### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Data Analysis
- 3. Summary and Discussion
- Acknowledgments
- References

[1] Using Cluster data, we investigate the electric structure of a dipolarization front (DF) – the ion inertial length (*c/ω*_{pi}) scale boundary in the Earth's magnetotail formed at the front edge of an earthward propagating flow with reconnected magnetic flux. We estimate the current density and the electron pressure gradient throughout the DF by both single-spacecraft and multi-spacecraft methods. Comparison of the results from the two methods shows that the single-spacecraft analysis, which is capable of resolving the detailed structure of the boundary, can be applied for the DF we study. Based on this, we use the current density and the electron pressure gradient from the single-spacecraft method to investigate which terms in the generalized Ohm's law balance the electric field throughout the DF. We find that there is an electric field at ion inertia scale directed normal to the DF; it has a duskward component at the dusk flank of DF but a dawnward component at the dawn flank of DF. This electric field is balanced by the Hall (**j** × **B**/*ne*) and electron pressure gradient (∇ *P*_{e}/*ne*) terms at the DF, with the Hall term being dominant. Outside the narrow DF region, however, the electric field is balanced by the convection (**V**_{i} × **B**) term, meaning the frozen-in condition for ions is broken only at the DF itself. In the reference frame moving with the DF the tangential electric field is almost zero, indicating there is no flow of plasma across the DF and that the DF is a tangential discontinuity. The normal electric field at the DF constitutes a potential drop of ∼1 keV, which may reflect and accelerate the surrounding ions.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Data Analysis
- 3. Summary and Discussion
- Acknowledgments
- References

[2] Magnetic reconnection in the Earth's magnetotail accelerates fast plasma jets. On the front edge of the earthward propagating jet a discontinuity forms, called the dipolarization front (DF). DFs have a typical scale comparable to the ion inertial length (*c*/*ω*_{pi}). They are characterized by the sharp increase of *Bz*, decrease of density and increase of plasma flow velocity [e.g., *Runov et al.*, 2009]. Strong gradients of *Bz* and density at the DF as well as the pileup of magnetic field behind the DF generate strong DC electric fields as well as electromagnetic waves in a broad frequency range [e.g., *Sergeev et al.*, 2009; *Zhou et al.*, 2009; *Deng et al.* 2010; *Khotyaintsev et al.*, 2011]. DFs are also associated with the acceleration of energetic electrons [e.g., *Fu et al.*, 2011; *Asano et al.* 2010]. Understanding the electric structure of DFs is one of the keys to understanding the DF dynamics and the associated particle acceleration.

[3] According to the generalized Ohm's law [e.g., *Khotyaintsev et al.* 2006],

the convection, **V**_{i} × **B**, Hall, **j** × **B**/*ne*, and electron pressure, ∇ ⋅ P_{e}/*ne*terms all balance the electric field at the sub-proton scale (*L* ≤ *c/ω*_{pi}). The electron inertia term is neglected in equation (1) as it becomes important only at scales comparable to the electron inertial length (*L*∼*c/ω*_{pe}) [e.g., *Birn and Priest*, 2007]. While it is straightforward to compute the convection term from the spacecraft measurements, in order to estimate the Hall and electron pressure terms, one should use some simplifying assumptions. For instance, *Zhou et al.* [2009] and *Zhang et al.* [2011] calculated the current as using single-spacecraft data and assuming constant DF speed. This is reasonable when one is crossing the central part (subsolar point) of a DF, where normal to the DF is along*X*_{GSM} [*Runov et al.*, 2009, Figure 4b] and the largest change is in the *Bz.* However, this assumption clearly does not work at the flank of a DF. *Zhang et al.* [2011] estimated *j*_{y} as a sum of contributions from the bulk ion motion, **E** × **B** drift and the electron diamagnetic current, for which the electron pressure term ∇ ⋅ P_{e} was computed using the two THEMIS probes which are separated mainly in the *Z*_{GSM} direction. This approach works well away from the DF, however, it will not work at the DF where plasma parameters are changing fast during a spacecraft spin period (several sec) which results in unreliable particle moments. Alternatively, one can estimate the current *j*at a DF using the multi-spacecraft curlometer technique [*Dunlop et al.*, 1988, 2002], as done for example by *Schmid et al.* [2011] and *Hwang et al.* [2011]. This method gives reliable results only in case the current sheet thickness exceeds the separation between the spacecraft, and thus can be applied only to some of the thickest DFs observed by Cluster.

