Profiles of electron temperature and B-z along Earth’s magnetotail

. We study the electron temperature distribution and the structure of the current sheet along the magnetotail using simultaneous observations from THEMIS spacecraft. We perform a statistical study of 40 crossings of the current sheet when the three spacecraft THB, THC, and THD were distributed along the tail in the vicinity of midnight with co-ordinates X B ∈ [− 30 R E , − 20 R E ] , X C ∈ [− 20 R E , − 15 R E ] , and X D ∼ − 10 R E . We obtain proﬁles of the average electron temperature h T e i and the average magnetic ﬁeld h B z i along the tail. Electron temperature and h B z i increase towards the Earth with almost the same rates (i.e., ratio h T e i / h B z i ≈ 2 keV/7 nT is approximately constant along the tail). We also use statistics of 102 crossings of the current sheet from THB and THC to estimate dependence of T e and B z distributions on geomagnetic activity. The ratio h T e i / h B z i depends on geomagnetic activity only slightly. Additionally we demonstrate that anisotropy of the electron temperature h T k /T ⊥ i ≈ 1 . 1 is almost constant along the tail for X ∈ [− 30 R E , − 10 R E ] .


Introduction
The current sheet (CS) of the near-Earth magnetotail (−10 R E < X < −30 R E , where R E is the Earth radius) can be approximately considered as a 2-D structure with the main gradient along the normal direction (along Z) and the weak gradient along x-direction (GSM coordinate system is used). Configuration of the magnetotail CS controls the rate of charged particle acceleration and following injection into the inner magnetosphere (see recent reviews by Baumjohann et al., 2007;Sergeev et al., 2012, and references therein). With the multi-spacecraft Cluster mission, transverse (along Z) structure of the CS was studied in detail (especially at the distances X ∼ −20 R E , where Cluster collected main statistics; see Runov et al., 2006;Artemyev et al., 2011b). Although the distribution of the magnetic field along the tail plays an important role in particle (mainly electron) acceleration, it is less investigated.
Electrons are magnetized in the magnetotail, and their motion is controlled by the global gradients of the magnetic field. One of the most effective mechanisms of electron energization is adiabatic heating in the course of the earthward convection (Lyons, 1984;Zelenyi et al., 1990). This mechanism predicts monotonic growth of electron temperature with B z (X) (i.e., with X). However, recent observations and the numerical modeling suggest that a substantial role in electron energization could be played by various transient processes (see Asano et al., 2010;Fu et al., 2011;Ashour-Abdalla et al., 2011;Fu et al., 2012;Birn et al., 2012, and references therein). To compare impacts of adiabatic heating (throughout the paper, we use this term for global betatron acceleration) and transient acceleration to the electron energization, one needs to study gradient ∂/∂X of the electron temperature and magnetic field B z along the magnetotail.
So far, estimates of the gradient ∂/∂X in the magnetotail have been obtained by using three independent approaches: (1) small-scale gradients of the near-Earth dipolarized CS can be estimated using direct measurements of Cluster mission in 2007(Nakamura et al., 2009 or THEMIS mission in the case of specific spacecraft configuration (Saito et al., 2010;Panov et al., 2012); (2) statistical investigation can give average profiles of the main CS parameters (B z component, plasma pressure and plasma density) along the tail (see statistics collected by AMPTE/IRM, Geotail and THEMIS spacecraft in Kan and Baumjohann, 1990;Wang et al., 2012) tracers of the magnetotail structure (Artemyev et al., 2011c). The distribution of electron temperature along the magnetotail was studied only using statistical investigation (e.g., Wang et al., 2012). Each of these methods has principal disadvantages for the determination of the gradient ∂/∂X in the magnetotail: (1) direct calculation of ∂/∂X is possible only for relatively strong gradients in the dipolarized CS; (2) average profiles cannot give snapshot-like information about the magnetotail structure (the latter is important because major tail parameters vary in a wide range on timescales from minutes to hours); and (3) to use electrons as tracers, one needs realistic models of the electron heating and detailed information about the transverse structure of CS.
Therefore, existing approaches cannot be used for investigation of the relation between T e (X) and B z (X) profiles and, as a result, cannot help us study the role of electron adiabatic heating for general electron energization in the magnetotail. To obtain distribution of CS parameters along the magnetotail, the only reliable way is to organize simultaneous observations at substantially different x-coordinates. In this paper we use the statistics of simultaneous observations of thin CSs from several THEMIS spacecraft to study the relation between T e (X) and B z (X) distributions.

