Hot electrons as tracers of large-scale structure of magnetotail current sheets



[1] This paper is devoted to the investigation of the vertical electron temperature distribution in thin currents sheets of the Earth magnetotail. The dependence of electron temperature on magnetic field Te(Bx) is examined for 62 thin currents sheet crossings by the Cluster spacecraft. The profiles Te(Bx) are approximated by the simple expression TeTemax(1 − α(Bx/Bext)2), where α is a constant parameter, Temax is the maximum value of electron temperature and Bext is the amplitude of Bx obtained from the vertical pressure balance. The mean value of the parameter 〈α〉 ≈ 1. The profiles Te(Bx) are described in the frame of the theory of adiabatic electron heating in the course of the Earthward plasma convection. The comparison between the observed values of α and the theoretical predictions allows to estimate the scale Lx of observed current sheets along the magnetotail (Bz ∼ (−x/Lx)h) and the ratio of Lx and current sheet thickness Lz. For the majority of the observed thin current sheets Lx ∈ [5RE, 20RE] and the average ratio of the scales 〈Lx/Lz〉 ≈ 25.

1. Introduction

[2] Thin current sheets (TCSs) are one of the key elements of the magnetosphere, in particular their formation and stability determine the dynamics of the entire magnetotail. The detailed investigation of TCS started with the observations of the dual ISEE mission [see Sergeev et al., 1993, and references therein], however, the most complete information was collected by the multispacecraft Cluster mission [Asano et al., 2005; Runov et al., 2006; Nakamura et al., 2006]. One of the main properties of TCS is it's embedding in the thick background plasma sheet with small current density. The maximum TCS magnetic field B0 is not equal to the magnetic field magnitude in the lobes Bext, and consequently current density profiles have strong narrow maxima embedded inside the more diffuse plasma density profiles [Asano et al., 2005; Artemyev et al., 2010].

[3] Up to now investigations were devoted mainly to the structure of TCS along the normal direction and to the properties of the their ion component. It was shown that the model profiles of current density approximate the observed profiles with reasonably good accuracy in a wide range of main TCS parameters [Sitnov et al., 2006; Artemyev et al., 2008]. The essential role of protons at transient orbits for TCS formation under quiet conditions has been revealed [Artemyev et al., 2010]. The spatial scale of TCS along the normal direction Lz was estimated to be about a few proton gyroradii in the B0 field basing on the statistical study of Petrukovich et al. [2011] in a good agreement with the theoretical estimates (see discussion and references of Artemyev et al. [2010]).

[4] Although, the electron component of TCS is included in the recent models [Zelenyi et al., 2004; Sitnov et al., 2006] and is shown to be of a primary importance, it is less studied in observations. Nevertheless, it was shown that electrons can carry most of the measured electric current in many TCSs due to the cross-field drift [see Zelenyi et al., 2010; Artemyev et al., 2010, and references therein]. The majority of electrons is bouncing between magnetic mirrors inside the plasma sheet. In the course of the Earthward convection magnetized electrons traverse the large distance along the Earth-Sun direction inside TCS and their characteristics depend on the spatial distribution of magnetic field Bz(x) [Zelenyi et al., 2010]. In particular, electrons are heated during the Earthward convection due to the presence of the dawn-dusk large-scale electrostatic field Ey and the gradient δBz/δx > 0 [Lyons, 1984; Zelenyi et al., 1990]. If the magnetic field lines in the magnetotail have the form of a very stretched parabolas, then a spacecraft at some distance from the neutral plane might observe electrons, coming from the far downtail region. Therefore, a vertical profile of the electron temperature obtained from a single TCS crossing can be used to reveal the large-scale structure of TCS related to the gradient ∂Bz/∂x. In this paper we develop the method for such an analysis.

2. Electron Temperature in TCS

[5] We use the statistics of 62 TCS crossings by Cluster during 2001, 2002 and 2004 years. We select crossings with at least several measurements of the electron temperature from the list of crossings, studied in previous papers [Zelenyi et al., 2010; Artemyev et al., 2010]. We use electron moments from the PEACE instrument on C2 [Owen et al., 2001] and the magnetic field from the FGM instrument from all spacecraft [Balogh et al., 2001]. All data are taken from the Cluster Active Archive ( We use the GSM coordinate system.