[4] Since computation of *j* in the previous studies depends on the location of a spacecraft relative to the DF [e.g., *Zhou et al.*, 2009; *Zhang et al.*, 2011] or cannot resolve the small-scale structure of DF [e.g.,*Schmid et al.*, 2011; *Zhang et al.*, 2011], we re-compute*j* and ∇ ⋅ P_{e}using two different techniques: single-spacecraft and multi-spacecraft technique (using 4 Cluster SC), and then compare the results from them. We perform the analysis in the local*lmn* coordinates, which is valid for both the center (subsolar point) and the flank of DF. Then we use the obtained *j* and ∇ ⋅ P_{e} to examine which terms in the generalized Ohm's law (equation (1)) balance the electric field throughout the DF.

### 2. Data Analysis

- Top of page
- Abstract
- 1. Introduction
- 2. Data Analysis
- 3. Summary and Discussion
- Acknowledgments
- References

[5] Figure 1 gives an overview of a DF observed from 1352:30 to 1353:30 UT on August 29, 2003, by the four Cluster spacecraft that are located at [−17.5, −1.8, 2.8] *R*_{E} in GSM coordinates. This DF was one of the events selected by *Schmid et al.* [2011] for investigating the statistical properties of DFs. We can see that, before 1353:10 UT, the magnetic field and plasma parameters are very stable. The *Bx* component is small (Figure 1b), and the plasma beta is larger than 0.5 (Figure 1d, blue dashed line). According to the criteria given by *Cao et al.* [2006], Cluster is located in the plasma sheet during this period. At 1353:15 UT, a sudden jump of *Bz* from 2 nT to 14 nT is observed (shadow region), indicating the arrival of a DF. Associated with this DF, there is an increase of ion bulk velocity from 50 to 200 km/s (Figure 1c). The electric field is also elevated at the DF; it has an earthward component up to 6 mV/m (C4, Figure 1f) and a dawnward component up to 6 mV/m (C4, Figure 1g). Since PEACE and CIS have low resolution (4 s) for the measurement of particles, we use the spacecraft potential measured by EFW that has a resolution of 0.2 s to deduce the plasma density [e.g., *Pedersen et al.*, 2008]. In Figure 1h, we see that the plasma density derived from the SC potential (solid lines) is consistent with the C2-PEACE measurement (dots). This density drops from 0.3 to 0.2 cm^{−3} at the DF. The parallel and perpendicular electron temperatures measured by C2 have a low resolution (4 s; see the dots in Figure 1i) during this period. We interpolate them to the time line of the density measurement to obtain the high-resolution temperature data (dashed lines inFigure 1i), which are then used to calculate the electron pressure based on the fact that the off-diagonal terms of the electron pressure tensor are close to zero during this period (not shown). The derived electron pressure drops from 0.1 to 0.06 nPa at the DF (Figure 1j). As there are no measurements of the electron temperature at C1, C3 and C4 during this period, we assume the electron temperatures at these locations to be the same as at C2. This assumption is reasonable since during this period the separation between the four spacecraft is very small (Figure 1d), and the electron temperature is relatively steady compared to the density (Figure 1h), so that the changes in the pressure are coming mainly from the changes in the density.

[6] Detailed observations of the DF from 1353:10 UT to 1353:20 UT are shown in Figure 2. The DF (change in *Bz*, Figure 2a) is observed sequentially by C3, C2, C4 and C1. The magnetic field profiles observed by the four spacecraft are very similar, meaning we can do the timing unambiguously. By time-shifting the*Bz* data from the various spacecraft (Figure 2b), we determine the normal velocity of the DF as *V*_{DF} = 197 * [0.61 −0.70 0.36] km/s in the GSM coordinates, with an uncertainty of . Considering that the DF lasts for ∼2.1 s (from 1353:13.3 to 1353:15.4 UT at C4), its thickness is about 420 km, which is very close to one ion inertial length (see Figure 1e). *V*_{DF} defines the *n* direction for the local *lmn* coordinate system, the *m* direction is obtained from , where B_{0} is the average magnetic field over the DF, and the *l*completes the right-handed system.

[8] Using the multi-spacecraft curlometer technique [*Dunlop et al.*, 1988, 2002], the current density *j* and electron pressure gradient ∇*P*_{e} can also be derived. In this event, the four Cluster spacecraft have a regular tetrahedron configuration [*Dunlop et al.*, 2002] and a separation of ∼200 km, which is smaller than the local ion inertial length (Figure 1e), indicating that the currents/gradients at scales >200 km can be estimated using this method. Since the DF propagates with a speed of 197 km/s, scales >200 km correspond to frequencies <1 Hz. The uncertainties for the curlometer technique are usually determined by |∇ ⋅ **B**|/|∇ × **B**| [e.g., *Robert et al.*, 1998; *Hwang et al.*, 2011], which has a statistical consistence with Δ*J*/*J* [e.g., *Robert et al.*, 1998]. In this event, |∇ ⋅ **B**|/|∇ × **B**| at the DF is below 40% (not shown).

