Adiabatic electron heating in the magnetotail current sheet: Cluster observations and analytical models

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117,, doi: /2012ja017513, 2012 Adiabatic electron heating in the magnetotail current sheet: Cluster observations and analytical models A. V. Artemyev, 1 A. A. Petrukovich, 1 R. Nakamura, 2 and L. M. Zelenyi 1 Received 9 January 2012; revised 22 March 2012; accepted 30 April 2012; published 12 June [1] We consider the electron distribution in current sheets observed by Cluster mission in the Earth magnetotail. We use the statistics of 70 fast (less than 20 minutes) and 12 slow (more than one hour) crossings of horiontal current sheets. We demonstrate that for both types electron temperature decreases with increase of magnetic field B x away from the current sheet center. We use the approximations T e? /T e? max 1 a T? (B x /B ext ) 2 and T ek /T ek max 1 a Tk (B x /B ext ) 2, where B ext is value of B x in the lobes. For statistics of thin current sheets (fast crossings) we obtain mean values a T? a Tk 1. For thick current sheets (slow crossings) we also obtain a T? a Tk, but a T?, a Tk >1. The electron temperature anisotropy is about T ek /T e? = and vertical profiles T ek /T e? const. Observed vertical distributions of T ek and T e? are described by the analytical model of electron heating in the course of the earthward convection in thin current sheets with (B x (), B (x)) and in thick current sheets with (B x (x, ), B (x, )). We also show that the observed electron temperature anisotropy is provided by the electron population in the energy range between 50 ev and 3 kev. The cold core of electron distribution (<50 ev) is isotropic and the hot tail (>5 kev) has T ek /T e? 1 or even T ek /T e? < 1. We consider electron pressure tensor in observed thin current sheets and demonstrate that electron velocity distribution is gyrotropic with high accuracy. Citation: Artemyev, A. V., A. A. Petrukovich, R. Nakamura, and L. M. Zelenyi (2012), Adiabatic electron heating in the magnetotail current sheet: Cluster observations and analytical models, J. Geophys. Res., 117,, doi: /2012ja Introduction [2] Owing to the multispacecraft analysis of the last decade various mesoscale structures in the Earth magnetotail have been revealed and investigated: thin current sheets, dipolariation fronts, etc. These structures have a transverse spatial scale about few local ion gyroradii and correspond to various dynamical processes with ion timescale (tens seconds) (see reviews by Baumjohann et al. [2007] and Sharma et al. [2008]). It was reasonable to believe that ions are responsible for the formation of these mesoscale structures as well as for their evolution. However, we still have poor information about a role played by electrons. [3] In the Earth magnetotail electrons can be most of the time considered as magnetied adiabatic particles. The difference of motion of adiabatic electrons and nonadiabatic ions leads to the charge separation and current sheet polariation (formation of E and E x components of the electrostatic field [see, e.g., Birn et al., 2004a; Zelenyi et al., 1 Space Research Institute, RAS, Moscow, Russia. 2 Space Research Institute, Austrian Academy of Sciences, Gra, Austria. Corresponding author: A. V. Artemyev, Space Research Institute, RAS, Profsounaya St., 84/32 GSP-7, Moscow , Russia. (ante0226@yandex.ru) American Geophysical Union. All Rights Reserved. 2004]). Formation of the earthward electrostatic field E x results in a substantial current redistribution and leads to the dominance of a electron current in current sheets with B / x >0[Zelenyi et al., 2010]. This effect is enhanced by the current sheet embedding [Artemyev et al., 2010] and is stronger at the dawn flank of the magnetotail [Wang et al., 2009; Artemyev et al., 2011b]. [4] Due to the adiabaticity of electrons, to describe their motion one can apply the guiding-center theory. The first invariant regarding to the gyrorotation (magnetic moment) and the second invariant regarding to the bounce oscillations are conserved for electrons at least in quiet ion-scale current sheet. Presence of the dawn-dusk electrostatic field provides the earthward convection of adiabatic electrons and results in an electron heating. According to the theoretical models this heating can be isotropic [Lyons, 1984] or anisotropic, when parallel and perpendicular temperature are growing with different rates [Tverskoy, 1969; Zelenyi et al., 1990; Artemyev et al., 2011a]. Both models are based on assumption that energy of adiabatic electrons grows with increase of the magnetic field (so called betatron and Fermi mechanisms). These mechanisms are widely applied for description of adiabatic particles energiation in the course of a dipolariation process [see Delcourt and Sauvaud, 1994; Apatenkov et al., 2007; Fu et al., 2011, and references therein]. Comparison of these two mechanisms, using the experimental data is the first topic of this paper. We demonstrate that 1of13

2 Table 1. The List of Slow Crossings Date B ext (nt) L (10 3 km) b n a Tk a T? anisotropic model can describe adequately the observed heating of electrons in current sheet of the Earth magnetotail. [5] Heating of electrons in the course of the earthward convection depends on the configuration of magnetotail current sheet. The comparison between model and observed vertical profiles of the electron temperature T e (B x ) could provide the estimates of important current sheet parameter - the spatial scale of the magnetic field inhomogeneity along the Sun-Earth direction, L x [Artemyev et al., 2011a]. The second topic of our research extends this approach by taking into account 2D effects ( B x / x 0) for adiabatic electron heating. We also compare vertical profiles of electron anisotropy T e k /T e? (B x ) obtained from analytical models and those observed by the Cluster mission. [6] Incorporation of the electron component into kinetic current sheet models can be accomplished by two ways. According to the first way (known as the Vlasov approach), one introduces the electron velocity distribution as a function of the local invariants of motion: the momentum along the current density direction and the total energy [see, e.g., Nicholson, 1963; Motte, 2003; Birn et al., 2004a, and references therein]. In this case to reproduce the observed electron temperature anisotropy it is necessary to include into the model a temperature anisotropy T xx T yy, where x and y-axis are directed along main component of the magnetic field and along current density, respectively. The alternative approach considers the nonlocal invariants of the electron motion: magnetic moment [Zelenyi et al., 2004, 2011] or even quasi-adiabatic invariant [Sitnov et al., 2006]. In this case temperature anisotropy can be incorporated as T ek T e?. The preference to one of these approaches can be given on the basis of experimental observations. This problem is the third topic of our paper. 