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.
This is a preview of subscription content, access via your institution
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 / 30 days
cancel any time
Subscribe to this journal
Receive 12 print issues and online access
$209.00 per year
only $17.42 per issue
Rent or buy this article
Prices vary by article type
Prices may be subject to local taxes which are calculated during checkout
All MMS data are available to the public via https://lasp.colorado.edu/mms/sdc/public/.
The code used to plot the MMS gradient distributions will be made available upon reasonable request.
Vlasov, A. A. On the kinetic theory of an assembly of particles with collective interaction. J. Phys. (USSR) 9, 25–40 (1945).
Nicholson, D. R Introduction to Plasma Theory (Wiley, 1983).
Landau, L. D. On the vibrations of the electronic plasma. J. Phys. (USSR) 10, 25–34 (1946).
Matthaeus, W. H. & Lamkin, S. L. Turbulent magnetic reconnection. Phys. Fluids 29, 2513–2534 (1986).
Daughton, W. Nonlinear dynamics of thin current sheets. Phys. Plasmas 9, 3668–3678 (2002).
Sundkvist, D., Retinò, A., Vaivads, A. & Bale, S. D. Dissipation in turbulent plasma due to reconnection in thin current sheets. Phys. Rev. Lett. 99, 025004 (2007).
Gershman, D. J. et al. Wave–particle energy exchange directly observed in a kinetic Alfvén-branch wave. Nat. Commun. 8, 14719 (2017).
Viñas, A. F. & Klimas, A. J. Flux-balance Vlasov simulation with filamentation filtration. J. Comput. Phys. 375, 983–1004 (2018).
Li, T. C., Howes, G., Klein, K., Liu, Y.-H. & TenBarge, J. M. Collisionless energy transfer in kinetic turbulence: field-particle correlations in Fourier space. J. Plasma Phys. 85, 905850406 (2019).
Chen, C., Klein, K. & Howes, G. Evidence for electron Landau damping in space plasma turbulence. Nat. Commun. 10, 740 (2019).
Moore, T. E., Burch, J. L. & Torbert, R. B. Magnetic reconnection. Nat. Phys. 11, 611–613 (2015).
Cassak, P. A. Inside the black box: magnetic reconnection and the magnetospheric multiscale mission. Space Weather 14, 186–197 (2016).
Burch, J. L. et al. Electron-scale measurements of magnetic reconnection in space. Science 352, aaf2939 (2016).
Torbert, R. B. et al. Electron-scale dynamics of the diffusion region during symmetric magnetic reconnection in space. Science 362, 1391–1395 (2018).
Webster, J. M. et al. Magnetospheric multiscale dayside reconnection electron diffusion region events. J. Geophys. Res. Space Phys. 123, 4858–4878 (2018).
Phan, T. D. et al. Electron magnetic reconnection without ion coupling in Earth’s turbulent magnetosheath. Nature 557, 202–206 (2018).
Hesse, M. & Cassak, P. A. Magnetic reconnection in the space sciences: past, present and future. J. Geophys. Res. Space Phys. 125, e2018JA025935 (2020).
Roth, M., De Keyser, J. & Kuznetsova, M. M. Vlasov theory of the equilibrium structure of tangential discontinuities in space plasmas. Space Sci. Rev. 76, 251–317 (1996).
Neukirch, T., Wilson, F. & Allanson, O. Collisionless current sheet equilibria. Plasma Phys. Control. Fusion 60, 014008 (2018).
Harris, E. G. On a plasma sheath separating regions of oppositely directed magnetic field. Nuovo Cimento 23, 115–121 (1962).
Chen, L.-J. et al. Evidence of an extended electron current sheet and its neighboring magnetic island during magnetotail reconnection. J. Geophys. Res. Space Phys. 113, A12213 (2008).
Egedal, J., Daughton, W. & Le, A. Large-scale electron acceleration by parallel electric fields during magnetic reconnection. Nat. Phys. 8, 321–324 (2012).
Bessho, N., Chen, L.-J., Shuster, J. R. & Wang, S. Electron distribution functions in the electron diffusion region of magnetic reconnection: physics behind the fine structures. Geophys. Res. Lett. 41, 8688–8695 (2014).
Shuster, J. R. et al. Spatiotemporal evolution of electron characteristics in the electron diffusion region of magnetic reconnection: implications for acceleration and heating. Geophys. Res. Lett. 42, 2586–2593 (2015).
Wang, S. et al. Electron heating in the exhaust of magnetic reconnection with negligible guide field. J. Geophys. Res. Space Phys. 121, 2104–2130 (2016).
Yamada, M. et al. Experimental investigation of the neutral sheet profile during magnetic reconnection. Phys. Plasmas 7, 1781–1787 (2000).
Pritchett, P. L. Collisionless magnetic reconnection in an asymmetric current sheet. J. Geophys. Res. Space Phys. 113, A06210 (2008).
Daughton, W., Nakamura, T. K. M., Karimabadi, H., Roytershteyn, V. & Loring, B. Computing the reconnection rate in turbulent kinetic layers by using electron mixing to identify topology. Phys. Plasmas 21, 052307 (2014).
Swisdak, M. et al. Localized and intense energy conversion in the diffusion region of asymmetric magnetic reconnection. Geophys. Res. Lett. 45, 5260–5267 (2018).
Alpers, W. Steady state charge neutral models of the magnetopause. Astrophys. Space Sci. 5, 425–437 (1969).
Channell, P. J. Exact Vlasov–Maxwell equilibria with sheared magnetic fields. Phys. Fluids 19, 1541–1545 (1976).