[9] Figures 2c–2g show the current density *j*, Hall term **j** × **B**, and electron pressure gradient term ∇*P*_{e}/*ne*derived using the two methods described above. Results from the single-spacecraft and multi-spacecraft methods are shown by the thin and thick lines respectively. Black, red and blue lines represent the*l*, *m*, *n* components. The black bar between Figures 2e and 2findicates the ion inertial length. Considering the uncertainties from the single- (<30%) and multi-spacecraft (<40%) methods, one can see that the results from these two methods generally agree with each other. At the DF, the substantial difference, as expected, may attribute to the current structure at scales below the spacecraft separation that can only be resolved by the single- but not multi-spacecraft method. The normal component of the current*j*_{n} (Figure 2e), as well as the tangential components of the pressure gradient (∇*P*_{e}/*ne*)_{m} and (∇*P*_{e}/*ne*)_{l} (Figure 2g) obtained from the multi-spacecraft technique are all close to zero. This supports our initial assumptions used to compute the single-spacecraft quantities, and thus we conclude that the parameters from the single-spacecraft method (thin lines) are reliable, and give additional confidence in our estimate of the boundary orientation and velocity.

[10] Having obtained **j** × **B** and ∇*P*_{e}/*ne*, we are now able to examine the relative contributions from the different terms in the generalized Ohm's law at the DF. Figures 3b and 3c show, respectively, the *l* and *m* components of the four terms of the generalized Ohm's law in the spacecraft reference frame. Here we use **j** × **B**/*ne* and ∇*P*_{e}/*ne*from the single-spacecraft method as it can resolve the small-scale structure of the boundary. The uncertainty of these two terms is <30%. Due to the asymmetry of the photoelectron emission, the EFW instrument (red lines) usually has an offset, ∼1 mV/m, along the sun-aligned direction. By requiring the tangential electric field*E*_{t} to be a constant across the DF boundary (see Figure 3e), we reduce this uncertainty to be *δE*≈0.3 mV/m in this event, which is approximately the standard deviation of the measurement of the tangential field (Figure 3e). In Figures 3c and 3d, one can see that the Hall and pressure gradient terms in the *l* and *m* direction are very small, and the electric field is approximately balanced by the convection term especially behind the DF. Ahead of the DF, the relatively striking difference between the convection term and the measured electric field may attribute to the uncertainty of the flow velocity. In the quiet plasma environment, this uncertainty is typically ∼9% [*Rème et al.*, 1997]. Just ahead of the DF, however, this uncertainty can be much larger (>9%) as the spin period (4 s) that used to compute the flow velocity covers both the plasma sheet and the DF region.

[11] To get rid of the motional electric field due to the propagation of the DF, we transform the data into the frame moving with the DF (Figures 3d and 3e). We notice that, in this frame, the median values of the tangential components, 〈*E*_{l}〉≈0.006 mV/m, 〈(−*V*_{i} × *B*)_{l}〉≈0.03 mV/m, 〈*E*_{m}〉≈−0.02 mV/m, and 〈(−*V*_{i} × *B*)_{m}〉≈0.25 mV/m are very small, and can be treated as zero within the experimental uncertainty of CIS and EFW. This means *V*_{n} ≈ *E*_{l}*B*_{m} − *E*_{m}*B*_{l} ≈ 0; there is no flow across the DF boundary, and thus the DF is a tangential discontinuity. The normal component of the electric field (Figure 3f) does not change during the transformation from the spacecraft to the DF frame. We find that the normal electric field is enhanced significantly at the DF (shadow region). This enhancement is contributed mainly from the Hall and electron pressure gradient terms. The four terms of generalized Ohm's law at the peak of *E*_{n} (shadow region) are, respectively, *E*_{n} ≈ 9 ± 0.3 *mV*/*m*, (− **V** × **B**)_{n} ≈ − 0.5 ± 0.04 *mV*/*m*, (**j** × **B**/*ne*)_{n} ≈ 11 ± 3.3 *mV*/*m*, (− ∇*P*_{e}/*ne*)_{n} ≈ − 2.5 ± 0.75 *mV*/*m.* The measured electric field and the sum of all other three terms agree well within the uncertainties during this period (Figure 3g). Behind the DF (flux-pileup region, FPR), the measured electric field (red line) is almost constant (0 ± 0.3 *mV*/*m*); the Hall term (black line), however, shows a peak (3.5 ± 1.0 *mV*/*m*) at 1353:16.3 UT, and a valley (− 4 ± 1.2 *mV*/*m*) at 1353:17.0 UT. This peak and valley in the FPR are real fluctuations and can be balanced by the pressure gradient term (blue line). The enhancement of electric field at the DF corresponds to a potential drop of ∼1 kV (not shown). Our analysis indicates that, within the uncertainties, the frozen-in condition for ions is valid before and after (flux-pileup region, FPR) the DF, but it is broken at the DF.

