼1

CNþn�n zð Þ ð44Þ

ef zð Þ ¼XNn¼1

C2Nþn�n zð Þ: ð45Þ

The specific set of basis functions used in our analysis areHermite functions [e.g., Daughton, 1999]

�n zð Þ ¼ffiffiffip

pn!2n

� ��1=2Hn zð Þ exp �z2=2

� �; ð46Þ

with the inner product h f i =R1�1 f(z) dz.

5.1. Benchmark Case: Harris Equilibrium

[25] The finite element analysis in itself is a complexnumerical procedure. This is why we first performed a setof benchmark runs to check the performance of this procedurefor the known case of the Harris equilibrium described byDaughton [1998, 1999, 2003]. Figures 4 and 5 show theresults of these test runs for the casemi/me=16,Ti=Te, andL=

Figure 4. Frequency (upper curve) and growth rate of thedrift-kink instability found as a solution of the eigenvalueproblem using the finite element method for the case of theHarris equilibrium with mi/me = 16, Ti = Te, and L = r0i.


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A03205

r0i. In particular, Figure 4 resembles the corresponding plotsfor the frequency and growth rate obtained by Daughton withthe same set of parameters but r0i/L = 0.7 [Daughton, 1998,Figure 8; Daughton, 1999, Figure 10]. The eigenfunctions atkL = 0.7 (Figure 5), corresponding to the drift kink mode, arealso consistent with Daughton [1998, Figure 6] and Daugh-ton [1999, Figure 9], although they revealmore diversity. Thiscan be explained by the fact that the eigenfunctions can bedetermined to within an arbitrary complex phase. We addi-tionally checked the convergency of eigenfunctions by plot-ting in the upper panel the component Ay found from thegauge condition (equation (14)) ikAiy = �@zA1z. The numberof basis functions used in this benchmark runs was N = 6.Another solution corresponding to the region of the relativelylong wavelength LHDI, which was recently considered byDaughton [2003], is obtained using a larger number of basisfunctions N = 20 and is shown in Figure 6. As one can seefrom this figure, the solution is basically consistent with theresults shown in Daughton [2003, Figure 5] except for a bitsmaller growth rate. Note also that in contrast to the case of theDKI (Figure 5) and similar to the classical LHDI case, theelectromagnetic component of the solution is rather smallcompared to its electrostatic part. This is why even though theelectromagnetic component Az is peaked at the center of thesheet, it will hardly perturb this region significantly.

5.2. Forced Current Sheet Stability Results

[26] The unstable modes of forced current sheets havebeen investigated for the parameters mi/me = 1836, d = 3,L = 0.69r0i, two different temperature ratios Ti/Te = 1, and

Ti/Te = 4 with the use of N = 6 basis functions. The basic setof linear equations (26)–(38) can be further simplified inthis region with the details given in Appendix B. Figure 7shows that even for realistic mass ratios the growth rate ofthe unstable modes is quite significant, and according toFigure 10, it depends weakly on the temperature ratio. Wehave found unstable modes of both parities (Figures 8, 9,10, 11, and 12). It is important to note that these modesdiffer significantly from the well-known DKI solutionsfound in the cases of small mi/me [Daughton, 1998, 1999]and the hypothetical DSI modes [e.g., Lapenta andBrackbill, 1997]. In fact, they resemble more closely thelong wavelength LHDI solutions found recently in the

Figure 5. Electrostatic potential ef and components Ay andAz of the vector-potential found for the parameters ofFigure 3 with kL = 0.7. Dash-dotted and dash-triple dottedlines show the profiles of the real and imaginary part of thepotential Ay inferred from the profiles of Az using the gaugecondition. To provide the proper resolution, the latterprofiles are additionally shifted along the ordinate axis bythe value dAy = 0.01.

Figure 6. An example of the LHDI solution for the HarrisTCS obtained using N = 20 basis functions. The solutionwith �w/W0i = 4.7 and g/W0i = 0.21 is found for the casemi/me = 512, Ti = Te, L = 0.5r0i, and kL = 2.

Figure 7. Growth rate (upper panel) and frequency (lowerpanel) of the unstable eigenmodes of odd (diamond symbol)and even (star symbol) parity in forced current sheets withmi/me = 1836, Ti = Te, d = 3, and L = 0.69r0i.