2. The Data and Methods [7] In this paper we use the statistics of 70 fast crossings of thin current sheets (TCS) by Cluster mission. The list of events can be found in [Artemyev et al., 2010, 2011b]. Also we use 12 slow crossings of thick current sheets for 2001 and 2004 years (Table 1). We distinguish these two types of current sheets using following criterion: a relatively large vertical velocity of current sheet motion (up to hundred km/s, see statistics collected by Sergeev et al. [2006], Petrukovich et al. [2006], and Zelenyi et al. [2009]) and a small thickness make it possible to cross current sheet from our statistics of fast crossings in 1 10 minutes, while more than one hour is needed to cross current sheet from our database of slow crossings. Difference of these two types is supported by comparison of magnetic field amplitudes at the boundary of current sheets. For fast crossings we can distinguish the value of B x at the TCS boundary B 0 (see details in Artemyev et al. [2010]) and the value of B x in the lobes B ext (using the vertical pressure balance). Schematic view of such magnetic field distribution in TCS is shown in Figure 1a. Different rates of B x variation across current sheet for B x < B 0 and for B 0 < B x < B ext are interpreted as a presence of TCS with a strong current density embedded into thick current sheet (the latter supports magnetic field variation from B 0 up to B ext ). For slow crossings we cannot separate B ext and B 0 and assume B 0 B ext. Thus, we interpret such situation as absence of TCS, when only thick current sheet with single transverse spatial scale (thickness) is observed (see scheme in Figure 1b). [8] Current sheet thickness L is defined as (c/4p)b 0 /j max, where amplitude of the current density j max is determined with the help of the curlometer technique [Paschmann and Schwart, 2000] and averaged over interval B x < 5 nt. Typical value of j max also confirms the difference between fast and slow crossings: for fast crossings we have j max > 5 na/m 2 and for slow crossings we obtain only j max 2nA/m 2.Value of the magnetic field component normal to current sheet is B n (for horiontal current sheets from our statistics we determine B n as averaged value of B in the region, where B x < 5 nt). Hereinafter we use the dimensionless parameter b n = B n /B 0. [9] Table 1 demonstrates that slow crossings correspond to the thick current sheet with L 10 4 km, while the typical transverse scale of TCS is L km [see Runov et al., 2006; Artemyev et al., 2010, and references therein]. [10] Based on our statistics we consider the electron content of current sheets. We use electron moments and (pitch angle, energy) distributions from PEACE instrument [Johnstone et al., 1997]. To determine the position of spacecraft relative the neutral plane of current sheet we use the magnetic field from FGM experiment [Balogh et al., 2001]. All data are obtained from the Cluster Active Archive ( [11] In Figure 2 we show one typical crossing of TCS to explain main parameters included in our statistics. TCS is crossed by spacecraft in 10 minutes and at the boundary of TCS we define B 0 value (alternative methods of B 0 determination can be found in Artemyev et al. [2010]). For this TCS we find B ext 26 nt, i.e. B ext is substantially larger than B 0 Figure 1. Schematic view of current sheet geometry and magnetic field distribution. See details in the text. 2of13

3 (first panel of Figure 2). Central region of TCS ( B x < 5 nt) is shown by grey color. Second panel demonstrates variation of B component of magnetic field and curlometer current density across TCS. We also indicate amplitude of current density j max and averaged value of the normal component B n. Third panel shows increase of electron temperature (both parallel and perpendicular components) in the central region. Electron density is close to proton density and weakly increases in the central region of the current sheet (see last panel of Figure 2). Figure 2. TCS crossing: first panel shows B x component from all spacecraft and B x at barycenter. Second panel presents normal component of magnetic field B and curlometer current density j y (also averaged values B n and j max are indicated). Two bottom panels show electron temperature and density along the crossing. 3. Temperature Distribution: Observations [12] In this section we consider the distributions of electron temperature T e ¼ 1 3 2T e? þ T ek and anisotropy Tek /T e? in current sheet. For 12 selected examples of fast TCS crossings, the profiles of T e /T emax and T ek /T e? are shown in Figure 3. Electron temperature in TCS decreases toward the boundary of current sheet in agreement with the statistical result obtained in previous observations [Artemyev et al., 2011a]. For majority of cases the drop of electron temperature between TCS center and boundary is about ( ) T emax, where T emax is defined as averaged value of temperature in the central region of TCS ( B x < 5 nt). [13] To obtain the quantitative estimates of the distribution of plasma parameters across current sheet one can use an approximation by the series n c n B n x, where c n = const. This approach was applied for electron pressure [Zelenyi et al., 2010], for ion [Artemyev et al., 2011b] and electron [Artemyev et al., 2011a] temperatures. As was shown by Artemyev et al. [2011a], profiles of T e can be approximated by the parabola T e /T emax = 1 a T (B x /B ext ) 2 with a Figure 3. Profiles of electron temperature and anisotropy for 12 fast crossings are shown versus normalied magnetic field B x /B ext. Grey curves are approximation of experimental data by parabola T e /T emax =1 a T (B x /B ext ) 2. Time interval of each crossing is indicated inside the corresponding panel. 3of13