Mottez, F. Exact nonlinear analytic Vlasov–Maxwell tangential equilibria with arbitrary density and temperature profiles. Phys. Plasmas 10, 2501–2508 (2003).
Aunai, N., Belmont, G. & Smets, R. First demonstration of an asymmetric kinetic equilibrium for a thin current sheet. Phys. Plasmas 20, 110702 (2013).
Allanson, O., Wilson, F., Neukirch, T., Liu, Y.-H. & Hodgson, J. D. B. Exact Vlasov–Maxwell equilibria for asymmetric current sheets. Geophys. Res. Lett. 44, 8685–8695 (2017).
Mottez, F. The pressure tensor in tangential equilibria. Ann. Geophys. 22, 3033–3037 (2004).
Newman, D. L., Lapenta, G. & Goldman, M. Dynamics of Electron-scale Current Sheet Wquilibria based on MMS Observations American Geophysical Union Fall Meeting, eLightning Presentation SM24B-13 (Earth and Space Science Open Archive, 2019); https://doi.org/10.1002/essoar.10502614.1
Newman, D. L., Sen, N. & Goldman, M. V. ‘Reduced’ multidimensional Vlasov simulations, with applications to electrostatic structures in space plasmas. Phys. Plasmas 14, 055907 (2007).
Holmes, J. C. et al. Structure of electron-scale plasma mixing along the dayside reconnection separatrix. J. Geophys. Res. Space Phys. 124, 8788–8803 (2019).
Palmroth, M. et al. Vlasov methods in space physics and astrophysics. Living Rev. Comput. Astrophys. 4, 1 (2018).
von Alfthan, S. et al. Vlasiator: first global hybrid-Vlasov simulations of Earth’s foreshock and magnetosheath. J. Atmos. Sol. Terr. Phys. 120, 24–35 (2014).
Hoilijoki, S. et al. Reconnection rates and X line motion at the magnetopause: global 2D-3V hybrid-Vlasov simulation results. J. Geophys. Res. Space Phys. 122, 2877–2888 (2017).
Vasyliunas, V. M. Theoretical models of magnetic field line merging. Rev. Geophys. 13, 303–336 (1975).
Viñas, A. F. & Gurgiolo, C. Spherical harmonic analysis of particle velocity distribution function: comparison of moments and anisotropies using cluster data. J. Geophys. Res. Space Phys. 114, A01105 (2009).
Torbert, R. B. et al. Estimates of terms in Ohm’s law during an encounter with an electron diffusion region. Geophys. Res. Lett. 43, 5918–5925 (2016).
Shuster, J. R. et al. Hodographic approach for determining spacecraft trajectories through magnetic reconnection diffusion regions. Geophys. Res. Lett. 44, 1625–1633 (2017).
Rager, A. C. et al. Electron crescent distributions as a manifestation of diamagnetic drift in an electron-scale current sheet: magnetospheric multiscale observations using new 7.5-ms fast plasma investigation moments. Geophys. Res. Lett. 45, 578–584 (2018).
Hesse, M., Neukirch, T., Schindler, K., Kuznetsova, M. & Zenitani, S. The diffusion region in collisionless magnetic reconnection. Space Sci. Rev. 160, 3–23 (2011).
Hesse, M., Aunai, N., Sibeck, D. & Birn, J. On the electron diffusion region in planar, asymmetric, systems. Geophys. Res. Lett. 41, 8673–8680 (2014).
Genestreti, K. J. et al. MMS observation of asymmetric reconnection supported by 3-D electron pressure divergence. J. Geophys. Res. Space Phys. 123, 1806–1821 (2018).
Shuster, J. R. et al. MMS measurements of the Vlasov equation: probing the electron pressure divergence within thin current sheets. Geophys. Res. Lett. 46, 7862–7872 (2019).
Pollock, C. et al. Fast plasma investigation for magnetospheric multiscale. Space Sci. Rev. 199, 331–406 (2016).
Phan, T. D. et al. MMS observations of electron-scale filamentary currents in the reconnection exhaust and near the X line. Geophys. Res. Lett. 43, 6060–6069 (2016).
Mahajan, S. M. Exact and almost exact solutions to the Vlasov–Maxwell system. Phys. Fluids B Plasma Phys. 1, 43–54 (1989).
Klimas, A. J. & Viñas, A. F. Open-boundary spectral and flux-balance Vlasov simulation. J. Plasma Phys. 85, 905850610 (2019).
Gershman, D. J. et al. Spacecraft and instrument photoelectrons measured by the dual electron spectrometers on MMS. J. Geophys. Res. Space Phys. 122, 11548–11558 (2017).
Gershman, D. J., Dorelli, J. C., F.-Viñas, A. & Pollock, C. J. The calculation of moment uncertainties from velocity distribution functions with random errors. J. Geophys. Res. Space Phys. 120, 6633–6645 (2015).
Gurnett, D. A. & Bhattacharjee, A. Introduction to Plasma Physics: With Space and Laboratory Applications (Cambridge Univ. Press, 2005).
Harvey, C. C. in Analysis Methods for Multi-Spacecraft Data ISSI Scientific Reports Series (eds Paschmann, G. & Daly, P. W.) 307–322 (ISSI/ESA, 1998).
Wetherton, B. A., Egedal, J., Montag, P., Le, A. & Daughton, W. A drift-kinetic method for obtaining gradients in plasma properties from single-point distribution function data. J. Geophys. Res. Space Phys. 125, e2020JA027965 (2020).
Hazeltine, R. D. & Meiss, J. D. Plasma Confinement (Addison-Wesley, 1992).
Graham, D. B. et al. Electron currents and heating in the ion diffusion region of asymmetric reconnection. Geophys. Res. Lett. 43, 4691–4700 (2016).
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.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
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