[12] Using the same analysis method, we have investigated another DF that is observed by Cluster at 0156:30 UT on September 1, 2003. As in the presented case, we find that the frozen-in condition is broken at the DF but still valid outside it. Both Hall and electron pressure gradient term contribute to the electric field at the DF. The Hall term has about 3 times larger magnitude than the pressure gradient term.

### 3. Summary and Discussion

- Top of page
- Abstract
- 1. Introduction
- 2. Data Analysis
- 3. Summary and Discussion
- Acknowledgments
- References

[13] A dipolarization front (DF) structure, which is observed by four Cluster satellites at 1353:15 UT on August 29, 2003, is analyzed at the sub-proton scale. The multi-spacecraft technique [*Dunlop et al.*, 1988, 2002] and the single-spacecraft discontinuity analysis technique [*Khotyaintsev et al.*, 2006] are used to compute the electric current and electron pressure gradient throughout the DF. The results from the two methods agree with each other well except that the single-spacecraft analysis can resolve the small-scale (∼10 km) structures, while the multi-spacecraft analysis works only for the large-scale (>200 km) structures. Comparing to the previous studies [e.g.,*Zhou et al.*, 2009; *Zhang et al.*, 2011], our methods do not depend on the location of spacecraft relative to the DF because they are performed in the local *lmn* coordinates.

[14] For the first time we are able to examine the generalized Ohm's law throughout the DF, and show which term balance the observed electric field. We find that, at the DF, a strong electric field normal to the DF is present at the ion inertia scale and that it is balanced by the Hall and electron pressure gradient terms, with the Hall term being dominant. The sum of the Hall, convection and electron pressure gradient term agrees with the total electric field from DC component up to 4 Hz, which corresponds to spatial scales down to ∼0.1 ion inertial length. The normal electric field constitutes a potential drop of ∼1 kV across the DF. We have also analyzed another DF event observed by Cluster at 0156:30 UT on September 1, 2003, and found similar boundary structures. The electric structure of the DF is similar as that of the reconnection separatrix region at the magnetopause, where the strong electric fields normal to the boundary are also observed at ion inertia scale [*Khotyaintsev et al.*, 2006; *Lindstedt et al.*, 2009]. The electric field at both the DF and the separatrix region can accelerate the surrounding ions, as reported by *Zhou et al.* [2010] and *Lindstedt et al.* [2010]. It is thus important for the ion dynamics near the boundary.

[15] We find from the two investigated cases that the electric field is nearly along the normal direction (*E*_{n}>>*E*_{m}) at the DF. This electric field and the associated current system are illustrated in Figure 4. Since the DF is structured like a sandal [*Runov et al.*, 2009, Figure 4b], the current in the tangential plane of the DF will result in a dawnward *j* × *B* electric field at the dawn flank and a duskward electric field at the dusk flank of the DF. At the subsolar point of the DF, the current is almost along *Y*_{GSM} and the electric field is along *X*_{GSM}, just as the situation observed by *Zhou et al.* [2009]. For the presented case (2003-08-29), the propagation velocity of the DF has a positive*X* and a negative *Y* GSM component (see Figure 2b). This means Cluster encounters the dawn flank of DF. The earthward (Figure 1f) and dawnward (Figure 1g) components of electric field thus should be expected. For the other case we have analyzed (2003-09-01), Cluster observes earthward and duskward components of the electric field, as the dusk flank of a DF is encountered. The sign of*Ey*_{GSM}, in this way, can be an indication of the relative location of the DF.

[16] Inside the flux-pileup region (FPR), which is located behind the DF, the electric field is primarily in the tangential plane (*E*_{n}≪*E*_{m}). This electric field is contributed by the convection term, −*V*_{i} × *B*. Thus, the frozen-in condition is valid inside the FPR, while it is broken at the DF. This tangential electric field vanishes in the frame moving with the DF. This means there is no plasma flow across the DF boundary (*V*_{n} ≈ *E*_{l}*B*_{m} − *E*_{m}*B*_{l} ≈ 0), confirming that DF is a tangential discontinuity, as suggested by *Sergeev et al.* [2009] and *Khotyaintsev et al.* [2011].