4 [17] Here we use the small statistics of 12 observations of 2DCS by the Cluster spacecraft. The duration of the most of crossings exceeds one hour. Corresponding profiles of electron temperature and anisotropy are presented in Figure 6: T e /T emax decreases more substantially, than in TCS, even down to 0.2. The profiles demonstrate the relatively stable behavior. Electron anisotropy T ek /T e? can be considered as a constant value. However, for some events one can observe a weak maximum of T ek /T e? in the central region of 2DCS. Figure 4. Ratio T ek /T e? as a function of B x /B ext for the whole statistics of TCS. The averaged curve and errors are shown. 4. Temperature Distribution: Models [18] In this paper we use two models of current sheet to describe observed profiles of electron temperature and anisotropy. The schematic view of parameters and models structure is presented in Figure 1c. It is assumed that reasonable accuracy. For wide statistics of observations the mean value is a T 1. [14] Distribution of electron anisotropy in TCS does not correspond to any function of B x. It is rather constant T ek /T e? const. For all TCS from our statistics we plot T ek /T e? versus B x /B ext (Figure 4). We also plot the averaged values of T ek /T e? as a function of B x /B ext. The figure demonstrates that the approximation T ek /T e? const indeed could be used as a first order approximation. The mean value is T ek /T e? 1.1 in agreement with the previous studies [Stiles et al., 1978; Zelenyi et al., 2010; Artemyev et al., 2011b]. [15] Here we apply the same technique of approximations for parallel and perpendicular electron temperature: T e? /T e?max = 1 a T? (B x /B ext ) 2 and T ek /T ekmax = 1 a Tk (B x /B ext ) 2. Coefficients a Tk and a T? are obtained by the least squares method. Distributions of obtained a Tk and a T? are presented in Figure 5. The mean value a Tk is similar to the mean value a T? [16] Generally speaking, the range of the temperature variation across current sheet strongly depends on the value of B / x [Artemyev et al., 2011a] and in TCS this gradient B / x is suppressed (according to the common conception TCS develops during growth phase, when magnetic field lines are stretched along the magnetotail and gradient of B component decreases, see discussion in Sergeev et al. [2011, and references therein]). Thus to observe more substantial decrease of T e one needs to consider relatively slow crossings of thicker current sheets. In these thicker current sheets gradient B / x should be larger and we will call them 2DCS. Difference between TCS and 2DCS can be quantitatively described as follows: TCS represents current sheet formed in the stretched magnetotail, when spatial scale along x axes substantially increases and gradient B / x is practically absent. 2DCS may represent current sheet observed in the magnetotail during dipolariation, when magnetic field lines have more dipole like structure. For such current sheet gradient B / x plays an essential role in pressure balance. Figure 5. Distribution of parameters a Tk and a T? for statistics of TCS. 4of13

5 Figure 6. Profiles of electron temperature and anisotropy for 12 slow crossings are shown versus normalied magnetic field B x / B ext. For each crossing the profile of averaged values is shown by back curve with the standard deviation. Time interval of each crossing is indicated inside the corresponding panel. spacecraft is crossing current sheet along the vertical slice with x = x obs. [19] The first model approximates TCS. In this model magnetic field B depends only on x coordinate and B x depends only on coordinate: B ðþ¼b x n gx ðþ¼b n x h L x ð1þ B x ðþ¼b 0 L Here x L x 1 and g(x obs ) = 1. According to observations the TCS thickness L is about one to three ion gyroradii in the field B 0 [Petrukovich et al., 2011] and B 0 /B ext 0.4 [Artemyev et al., 2010]. Because for TCS model condition divb = 0 is satisfied automatically we have two free parameters, h and L x. Other parameters can be obtained directly from spacecraft observations (see Artemyev et al. [2010, 2011a] for details). [20] The second model describes two dimensional thick current sheet. In this case both components of the magnetic field depend on both coordinates: h B ðx; Þ ¼ B n 1 2 gx ðþ¼b n 1 2 B x ðx; Þ ¼ B 0 L L 2 g h 1 L 2 x L x h ¼ B 0 x 1 h : ð2þ L L x [21] The condition divb = 0 gives h ¼ 2B nl x B 0 L þ 1. Thus for this model we have only the single free parameter, L x. For 2DCS we assume B 0 B ext. Formally, models (1) and (2) can be considered as expansions of the model by Zwingmann [1983] in the central region, where B x. [22] Expressions for field lines of both models are derived in Appendix A and corresponding curves are presented in Figure 7. Dependencies of B x on B along the field lines are also incorporated into the figure. In the both models field lines have a similar structure if h is close to one. 5of13

6 [25] For both models q e? (g ) grows with g * almost linearly according to the conservation of the magnetic moment. For TCS parallel temperature q e k (g ) decreases toward the deep tail slower than q e? (g ). This effect provides the growth of anisotropy q e k /q e? with the decrease of g *.In 2DCS q e k (g ) decreases with the decrease of g * more rapidly, because away from the neutral plane the decrease of magnetic field amplitude toward the deep tail is comparable with g * (B x g (1 h)/h ). As a result, the variation of the anisotropy q e k /q e? along the tail for 2DCS is not as essential as for TCS. [26] Profiles along the magnetotail obtained above can be presented as profiles relative to B x (x obs, ) in the vertical slice Figure 7. Field lines for TCS and 2DCS models. Here L TCS ¼ ð1 hþ B 0L 2B n L x. [23] We obtain electron temperature profiles in current sheet for both models according to the method proposed by Artemyev et al. [2011a]: (1) we obtain electron temperature profiles along the neutral plane; (2) we map the electron velocity distribution along the field lines; (3) we integrate the velocity distribution and obtain vertical profiles of electron parallel and perpendicular temperatures. [24] To obtain profiles of electron temperature along the neutral plane where B x 0 we take into account the conservation of both the magnetic moment and the second adiabatic invariant of electrons in the course of the earthward convection. At the observation point in the neutral sheet (where magnetic field B = B n and B x = 0) we set the anisotropic Maxwellian velocity distribution ( v 2 f 0 v? ; v k exp? m e v2 k m ) e 2T e?max 2T ekmax Here T e?max and T ekmax are values of electron perpendicular and parallel temperatures which can be observed directly by spacecraft. By using the conservation of the invariants we obtain functions v? = v? (u?, u k ) and v k = v k (u?, u k ), where (u?, u k ) are particle velocity components in the point with B (x,0)=b n g and B x = 0 (see Appendix B). Integration of this new velocity distribution over the coordinates (u?, u k ) gives electron temperature as a function of the dimensionless magnetic field g = B (x, 0)/B n along the neutral plane, q e? (g ) and q e k (g ) (see Figure 8). We use notations q e? (g ), q e k (g ) for electron temperature along the neutral plane (i.e. = 0), while notations T e? and T ek are used for electron temperature along the current sheet crossing (i.e. at g * = 1). Figure 8. Profiles of electron temperature and anisotropy along the magnetotail for TCS and 2DCS models (smaller g * corresponds to the downtail direction). In the point with g * = 1 we set T e k max /T e? max = of13

7 model we estimate the degree h 1.5. Therefore, for typical values B 0 B ext nt and B n 2 3 nt we have L x 5L, which is much smaller than L x 25L obtained for TCS [Artemyev et al., 2011a]. Figure 9. Vertical profiles of electron temperature (black color) and anisotropy (grey color) for TCS and 2DCS models. where B (x, 0)=B n. For this purpose we use the conservation of the magnetic moment and the total energy during the given bounce oscillation. With this assumption we map the velocity distribution from the neutral plane along the field lines toward the slice with B x (x obs, ) and B x = B n and integrate it (see Appendix C). Obtained temperature T e ¼ 1 3 2T e? þ T ek profiles as well as profiles of electron anisotropy T ek /T e? are presented in Figure 9. As one can see, anisotropy T ek /T e? has a small maximum in the central region of current sheet for both models. The decrease of electron temperature toward the current sheet boundary is more substantial for 2DCS in agreement with our observations (compare with Figures 3 and 6). [27] The temperature and anisotropy obtained in our models have profiles rather similar to observed ones (Figures 3 and 6). Direct comparison of the developed model with the experimental data could provide the estimates of the free parameter, L x, in case of TCS for any given model value of h (Artemyev et al. [2011a] used h = 0.8). [28] For 2DCS h should be larger than one. Based on the comparison of observed temperature profiles and 2DCS 5. Multicomponent Electron Population [29] As it was shown above, temperature anisotropy of electron velocity distribution is present in TCS in accordance to the considered mechanism of the heating. If h is close to one, then for 2DCS parallel temperature could decrease with the distance from the Earth faster (or comparable) than perpendicular temperature. In any case, at x 20R E, where current sheets are observed by Cluster, we have T ek /T e? > 1 (see Figure 4). To describe electron heating we apply the model of anisotropic acceleration due to the conservation of adiabatic invariants [Tverskoy, 1969; Zelenyi et al., 1990]. However, one can also consider alternative model developed by Lyons [1984]. According to this model particle energy u 2 k + u 2? grows as g 2/3 while pitch angle scattering keeps the distribution isotropic. In this section we show how these two models can be combined to explain the available observations. [30] We consider energy distributions for three different pitch angles a =0,90, 180 collected in the central region of TCS (where B x < 5 nt). For six examples of TCS crossings these distributions are presented in Figure 10. We also plot the difference F(ɛ) =Df/f =1 f a =0 (ɛ)/f a =90 (ɛ), where ɛ is the particle energy. As one can see, the electron anisotropy (Df/f < 0) is associated with a certain energy range: 3 kev > ɛ > 50 ev. Distributions can be separated in three parts: the cold core (ɛ < 50 ev) without any significant anisotropy (this population could be partially related to the effect of photoelectrons and we do not discuss it in this paper), the hot tail (ɛ > 5 kev) with variable small anisotropy (even sometimes with Df/f > 0) and the intermediate region with substantial parallel anisotropy Df/f < 0. Particles of the latter population are responsible for the formation of the observed temperature anisotropy in TCS (T ek /T e? > 1). [31] We use our statistics of TCS crossings to determine energy ɛ* with maximal anisotropy F(ɛ*) = min ɛ F(ɛ). We also find ɛ max as an upper boundary value of the anisotropic population, F(ɛ max ) = 0.1F(ɛ*), ɛ max > ɛ*. The distributions of obtained values ɛ* and ɛ max are presented in Figure 11. The figure demonstrates that for several cases ɛ* < 100 ev. However, this energy region is related to the cold core and ɛ* could be overestimated due to photoelectrons. For majority of events we have ɛ* ev. This anisotropic population can be clearly seen in the distribution functions (Figure 10). The heating and anisotropiation of this population corresponds to the mechanism considered above. [32] To describe the energy width of this population we use ɛ max. As one can see from Figure 11, ɛ max is often 1 kev. Thus the energy range 100 ev ɛ 1000 ev is filled by anisotropic electrons. Energies higher than 5 kev correspond to the isotropic population. The heating of this population could be related to pitch angle scattering, for example due to the mechanism, proposed by Lyons [1984]. [33] Because only a part of whole electron content supports electron temperature anisotropy T ek /T e? > 1, one 7of13

8 Figure 10. Energy distributions of electrons for three pitch angles. Also the ratio Df/f is presented in the inset panels. can expect that the real anisotropy of this population (ɛ [100, 1000] ev) is larger than the observed averaged value. 6. Anisotropy and Nongyrotropy of Electrons: Observational Data and Model Approaches [34] This section is devoted to study of the electron pressure tensor in current sheet. Observations demonstrate that electron anisotropy is always present in current sheet (see Figure 4) [Zelenyi et al., 2010; Artemyev et al., 2011b]. Here we consider two theoretical approaches of including this anisotropy in current sheet models. We use the coordinate system, where x-axis is directed along the main component of the magnetic field, B x () is changing sign in the plane = 0, y-axis is directed along the current density j y B x / and -axis is directed along the normal to the current sheet plane. [35] The first approach was proposed by Nicholson [1963] and further developed by Motte [2003]; Birn et al. [2004a] is based on consideration of the electron velocity 8of13

