Modern investigations of dynamical space plasma systems such as magnetically complicated topologies within the Earth's magnetosphere make great use of supercomputer models as well as spacecraft observations. Space plasma simulations can be used to investigate energy transfer, acceleration, and plasma flows on both global and local scales. Simulation of global magnetospheric dynamics requires spatial and temporal scales currently achievable through magnetohydrodynamics or hybrid-kinetic simulations, which approximate electron dynamics as a charge-neutralizing fluid. We introduce a novel method for Vlasov-simulating electrons in the context of a hybrid-kinetic framework in order to examine the energization processes of magnetospheric electrons. Our extension of the Vlasiator hybrid-Vlasov code utilizes the global simulation dynamics of the hybrid method whilst modelling snapshots of electron dynamics on global spatial scales and temporal scales suitable for electron physics. Our eVlasiator model is shown to be stable both for single-cell and small-scale domains, and the solver successfully models Langmuir waves and Bernstein modes. We simulate a small test-case section of the near-Earth magnetotail plasma sheet region, reproducing a number of electron distribution function features found in spacecraft measurements.

Physical processes in near-Earth space are dominated by plasma effects such as non-thermal particle distributions, instabilities, plasma waves, shocks, and reconnection. Modern research into space phenomena utilizes both spacecraft measurements and supercomputer simulations, investigating how ions, electrons, and electric and magnetic fields interact in the vicinity of plasma structures. Spacecraft provide point-like observations, limited in their ability to investigate spatial structures, although modern constellation missions can have multiple satellites close by allowing for multipoint analysis to decipher, e.g. current sheet directions

Simulations capable of modelling the whole near-Earth geospace have historically used magnetohydrodynamics, neglecting kinetic effects and implementing electrons only as the Hall term correction to Ohm's law for example. These models can be run for extended periods of time, but as they model plasma motion as a fluid, they use coarse grids, e.g.

In order to understand electron physics, kinetic modelling of electrons has been investigated by a number of methods such as full-PIC (ions and electrons as interacting particles, e.g.

More recent simulation studies of electron physics in the magnetosphere such as the PIC simulations by

An example of small-scale global electromagnetic implicit PIC modelling for a weak comet has been performed by

Another approach compared to PIC simulations is to represent particle velocity distributions with moments beyond the MHD approach

Several processes that occur in the magnetosphere that depend on electron behaviour are still poorly understood. Recently, missions such as Magnetospheric MultiScale

This paper introduces an alternative, novel method for simulating electron distribution function physics in the context of global ion-determined fields. The aim is to investigate how much of the global electron physics and distribution functions can be understood by utilizing ion-generated field as modelled by hybrid-kinetic codes, as opposed to a numerically unfeasible global full-kinetic approach. The paper is organized as follows. In Sect.

Vlasiator

In the spatial domain, Vlasiator can be run in 1D, 2D, or 3D, with 2D the most usual choice in order to evaluate global dynamics. Simulations have used spatial resolutions of

Vlasiator models standard collisionless space plasmas dominated by protons but can also model other particle species in the same self-consistent simulation. However, until now, the electron population has been treated in the usual way of implementing it as a massless charge-neutralizing fluid. The method does not track the evolution of electrons beyond assuming charge neutrality, and therefore, these standard Vlasiator simulations cannot be used to infer electron dynamics. This paper presents a novel approach for investigating how a global plasma current structure can influence electrons with limited self-consistency enforced through plasma oscillation and current densities.

Vlasiator uses the hybrid-Vlasov ion approach to model the near-Earth space plasma environment. The full six-dimensional (6D) phase space density

The generalized Ohm's law providing closure for the Vlasov system is

As Vlasov methods do not propagate particles but rather evolve distribution functions, we now briefly explain the semi-Lagrangian method employed by Vlasiator (for a full description, see chapter 5.3.1 in

In this section we introduce a novel method of simulating electron dynamics within the Earth's magnetic domain by building on the strengths of Vlasiator simulations. The method, called eVlasiator, focuses on the evolution of accurately modelled velocity distribution functions based on global plasma dynamics and structures evolved by the hybrid model. The spatial scales used in Vlasiator are not sufficient to resolve in detail small-scale phenomena such as electron-dominated reconnection, but this balances out with a realistic representation of global structures and asymmetries of the whole magnetosphere. The eVlasiator model solves the Vlasov equation for electron distribution functions using mostly the same methodology as Vlasiator itself but applies a simplified field solver, neglecting magnetic field evolution.