9 of 15

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Vlasov stability analysis of the Harris sheet [Daughton,2003] and earlier two-fluid analysis [Yoon et al., 2002]. Inparticular, similar to the LHDI, which is driven by theplasma density gradient at the edges of the sheet, the FCSinstability appears to be driven by the bulk flow velocityshear, which is also located off the sheet center. This explainsthe similar growth rates for even and odd parity modes thatare also characteristic of the LHDI (see, for instance,

Daughton [2003, Figure 5]). Since in both these cases theinstabilities develop on either side away from the center ofthe sheet they are fairly independent of each other. Thus onecan expect that the corresponding global perturbations ofeven and odd parity will be almost equally probable.[27] The relatively large growth rate of even parity modes

does not necessarily mean that these modes will dominatethe FCS evolution beyond the linear growth stage. Incontrast to the kink perturbations, the amplitude of the evenparity modes is limited to the current sheet thickness, which

Figure 8. Electrostatic potential ef and components Ay andAz of the vector-potential found for the parameters ofFigure 7 for the mode with the odd parity and kL = 0.18.

Figure 9. Potentials ef, Ay, and Az found for the parametersof Figure 7 for the mode with the even parity and kL = 0.18.

Figure 10. Growth rate (upper panel) and frequency(lower panel) of the unstable eigenmodes of odd (diamondsymbol) and even (star symbol) parity in forced currentsheets with the parameters mi/me = 1836, Ti = 4Te, d = 3, andL = 0.69r0i.

Figure 11. Potentials ef, Ay, and Az found for theparameters of Figure 9 for the mode with the odd parityand kL = 0.18.


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is fairly small. Thus one can expect observing the lattermodes as transient phenomena before the global currentsheet kinking. Note that these transient phenomena havebeen actually observed in particle simulations of anotherTCS type, namely the bifurcated sheet, which implies evenstronger separation of the instability driving sources off thesheet center [Sitnov et al., 2003b].[28] The bulk flow velocity shear may be one of the

driving forces in another class of current sheet equilibriacomposed of the Harris sheet and a background plasma. Inthese equilibria the stability picture is drastically differentfrom the case of the FCS. In particular, only the instabilityof the odd parity mode similar to the DKI develops,although it does have relatively large growth rate forrealistic mass ratio [Daughton, 1999; Karimabadi et al.,2003a, 2003b]. However, the equilibria with a backgroundplasma are drastically different from the FCS equilibrium.In particular, the background population of ions results in atwo-humped ion distribution in velocity space [Karimabadiet al., 2003b]. Such distributions have an additional free-energy source similar to that in the beam-plasma systems.To avoid the corresponding artificial ion-ion instabilities,some authors [e.g., Shinohara et al., 2001] attempted toreduce the background plasma density inside the currentsheet. This results, however, in a non-self-consistentdescription of the current sheet equilibrium. In contrast,the current sheet equilibrium studied in our paper does nothave a two-humped velocity distribution of ions in thedirection of the current. As shown in Figure 1 [see alsoBurkhart et al., 1992, Figure 3h], the distribution inside thesheet has instead a bean-shaped structure in the sheet plane.Therefore it should be stable with respect to ion-ioninstabilities inside the sheet, and our present study high-lights the effect of the shear flow.

[29] On the other hand, the unstable FCS modes have anumber of properties, which distinguish them from those ofthe lower-hybrid drift modes and may be important forapplications. They have in particular much lower frequency,which is of the order of the ion gyrofrequency w0i outsidethe sheet, and much larger electromagnetic component,similar to the DKI. Most noteworthy is the profile of theFCS modes across the sheet, which is much less structuredas compared to that of the LHDI (compare, for instance,Figure 6 and 8). As a result, based on the arguments ofGladd [1976] and Pritchett and Coroniti [2001], one cannotexpect any significant stabilization of these modes by thenormal magnetic field Bn, which is characteristic of themagnetotail.[30] The wavelength l of the most unstable waves with

kL � 0.2 is quite large. However, taking into account thatthe current sheet itself may be very thin, the value of l isquite consistent, for instance, with recent Cluster observa-tions [Runov et al., 2003; Sergeev et al., 2003]. In particular,with r0i = 600 km, kL � 0.2, and L = 0.69r0i we find l �2RE, which coincides with the estimate of l for the flappingmotions of the magnetotail reported by Sergeev et al.[2003].