9 develop an equilibrium current sheet with T ek T e? without any nongyrotropy. This approach was applied to the electron component in TCS models (see review by Zelenyi et al. [2011 and references therein]). [37] To give the preference to one of these approaches we consider the electron pressure tensor in observed TCS from our statistics. For each crossing we obtain three main components of the pressure tensor p xx, p yy and p. We also obtain two components p k and p?. The ratios of these components are presented in Figure 12. As one can see, p p yy p? and p xx p k. Therefore, in the observed TCS the electron nongyrotropy is practically absent. Moreover, the averaged electron gyroradius is much smaller than the vertical spatial scale of TCS, L. As a result, p xx almost everywhere can be approximated by p k. 7. Discussion [38] For all TCS from our statistics electron temperature decreases with increase of B x, i.e. a T? > 0 and a Tk > 0 (the similar result for a T > 0 can be found in Artemyev et al. [2011a]), which corresponds to the gradient B / x > 0. This result is in agreement with the observation of the electron temperature distributions in TCS during growth phase [Sergeev et al., 2011]. However, individual case studies demonstrate possibility of the existence B / x <0in the magnetotail current sheets [Nakamura et al., 2009; Saito et al., 2010; Panov et al., 2012]. The sign of the gradient Figure 11. Distributions of ɛ* and ɛ max. distribution f e as a function of local integrals of motion: the momentum p y ¼ m e v y jej c A yðx; Þ and the total energy H ¼ 1 2 m e v 2 x þ v2 y þ v2 jejfðx; Þ, where A y (x, ) and f(x, ) are vector and scalar potentials. In this approach f e = f e (p y, H) depends on v 2 x + v 2 and on v y. Therefore, one can develop only an equilibrium current sheet with anisotropy T xx = T T yy. The direction along y is perpendicular to the magnetic field B ¼ B x ðx; Þe x þ B ðx; Þe and T yy could be considered as T e?. However, according to this approach the existence of anisotropy leads to a (1) (2) nongyrotropy. One can write T e? = T yy and T e? = T xx (B /B) 2 + T (B x /B) 2. Then if we introduce anisotropy (1) T xx = T T yy we obtain nongyrotropy T e? T (2) e?. [36] The second approach is based on the nonlocal quasi-adiabatic invariant I ¼ v d [Zelenyi et al., 2004; Sitnov et al., 2006]. For adiabatic electrons I m, where m is the magnetic moment. In this case velocity distribution has a form f e = f e (m,h), where H ¼ 1 2 m e v 2? þ v2 k jejfðx; Þ and m = m e v 2? /2B. The moments of ero and second orders (density and pressure) can be obtained directly by the integration of f e and current density is determined from the well known drift equations [see, e.g., Whipple et al., 1991, and references therein]. Therefore, distribution function depends on (v?, v k ) and one can Figure 12. Distribution of ratios of electron pressure tensor components for TCS statistics. 9of13

10 B / x is important for the current sheet structure, as well as for the current sheet stability. For example, oscillations of current sheet (so-called flapping oscillations [see Sergeev et al., 2006; Petrukovich et al., 2006, and references therein]) are controlled by the sign of B / x. Theories of the double-gradient instability [Erkaev et al., 2009] and the interchange instability [Pritchett and Coroniti, 2010] predict current sheet oscillations if B / x < 0. Theory of the eigenmodes of TCS with B / x 0 describes relatively fast TCS oscillations [Zelenyi et al., 2009]. In our statistics we were unable to find conditions necessary for the development of the double-gradient or interchange instabilities ( B / x < 0). However, we studied only current sheet at the distance x 20R E, while the observations of reverse B gradient correspond to the near-earth tail [Nakamura et al., 2009; Saito et al., 2010; Panov et al., 2012]. In any case, further investigations of the influence of B / x gradient on TCS stability and structure in theory and observations are required. [39] The various groups of electrons should interact with strong gradients of the magnetic field in TCS in different ways. In the central region of current p sheet B 2 nt and electron gyroradius r en 3km ffiffiffiffiffiffiffiffiffiffiffi ɛ½evš. The TCS thickness L varies from 1000 km to 3000 km and parameter B n /B [Artemyev et al., 2010; Petrukovich p et al., 2011]. Particle motion is controlled by k e ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b n L =r en parameter [Büchner and Zelenyi, 1989]. Thus, for TCS we have k e 5 10(ɛ[eV]) 1/4. For electrons with ɛ > 5 kev we have k e < 1, while for ɛ < 50 ev we have k e > 2. Thus, electrons with ɛ > 5 kev could be considered as nonadiabatic particles according to Büchner and Zelenyi [1989]. For these particles perpendicular and parallel motion could not be separated and identified as it happens with magnetied particles. For this population it is more relevant to use the full pressure tensor rather than the Chu-Goldberger-Low tensor with parallel-perpendicular pressure[ashour-abdalla et al., 1994]. [40] For electrons with energy >5 kev the considered adiabatic mechanism of acceleration cannot play any significant role because these particles should be scattered in TCS, where their adiabatic invariants are not conserved. However, for these particles the first adiabatic invariant (magnetic moment) transforms into the first quasi-adiabatic invariant I ¼ v d [Büchner and Zelenyi, 1989]. The second invariant transforms into the second quasi-adiabatic invariant I x ¼ v x dx [Zelenyi et al., 1990; Vainchtein et al., 2005]. The approximate conservation of these invariants is provided due to existence of a small parameter k e [see Vainchtein et al., 2005, and references therein]. The conservation of I and I x leads to particle heating in the course of earthward convection with the following laws: energy of particles with distant mirror points grows as g 2/5 and energy