Resolving the Force Term of the Electron Vlasov Equation with MMS

Widely employed to model collisionless plasma phenomena occurring
naturally in Earth’s magnetic environment, throughout the heliosphere,
and in laboratory fusion devices, the Vlasov equation self-consistently
describes the fundamental kinetic dynamics of plasma particles as they
are accelerated through phase space via electric and magnetic forces.
The Fast Plasma Investigation (FPI) onboard NASA’s Magnetospheric
Multiscale (MMS) four-spacecraft mission sufficiently resolves the seven
spatial, temporal, and velocity-space dimensions of phase space needed
to directly observe terms in the Vlasov equation, as recently
demonstrated by *Shuster et al.* [2021] in the context of
electron-scale current layers at the reconnecting magnetopause. These
results motivate novel exploration of the types of distinct kinetic
signatures in ∂*f*_{e}/∂*t*,
v⋅∇*f*_{e}, and
(F/*m*_{e})⋅∇_{v}*f*_{e}
which are associated with the magnetic reconnection process, where F =
−*e*(E + v×B) represents the Lorentz force on an electron, and
*f*_{e} specifies the electron phase space density. We
apply this approach to characterize the structure of the velocity-space
gradient terms in the electron Vlasov equation measured by MMS.
Discussion of the uncertainties which arise when computing the
velocity-space gradients of the FPI phase space densities is presented,
along with initial validation of the
(F/*m*_{e})⋅∇_{v}*f*_{e}
measurements by comparison to the ∂*f*_{e}/∂*t*
and v⋅∇*f*_{e} terms. Successful measurement of the
force term
(F/*m*_{e})⋅∇_{v}*f*_{e}
in the Vlasov equation suggests a new technique for inferring spatial
gradients from single spacecraft measurements which may be applied to
improve the spatial resolution of the electron pressure divergence
∇⋅P_{e} necessary to understand the microphysics of the
electron diffusion region of magnetic reconnection. Reference: Shuster,
J. R., et al. (2021), Structures in the terms of the Vlasov equation
observed at Earth’s magnetosphere, *Nature Physics*,
doi:10.1038/s41567-021-01280-6.