[6] All TCS from our statistics are crossed under relatively quiet conditions. We consider only almost horizontal TCS: the normals are directed along z, the current density vectors jcurl = (c/4π)curlB are directed along y and the main magnetic field component is Bx. For each TCS we estimate the magnitude of the magnetic field B0 (the method of B0 determination is described by Artemyev et al. [2010]) and the value of Bext = equation image (here p is the total plasma pressure in the region with Bx ∼ 0). We determine the TCS thickness as Lz = cB0/(4πmax∣jcurl∣). For each TCS we plot the profile of the relative electron temperature Te/Temax as a function of the dimensionless magnetic field Bx/Bext (six profiles from our statistics are presented in Figure 1). Since electron temperature Te decreases towards the TCS boundaries and has a symmetric profile we can approximate it by the simple expression Te = Temax(cTα(Bx/Bext)2) and determine coefficients cT and α by the least squares method. For each TCS we determine the electron anisotropy Te/Te. The range for our statistics Te/Te ∈ [1, 1.6] and the average value 〈Te/Te〉 ≈ 1.2 according to the previous estimates [Zelenyi et al., 2010]. For almost all observations Te/Te is only slightly dependent on Bx inside TCS (where ∣Bx∣ < B0) and could be considered as a constant.

Figure 1.

Profiles of electron temperature versus normalized magnetic field (grey circles) and the approximation Te = Temax(cTα(Bx/Bext)2) (black curves) for six samples of TCS crossings.

3. Heating of Electrons During Earthward Convection

[7] For all TCS in our statistics electron temperature decreases with the increase of the local magnetic field (see Figure 1): at some distance from the neutral plane electrons are cooler than in the neutral plane, where Bx ∼ 0. Because at the certain Bx ≠ 0 the spacecraft observes electrons coming along field lines from the downtail region, the vertical profile of the electron temperature can be understood as a mapping of the electron temperature profile along the magnetotail (see Figure 2). To describe the observed profiles Te(Bx) we should take into account the mechanism of electron heating during the steady Earthward convection in the magnetotail [Lyons, 1984; Zelenyi et al., 1990]. This mechanism is based on the conservation of the first and second adiabatic invariants of motion (so-called betatron and Fermi accelerations).

Figure 2.

The scheme of the system geometry.

[8] To obtain model vertical profiles Te(Bx) for TCS with a finite Bz depending on x and ∣Bx∣ < B0 we use the following procedure:

[9] 1. We assume that the magnetic field component normal to the current sheet is distributed along the magnetotail as equation imagez(x) = Bzg(x). We use g(x) = (−x/Lx)h, where Lx is the gradient scale along magnetotail axis and the index h ≈ −0.8 is chosen according to the earlier observations for x < −15RE [see Birn et al., 1977, and references therein]. Value x = xobs corresponds to the TCS crossing by spacecraft and equation imagez(xobs) = Bz.

[10] 2. We use the approximation of TCS Bx(z)/B0z/Lz embedded inside the thick plasma sheet. The equation of field lines is dx/Bx(z) = dz/equation imagez(x) or

equation image

After integration we get equation image(z/Lz)2 = (BzLx/LzB0)((−x/Lx)h+1 − 1) and, as a result:

equation image

Here bx = Bx(z)/Bext, s = equation imagebe2(B0Lz/BzLx), and be = Bext/B0. For each point (xobs, bx) equation (2) determines the conjugate point (x, 0) at the neutral plane. The TCS boundary (x = xobs, Bx = B0) is conjugate to the point −x*/Lx = (1 + sbe−2)equation image, see Figure 2.

[11] 3. We need to obtain the electron distribution function fe at each point x ∈ [xobs, x*]. We assume that at x = xobs, Bx = 0 electrons have the anisotropic Maxwellian distribution fe(v0⊥, v0∥) ∼ exp(−mev0⊥2/(2Te) − mev0∥2/(2Te)), where Te and Te are directly measured electron temperature in the center region of TCS.