6. Discussion and Conclusion

[31] In this paper we have reported on the nonlocalkinetic linear stability analysis, which is similar to theanalysis made by Daughton [1998, 1999, 2003] for theHarris sheet. The new analysis is done for one of a fewknown self-consistent current sheet equilibria different fromthe Harris model, namely the so-called forced current sheet[Sitnov et al., 2000a, 2000b]. In contrast to the Harrismodel, the FCS equilibrium exists due to the plasmaanisotropy outside the sheet and characteristics of the ionorbits in thin current sheets that cannot be described interms of the conventional magnetic moment. The FCSmodels describe thin current sheets with the thicknesscomparable to the thermal ion gyroradius in the field outsidethe sheet and shear of the bulk flow velocity of the ionspecies. They can be readily generalized to the case of asmall finite normal magnetic field Bn. Like the case of themodified Harris model with Bn 6¼ 0, the more general theorydoes not contain the parameter Bn explicitly. Moreover, incontrast to the Harris case, the model remains 1-D as themagnetic tension is balanced by the ion inertia rather thanthe pressure gradient. Such generalized FCS models areoften considered as a basis for the models of magneticmerging [Hill, 1975; Francfort and Pellat, 1976; Kropotkinet al., 1997; Sitnov et al., 2002], which may be analternative to the X-line reconnection as a mechanism ofthe transformation of the magnetic field energy into particlekinetic energy [Biskamp, 1986, 2000]. In the physics ofmagnetospheric substorms this alternative mechanism isknown as current disruption [Lui, 1996].[32] We have found that the stability picture in the FCS

drastically differs from the case of the Harris sheet. Theunstable FCS modes resemble the long wavelength(ky

ffiffiffiffiffiffiffiffiffiffiffir0ir0ep �1) LHDI modes, which were recently studied

by Yoon et al. [2002] and Daughton [2003]. Their growthrates are quite significant even for realistic mass ratiomi/me = 1836 and are comparable for the modes with odd

Figure 12. Potentials ef, Ay, and Az found for theparameters of Figure 9 for the mode with the even parityand kL = 0.18.


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and even parity. In terms of the structure of the eigenmodesacross the sheet, they differ from the DKI modes andresemble the higher-order LHDI-like analogs of the groundstate solutions corresponding to the DKI and hypotheticalDSI. On the other hand, the unstable FCS modes furtherstrengthen some features of the long wavelength LHDImodes that distinguish themselves from the classical LHDI.They have in particular a lower frequency and a largerelectromagnetic component, similar to the DKI. Finally,they are much less structured across the sheet as comparedeven to the long wavelength LHDI. This gives us hope thatlike the cases of the DKI at low mass ratios or with abackground plasma, the newly found unstable FCS modeswill survive in the presence of the finite normal magneticfield typical for the Earth’s magnetotail and the outflowregions of the reconnections patterns.[33] Our results reveal the important role of the bulk

flow velocity shear in destabilizing thin current sheets.They are consistent with recent PIC simulations ofcurrent-driven instabilities in the models of Harris sheetsassuming a bulk flow velocity shear due to an additionalnon-self-consistent background plasma [Shinohara et al.,2001], the models where the bulk flow velocity sheararises as a consequence of the nonlinear saturation of theLHDI [Horiuchi and Sato, 1999; Daughton, 2002;Lapenta and Brackbill, 2002], and the Harris modelswith a uniform background plasma [Daughton, 1999;Karimabadi et al., 2003a, 2003b]. However, in the lattercase the shear flow effects may be strongly masked bythe more conventional two-stream instability. In contrastto the Harris sheet, the FCS equilibria appear to assume awider spectrum of the unstable modes, including modesof even parity. This is consistent with recent observationsof the electromagnetic fluctuations in laboratory plasmas(Ji et al., submitted manuscript, 2004) and similar fluc-tuations in the geomagnetotail during current disruptions[Lui and Najmi, 1997; Sigsbee et al., 2002], whichclearly reveal a wide variety of excited modes.