Modelling the evolution of electron distribution functions in response to global magnetic field structures requires input from the large-scale fields and moments of a Vlasiator simulation of near-Earth space. In the eVlasiator approach, we read magnetic field vectors and proton plasma moments for the chosen simulation domain and apply user-defined temperature scaling to generate initial Maxwellian electron velocity distribution functions. We do not model electrons throughout the whole global domain, choosing instead a region of interest to reduce the computational cost, though our method is designed to work with any subset of and up to the whole global domain. For the selected domain, we read in the Vlasiator ion-hybrid simulation proton moments, cell-face-average magnetic field components and cell-edge-average electric field components (the latter being used by the staggered-grid field solving algorithm from

eVlasiator solves the evolution of electron eVDFs similar to how Vlasiator simulates proton VDFs (for a detailed explanation, see

During each translation step, as depicted in Fig.

During each acceleration step, as depicted in Fig.

Due to the inherent connection between rapid electron motion and the local electric field response, we update electric fields in tandem with electron acceleration. The approach is detailed in the next subsection.

In the eVlasiator field solver we maintain static magnetic fields as read from the input Vlasiator simulation, only calculating electric field evolution. We model the electric field by including additional terms in Ohm's law (Eq.

As we keep magnetic fields static, we do not implement Faraday's law (Eq.

Collisionless plasma physics assumes that electrons are fast enough to balance out any charge imbalance, and in hybrid-kinetic simulations this holds true. We do not implement Gauss's law (Eq.

The last term in Ampère's law (Eq.

As our plasma remains collisionless, we maintain our assumption of infinite conductivity, and thus the

The Hall term,

As eVlasiator models electrons with full distribution functions, we include the full electron pressure tensor

The final term of the general Ohm's law is the electron inertia term. Much like with our choice of including the displacement current, we now include the electron inertia term in our solver.

For electron dynamics to be modelled, electron gyration and plasma oscillation must both be considered. We choose to limit the acceleration time step

The electron oscillation and electric field feedback loop is handled in parallel with gyration by tracking a cell-volume-averaged electric field

Electron solver procedure including substepping. At simulation start, a half-length acceleration step (0.) is performed. After that, translation (1, 3, … ) and acceleration (2, 4, … ) steps alternate in a leapfrog approach. Each acceleration step applies a transformation matrix which is generated in substeps, each of which updates electron acceleration

With each substep, the transformation matrix is evolved by compounding the following transformations:

Apply the acceleration

Accelerate electrons by

Transform to the frame of reference of the electron bulk motion.

Rotate the eVDF around the magnetic field direction for

Transform back from the frame of reference of the electron bulk motion to the simulation frame.

Accelerate electrons by

After substepping is completed, the transformation matrix describing Vlasov acceleration is passed to the SLICE-3D algorithm, which decomposes the transformation into three Cartesian shears and updates the eVDF.

In this method introduction, we use a noon–midnight meridional-plane 2D-3V Vlasiator simulation as our test-case input data. This 2D-3V Vlasiator simulation has been used to investigate global and kinetic magnetospheric dynamics in multiple studies such as

For this eVlasiator sample run, we choose a region from the magnetotail with

Simulation box initialization values.

The electron distributions are discretized onto eVlasiator velocity meshes, with the electron velocity mesh consisting of

To validate the performance of our electron solver, we performed single-cell tests, with resultant electron bulk velocities

Graphs of solver stability in relation to electron plasma oscillation and gyromotion in a single-cell simulation. Note the different time axes used.

Although our method is geared towards solving electron motion at coarse spatial resolutions, to further validate the solver, a wave dispersion test was run

Two 1D-simulation set-ups with a spatial grid resolution of

Figure

Dispersion analysis of the electron solver in a 1D test case with an axis-parallel

We also evaluate the stability of our solver over the larger simulated domain described in Sect.

Panel (f) shows the agyrotropy measure

In panels (h) through (k) of Fig.

