Velocity-Space Structure of Terms in the Electron Vlasov Equation: MMS Magnetopause Observations and Model Results

The Vlasov equation describes collisionless plasmas in the continuum limit and applies to fundamental plasma energization phenomena occurring in Earth's magnetosphere, throughout the heliosphere, and in laboratory fusion experiments. Because this equation governs the evolution of plasmas in six-dimensional phase space, studies of its structure typically rely on numerical or analytical approaches. In this work, each term of the Vlasov equation is determined from direct observations of electron phase-space density gradients measured by the Magnetospheric Multiscale (MMS) spacecraft in the vicinity of magnetic reconnection occurring at Earth's magnetopause. The unprecedented temporal, spatial, and velocity-space resolution offered by the MMS tetrahedron enables us to identify the electron velocity-space distribution that supports the pressure divergence within electron-scale current layers. We characterize the relationship between the distribution's velocity-space structure and spatial gradients in the bulk plasma moments: unipolar, bipolar, and ring structures in the electron phase-space density gradient terms are compared to a simplified Maxwellian model and correspond to gradients in density, velocity, and temperature, respectively. We compare the MMS observations to exact Vlasov-Maxwell solutions and particle-in-cell simulations of asymmetric reconnection suitable for modeling Earth's magnetopause. Our results provide a perspective relevant to how the electron pressure divergence develops to violate the frozen-in condition and sustain electron-scale energy conversion processes, such as the reconnection electric field, in collisionless plasmas. This work is immediately relevant to the study of fundamental energy conversion processes, including electron diffusion regions fueling magnetic reconnection, kinetic-scale turbulence, and wave-particle interactions consistent with Landau damping that were recently reported using MMS data from Earth's turbulent magnetosheath.