Proton velocity distribution in thin current sheets: Cluster observations and theory of transient trajectories

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi: /2010ja015702, 2010 Proton velocity distribution in thin current sheets: Cluster observations and theory of transient trajectories A. V. Artemyev, 1,2 A. A. Petrukovich, 1 R. Nakamura, 3 and L. M. Zelenyi 1 Received 20 May 2010; revised 13 September 2010; accepted 27 September 2010; published 30 December [1] The structure of thin current sheets in the Earth magnetotail is studied on the basis of 43 crossings by Cluster spacecraft. Current sheet embedding is described as the ratio of the magnitudes of magnetic field at the thin current sheet boundary and in the lobe. This ratio is found to be 0.4 on average. The proton population consists of the current carrying particles with density 10% 20% of the total density, and the background plasma. This current carrying distribution is found to have a half ring structure in the (v x, v y ) plane, which is consistent with the theory of transient particle motion (so called Speiser type trajectories). The main parameter of the thin current sheet models, the flow anisotropy ", is estimated to be " The E B drift of the background plasma due to the presence of an earthward ambipolar electric field can weaken the total proton current density (in the magnetospheric rest frame) by a factor of (1 T e n p /T p n D ), where T e and T p are electron and proton temperatures and n D and n p are the density of the current carrying particles and the total plasma density, respectively. Citation: 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,, doi: /2010ja Introduction [2] Thin current sheets (TCSs) with a typical scale of about 1000 km are a remarkable feature of the Earth s magnetotail. In particular, TCSs form during the substorm growth phase and are disrupted at substorm onset. Such a structure was first directly observed with ISEE spacecraft [Mitchell et al., 1990; Sergeev et al., 1993; Pulkkinen et al., 1993]. However, detailed experimental investigation of TCSs became possible only with multispacecraft Cluster observations [e.g., Asano et al., 2005; Runov et al., 2006; Nakamura et al., 2006; Israelevich et al., 2008]. TCSs are often detected owing to the large scale motions (flapping) of the sheet or of the entire magnetotail. This allows registration of a cross sheet profile within a short time interval [Sergeev et al., 1998; Runov et al., 2005]. [3] The first theoretical description of TCS was suggested by Eastwood [1972]. It was shown that TCS formation could be self consistently supported by the currents related to transient proton trajectories in the magnetotail (also known as Speiser orbits [Speiser, 1965]). Using a quasi adiabatic approach for the description of particle motion, one can identify transient, cucumber, and ring types of proton 1 Space Research Institute, Russian Academy of Sciences, Moscow, Russia. 2 Also at Nuclear Physics Institute, Moscow State University, Moscow, Russia. 3 Space Research Institute, Austrian Academy of Sciences, Graz, Austria. Copyright 2010 by the American Geophysical Union /10/2010JA trajectories in TCSs [Chen and Palmadesso, 1986; Buechner and Zelenyi, 1989]. The later models have shown that thin sheets could be formed mainly by transient protons, whereas particles on cucumber and ring orbits are less efficient in the maintenance of the cross tail currents [Holland and Chen, 1993; Kropotkin and Domrin, 1996; Sitnov et al., 2000; Zelenyi et al., 2000]. The typical spatial scale (thickness) of such sheets is of the order of a proton gyroradius [Francfort and Pellat, 1976; Pritchett and Coroniti, 1992; Ashour Abdalla et al., 1994]. [4] Reconstruction of TCSs with the four point Cluster data revealed that the thickness of the current sheet (along the direction of the normal) is almost always much narrower than that of the plasma sheet [Runov et al., 2006]. Such a difference between spatial profiles of current density and plasma density is in sharp contrast with the classical Harris model [Harris, 1962], in which these scales coincide. [5] Further comparison of observations with the models, allowing embedding of TCS in the thicker plasma sheet, was more successful [Baumjohann et al., 2007; Artemyev et al., 2008, 2009; Zhou et al., 2009a]. In particular, the necessity of two independent plasma components was shown by Zhou et al. [2009a]. The first component creates the main peak of current density in a TCS, while the second one contains the main part of the number density (this component is responsible for the formation of the plasma sheet). Thus, the plasma sheet can be considered as a background for the TCS. Current density, created by this plasma sheet component, is usually too small to be detected by the Cluster tetrahedron when it has the proper separation to resolve TCSs (of the order of 1000 km during 2001, 2002, and 2004). However, the 1of10