Appendix A: Approximation of the EquilibriumForced Current Sheet Profiles

[34] As shown in Figure 2, the magnetic field profile(equation (9)) in the region of strong anisotropy case can beapproximated as

b zð Þ ¼ tanhn mz1=n� �

; ðA1Þ

where z = zd4/3w0i/vD = ezd1/3(r0a/r0i), and the coefficientsm = 0.517 and n = 0.897 are found by minimazing thestandard deviation between equation (A1) and the exactsolution of the Grad-Shafranov equation [Sitnov et al.,2000b]. For computation of the dimensionless sheetinvariant one needs also the dimensionless function eb(ea) =b(ez(ea)) whereea ezð Þ ¼

Z ez0

b ez0ð Þdez0 ¼ d�1=3 r0ir0a

Z z ezð Þ0

b z0ð Þ dez0dz

dz0

¼ d�1=3 r0ir0a

a z ezð Þð Þ ðA2Þ

Therefore one can represent eb (ea) in the form eb(ea) = b(z(ea)) =b(a�1(a)), where a = d1/3(r0a/r0i) ea. This scaling is similar toequation (A1).[35] Let us now consider the limit z ! 0. Then we have

b(z) = mnz and a(z) = mnz2/2, and therefore eb(ea) = ffiffiffiffiffiffiffiffiffiffi2mna

p.

Thus the finction eb(ea) should be approximated as

eb eað Þ ¼ tanhn1 m1=21 2að Þ1=2n1� �

: ðA3Þ

Here however the coefficients m1 and n1 do not necessarilycoincide with the pair m and n used in the approximation(A1) because the only relation, which provides consistencyof equations (A1) and (A3) is mn = mn11 . Indeed,approximating equation (A3) yields m1 = 0.449 and n1 =0.748 with mn11 = 0.549, whereas mn = 0.553.[36] For completeness we need now to find the electro-

static potential. This can be done using the quasi-neutralityrelation

n0e ¼ expef0

Te

� �¼Z ef0id3ev; ðA4Þ

which in the limit of strong anisotropy takes the form

expef0

Te

� �¼ d

2p

Z 1

�1devz Z 2p

0

exp �ei evz;jð Þð Þdj: ðA5Þ

One of our main tasks here is to find the scaling of f0 as afunction of d. To reveal this scaling we rewrite the invariantei in the right-hand side of equation (A5) as follows

ei ¼ 2

p

Z ea ez1ð Þ

ea ez0ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiev2y þev2z � evy þ ea0 � ea ezð Þ" #2q

dea0eb ea0ð Þ

¼ 2

pd�2=3

Z a ez1ð Þ

a ez0ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid8=3 sin2 jþ d2=3ev2z � d4=3 sinjþ a0 � a ezð Þ

h i2rda0

b a0ð Þ

� 2

p

ffiffiffiffiffiffiffiffiffiffiffiffiffi2 sinj

p Z a ez1ð Þ

a ez0ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid8=3 sin2 jþ w2

z

q� d4=3 sinj� a0 þ a ezð Þ

rda0

b a0ð Þ ;

where wz = d1/3evz. Expanding the expression under thesquare root we have

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid8=3 sin2 jþ w2

z

q� d4=3 sinj� a0 þ a ezð Þ¼ w2

z

2d4=3 sinj� a0 þ a ezð Þ

ðA7Þ

Now we introduce the new variable in the integral on theright-hand side of equation (A5)a00 = wz

2/(2d4/3sin j) + a(ez)to get

expef0

Te

� �¼ d4=3ffiffiffi

2p

p

Z 1

a

da00ffiffiffiffiffiffiffiffiffiffisinj

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia00 � a ezð Þ

p Z 2p

0

exp �ei evz;jð Þð Þdj;

ðA8Þwhere

ei ¼ 2

p

ffiffiffiffiffiffiffiffiffiffiffiffiffi2 sinj

p Z a00

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffia00 � a0

p da0

b a0ð Þ : ðA9Þ

(A6)


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This shows that the right-hand side of equation (A5) scalesas d4/3. However, this scaling cannot be merely introducedin the approximating formula for electrostatic potential,which has to disappear outside the current sheet jzj ! 1.One can propose nevertheless the approximating formula,which reconciles the above two requirements. It has theform

f0 zð Þ ¼ Tee�1sj0 exp �j1 zs�1=2

� �j2h i

; ðA10Þ

where j0 = 2.886, j1 = 0.08, j2 = 2.2, and s = 1 � (4/3 j0)log (10/d). This approximation is based on the numericalsolution of the Grad-Shafranov equation for thin currentsheets [Sitnov et al., 2000b] for d = 10, and confirmed thenfor d = 7, 5, and 3. In particular, j0 = (ef0/Te)jd=10;z=0, and(ef0/Te)jz=0 � j0 = (4/3) log (d/10).