of particles moving in the vicinity of the neutral plane grows as g * [Zelenyi et al., 1990; Vainchtein et al., 2005]. However, this mechanism could be involved only if convection is fast enough (faster than particle scattering due to the diffusion of invariants I and I x (for details see Vainchtein et al. [2005]). So, although magnetic field does not directly control the motion of such particles, their energy gain occurs similarly to the adiabatic ones. In addition, the effect of perpendicular energy growth can be enhanced due to the betatron electron acceleration by various transient increases of B [Delcourt and Sauvaud, 1994; Birn et al., 2004b; Ashour-Abdalla et al., 2011]. The parallel acceleration of high energy particles could be also associated with the magnetic reconnection [see, e.g., Asano et al., 2010]. [41] The relations between components of the pressure tensor suggest that the consideration based on the adiabatic invariant conservation should provide the more realistic current sheet model for electrons, than the one obtained by the Vlasov approach. However, in order to use m (or I )as integrals of motion one needs to develop the space bounded models. Such current sheet models should have the top and bottom boundaries where magnetic field has the constant values and where the distribution functions are specified. Conservation of invariants should be taken into account to trace these distributions into the central region of current sheet from top and bottom boundaries. This is the drastic difference from models developed based on the Vlasov approach, where velocity distributions are specified in the entire spatial domain. Models of TCS with transient trajectories are well known for ions [see Zelenyi et al., 2004; Sitnov et al., 2006] (and references to previous studies therein). Quasi-adiabatic description of electrons was before applied for TCS model by Sitnov et al. [2006], while in the model of Zelenyi et al. [2004] adiabatic invariant (magnetic moment) was used. It was shown, that even small population of quasi-adiabatic electrons (or adiabatic, but anisotropic electrons) produces strong current and modifies inner structure of current sheets. Because quasi-adiabatic electrons seem to be underestimated as a source of current density and corresponding thinning of current sheet, further investigation of their role is necessary. [42] We need also to mention the important role of geomagnetic activity in breaking down the electron adiabaticity. Indeed, magnetic field fluctuations related to various mesoscale transient processes (BBF, plasmoids, etc.) in the magnetotail should result in violation of conservation of adiabatic invariants. Although, this effect seems to be the most important for ions (because spatial and timescales of ion motion are comparable with corresponding scales of transient processes), influence on electron dynamics and acceleration also could be substantial. Variations of magnetic field in mesoscale structure have timescales around tens of seconds and, thus, can be considered as adiabatically slow perturbation for fast electrons. In this case electron temperature profiles collected for several minutes during fast crossings would have slightly nonmonotonous character (indeed, see Figure 3). This effect is not so important for slow crossings, where averaging over a large number of points results in more regular profile. These mesoscale structures often change the topology of magnetic field in the magnetotail and thus should significantly change the electron convection as well as electron adiabatic heating. We plan to develop model defining profiles of electron temperature for the magnetotail with local closed field line structures in some future study. 8. Conclusions [43] In this paper we consider the electron component of current sheet observed by Cluster mission. We demonstrate that 10 of 13

11 [44] 1. Both components of electron temperature are described by parabolic dependence on B x, T ek,? = T ek,?,max (1 a Tk,? (B x /B ext ) 2 ) with averaged values a T? = , a Tk = The vertical profiles of electron anisotropy can be approximated by a constant T ek /T e? without dependence on B x. [45] 2. Electron component could be divided into three populations: the cold isotropic core (ɛ < 50 ev), the anisotropic intermediate population with T ek /T e? > 1 (3 kev > ɛ > 50 ev), the hot tail with small variable anisotropy (even T ek /T e? < 1). [46] 3. The model of adiabatic electron heating for intermediate-energy electrons qualitatively describes dependence of T e on B x as well as constant profiles T ek /T e? (B x ) both in TCS and in 2DCS. [47] 4. Ratio of spatial scales in 2DCS L x / L 5 is much smaller than in TCS, where L x / L 25. As a result, 2DCS have more substantial decrease of temperature with increase of B x in comparison with TCS. [48] 5. Electron pressure tensor in TCS is gyrotropic and p k p xx. Appendix A: Field Lines Equations [49] In this appendix we derive the expressions for field lines for two current sheet models (TCS and 2DCS). We start with the TCS model (see equation (1)). A field line is defined by the standard expression (d/b (x) =dx/b x ()): B n L x B 0 L xz obs=l x x=l x ð x Þ h dx ¼ Z*=L ð Þd =L Here the point (*, x obs ) is magnetically conjugated to the point (0, x*) in the neutral plane. After integration one can obtain ð x=l x Þ 1 h ¼ 1 þ ð1 hþ B 0L 2B n L x * 2 =L ð =L Þ 2.The corresponding relation between the field components is B* x = B x (*) = B 0 (g h/(1 h) * 1) 1/2 /L 1/2 TCS, where L TCS ¼ ð1 hþ B 0L 2B n L x and g * = B* /B n = B (x*)/b n. We also introduce the following coefficient B * 0 1 1=2 g 1 h h r TCS ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðb * ¼ g * * x Þ2 þ B 2 b 2 n L þ 1A TCS n For 2DCS we have B x (x, ) andb (x, ) from equation (2). In this case the field line is defined by the expression (d/b (x, ) =(dx/b x (x, )): B n L x B 0 L xz obs=l x x=l x ð x Þ 1 dx ¼ Z*=L =L After integration we obtain x ¼ 1 2 L x L L 2 ¼ L L 2 B 0 L 2BnLx h d where x* L x ¼ h 1 L. Therefore, one can write the 2 relation between magnetic field components at points of its intersection with plane (x*, 0) and with vertical slice (x obs, *): B * h x* x*; 0 ¼ Bn ¼ B n g L x * vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! B * u x x obs ; * ¼ B0 1 x* 1 h rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t ¼ B 0 1 g h 1 h : ða1þ * L x [50] From equation (A1) one can obtain the ratio rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B * x =B* ¼ ðb 0 =B n Þ 1 g h 1 h * g * 1 We introduce following parameter: B * 0 1 r 2DCS ¼ r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B * ¼ g 1 g 2h 1 h * * 2 b 2 þ 1A x þ B 2 n n 1=2 Finally the expression for magnetic field components along field line can be written as: B ¼ B n 1 2 L 2 gx ðþ¼b n 1 2 L 2 g 1 h h 1 ¼ B n gg * =g h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B x ¼ B 0 g h 1 h 1 h h ¼ B 0 1 g L * =g g h 1 h 1 ¼ B 0 g h 1 h 1 2 * L L 2 ða2þ And the ratio is B x =B ¼ ðb 0 =B n Þg 1 h * L 1 2 L 2 2 h h 1. Appendix B: Tracing Along the Neutral Plane [51] In this appendix we derive the expressions for particle velocity components as a function of the magnetic field in the neutral plane. In the point with coordinates x = x obs and =0 magnetic field is B x =(x obs,0) = B n and velocity components are (v?, v k ). At any other point in the neutral plane we have magnetic field B (x*,0) = B* = B n g * and velocity components (u?, u k ). The convection of particles can be considered as a slow process (in comparison with particle thermal motion). In the course of convection two invariants are conserved: the magnetic moment m v 2? /B and the second adiabatic invariant J k ¼ v k ds (ds is an element of the field line length). For TCS we can write two equations for (u?, u k ): 8 >< u 2? ¼ g * v2?!! F TCS 1 þ v2 k v 2 ¼ g 3=2 * F TCS 1 þ u2 k ðb1þ >:? u 2? Here F TCS is defined as F TCS ðaþ ¼ Z a b min pffiffiffiffiffiffiffiffiffiffiffi a bgb ð Þdb ðb2þ where b = B / B* and function G(b)db = B 0 ds/b *L is defined along field lines. The bottom limit of the integration can be 11 of 13

12 We write magnetic field components as functions of coordinate (see equation (A2)) and use ds =(B/B )d to obtain equations for (u?, u k ): 8 >< u 2? ¼ g * v2?!! F 2DCS 1 þ v2 k v 2 ¼ g 1=2 * F 2DCS 1 þ u2 k ðb4þ >:? u 2? Here F 2DCS (a) is defined as where Z max F 2DCS ðaþ ¼ 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a h 1 Q ð ÞQ ð Þd sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 h Q ðþ¼ L 1 h g * 2 ð1 2 Þ 2 h 1 þ 1 2 : Q ð max Þ 1 2 h 1 max ¼ a Figure B1. Functions F 2DCS and F TCS for various values of b n and h. found from equation G 1 (b min ) = 0. Following to Tverskoy [1969] and Zelenyietal.[1990] we assume that for a given bounce oscillation B changes weakly and one can use B B*. The combination of equation B ds = Bd (ds = bd) and equation db =( b/ )d gives G(b) =(B 0 /B *)(L b/ ) 1 b. Expression for G(b) acquires the form: Gb ð Þ ¼ B 0 b 1 B * ð=l Þ b ¼ 0 fl 1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B x B 2 B u ð=l Þ B x t B * 2 þ 1C A 0 1 fl B1 B x b B * 2 A 1 fl 1 b B * ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi b 2 1 Eventually we can rewrite equation (B2): Z a F TCS ðaþ ¼ 1 b 2 B 0 B * pffiffiffiffiffiffiffiffiffiffiffi a b pffiffiffiffiffiffiffiffiffiffiffiffiffi b 2 db b 2 1 b ¼ ðb3þ This is a well known expression, derived by Tverskoy [1969]. One can easily obtain simple estimates for two limits. For particles with a large pitch angle (these particles drift in the neutral plane) we have a 1andF a 1. Substituting this expression into equation (B1) we obtain the expression for energy u? 2 g. Therefore, perpendicular temperature would grow approximately as g *. For particles with distant mirror points we have a 1andF a 5/2.Inthiscaseweobtain u k 2 g 2/5 according to Tverskoy [1969] and Zelenyi et al. [1990]. [52] The expressions for 2DCS is defined by equation (2). In this case we cannot use variable b = B/B* because component B changes essentially during each bounce oscillation. [53] We plot F TCS and F 2DCS as functions of a for various values of b n and h (Figure B1). Profile F TCS has an asymptote a 5/2. Function F 2DCS grows with a slower than F TCS. This effect results in a more substantial dependence of the parallel temperature on g * for 2DCS. Appendix C: Method of Mapping [54] To obtain vertical profiles of electron temperature T e (B x ) one needs to map profiles of q e? (g ), q e k (g ) along field lines. We use the method described by [see also Whipple et al., 1991, and references therein]. The point with B x *=B x (x obs, *) and B = B n is conjugate to some point at the neutral plane with B x = 0 and B *=B n g. In this point at the neutral plane we have the anisotropic Maxwellian velocity distribution: ( u 2 f np u? ; u k exp? m e u2 k m ) e : 2q e? 2q ek [55] This distribution can be rewritten as a function of two invariants: the magnetic moment m u? 2 /B * w? 2 / (B n 2 +(B x *) 2 ) 1/2 and the total energy u? 2 + u k 2 = w? 2 + w k 2, where (w?, w k ) are velocity components in the point with B x (x obs, *) and B = B n : ( w 2 f w? ; w k exp? m e r þ q e? ð1 rþ 2q e? q ek w2 k m e 2q ek where r = B */((B x *) 2 + B n 2 ) 1/2. After integration we have: T e? ¼ q e? r þ q e? ð1 rþ q ek 1 T ek ¼ q ek The ratio is T ek =T e? ¼ 1 þ r q ek q e? 1. Coefficient r was determined in Appendix A for TCS (r TCS ) and for 2DCS (r 2DCS ). 12 of 13

13 [56] Acknowledgments. Authors would like to acknowledge Cluster Active Archive and Cluster instrument teams, in particular FGM and PEACE for excellent data. Comment and suggestions of both reviewers are appreciated. The work of A.V.A., A.A.P. and L.M.Z. was supported in part by the RF Presidential Program for the State Support of Leading Scientific Schools (project NSh ), the Russian Foundation for Basic Research (projects , ). The work of R.N. was supported by Austrian Science Fund (FWF) I429-N16. A.V.A. would like to acknowledge hospitality of IWF, Gra, Austria. [57] Masaki Fujimoto thanks the reviewers for their assistance in evaluating this paper. References Apatenkov, S. V., et al. (2007), Multi-spacecraft observation of plasma dipolariation/injection in the inner magnetosphere, Ann. Geophys., 25, Artemyev, A. V., A. A. Petrukovich, R. Nakamura, and L. M. Zelenyi (2010), Proton velocity distribution in thin current sheets: Cluster observations and theory of transient trajectories, J. Geophys. Res., 115, A12255, doi: /2010ja Artemyev, A. V., L. M. Zelenyi, A. A. Petrukovich, and R. Nakamura (2011a), Hot electrons as tracers of large-scale structure of magnetotail current sheets, Geophys. Res. Lett., 38, L14102, doi: / 2011GL Artemyev, A. V., A. A. Petrukovich, R. Nakamura, and L. M. Zelenyi (2011b), Cluster