Pressure Tensor Elements Breaking the Frozen-In Law During Reconnection in Earth’s Magnetotail

Aided by fully kinetic simulations, spacecraft observations of magnetic reconnection in Earth’s magnetotail are analyzed. The structure of the electron diffusion region is in quantitative agreement with the numerical model. Of special interest, the spacecraft data reveal how reconnection is mediated by off-diagonal stress in the electron pressure tensor breaking the frozen-in law of the electron fluid.

Figure 1

(a)–(k) MMS1 (red lines) and MMS3 (blue lines) measurements of a range of quantities during the July 11, 2017 EDR encounter, with $t=0s$ corresponding to 22∶34:00 UT. The raw data provided by MMS are translated to the $LMN$ coordinates of the event determined in the Supplemental Material [22]. In geocentric solar ecliptic (GSE) coordinates, our unit vectors $[\mathbf{L};\mathbf{M};\mathbf{N}]$ are $[0.94,-0.35,-0.03;0.32,0.90,-0.33;0.15,0.30,0.94]$. (l)–(v) Matching simulation quantities evaluated along the trajectories shown in Fig. 2. Error estimates are shown in pink and the vpic results are translated to MMS units and plotted with axis settings identical to those of (a)–(k).

Figure 2

Numerical profiles of $e{\mathrm{\Phi }}_{\parallel }/T_e$, $B_L$, $B_N$ $T_{e\parallel }/T_{e\perp }$, and $E_N$. Trapped electron trajectories are shown in (a), while the reconstructed path of the MMS spacecraft through the simulation is shown by the red (MMS1) and black (MMS3) lines in (b)–(e). Length scales are normalized by $d_e\simeq 29\mathrm{km}$ calculated using the vpic density $n/n_0=0.75$ observed at the $x$ line. Here $n_0$ is the initial density in the center of the simulation domain. For the simulation, open boundary conditions were applied [24] on a domain of total size of $4032\times 4032$ cells = $10d_i\times 10d_i\simeq 428d_e\times 428d_e$, initialized with approximately 800 particles per species per cell. Other parameters determined to be consistent with the observations [22] include ratios of upstream electron and ion pressure to magnetic pressure of ${\beta }_{e\infty }=0.045$ and ${\beta }_{i\infty }=0.45$. Also, our optimization scheme [22] led to the inclusion of a small out of plane guide magnetic field $B_g/B_0=0.006$, here normalized by the initial upstream reconnection magnetic field.

Figure 3

In (a) and (b) the MMS and vpic profiles of $P_{LM}$ are shown in the format of Fig. 1. In (c) the blue line is the raw $P_{MN}$ signal provided by MMS3, while the red line is the same signal low-pass filter at $\lesssim 4\mathrm{Hz}$ [the same filter was also for the lines in (a)]. In (d) low-pass filtered signals of $P_{MN}$ are shown for all MMS spacecraft. Given the vpic structure of $\partial P_{MN}/\partial L$ in (e) and the MMS constellation in (f), from the lines in (d) we can deduce a MMS profile of $\partial P_{MN}/\partial N$ and estimate its error range [see text and Fig. 4].

Figure 4

(a)–(d) Color contours of terms important for the electron momentum balance in the $M$ direction. (e),(f) Matching profiles of MMS and vpic pressure tensor term $(\partial p_{eMN}/\partial N)/ne$, where the MMS profile is obtained from the curves in Fig. 3. The ranges of error are shown in pink, determined as described in the text, and are consistent with quantitative agreement between MMS and vpic signals within the inner EDR.