2 Table 1. List of Used TCS Crossings Date UT X(R E ) Y(R E ) Z(R E ) B 0 (nt) B ext (nt) 24 Jul Aug Aug Aug Sep Sep Sep Sep Sep Sep Oct Oct Oct Oct Oct Oct Oct Oct Oct Oct Oct Oct Nov Aug Aug Sep Sep Oct Jul Jul Aug Aug Aug Aug Sep Sep Sep Oct Oct Oct Oct Oct Oct number density of the first component is small, because only weak variations of density across TCS are observed. Therefore, its bulk velocity in the cross tail direction should be sufficiently large to support the peak of current density. [6] According to theoretical models, protons carry most of the current in the reference frame where the electric field is absent [Harris, 1962; Zelenyi et al., 2000] (except electronscale current sheets, which are not considered here). However, observations show that electron currents are often larger than the proton ones, even in ordinary TCSs [Asano et al., 2003; Runov et al., 2006; Israelevich et al., 2008; Artemyev et al., 2009]. Such a redistribution between electron and proton flows could be principally achieved by the electric field, arising owing to a difference in behavior between quasiadiabatic protons and magnetized electrons [Zelenyi et al., 2010]. [7] In this paper we study the formation of proton currents in embedded current sheets with all available details. We analyze magnetic field, plasma moments, as well as proton velocity distribution functions observed by Cluster and compare those with the theoretical prediction of transient particle distribution. In section 2 we describe the Cluster data that have been used in this paper. Section 3 presents general statistics of TCS embedding. In section 4 several TCS crossings with mostly proton currents are studied to reveal the structure of velocity distribution functions of currentcarrying protons. The comparison of observed velocity distribution with theoretical predictions can be found in section 5. In section 6 we investigate the joint effect of the proper TCS electric field from the model [Zelenyi et al., 2010] and the relatively dense background plasma on proton currents. 2. The Data and Methods [8] In this investigation we use a selection of 43 TCS crossings detected by Cluster in 2001, 2002, and 2004 (Table 1). All crossings from 2001 and 2004 are also listed in a database ( runov/). Some crossings have been studied in several earlier papers [Runov et al., 2006; Artemyev et al., 2008, 2009]. [9] The coordinate system of each TCS crossing is constructed as follows: n is normal, l is the direction of magnetic field maximum variation, and m =[l n]. The current density is calculated as j curl = m(4p/c)curl B [Runov et al., 2005, 2006]. All cases studied in this paper represent almost horizontal sheets with current flowing nearly along the dawndusk direction and (ne z ) > 0.8. [10] We use proton moments and distribution functions from the Cluster Ion Spectrometry (CIS)/Composition and DistributionFunction(CODIF) instrument [Reme et al., 2001] and magnetic field from an FGM instrument [Balogh et al., 2001]. For 2001 and 2002 cases, proton moments are taken from C4, whereas for 2004 cases they are taken from C1. All data are taken from the Cluster Active Archive. All vectors and coordinates are in the GSM system. 3. Embedding of TCS [11] The statistics of 43 crossings allows us to analyze the occurrence distribution of the embedding parameters. The concept of embedding geometry is illustrated in Figure 1. A thin current sheet with a narrow maximum of current density and a scale L CS is embedded into a thicker plasma sheet with much weaker current density and scale L ext. The current density in a TCS could be an order of magnitude Figure 1. Scheme of an embedded thin current sheet (adapted from Burkhart et al. [1992b]). 2of10

3 Figure 2. Occurrence distribution of (left) B 0 /B ext, (middle) n p /n out and (right) j curl /j out for Cluster statistics of a TCS. higher than in the surrounding plasma sheet [Runov et al., 2006; Nakamura et al., 2006]. The main characteristic of an embedded sheet is magnetic field B 0 at its boundary, which can be about 30% 50% of the total (lobe) magnetic field B ext [Artemyev et al., 2008]. Here we calculate B ext, assuming vertical pressure balance and knowing the total pressure (magnetic plus plasma) in the plasma sheet. [12] Estimation of B 0 can be carried out by several methods, with rather small differences in the resulting values. In our paper we determine B 0 as a value of Bl at the TCS boundary (details can be found in Appendix A). The distribution of the ratio B 0 /B ext for our statistics is presented in Figure 2 (left) (the average value is 0.4). These data can be used to estimate the thickness of TCS in the units of proton gyroradius, r. Runov et al. [2005] estimated the sheet scales as 5 < L CS /r < 15 for the majority of the observed TCSs. Scales were calculated as L CS B ext /j max ; B ext was also used to calculate r 1/B ext. However, for an embedded TCS magnetic