[12] 4. To recalculate fe for x < xobs we use adiabatic invariants. The conservation of magnetic moment μ = mev2/2equation imagez(x) gives the expression for v2:

equation image

To take into account the conservation of J = ∮vds (ds is the element of a length along the field line), we introduce the function Φ(a) = equation imagey2dy according to [Tverskoy, 1969]:

equation image

From equations (3) and (4) we obtain v0∥, v0⊥ as functions of v, v and g(x). Therefore, the velocity distribution at points of the neutral plane with x < xobs acquires the form:

equation image

[13] 5. Electron motion away from the neutral plane along a field line during a given bounce period occurs with the constant energy (v2 + v2 = const) and magnetic moment μ [Zelenyi et al., 1990]. Taking into account the dependence x = x(bx) from equation (2) we present velocity components v, v as functions of velocity components equation image, equation image at the point (xobs, bx): v(equation image, equation image, bx) = g(x(bx))equation image/equation image and v(equation image, equation image, bx) = equation image, and substitute these expressions into equation (5). Integration of the obtained velocity distribution over (equation image, equation image) gives the required profiles Te(Bx) = equation image(Te(Bx) + 2Te(Bx)) (Figure 3).

Figure 3.

Theoretical profiles of electron temperature versus magnetic field for various values of the system parameters.

[14] Figure 3 shows that the electron temperature derived theoretically has the profile similar to parabolic one and decreases to the TCS boundary in an agreement with the Cluster observations shown in Figure 1.

4. TCS Spatial Scales

[15] For all TCS from our statistics we use the approximation Te = Temax(cTα(Bx/Bext)2). The obtained coefficients cT are close to unity (Temax corresponds to a value of Te near the field reversal, where Bx ∼ 0). The distribution of the coefficient α is presented in Figure 4a. To estimate Lx for each TCS we obtain α from our model using the same approximation of Te(Bx) as for observations. For the model profiles we take the values of be, Bz, Te, Te and Lz determined from the observations. We vary the model value of Lx until the difference between the model and the observed values of α is smaller than α/10 (Lx does not change significantly if we take this difference less than α/10). The distribution of obtained values of Lx is presented in Figure 4b. The majority of observed TCS have the scales in the range 5RE < Lx < 20RE and the average ratio Lx/Lz ∼ 25 (Figure 4c).

Figure 4.

(a) The observed distribution of the coefficient α; (b) the spatial scale Lx in RE; (c) the ratio Lx/Lz; (d) the parameter λ = 2BzLx/B0Lz.

5. Discussions and Conclusions

[16] We can conclude that the model of electron adiabatic heating in the magnetotail could be used for the description of electron temperature profiles obtained by Cluster spacecraft. This model enables to estimate the spatial scale of Bz variation, Lx, using the data from a single TCS crossing. The scale Lx is an important characteristic of TCS, useful for the validation of theoretical models as well as for general understanding of the magnetotail structure and evolution. The first estimates of Lx could be obtained also with an even simpler approach. We always can expand g(x) around xobs and use only a linear term: g(x) = 1 − Δx/Lx = 1 − (Δxx*)(1 − (Bz*/Bz)), where Δx = xobsx, Δx* = xobsx* and Bz* = equation imagez(x*). Here x = x* is the conjugate point of TCS boundary z = Lz. The corresponding expression of field line can be obtained from equation (1): equation image(z2/Lz)B0 = BzΔx(1 − equation imagexx*)(1 − (Bz*/Bz))). Specifically for x = x*, we obtain Δx*/Lz = (B0/Bz)(1 + (Bz*/Bz))−1 and Lx/Lz = (B0/Bz)(1 − (Bz*/Bz)2)−1. Value of Bz*/Bz could be estimated as ∼Te/Temax = 1 + αbe−2 according to μ conservation. For the majority of observed TCS Δx* ∼ 3 − 5RE and Lx ∼ 15 − 25RE in agreement with Figure 4.

