Multi-scale structure of the electron diffusion region



[1] Kinetic simulations of magnetic reconnection indicate that the electron diffusion region (EDR) can elongate into a highly stretched current layer with a width on the electron scale and a length that exceeds tens of ion inertial lengths. The resulting structure has no fluid analogue and consists of two regions in the exhaust direction. The inner region is characterized by the locale where electrons reach a peak outflow speed near the electron Alfvén velocity. Ions also approach ∼80% of their peak velocity in this inner region but remain sub-Alfvénic. There exists a large electrostatic potential that can temporarily trap electrons within this inner region. The electron frozen-in condition is violated over a wider outer region characterized by highly collimated electron jets that are gradually decelerated and thermalized. Reconnection proceeds continuously but the rate is modulated in time as the EDR elongates into an extended layer. The elongation of the EDR is controlled by the competition between the outward convection of magnetic flux and the non-ideal term involving the divergence of the electron pressure tensor. The occasional balance between these two terms leads to periods of quasi-steady reconnection. However, over longer time scales, a natural feature of the reconnection process appears to be frequent formation of plasmoids due to the instability of the elongated EDR which leads to larger variations in the reconnection rate. These new findings provide testable predictions and indicate the need to reconsider the diagnostics for identification of the diffusion region and interpretation of observational data.

1. Introduction

[2] The basic structure and role of the electron diffusion region (EDR) in magnetic reconnection are the subject of recent controversy. While it is generally agreed that the onset of reconnection is controlled by the details of electron physics, the view articulated in the GEM challenge and related studies has been that the subsequent nonlinear evolution of the system and the resulting reconnection rate are controlled by the ions [e.g., Birn et al., 2001; Shay et al., 2001; Rogers et al., 2001]. According to these models, the EDR remains stable and is localized on the electron scale in both the inflow and outflow directions. In this scenario, the non-ideal electron region extends ∼5de in the outflow direction while the ion diffusion region extends ∼10di, where de and di are the electron and ion inertial lengths. However, recent simulations have brought these results into question [Daughton et al., 2006; Fujimoto, 2006]. Using large-scale fully kinetic simulations with both periodic and open boundary conditions, the structure of the EDR was observed to elongate in time along the exhaust direction with scales ∼5–20di while maintaining an electron scale thickness in the inflow direction [Daughton et al., 2006].

[3] In this work, the structure of the EDR is examined in detail using new diagnostics to objectively quantify the spatial region in which the electrons are demagnetized. The results indicate that the EDR consists of two distinct regions in the exhaust direction. The inner layer is characterized by a strong out-of-plane electron current that is produced by the reconnection electric field. This electron flow is turned into the outflow direction by the Lorentz force leading to an intense in-plane current layer or “outflow jet”. These electron jets are found to extend to ±10di before they are decelerated and thermalized. We demonstrate that the elongation of the EDR is a natural consequence of the rapid outward convection of magnetic flux due to the electron outflow jets. Reconnection proceeds continuously but the rate is modulated in time due to the elongation of the EDR. The expansion is counteracted by a localizing effect arising from the divergence of the electron pressure tensor and this leads to occasional periods of quasi-steady reconnection during which variations in the rate are relatively small. However, over longer time scales the elongated electron layers are unstable to plasmoid formation leading to larger variations in the rate. These findings are at odds with previous theoretical expectations and require rethinking of data analysis for existing missions as well as planning of the upcoming Magnetospheric Multiscale mission. One of the clear predictions of this work is the presence of thin but highly elongated, non-gyrotropic electron jets associated with the reconnection site.

2. Simulation Setup and Parameters

[4] We consider a Harris equilibrium with zero guide field using a coordinate system where the initial magnetic field is Bx = Botanh(z/L), where L is the half-width of the layer. The density profile is n(z) = nosech2(z/L) + nb where no is the peak Harris density and nb is the background density. Results are presented from two fully kinetic simulations with open boundary conditions [Daughton et al., 2006] and larger system sizes than considered previously. The first case (run 1) is 50di × 50di where di is an ion inertial length based on no. The simulation parameters are L/di = 0.91, mi/me = 100, Ti/Te = 5, nb/no = 0.3, ωpece = 3, 2560 × 2560 cells with 3 × 109 particles. The second case (run 2) has a larger system size 100di × 100di with parameters L/di = 0.35, Ti = Te, ωpece = 2, 2048 × 2048 cells with 2.5 × 109 particles and other parameters the same as run 1. Both simulations employ a weak long wavelength perturbation [Daughton et al., 2006].