field, B ext should be replaced by B 0 in all estimates. Thus, L CS /r should be multiplied by the factor (B 0 /B ext ) 2, equal on average to 0.16, and the typical normalized thickness is actually closer to unity (L CS /r < 2). This estimation is in agreement with the TCS theory [Francfort and Pellat, 1976; Ashour Abdalla et al., 1994; Zelenyi et al., 2000]. [13] Since (B 0 /B ext ) , the increase of plasma pressure inside the TCS (relative to its edge) is rather small and often marginally visible. The increase of plasma density is even smaller due to the usual growth of temperature in the center of the current sheet [Runov et al., 2006]. The difference in proton density between the center of TCS, n p (taken where B l < 5 nt), and the outer region, n out (for B l > 0.9 B 0 ), in our statistics is about 15% on average (Figure 2, middle). We assume in this paper that electric current is created by protons with the density n p n out and the TCS is embedded into a thick sheet with the density n out. The embedding of the TCS is revealed with the narrow maximum of current density in the central region [Asano et al., 2005]. The current density decreases several times, while the magnetic field increases to B 0 [see Runov et al., 2006; Artemyev et al., 2008]. For our statistics, we plot the distribution of the ratio of current density in the center of the TCS, j curl, and current density in the outer region, j out, for B 0 > B l > 0.9 B 0 (Figure 2, right). This ratio is about 5 on average and almost all TCSs from our statistics are embedded. 4. Velocity Distribution of Current Carriers [14] In this section we analyze velocity distributions (VDs) f(v) of the current carrying protons in TCSs. We select three crossings that have relatively large proton current densities. Plots of magnetic field, proton density, and bulk velocity are shown in Figures 3, 4, and 5, respectively. The centers of TCSs (the regions with the significant positive bulk velocity u y ) are marked by vertical lines. In contrast with the change of u y, the increase of proton density does not exceed 20% in these regions. As a first step we plot one dimensional distributions f(v y ) for each crossing averaged over the two intervals at the boundaries (before and after the crossing) and the center of the current sheet (Figure 6). The left wing of f(v y ) does not change in the course of crossings. In the center of the TCS, where u y increases, the right wing is higher than the left one, and the distribution function is asymmetric. Therefore, we can conclude that the increase of u y corresponds to the appearance of a small population of protons with positive velocity v y, but not due to the shift of the entire VD (as in the case of Harris s [1962] model). The remaining symmetrical part of the distribution corresponds to the background plasma. [15] To restore the two dimensional (v x, v y ) VD of current carriers, we assume that the VD of the background plasma does not change significantly in the course of the TCS crossing. Figures 3 5 (bottom) demonstrate the differences between the distribution functions inside and outside the sheets: D n = f n (v x, v y ) f 0 (v x, v y ). The data in Figure 5 are shown in two coordinate systems ((x, y) and (l, m)) because significant variation of the B y component during the crossing suggests a substantial difference between the x and l axes. Distribution functions f n (v x, v y ) are obtained by averaging over the intervals inside the TCS (where n is the number of the interval; see numbering in Figures 3 5) and integrating over v z. The reference distribution f 0 (v x, v y ) corresponds to the interval with n = 0 outside each TCS. Since D n can be both positive and negative, we cannot use the logarithmic scale, so we choose another approach to visualize the main part of D n. 3of10

4 center of the TCS (Figures 3 5). We introduce the following simple approximation, which is the generalization of one suggested by Burkhart et al. [1992b]: fd ¼ C 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > 2 2 þ v2 v < exp v 2 ; vy > 0; þv v 0 x y T z > : ð1þ 0; vy < 0; where C is a normalization constant. This distribution in the vt and radius space (vx, vy) is a half ring with a thickness R 3 v0. The current density R jd3(v0, vt ) = vy fdd v and plasma density nd(v0, vt ) = fdd v are functions of vt and v0 (the /en v20ffiffiffi/vt +, and ratio jdp ffiffiffi D = ud(v0, vt ) = a1vt + a2v0 + a3p a1 = 1/, a2 = (4 p)/p, and a3 = (p 3)/ ). [17] The shift v0 appears due to the parallel (relative to the magnetic field) velocity at the boundary of the TCS, v0 vk0 [Burkhart et al., 1992a; Pritchett and Coroniti, 1992]. The ratio vt /vk0 = " (the degree of a flow anisotropy) is the key parameter of all models based on the concept of transient trajectories [Burkhart et al., 1992a; Holland and Chen, 1993; Figure 3. Magnetic field, proton bulk velocity uy, proton density np, and differences of 2 D proton VD for