The Vlasov equation describes collisionless plasmas in the continuum limit and applies to many fundamental plasma energization phenomena. Because this equation governs the evolution of plasma in six-dimensional phase space, studies of its structure have mostly been limited to numerical or analytical methods. Here terms of the Vlasov equation are determined from observations of electron phase-space density gradients measured by the four Magnetospheric Multiscale spacecraft in the vicinity of magnetic reconnection at Earth’s magnetopause. We identify which electrons in velocity space substantially support the electron pressure divergence within electron-scale current layers. Furthermore, we isolate and characterize the effects of density, velocity and temperature gradients on the velocity-space structure and dynamics of these electrons. Unipolar, bipolar and ring structures in the electron phase-space density gradients are compared to a simplified Maxwellian model and correspond to localized gradients in density, velocity and temperature, respectively. These structures have implications for the ability of collisionless plasmas to maintain kinetic Vlasov equilibrium. The results provide a kinetic perspective relevant to how the electron pressure divergence may develop to violate the electron frozen-in condition and sustain electron-scale energy conversion processes, such as the reconnection electric field, in collisionless space plasma environments.
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We especially thank the MMS instrument teams for their dedication and commitment to providing unprecedented, high-quality datasets. J.R.S. thanks L. Morrison for helpful discussions regarding the intricacies of phase space. This research was supported in part by NASA grants to the Fast Plasma Investigation, FIELDS team and Theory and Modeling programme of the MMS mission. J.R.S. was supported by NASA grants 80NSSC19K1092 and 80NSSC21K0732. S.W. was supported by NASA grant 80NSSC18K1369 and DOE grant DE-SC0020058. P.A.C. was supported by NASA grants NNX16AG76G and 80NSSC19M0146, NSF grants AGS-1602769 and PHY-1804428 and DOE grant DE-SC0020294. R.E.D. was supported by NASA grant 80NSSC19K0254. V.M.U. was supported by NASA grant NNG11PL10A.
The authors declare no competing interests.
Peer review information Nature Physics thanks Jan Egedal and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
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Shuster, J.R., Gershman, D.J., Dorelli, J.C. et al. Structures in the terms of the Vlasov equation observed at Earth’s magnetopause. Nat. Phys. 17, 1056–1065 (2021). https://doi.org/10.1038/s41567-021-01280-6