3. Multi-Scale Structure

[5] The structure of the reconnection layer is shown in Figure 1 at tΩ ci = 80 for run 1. In the first two panels, the ion and electron outflow velocities are normalized by their respective Alfvén speed VAs = Bs/(4πmsnb)1/2 based on the magnetic field upstream of the ion and electron diffusion regions (Bi ≈ 0.78 Bo for ions and Be ≈ 0.36 Bo for electrons). The non-ideal electric field (E + Ue × B)y is shown in the third panel, followed by the electron agyrotropy Ae ≡ 2∣Pe1Pe2∣/(Pe1 + Pe2) which ranges from a maximum of 2 down to 0 in the gyrotropic limit. Nonzero Ae indicates demagnetization of thermal electrons. In the last panel, the electrostatic potential ϕ is normalized by the initial uniform electron temperature Te. Below each quantity is a 1D cut along the horizontal line indicated. Figure 1 reveals interesting features as well as a number of surprises.

Figure 1.

Results from run 1 at tΩci = 80 showing ion outflow Uix, electron outflow Uex, non-ideal electric field (E + Ue × B)y (normalized by Ey at x-point), electron agyrotropy Ae and electrostatic potential normalized to the initial electron temperature eϕ/Te. Black lines correspond to flux surfaces and 1D line plots are along the horizontal white line indicated in each panel.

[6] The 1D cuts clearly show the multi-scale nature of the EDR. The length De of the inner region is defined by the location where the electrons reach their peak outflow velocity (shaded red). The ions going through this layer are also accelerated over a similar scale to ∼80% of their peak velocity. The acceleration of ions within this region is due to the presence of the electrostatic potential. In contrast, the electrons are accelerated by the reconnection electric field Ey and redirected into the outflow by the Lorentz force. Notice that the electrostatic potential acts to counteract this turning process and thus may play a key role in determining the structure of the layer. The length Δe of the outer region is identified by the location where the electrons become fully magnetized Ae → 0, marked by the vertical lines. This occurs much farther out and coincides with the termination of the electron jets. The electron bulk flow slows down in the outer layer, converted partially to thermal energy. In most of the inner layer the electrons lag the field (E + Ue × B)y > 0 whereas in the outer layer (E + Ue × B)y < 0 indicating that electrons are outrunning the field. The electron pressure tensor balances this non-ideal electric field in the vicinity of the x-point and continues to play a dominant role along the entire length of the EDR.

[7] The full length of the electron jets are seen to extend to Δe ∼ 20di in the exhaust direction rather than the expected 10de [e.g., Shay et al., 2001]. In the previous studies, the location of the peak electron outflow velocity was argued to be where the electrons start to become fully magnetized. In the present study, the elongation of De to ion scales drives an intense electron outflow jet that forces the point of full magnetization Δe farther downstream, giving rise to a two-scale structure in the outflow direction.

[8] The electron outflow velocity is near the theoretical limit of the electron Alfvén velocity UexVAe based on conditions upstream of the EDR. This is not sufficient to overcome the bottleneck that arises from the elongation of De, and thus variations in the reconnection rate are observed.

[9] The ion outflow velocity is significantly less than the Alfvén speed Uix ≈ 0.4VAi using the proper normalization based on upstream conditions. This is considerably below the expected value based on the traditional understanding [e.g., Shay et al., 1999]. This may be partly due to the presence of significant pressure gradients in these solutions.

[10] There exists a large in-plane electrostatic field associated with the inner layer and also along the separatrices. The potential difference between the x-point and the edge of the inner layer (red region) is nearly eϕ/Te ∼ 4 in Figure 1. This implies that most electrons are temporarily trapped within the inner region until they gain enough energy from the reconnection electric field to escape. This process may play a crucial role in determining the structure and energy exchange within the layer.