the crossing UT, 12 September Vertical lines mark the center of TCS and intervals of VD averaging. Values of Dn are divided into three domains: (1) large positive, Dn > Dsup (gray); (2) small, Dn < Dsup (white); (3) large negative, Dn < Dsup (black). Here Dsup = max Dn / 100. Inside the TCS, the excess particle population (which is absent at the TCS boundary) has essentially an asymmetric distribution: the phase volume with vy < 0 is nearly empty. Owing to this population, the value of uy increases substantially in TCS. 5. Theory of Transient Trajectories [16] To describe the observed VD we use the concept of transient particle motion. These particles enter the TCS along field lines, turn around the normal magnetic field Bz, and leave the sheet [Speiser, 1965]. Transient particles carry large current because of the openness of their trajectories. The VD of transient particles in the center of a TCS has a typical half ring structure [Hamilton and Eastwood, 1982; Ashour Abdalla et al., 1991; Burkhart et al., 1992a]. We use the analytical expression of the distribution function of transient particles from the TCS model [Zelenyi et al., 2000] to plot the velocity distribution in the center (Figure 7). The shape of this distribution looks similar to the VD observed in the Figure 4. Same as in Figure 3 but for the crossing UT, 12 September of 10

5 Figure 7. VD of transient protons from the model of a TCS [Zelenyi et al., 2000]. Figure 5. Same as in Figure 3 but for the crossing UT, 8 October Zelenyi et al., 2000, 2004]. Even if v 0 is small, u D could be sufficiently large due to the asymmetry of the VD (j D /(en D ) v T ). Thus, even a relatively small population of particles, described by f D, is sufficient to account for the observed current density. The values of v 0 and " can be estimated from observations by using the dependence of j D (v 0, v T )onv 0 and assuming that in some reference frame (close to the de Hoffman frame without electric field) the observed current density is almost entirely supported by transient protons. In this case j D (v 0, v T ) should be equal to j curl (averaged over the central region of the TCS). Keeping n D as a free parameter, the equation j D (v 0, v T )=en D u D (v 0, v T )=j curl is solved to find v 0 (and the thermal velocity of transient particles, v T, is taken equal to that of the entire distribution function). For some values of n D this equation might have no solution because j D (0, v T )>j curl ; that is, even the model with v 0 = 0 gives a larger current density than the observed one. Thus, for each value of n D, we find the number of TCS crossings in our statistics for which the equation j D (v 0, v T )=j curl has at least some solution (Figure 8). [18] According to Figure 8, with the density of transient protons n D at about 10% 20% of the total density, current carriers with f D and some v 0 can account for the measured values of current density in all our TCS crossings. These estimates are consistent with the observed increase of density inside the TCS, since n D n p n out. For two typical Figure 6. centers. One dimensional VD f(v y ) for each TCS crossing from Figures 3 5 compared at boundaries and 5of10

6 Figure 8. Number of TCS crossings that can be described by the model with f D as a function of relative density of transient particles n D /n p. values of n D /n p equal to 0.1 and 0.2, we obtain values of v 0 and " = v T /v k0 v T /v 0 (Figure 9). Values of " are typical for analytical models of TCSs [Zelenyi et al., 2000, 2004] and numerical simulations of TCSs [Burkhart et al., 1992a; Pritchett and Coroniti, 1992; Holland and Chen, 1993; Mingalev et al., 2007]. Values of " could be larger in reality (corresponding to a smaller flow anisotropy of transient population) if one takes into account the possible contribution of electrons to j curl, probably a larger temperature of the transient population, and so on. 6. Current Redistribution [19] Now let us return to a more general case in the reference frame corresponding to the observed current sheet, when proton current density is often small [see Runov et al., 2006; Israelevich et al., 2008; Artemyev et al., 2009]. We describe the role of embedding in this decrease of proton current (proton bulk flow along the electric current direction). We use the model of the proper electrostatic field in a TCS, arising from the principal difference between proton and electron motion [Zelenyi et al., 2004, 2010]. According to this model the potential drop across the TCS is of the order of electron temperature D T e /e [Zelenyi et al., 2004]. The electric field can be estimated as E x E z (L z /L x ), E z = D /L z for configurations with small B z / x (where L z and L x are typical spatial scales across and along a sheet) [Zelenyi et al., 2010]. The drift v E B = ce x /B z is the only one in the central region pof ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the TCS, because another