Fig. 2: Multi-spacecraft observations and kinetic simulations of smile-shaped electron gradient distributions at Earth’s magnetopause.

a–c 10-second overview of Magnetospheric Multiscale (MMS) 2 measurements with a vertical dashed line and green star indicating the time of the electron distributions (f, g) on the right. a Magnetic field data from the fluxgate magnetometer (FGM) in boundary-normal (LMN) coordinates. The coordinate transformation is provided by ref. ². b Electron omni-directional energy spectrogram from the dual electron spectrometers (DES). c Electron temperatures: total T_e (black), perpendicular T_e⊥ (red), and parallel T_e∥ (blue). d Magnitudes of the electron pressure divergence ∣ ∇ ⋅ P_e∣ (black), its decomposition into the temperature divergence ∣n_e ∇ ⋅ T_e∣ (orange) and density gradient ∣T_e ⋅ ∇ n_e∣ (green) contributions computed from the four spacecraft, along with the inertial term’s spatial divergence ∣m_e ∇ ⋅ (n_eU_eU_e)∣ (red) and the time derivative term ∣m_e∂_t(n_eU_e)∣ (blue) computed from MMS 2 for reference. e Zoom-in view (0.3 seconds) of the high cadence parallel electric field E_∥ from the electric field double probes (EDP) on MMS 2. f The four crescent-shaped electron velocity distributions f_e from the four MMS spacecraft used to compute (g) the smile-shaped electron gradient distribution oriented parallel to the magnetic field ∇_∥f_e at 13:07:02.160 UT indicated by the green star in (a–d). h–j Particle-in-cell (PIC) simulation results comparable to the MMS observations. h The 2D color plot shows the parallel electric field arising from the electron pressure divergence term \({(\nabla \cdot {{{{\bf{P}}}}}_{e})}_{\parallel }/{n}_{e}e\). The 1D cut panels below taken along the dashed lines at z = 8 d_e and z = 0 d_e show \({(\nabla \cdot {{{{\bf{P}}}}}_{e})}_{\parallel }/{n}_{e}e\) (black) balanced well by  − E_∥ (red), and the dominant temperature divergence (orange) and density (green) gradient contributions. For the z = 8 d_e cut, these gradient components are plotted for x < 320d_e and x > 420d_e to focus on the coherent separatrix structures in \({(\nabla \cdot {{{{\bf{P}}}}}_{e})}_{\parallel }/{n}_{e}e\) rather than the noisier signal on the low-density (z > 0) side between 320 < x < 420. i Smile-shaped electron gradient distribution in the parallel direction ∇_∥f_e taken at (x, z) = (375, 2) d_e in the PIC domain indicated by the right yellow box in (h) to compare with the MMS gradient distribution above. j Prominent smile-shaped electron gradient distribution ∇_zf_e oriented perpendicular to the magnetic field taken at (x, z) = (365, 2) d_e and indicated by the left yellow box in (h). Two electron crescent distributions are shown to the right from z = 3 d_e and z = 2 d_e, illustrating how the crescent population varies spatially to produce the smile-shape.