Appendix B: Basic Set of Equations in the Case ofStrong Ion Anisotropy

[37] Here we consider further simplifications of the ma-trix elements (equations (32)–(37) in the region d � 1. Inthis limit one can neglect first of all the trapped ionpopulation and represent the distribution function in theform

ef0i � 1

2p3=2exp �

ffiffiffiffiewp� d

� �2�ei �

; ðB1Þ

while @ef 0i/@ew = (d/ffiffiffiffiewp

�1) ef 0i and @ef 0i/@ei = �ef 0i.Moreover, in this limit d � 1 we have ef 0i � (1/2p) exp (�ei)d (ev � d), and therefore

Z ef0id3ev ¼ Z ef0idvzvdvdj � d2p

Z 1

�1devz Z 2p

0

exp �ei evz;jð Þð Þdj;

ðB2Þ

where vx = d cos j and vy = d sin j.[38] Using these simplifications the matrix elements

(equations (32)–(37) can be rewritten in the form

L21 Ay

� �¼� 2eg w2

pi

k2c2exp

ef0

Te

� �Z evzef0ebSe z;ev; ev0yAy z0ð Þn o� �

devydevz� d2p

w2pi

k2c2

Z 1

�1evzdevz Z 2p

0

exp �ei evz;jð Þð Þdj

� bSi z;ev; @ei0@ev0y egAy z0ð Þ þ evz e@zAy z0ð Þh i


( ) !ðB3Þ

L22 Azð Þ ¼ d2Az

dz2� Az � 2eg w2

pi

k2c2exp

ef0

Te

� �Z evzef0e� bSe z;ev; ev0zAz z

0ð Þ� ��

devydevz � d2p

w2pi

k2c2

Z 1

�1evzdevz Z 2p

0

� exp �ei evz;jð Þð Þdj� bSi�z;ev; � ikr0i@ei0@ev0yevz

þ @ei0@ev0z egþ ikr0ievy� ��

Az z0ð Þ�

ðB4Þ

L23 ef� �¼ 2eg w2

pi

k2c2exp

ef0

Te

� �Z evzef0ebSe z;ev; vTi

vTeef z0ð Þ

� � �devydevz

� d2p

w2pi

k2c2

Z 1

�1evzdevz Z 2p

0


� bSi z;ev; ikr0i@ei0@ev0y ef z0ð Þ þ @ei0

@ev0z e@zef z0ð Þ( ) !

ðB5Þ

L31 Ay

� �¼� eg vTi

vTeexp

ef0

Te

� �Z ef0ebSe z;ev; ev0yAy z0ð Þn o� �

devydevz� d4p

Z 1

�1devz Z 2p

0


� bSi z;ev; @ei0@ev0y egAy z0ð Þ þ evz e@zAy z0ð Þh i


( ) !ðB6Þ

L32 Azð Þ ¼ � eg vTi

vTeexp

ef0

Te

� �Z ef0ebSe z;ev; ev0zAz z0ð Þ

� �� devydevz

� d4p

Z 1

�1devz Z 2p

0


� bSi z;ev; �ikr0i@ei0@ev0y evz þ @ei0

@ev0z egþ ikr0ievy� �" #Az z

0ð Þ !

d3evðB7Þ

L33 ef� �¼� ef Ti

Teexp

ef0

Te

� �þ ef Z @ef0i

@ew d3evþ eg vTi

vTeexp

ef0

Te

� ��Z ef0ebSe z;ev; vTi

vTeef z0ð Þ

� � �devydevz � d

4p

Z 1

�1devz

�Z 2p

0

exp �ei evz;jð Þð ÞdjbSi�z;ev;�ikr0i @ei0@ev0y ef z0ð Þ

þ @ei0@ev0z e@zef z0ð Þ

�d3ev: ðB8Þ

[39] Acknowledgments. The research of M. I. Sitnov and P. N.Guzdar was supported by NASA grant NAG513047 and by NSF/DOEgrant ATM0317253 to the University of Maryland at College Park. M. I.Sitnov and A. T. Y. Lui acknowledge also the NASA grant NAG5-10475 tothe Johns Hopkins University Applied Physics Laboratory.[40] Arthur Richmond 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]; [email protected])A. T. Y. Lui, Applied Physics Laboratory, Johns Hopkins University,

Laurel, MD 20723-6099, USA. ([email protected])P. H. Yoon, Institute for Physical Science and Technology, University

of Maryland, College Park, MD 20742-2431, USA. ([email protected])


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