Evolution of electron and solver parameters over the whole simulation domain. (

As part of our evaluation of solver stability, we performed a comparison run where our electron solver performed the rotation transformation corresponding with gyromotion in the Hall frame instead of in the substep-associated electron bulk frame. This transformation choice resulted in unstable growth of, in particular,

Results from the electron simulation after

Figure

Electron distribution properties within the test domain after 1.0 s of simulation.

Figure

Figures

In Fig.

Electron properties and velocity distribution functions after

Figure

The maxima of instantaneous values of

The temperature anisotropies found in the near-Earth tail region of our simulation are mostly in the

Parallel heating near the magnetotail plasma sheet has been reported to coincide with bi-directional electron distributions

Observations of perpendicular crescents are shown in MMS data in

In this method paper we have presented a novel approach to investigating electron distribution function dynamics in the context of global ion-hybrid field structures. Our method exploits global dynamics provided by hybrid-Vlasov simulations in order to evaluate the response of gyrating and plasma oscillating electrons to global magnetic field structures.

We have shown our solver to behave in a stable manner, resolving electron inertia and updating a responsive electric field

Our model has several built-in limitations as it does not treat electrons as a fully self-consistent species. Magnetic fields gathered from the Vlasiator simulation are kept constant and thus force electron bulk motion to adhere to the required current density structure. As the initialization information is gathered from a hybrid-Vlasov simulation, it has a spatial resolution far below that required for resolving electron-scale waves such as whistlers, Bernstein waves and chorus waves. Scattering of electrons via these missing waves is somewhat accounted for by initializing every simulation from a Maxwellian isotropic distribution. These features together limit the applicability of the model to short periods of time. On the other hand, our model is efficient, and much larger spatial domains of investigation are easily achievable. Also, multiple eVlasiator runs can be performed from a single Vlasiator magnetosphere run to evaluate different driving conditions such as temperature ratios and anisotropies. The method builds on the efficiently parallelized Vlasiator codebase and will benefit from future numerical and computational improvements to Vlasiator solvers.

Our model can be applied to investigate electron dynamics on global spatial scales, with the current version applicable to 2D investigations, e.g. in the noon–midnight meridional plane. Electron velocity distribution functions generated by the model can be used to investigate, for example, energetic electron precipitation into the Earth's auroral regions. The generated electron anisotropies can be used to infer regions where, for example, whistler waves can be expected to grow. The model can be run for several different initialization time steps to evaluate long-term evolution of precipitating electron distributions. This could be used to, for example, evaluate electron distribution changes as bulk flows and dipolarization fronts in the Earth's magnetotail propagate earthward.

Future improvements to our model will allow simulation initialization from non-uniform 3D-3V Vlasiator meshes, allowing investigation of spatially three-dimensional topologies including tail plasma sheet clock angle tilt. A possible path of future investigation would be to upsample the initialization fields and moments in order to achieve better resolution, but we emphasize that the model does not attempt to solve electrons in a fully self-consistent manner, as magnetic fields are still kept constant. Increasing resolution by interpolating the input moments to a finer grid might not significantly improve plasma sheet density and temperature profiles. Increasing spatial resolution introduces numerous caveats including increased computational cost and possible charge imbalance resulting from spatially resolved electron oscillations, though our dispersion tests did not indicate such problems. If such imbalances arise from a future model, some method of solving Gauss's law such as a Poisson solver should be implemented. A more detailed investigation into comparing electron eVDFs and dynamics with observations is expected in a future study.

Vlasiator

MB wrote the paper and code description. MB and TB devised the solver method. MA assisted with data analysis, model development, and comparisons with observations. UG performed the dispersion tests. MP is the principal investigator of Vlasiator and leads the investigation. YPK, MG, KP, AJ, LT, MD, and MP participated in the discussion and finalization of the paper.

The authors declare that they have no conflict of interest.

We acknowledge the European Research Council for starting grant 200141-QuESpace, with which the Vlasiator model (

This research has been supported by the European Research Council (grant nos. 682068 and 200141) and the Academy of Finland, Luonnontieteiden ja Tekniikan Tutkimuksen Toimikunta (grant nos. 312351, 309937, and 322544).Open-access funding was provided by the Helsinki University Library.

This paper was edited by Wen Li and reviewed by two anonymous referees.