component of the drift, E z B x / B 2 x þ B2 z, is vanishing when B x 0 due to nonzero B z. E x is directed toward Earth; v E B is then directed eastward, reducing the proton flow, which creates the electric current in the magnetospheric rest frame [Zelenyi et al., 2010]. Substituting E x in the equation for drift velocity, one obtains v E B = ct e /(aeb 0 L z ) (a = L x B z /L z B 0 ). The parameter a determines the general configuration of a current sheet. Twodimensional plasma equilibria with the pressure balance established by magnetic field gradients should have a 1 [Schindler, 1972; Lembege and Pellat, 1982]. If some fraction of protons has transient trajectories, then a part of the pressure along the x direction is balanced by the inertia of proton motion along their transient orbits and a 1[Burkhart and Chen, 1993]. [20] The equation for the total proton current density in the TCS includes the main current carriers with density n D and bulk velocity u D + v E B v T + v E B, and the background plasma with density n p n D and bulk velocity v E B : j p ¼ en D ðv T þ v EB Þþen p n D veb : ð2þ Here we neglect background proton flows in the absence of the electric field since they are small. Substituting the value of v E B, L z r (proton Larmor radius), and a 1, we obtain j p j 0 1 T e n p ; ð3þ T p n D where j 0 = ev T n D j D is the proton current density in the absence of electric field and T p is the proton temperature. Therefore, the proper electric field of the TCS can almost vanish the observed proton current density if the TCS is embedded into the plasma sheet such that n D n p (T e /T p ) n p /5. A larger than average ratio T e /T p would make such a Figure 9. Distribution of " for two values of relative density of transient particles. 6of10

7 situation more probable. The whole proton velocity distribution in such cases should be shifted by v E B in the y direction [Zelenyi et al., 2010]. 7. Discussion [21] Using the relatively rich statistics of thin current sheets, observed by Cluster spacecraft, we determine typical parameters of their embedding in the background plasma sheet. The proton population, responsible for TCS formation, makes up just 10% 20% of the total plasma density but creates a sharp current density peak, an order of magnitude higher than that supporting the background sheet. The velocity distribution of current carriers is consistent with the distribution of transient protons. [22] However, such transient protons should not be considered a distinct population superimposed on the background plasma. The clarification of the mechanisms of formation of transient particle distribution comes from the analysis of proton motion within the TCS. In the frame of the theory of quasi adiabatic particle motion, there exists a quasiadiabatic invariant I z = H v z dz [Sonnerup, 1971; Buechner and Zelenyi, 1989]. I z is conserved far away from the TCS central region. The value of I z determines the type of proton trajectory both inside and outside the TCS. Transient (or Speiser) trajectories are characterized by small value of I z [Ashour Abdalla et al., 1991]. Ions with regular ring trajectories and with quasi regular cucumber trajectories have larger I z [Buechner and Zelenyi, 1989; Zelenyi et al., 2000]. Scattering of protons near the neutral sheet plane results in changes of I z (violation of adiabaticity of particle motion) and leads to the exchange between populations at different trajectories [Chen and Palmadesso, 1986; Buechner and Zelenyi, 1989]. In particular, I z experiences two jumps at the entry to and the exit from the TCS [Neishtadt, 1986; Buechner and Zelenyi, 1989]. In a general case, these jumps do not compensate each other and a particle changes the transient trajectory to the quasiregular one. However, for some particles two jumps of I z compensate each other and only these particles remain on transient trajectories [Zelenyi et al., 2000]. Thus, the relative amount of Speiser protons depends on the competition between scattering and resonant effects [Chen and Palmadesso, 1986; Buechner and Zelenyi, 1989; Ashour Abdalla et al., 1991] and is usually rather small. [23] Therefore, the majority of TCS particles apparently belongs to the population on ring and cucumber orbits. In addition to scattering of transient protons, many protons with large I z values enter the TCS due to the earthward plasma convection in the magnetotail. Particles at such closed trajectories can carry current only because of the relatively slow diamagnetic drift motion (caused by the presence of pressure gradients) and are likely responsible for the current in the background sheet. [24] As a result, important peculiarities of distribution functions related to the quasi adiabatic nature of proton motion (pressure anisotropy, nongyrotropy, plasma flows, and nondiagonal components of pressure tensor [Burkhart et al., 1992a; Holland and Chen, 1993; Mingalev et al., 2007]) are masked