Dataset
In this study we use two datasets, which consist of THEMIS and Cluster observations, respectively. THEMIS data provide us with simultaneous observations of the magnetotail CS at different x-coordinates, while the Cluster dataset contains observations in the magnetotail for 2001-2009 from the C2 spacecraft and gives average values of T e for several ranges of X. Due to small separation of Cluster spacecraft, we cannot use Cluster measurements for simultaneous observations of CS at different x-coordinates. The Cluster dataset includes measurements only with |B x | < 10 nT and |Y | < 5 R E (this is essentially the same dataset as the one used by Artemyev et al., 2011a).
The THEMIS statistics includes 102 events, when two spacecraft THB and THC crossed the magnetotail CS within 30 min (we consider 2008 and 2009). For 40 events from this list, THD also crossed the magnetotail CS within the same time interval. Thus, we have substatistics of 40 events with simultaneous observations at three different x-coordinates. All 102 events can be attributed to CSs with the duration of crossing varying from tens of seconds (for THB) up to tens of minutes (for THD). We cannot estimate thickness of observed CSs directly and use a criterion of rapid crossing for THB to select mainly thin CSs. For this dataset we use the magnetic field (Auster et al., 2008) and the electron measurements (McFadden et al., 2008). All data are obtained from the CDAWeb database (http://cdaweb.gsfc.nasa.gov/). For all crossings |Y | coordinate is smaller than |X| (i.e., all crossings occur approximately around the midnight).
A typical example of an event from THEMIS statistics is shown in Fig. 1. Grey color denotes CS crossings by the four spacecraft. THB spacecraft crossed CS; then THC and THE (as well as THD) crossed B x = 0 in agreement with the distribution of z-coordinates of spacecraft. Synchronous crossings correspond to the vertical flapping motions of the CS (see the detailed study of CS flapping observed by several THEMIS spacecraft in Runov et al., 2009).
For each CS crossing from THEMIS dataset, we use the average values of T e , T /T ⊥ , and B z component of the magnetic field computed in the central region of CS (|B x | < 10 nT), where T e = (T + 2T ⊥ )/3, and T and T ⊥ represent diagonal components of the electron temperature tensor (smaller than |B x | < 10 nT region of averaging sometimes leads to only a few points of electron data).

Distribution of CS parameters along the tail
We start with the statistics of 40 crossings of the magnetotail CS by three THEMIS spacecraft. For each event we have three values of T e and B z measured at corresponding coordinates X. Four examples of T e and B z distributions along the tail are shown in Fig. 2. One can see that the profiles T e (X) almost coincide with the profiles B z (X).
The profiles T e and B z along the tail for the whole THEMIS dataset are shown in Fig. 3a and c. There is a general increase of the magnetic field and the electron temperature with X (i.e., in the earthward direction). At the large distance where THB crossed the CS (X ∼ −25 R E ), the value of B z is very small (B z ∼ 1 nT). Therefore, any deformations of the CS could result in substantial errors in estimation of B z , Ann. Geophys., 31, 1109-1114, 2013 www.ann-geophys.net/31/1109/2013/  when the normal direction to the CS is tilted in the (X, Z) plane. This effect is clearly seen in Fig. 3f, where profiles T e as function of B z are shown. T e mainly varies from 200 to 500 eV up to 1.5-2 keV, while B z changes within the range 2-10 nT. However, there are several observations of small B z with relatively large T e shown by the grey circle in Fig. 3f. As a result, we cannot use the relation T e / B z ≈ const for the individual CS crossings.
To obtain a more reliable presentation describing profiles T e and B z , we average our statistics for four intervals of X ∈ [−10, −25] R E (see Fig. 3b, d). Although averaging results in a loss of information regarding the peculiarities of individual cases, this operation is more accurate than collection of Cluster statistics because it takes into account only monotonic profiles of T e and B z along the tail. For B z we obtain the following approximation based on the averaging at four intervals of X: B z ≈ 7 nT · (−1 − X/5 R E ) −1 . The corresponding approximation of the electron temperature is T e ≈ 2 keV · (−1 − X/5 R E ) −1 . These two approximations give the empirical proportionality T e / B z ≈ www.ann-geophys.net/31/1109/2013/ Ann. Geophys., 31, 1109-1114, 2013 2 keV/7 nT (exact value is 1.9(±0.5) keV/7.4(±1.2) nT). Due to the above-mentioned problems with small B z measured by THB, the obtained temperature growth is weaker than observed growth of B z . Thus, the approximation T e / B z ≈ 2 keV/7 nT slightly overestimates growth of T e . To correct the approximation of B z , we need to introduce one additional free parameter responsible for constant shift: We plot profiles T e as function of B z in Fig. 3f, where red and blue curves show both our approximations. One can see that both approximations describe observations well and actually are not that different (corresponding correlation coefficients are 0.62 and 0.7). Proportionality T e ≈ (1.7 keV/8 nT)·( B z +1 nT) approximates data slightly better than T e / B z ≈ 2 keV/7 nT (see Fig. 3g, where we combine the profiles from Fig. 3b, d and show T e as function of B z ). Moreover, both approximations of B z coincide with AMPTE/IRM statistics presented by Kan and Baumjohann (1990) (not shown here).
We also compare measurements of electron temperature from the THEMIS mission with the Cluster dataset. Figure 3d shows Cluster C2 data of the electron temperature collected in the central region of the magnetotail CS. Cluster measurements demonstrate the growth of T e with X (i.e., in the earthward direction) as well. However, the absolute values of the temperature are larger than data obtained by THEMIS. This discrepancy could be explained by different solar activity for the seasons, when Cluster and THEMIS operated in the magnetotail. Smallness of the THEMIS statistics also can be responsible for this effect.
The profile of the average electron anisotropy T /T ⊥ shown in Fig. 3e demonstrates that T /T ⊥ is almost constant along the tail and T /T ⊥ ≈ 1.1 (this is a typical value of electron anisotropy; see Sergeev et al., 2001;Artemyev et al., 2011b). The same results are derived from the Cluster statistics (see large crosses in Fig. 3e). Therefore, the combination of T e ∼ B z and T /T ⊥ ≈ const leads to the surprising conclusion that the parallel component of the temperature increases with the same rate as the perpendicular one (∼ B z ). However, we should note that the variance of T /T ⊥ is large. Thus, we only can conclude that temperature ratio is T /T ⊥ ∈ (1, 1.2) without strong variation along the magnetotail.
We use full THEMIS statistics of 102 events when THB and THC crossed the magnetotail CS within short time interval. The large amount of two-point events allows us to divide these events in three groups depending on geomagnetic activity (Kp ≤ 1+, Kp = 2, and Kp ≥ 3−). We do not consider the same statistics for X > −15 R E because of the small of number of events when THB/THC and THD crossed the magnetotail CS within 30 min. For each range of Kp, we obtain average ratios B TC z /B TB z and T TC e /T TB e (Table 1). For small Kp, the increase of B z is weaker than for Kp ≥ 2. The ratio of temperatures is T TC e /T TB e ∼ 1.8 independent of Kp.
We calculate T e = T TC e − T TB e and B z = B TC z − B TB z (distance between THB and THC is about ∼ 6 R E for events from our statistics). In this case, the approximation T e = const 0 B z + const 1 gives T e / B z = const 0 . For small Kp ≤ 1, the temperature difference T e is smaller than for Kp ≥ 2. The ratio T e / B z is similar to the value 2 keV/7 nT ∼ 290 eV/nT obtained for X ∈ [−10, −30] R E with three-point statistics. For Kp ≤ 1 this ratio is one and half times smaller. However, this can be the effect of small statistics. Therefore, dependencies of ratios T e / B z and T TC e /T TB e on geomagnetic activity seem to be weak.