statistics of thin current sheets in the Earth magnetotail: Specifics of the dawn flank, proton temperature profiles and electrostatic effects, J. Geophys. Res., 116, A09233, doi: /2011ja Asano, Y., et al. (2010), Electron acceleration signatures in the magnetotail associated with substorms, J. Geophys. Res., 115, A05215, doi: / 2009JA Ashour-Abdalla, M., L. M. Zelenyi, V. Peroomian, and R. L. Richard (1994), Consequences of magnetotail ion dynamics, J. Geophys. Res., 99, 14,891 14,916. Ashour-Abdalla, M., et al. (2011) Observations and simulations of nonlocal acceleration of electrons in magnetotail magnetic reconnection events, Nat. Phys., 7, Balogh, A., et al. (2001), The Cluster magnetic field investigation: Overview of in-flight performance and initial results, Ann. Geophys., 19, Baumjohann, W., et al. (2007), Dynamics of thin current sheets: Cluster observations, Ann. Geophys., 25, Birn, J., K. Schindler, and M. Hesse (2004a), Thin electron current sheets and their relation to auroral potentials, J. Geophys. Res., 109, A02217, doi: /2003ja Birn, J., M. F. Thomsen, and M. Hesse (2004b), Electron acceleration in the dynamic magnetotail: Test particle orbits in three-dimensional magnetohydrodynamic simulation fields, Phys. Plasmas, 11, Büchner, J., and L. M. Zelenyi (1989), Regular and chaotic charged particle motion in magnetotaillike field reversals: 1. Basic theory of trapped motion, J. Geophys. Res., 94, 11,821 11,842. Delcourt, D. C., and J. A. Sauvaud (1994), Plasma sheet ion energiation during dipolariation events, J. Geophys. Res., 99, Erkaev, N. V., V. S. Semenov, I. V. Kubyshkin, M. V. Kubyshkina, and H. K. Biernat (2009), MHD model of the flapping motions in the magnetotail current sheet, J. Geophys. Res., 114, A03206, doi: / 2008JA Fu, H. S., Y. V. Khotyaintsev, M. André, and A. Vaivads (2011), Fermi and betatron acceleration of suprathermal electrons behind dipolariation fronts, Geophys. Res. Lett., 38, L16104, /2011GL Johnstone, A. D., et al. (1997), Peace: A plasma electron and current experiment, Space Sci. Rev., 79, Lyons, L. R. (1984), Electron energiation in the geomagnetic tail current sheet, J. Geophys. Res., 89, Motte, F. (2003), Exact nonlinear analytic Vlasov-Maxwell tangential equilibria with arbitrary density and temperature profiles, Phys. Plasmas, 10, Nakamura, R., et al. (2009), Evolution of dipolariation in the near-earth current sheet induced by Earthward rapid flux transport, Ann. Geophys., 27, Nicholson, R. B. (1963), Solution of the Vlasov equations for a plasma in an externally uniform magnetic field, Phys. Fluids, 6, Panov, E. V., et al. (2012) Kinetic ballooning/interchange instability in a bent plasma sheet, J. Geophys. Res., doi: /2011ja017496, in press. Paschmann, G., and S. J. Schwart (2000), ISSI book on analysis methods for multi-spacecraft data, in Cluster-II Workshop Multiscale/Multipoint Plasma Measurements, Eur. Space Agency Spec. Publ., ESA SP-449, Petrukovich, A. A., et al. (2006), Oscillatory magnetic flux tube slippage in the plasma sheet, Ann. Geophys., 24, Petrukovich, A. A., A. V. Artemyev, H. V. Malova, V. Y. Popov, R. Nakamura, and L. M. Zelenyi (2011), Embedded current sheets in the Earth magnetotail, J. Geophys. Res., 116, A00I25, doi: / 2010JA Pritchett, P. L., and F. V. Coroniti (2010), A kinetic ballooning/interchange instability in the magnetotail, J. Geophys. Res., 115, A06301, doi: / 2009JA Runov, A., et al. (2006), Local structure of the magnetotail current sheet: 2001 Cluster observations, Ann. Geophys., 24, Saito, M. H., L.-N. Hau, C.-C. Hung, Y.-T. Lai, and Y.-C. Chou (2010), Spatial profile of magnetic field in the near-earth plasma sheet prior to dipolariation by THEMIS: Feature of minimum B, Geophys. Res. Lett., 37, L08106, doi: /2010gl Sergeev, V. A., et al. (2006), Survey of large-amplitude flapping motions in the midtail current sheet, Ann. Geophys., 24, Sergeev, V. A., V. Angelopoulos, M. Kubyshkina, E. Donovan, X.-Z. Zhou, A. Runov, H. Singer, J. McFadden, and R. Nakamura (2011), Substorm growth and expansion onset as observed with ideal groundspacecraft THEMIS coverage, J. Geophys. Res., 116, A00I26, doi: /2010ja Sharma, A. S., et al. (2008), Transient and localied processes in the magnetotail: A review, Ann. Geophys., 26, Sitnov, M. I., M. Swisdak, P. N. Gudar, and A. Runov (2006), Structure and dynamics of a new class of thin current sheets, J. Geophys. Res., 111, A08204, doi: /2005ja Stiles, G. S., E. W. Hones Jr., S. J. Bame, and J. R. Asbridge (1978), Plasma sheet pressure anisotropies, Geophys. Res. Lett., 83, Tverskoy, B. A. (1969), Main mechanisms in the formation of the Earth s radiation belts, Rev. Geophys., 7, Vainchtein, D. L., J. Büchner, A. I. Neishtadt, and L. M. Zelenyi (2005), Quasiadiabatic description of nonlinear particle dynamics in typical magnetotail configurations, Nonlinear Processes Geophys., 12, Wang, C., L. R. Lyons, R. A. Wolf, T. Nagai, J. M. Weygand, and A. T. Y. Lui (2009), Plasma sheet PV 5/3 and nv and associated plasma and energy transport for different convection strengths and AE levels, J. Geophys. Res., 114, A00D02, doi: /2008ja Whipple, E., R. Puetter, and M. Rosenberg (1991), A two-dimensional, time-dependent, near-earth magnetotail, Adv. Space Res., 11, Zelenyi, L. M., D. V. Zogin, and J. Büchner (1990), Quasiadiabatic dynamics of charged particles in the tail of the magnetosphere, Cosmic Res., 28, Zelenyi, L. M., H. V. Malova, V. Y. Popov, D. Delcourt, and A. S. Sharma (2004), Nonlinear equilibrium structure of thin currents sheets: Influence of electron pressure anisotropy, Nonlinear Processes Geophys., 11, Zelenyi, L. M., A. V. Artemyev, A. A. Petrukovich, R. Nakamura, H. V. Malova, and V. Y. Popov (2009), Low frequency eigenmodes of thin anisotropic current sheets and Cluster observations, Ann. Geophys., 27, Zelenyi, L. M., A. V. Artemyev, and A. A. Petrukovich (2010), Earthward electric field in the magnetotail: Cluster observations and theoretical estimates, Geophys. Res. Lett., 37, L06105, doi: / 2009GL Zelenyi, L. M., H. V. Malova, A. V. Artemyev, V. Y. Popov, and A. A. Petrukovich (2011), Thin current sheets in collisionless plasma: Equilibrium structure, plasma instabilities, and particle acceleration, Plasma Phys. Rep., 37, Zwingmann, W. (1983), Self-consistent magnetotail theory: Equilibrium structures including arbitrary variation along the tail axis, J. Geophys. Res., 88, of 13