[17] The obtained values of Lx can be used for the estimates of the TCS pressure balance. Figure 4d demonstrates that for the most of the observed TCS the parameter λ = 2BzLx/B0Lz > 1. This parameter determines the ratio between the two components of the plasma pressure in the stress balance equation [see Burkhart and Chen, 1993, and references therein]. If a two-dimensional current sheet is balanced by the gradient along x, then λ ≈ 1. If some part of the plasma pressure is related to the inertia of the transient ions moving along field lines, then λ > 1. The estimates of Lx point out that the role of transient ions in the TCS pressure balance is quite substantial. This conclusion is also independently substantiated by the successful comparison of TCS models including transient particles [Zelenyi et al., 2004; Sitnov et al., 2006] and the observations of TCS in the magnetotail [Artemyev et al., 2008].

[18] The applicability of our approach should strongly depend on three main factors: the absence (or weakness) of other mechanisms of electron heating for given TCS, the absence of a magnetic loop structures within the interval x ∈ [xobs, x*] (weakness of magnetic field fluctuations) and the Earthward convection of electrons. The latter condition could be violated during the late growth phase when the convection in the TCS is significantly reduced in the near Earth region [see Sergeev et al., 2011, and references therein]. However, this effect likely does not play an essential role in more distant tail region with x < −15RE, where our statistics are observed.

[19] In this paper we do not consider the model description of the electron anisotropy (ratio Te/Te is used only as an input parameter). In sheets from our statistics Te/Te almost does not depend on Bx when ∣Bx∣ < B0 and 〈Te/Te〉 ≈ 1.2. This fact suggests the role of the magnetic field fluctuations in electron heating for observed TCS could not be significant in a global context, because the presence of fluctuations should necessarily lead to the isotropozation [Lyons, 1984]. However, in many other situations particle acceleration due to the interaction with electromagnetic turbulence and mesoscale transient structures (magnetic islands, vortices, depolarization fronts etc.) can be locally significant, especially for production of energetic particle bursts.

[20] Theoretical models of electron heating during Earthward convection can be modified to reproduce not only profiles Te(Bx) but also profiles Te/Te(Bx). According to the model [Zelenyi et al., 1990], electrons with different pitch-angles gain different energies during the Earthward convection: for electrons in the neutral plane TeTeBz and for electrons with distant mirror points TeTeBz2/5 [Tverskoy, 1969]. The alternative approach assuming strong pitch-angle scattering suggests the universal law TeBz2/3 [Lyons, 1984]. The further comparison between these two theoretical approaches and observed Te/Te could clarify the relative roles of adiabatic effects versus non-adiabtic pitch-angle scattering.

[21] The main results of our paper can be summarized as: (1) electron temperature profiles Te(Bx) for a set of TCS observations could be approximated as TeTemax(1 − α(Bx/Bext)2), where the mean value 〈α〉 ≈ 1; (2) the observed distributions Te(Bx) can be explained by the theory of the electron adiabatic heating in the course of the Earthward magnetotail convection; (3) the comparison between the theory and the observations allows to obtain the following estimates of x spatial scale: Lx ∈ [5RE, 20RE] and the ratio 〈Lx/Lz〉 ≈ 25. We finally suggest that the developed method of the scale estimation based on electron temperature observations can be applied not only for TCS but also for other configurations in which adiabatic effects are dominating.


[22] Authors would like to acknowledge Cluster Active Archive and Cluster instrument teams, in particular FGM and PEACE, for excellent data. The work of L.M.Z, A.V.A. and A.A.P was supported by the RF Presidential Program for State Support of Leading Scientific Schools (project NSh-3200.2010.2.), the Russian Foundation for Basic Research (projects 10-05-91001, 10-02-93114). The work of R.N. was supported by Austrian Science Fund (FWF) I429-N16. A.V.A. would like to acknowledge hospitality of IWF, Graz, Austria. We would like to thank both referees for useful comments.

[23] The Editor thanks Antonella Greco and Victor Sergeev for their assistance in evaluating this paper.