by the dominant background population and often cannot be observed directly [Cully et al., 2006]. The situation could be different for sheets located close to active (reconnection) regions or for the sheets observed during substorms close to the end of their growth phase. In such cases the background population of plasma could be essentially decreased, for example, due to the small normal magnetic field B z (the scattering of I z is proportional to B z /B 0 [Buechner and Zelenyi, 1989]). The VD of transient protons under such conditions can be observed more readily [Raj et al., 2002; Runov et al., 2008; Nakamura et al., 2008; Zhou et al., 2009b]. [25] Here we should notice that one can likely offer the alternative interpretation of obtained velocity distributions (Figures 3 5) in the frame of the models with multicomponent plasma (e.g., that of Schindler and Birn [2002] and Yoon and Lui [2004]). It was shown [see Artemyev et al., 2009] that these models as well as the TCS model [Zelenyi et al., 2004] can describe the observed current density profiles of an embedded TCS with a small population of current carrying particles. The main difference between these alternative models (based on shifted Maxwellian distribution or Speiser particles) is the nature of the pressure balance along the x direction [Burkhart and Chen, 1993]. Therefore, only further investigation of the 2 D structure of TCSs can give a preference to one of these models. [26] It is important to mention several possible effects of embedding for current sheet stability. An observed embedded TCS essentially differs from the simple models used in the previous studies of stability [Harris, 1962; Schindler, 1972; Lembege and Pellat, 1982]. Investigation of the tearing mode as a main candidate in the role of a substorm trigger was stopped, because a Harris sheet with B z 0 was found to be stable [Pellat et al., 1991; Brittnacher et al., 1998]. However, the influence of a finite population of transient protons [Burkhart et al., 1992b; Zelenyi et al., 2008] as well as the effect of current sheet embedding [Zelenyi and Krasnoselskikh, 1979; Krallmann et al., 1994] change the criteria of TCS stability and open the door for tearing onset again. [27] Kink instability of a current sheet is also affected by embedding and the presence of transient protons. The Harris sheet with background plasma is found to be more unstable for the kink mode than that without background [Daughton, 1999; Karimabadi et al., 2003]. Theoretical predictions of kink instability of embedded TCSs with transient protons are in good agreement with observations of quasi periodic (period 100 s) sheet oscillations [Zelenyi et al., 2009]. [28] However, the problem of a theoretical description of the slower kink type sheet motions with periods more than several minutes (so called flapping) is still unsolved. Existing theories declare that the real part of the kink mode frequency should be related to the proton drift velocity [Pritchett et al., 1996; Lapenta and Brackbill, 1997; Buechner and Kuska, 1999]. Therefore, the direction of kink waves propagation in theory should be close to the direction of the cross tail current. However, observations at the dawn flank frequently detect flapping waves propagating eastward, opposite to the direction of the current [Sergeev et al., 2004; Runov et al., 2005; Zhang et al., 2005; Petrukovich et al., 2006]. This controversy could be resolved by taking into account the presence of an electric field E x and E B particle drift, caused by it. If the E x field is sufficiently strong to change the net direction of a proton drift (j D < ev E B n p ), it could stop or even reverse the direction of kink waves. Indeed, Erkaev et al. 7of10

8 responsible for TCS asymmetry along the normal because E x B y has the same sign for positive and negative B x. Figure 10. General structure of the velocity distribution of protons in a TCS. The small asymmetric transient population is embedded in a dense quasi Maxwellian background. Electric field drift shifts the whole distribution by a small value. [2009] suggested that kink instability in the current sheet without proton drift is caused by weak inhomogeneity of the normal magnetic field, which is also the primary reason for E x formation. However, the reason for dawn dusk asymmetry of E x, necessary for such an explanation, should be investigated in future papers. [29] The earthward electric field is important not only for flapping but also for the TCS structure in general. Embedding of a TCS in a relatively dense background combined with E x allows explanation of the relatively small proton currents frequently observed in experiment by an internal balance of TCSs and the surrounding plasma sheet. In general, the described configuration illustrates the strong influence of weak x gradients and electron dynamics on primary TCS properties. Thus, despite that these factors are usually of secondary