[11] From these results, we have developed a new picture of the EDR as illustrated in Figure 2. The resulting electron layer has a width on the order δe ∼ 2 − 4de and a total length that can exceed Δe > 20di in the exhaust direction. The inner portion of the EDR centered about the x-point (shaded red) corresponds to a nearly uniform inflow as indicated by the electron streamlines. The length De of this region coincides with the location where electrons reach their maximum outflow speed near VAe. Beyond the inner region, the electron jets start to slow down whereas ions continue to accelerate. The outer edge of EDR is defined by the location where electrons become magnetized as determined by the abrupt drop in the agyrotropy Ae as shown in Figure 1. This also corresponds to the termination of the electron jet and where the frozen-in condition is again satisfied E + Ue × B ≈ 0. These results point to a more complex structure of the electron diffusion region than previously expected. In the traditional picture, the length of the non-ideal region is well marked by the region of inflowing plasma. In contrast, the new results indicate a significant inflow of electrons only in the inner region while strong deviations from the frozen-in condition persist to much larger distances exceeding 10di.

Figure 2.

The multi-scale structure of electron diffusion region. The black lines are the electron streamlines for run 1 at tΩci = 80. The inner region (red) corresponds to nearly uniform electron inflow and strong out-of-plane current while the yellow regions correspond to the extended electron outflow jets.

4. Physical Mechanism for the Elongation

[12] We start with the generalized Ohm's law

display math

where G denotes the non-ideal terms. Combining this equation with Faraday's law along the centerline of the outflow (z = 0) yields:

display math

The first term on the right-hand side is the convective term that in the frozen-in limit expresses the outward convection of magnetic flux (e.g., due to electron jets). Since the profiles of Uex and Bz are both antisymmetric about the x-point, the product must be of the form UexBzx2 for some region about the x-point. Simulations indicate this corresponds approximately to the shaded red region in Figures 1 and 2. In the limit Gy → 0, the convection term alone implies ∂∣Bz∣/∂t < 0 within this region, indicating a collapse of the 2D structure into an elongated 1D current layer. Clearly at the x-point, a finite Gy is required to balance the reconnection electric field Ey. However, the non-ideal terms also must play an essential role in determining the length of the layer. While it has been argued that Hall physics gives rise to a constant UexBz in the outflow direction [Rogers et al., 2001], this clearly does not apply within the EDR. Instead, the dynamical evolution of the inner region (red in Figures 12) is controlled by the competition between the outward convection UexBz which elongates the layer and the localizing influence of Gy.

[13] In the outer region (yellow in Figures 1 and 2) the convection term UexBz is a decreasing function of x as the electron outflow jets slow down. The deceleration of the jets is accompanied by a large increase in electron pressure (factor of 7 increase between the x-point and the end of jet in Figure 1) which may partially regulate the length Δe. However, to achieve a steady-state balance for this region the non-ideal terms in (2) are again essential.

[14] These arguments imply that the expansion of the EDR will continue unless it can be balanced by the second term and/or interrupted by some other dynamic process such as the formation of a secondary island which can interrupt the expansion by forming two competing x-points [Daughton et al., 2006; Daughton and Karimabadi, 2007].

[15] To determine which terms dominate the balance in (2), Figure 3 shows the cut along the outflow of Ey, UexBz, and the two terms comprising Gy, namely, dUey/dt and (∇ · Pe)y at three different times for run 1. The spatial profile of Gy provides a measure of the region where the frozen-in condition is broken and is dominated by (∇ · Pe)y at most locations along the outflow. It is useful to draw a parallel to fluid models and think of the spatial extent of Gy as the region of finite “effective resistivity”, even though the physics of ∇ · Pe is very different than classical resistivity. It is apparent from Figure 3 that the mechanism that breaks the frozen-in condition is spatially localized, with significant gradients in the inner region, and with both its magnitude and spatial extent changing in time. At tΩci = 42, the two terms on the right hand side of (2), UexBz and Gy, are clearly out of balance, with Gy more localized than UexBz indicating the EDR is still expanding. At tΩci ∼ 48, a secondary island forms which leads to a temporary halt in the elongation. Between tΩci = 70–120, however, the UexBz term is nearly balanced by the non-ideal terms giving rise to a uniform Ey. During this interval, both the structure of the EDR and the observed reconnection rate are nearly stationary. However, the total length Δe ≈ 20di is comparable to what previous researchers had predicted for the ion diffusion region!

Figure 3.