Conclusions
In conclusion, we have shown the following: (1) electron temperature in the vicinity of the neutral plane increases with B z almost linearly, and (2) anisotropy of the electron temperature is almost constant along the tail for X ∈ [−25 R E , −10 R E ]. Here, we note that our results can be obtained only owing to simultaneous observations of the magnetotail current sheet from several spacecraft in different X. These two effects provide some difficulties for the simple explanation of electron energization via the adiabatic mechanism as well as via models of the transient acceleration. Profiles of magnetic field along the magnetotail do not contain any artificially (transient) large values of B z . Therefore, we observe in our statistics the averaged profiles of B z (X) corresponding to the quiet magnetotail configuration. If transient processes had substantial impact on the electron energization, electron temperature would not correlate with averaged B z profiles in the magnetotail. Moreover, in this case T e in the near-Earth region (X ∼ −15 R E ) would be larger than a value obtained from a simple approximation T e / B z ≈ const with T e measured at X ∼ −25 R E . In contrast, the approximation T e / B z ∼ const slightly overestimates growth of T e , and real energization is even weaker than the increase of B z . Thus, we can conclude that there is no evidence (direct or indirect) of a substantial role of any transient process (like reconnection and/or dipolarization) in additional energization of the thermal electron Ann. Geophys., 31, 1109-1114, 2013 www.ann-geophys.net/31/1109/2013/ population. Strictly speaking, this conclusion can only be applied to quiet time conditions when the averaged B z profile is monotonous along the magnetotail. However, transient mechanisms still can be important for acceleration of highenergy electrons (see Asano et al., 2010;Fu et al., 2011;Ashour-Abdalla et al., 2011;Fu et al., 2012;Birn et al., 2012, and references therein). Although the relation T e / B z = const can be considered as evidence of the adiabatic heating, we also cannot use this model to describe the observed electron energization. Adiabatic heating predicts a variation of the electron temperature anisotropy. T ⊥ increases linearly with B z according to the betatron mechanism, while the rate of T growth provided by Fermi mechanism is smaller: T ∼ B z 2/5 (see Lyons, 1984;Zelenyi et al., 1990;Artemyev et al., 2011c, and references therein). Therefore, in the simplified model of the magnetotail, one should obtain the decrease of the anisotropy T /T ⊥ with the increase of B z (see Artemyev et al., 2012).