importance in the stretched magnetotail, they should not be ignored during either experimental analysis or modeling. [30] An important feature, which appears to be beyond the scope of this paper, is the effect of B y 0. Nevertheless, B y is always present in the magnetotail and often B y > B z (as in our Figures 3 and 4). As soon as B y /B 0 < 1, the ion dynamics remains quasi adiabatic, but the presence of B y increases the scattering of ions [Buechner and Zelenyi, 1991]. Currently there is no kinetic model of a TCS with B z 0 and B y 0. However, models without a normal magnetic field and with B y can be constructed as a generalization of the Harris model (see review by Roth et al. [1996]). Here we only estimate the effect of B y (and these estimations are not based on any selfconsistent model). Equation (3) should be modified as j p /j 0 = 1 (T e n p /T p n D )(1 + B y 2 /B z 2 ) 1 and, if B y /B z 1 2, the effect of E x decreases by two to fourfold. The asymmetry of the B y distribution across the magnetotail [Petrukovich, 2009] should lead to asymmetry of E x B z drift, which can be useful to describe the observable asymmetry of the flapping motion. The B y field is also responsible for two drifts, v x E z B y and v z E x B y. The first one does not play an essential role because E z 0 in the center of a sheet. The second one could be 8. Conclusions [31] In this paper we studied velocity distributions of protons carrying currents in TCSs. The statistics of 43 TCS crossings allowed us to determine the average embedding of current density profiles. [32] The main results are summarized in Figure 10 and can be formulated as follows: [33] 1. A thin current sheet has the following typical embedding: 0.3 < B 0 /B ext < 0.5, n p /n out < 1.2 at the downtail distances, visited by Cluster spacecraft. The estimate of TCS thickness is 1 2 proton gyroradii. [34] 2. The velocity distribution of protons has a dual structure: the small population of particles (10% 20% of the total density) carries almost all current, while most of the protons represent just a background plasma for a TCS. [35] 3. The velocity distribution of current carrying protons in a TCS has a half ring structure asymmetric with respect to v y and is consistent with the theory of transient (or Speiser) motion. The main parameter (flow anisotropy) of TCS models estimated from experimental data is " [36] 4. The earthward electrostatic field E x > 0, appearing in a TCS due to the sheet charging and the small x gradient [Zelenyi et al., 2010], is sufficient to decrease the proton current density in an embedded TCS by a factor of (1 T e n p /T p n D ). Appendix A: Determination of Boundary Magnetic Field B 0 of a Thin Embedded Sheet [37] The magnetic field at the boundary of a thin current sheet B 0 is the main observational characteristic of embedded sheets. Although in many cases its determination looks rather Figure A1. Single spacecraft B 0 estimation for TCS crossings from Figures 3 5, based on B x profiles. 8of10

9 Figure A2. Alternative method of B 0 determination based on current density profiles with respect to B l.the diamonds show experimental profiles and the shaded curves denote the approximation. See text for details. straightforward, there is no single rule to do it. First of all, it depends on the definition of a TCS boundary. Available theory models suggest spatially infinite profiles (e.g., Harris model) with exponentially decreasing current density. Thus, determination of the boundary is based on the ad hoc criterion of significant reduction of current. The first possible method (mainly used by us) is based on single spacecraft observations of magnetic field. We assume that the variation of the B l (B x ) component is due to the vertical motion of the TCS with some constant velocity v z. In this case, B 0 is determined as a B l (B x ) value at the moment of time when the rate of B l (B x ) variation substantially decreases. Application of this method to TCS examples from Figures 3 5 as well as respective values of B 0 are presented in Figure A1. [38] The alternative approach is based on analysis of current density profiles j m as a function of magnetic field B l. (A similar approach was used by Sergeev et al. [1993].) The dependence j m (B l ) is approximated by the function j = j max (1 sb l 2 ) and parameter s is found by the least squares method (Figure A2). In this case, B 0 can be obtained as s 1/2. However, at the periphery of the TCS near B 0, current density is often not well resolved by the Cluster tetrahedron due to its smallness. Thus, B 0 tends to be underestimated. [39] Altogether, differences in methods and uncertainty of the boundary definition results in 10% 20% error in B 0 determination. With such an error, individual B 0 values should be used with care, whereas analysis of B 0 statistics should be more reliable. [40] Acknowledgments. 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