Various terms in (1) along outflow z = 0 for run 1 at three selected time slices. Terms in each panel are time averaged over an interval ΔtΩci = 2 and then normalized by the electric field Ey at time tΩci = 80.

5. Time Dependence of Reconnection

[16] A clear correlation between the elongation of the inner region De and the reconnection rate was demonstrated by Daughton et al. [2006]. The elongation of De, if continued unchecked, would lead to a significant reduction of the reconnection rate. However, as demonstrated in Figure 3, the divergence of the electron pressure tensor can at times balance the convection term and temporarily halt the expansion of the EDR. This leads to quasi-steady behavior of the reconnection rate over some time interval.

[17] The normalized reconnection rate is computed from

display math

where 〈Ey〉 denotes the average electric field at the x-point over a time interval 2Ωci−1 and both Bi and VAi are evaluated upstream of the ion diffusion region (at z ≈ 2.5di). In the presence of multiple x-points, this expression is evaluated for each and the maximum rate is selected. Figure 4 shows time evolution of ER for runs 1 and 2. The peak rate is the same for both cases followed by a decrease in time as De elongates. Both runs show periods of quasi-steady ER during which the terms in (2) approximately balance. The larger system (run 2) transitions into the continual formation of secondary islands for tΩci > 90 leading to larger variations in the reconnection rate. This suggests that reconnection may be inherently unsteady in large-scale systems.

Figure 4.

Reconnection rate normalized to conditions upstream of the ion diffusion region for run 1 (dashed line) and run 2 (solid line).

6. Conclusions

[18] Our results indicate that the electron diffusion region (EDR) forms a highly elongated layer with multiple-scales and a structure that is very different than previous expectations. The scale of the inner region of the EDR is in the range De ≈ 7–11di for the two simulations considered with mass ratio mi/me = 100. A more recent simulation with parameters similar to run 2 but with mi/me = 400 resulted in De ∼ 6.5 versus 9di for run 2 measured at the same time tΩci = 50. This is consistent with the weak mass ratio scaling of De/di ∼ (me/mi)1/4 first reported by Daughton et al. [2006]. For the physical mass ratio of hydrogen, this implies a scale of De ∼ 3 − 5di for the inner region of the EDR, but more work is needed to accurately determine the scaling with electron mass and temperature. The outer region of the EDR is longer and more dynamic with scales as large as Δe ∼ 20di in run 1 and Δe ∼ 40di in run 2. Thus even at realistic mass ratio it is expected that Δe may extend to tens of di.

[19] These findings have important consequences for identification of the EDR and interpretation of observational data for both existing and upcoming missions. For example, the full length of the non-ideal electron region would be a factor of 40–100 larger than previous estimates. One clear prediction that may be observationally testable is the presence of a thin (δe ∼ 4de) super-Alfvénic agyrotropic electron jet that may extend large distances from the x-line.

[20] The dynamical evolution of the EDR is controlled by a competition between the outward convection of magnetic flux and the localizing effect of ∇ · Pe. These two processes can balance at times, giving rise to a quasi-steady reconnection rate, although the system as whole remains time-dependent. One of the controversial issues regarding reconnection at the magnetopause has been whether reconnection occurs in a continuous or intermittent (switching on and off) fashion. Our results indicate that the number and location of x-lines can change in time but there always remains at least one x-line in the system and the macroscopic reconnection rate remains continuous but with modulations in time. This appears consistent with a recent Cluster study by Phan et al. [2004]. In the present simulations, the duration of quasi-steady reconnection is ∼50Ωci−1 corresponding to ∼15 seconds at the magnetopause, but longer durations cannot be ruled out. One natural feature of the reconnection process appears to be the frequent formation of plasmoids resulting from the instability of the elongated EDR. This may have relevance to observations of plasmoids in the magnetotail and flux transfer events (FTEs) in the dayside magnetopause.

[21] Finally, these results are based entirely on 2D simulations which do not permit instabilities in the out-of-plane direction such as the lower-hybrid drift mode. The influence of these instabilities on the structure of the reconnection layer remains an open question.


[22] This work was supported by NASA grants NNG05GJ25G, NNG05GJ01G, and IGPP grant, NSF grant 0447423, and by the DOE under award DE-FG02-06ER54893. Simulations were partially performed at LANL.