Vlasiator uses the hybrid-Vlasov ion approach to model the near-Earth space plasma environment. The full six-dimensional (6D) phase space density fs(x,v,t), with x the ordinary space variable, v the velocity space variable, and t the time variable, for each ion species s of charge qs and mass ms is evolved in time using the Vlasov Eq. (). The electric and magnetic fields, denoted E and B respectively, are evolved using Maxwell's equations: Faraday's law (Eq. ), Gauss's law (Eq. ) and Ampère's law (Eq. ), in which μ0 and ε0 are the vacuum permeability and permittivity, respectively, and j is the total current density. The solenoid condition in Gauss's law (Eq. ) is ensured via divergence-free magnetic field initial-condition reconstruction . In the hybrid approach, electrons are assumed to maintain plasma neutrality, resulting in the charge density ρq in Gauss's law vanishing. In the Darwin approximation, also used in many hybrid codes, propagation of light waves is neglected by removing the displacement current term ε0∂E∂t in Ampère's law (Eq. ). The Vlasiator field solver follows the staggered-grid approach of and is described in detail in .

1∂fs∂t+v⋅∂fs∂x+qsmsE+v×B⋅∂fs∂v=02∇×E=-∂B∂t3∇⋅B=0and∇⋅E=ρqε04∇×B=μ0J+ε0∂E∂t

The generalized Ohm's law providing closure for the Vlasov system is 5E+V×B=Jσ+J×Bnee-∇⋅Penee+menee2∂J∂t, where V is the plasma bulk velocity, σ is the conductivity, e is the elementary charge, ne is the electron number density, and Pe is the electron pressure tensor. In hybrid approaches of collisionless plasma, we can assume high conductivity, neglecting the first term on the right-hand side. In the limit of slow temporal variations, the electron inertia term (the last term on the right-hand side) also vanishes. The remaining two terms on the right-hand side of the equation are the Hall term, J×B/(nee), and the electron pressure gradient term, ∇⋅Pe/(nee). In hybrid models, a true description of electron pressure is unavailable so it must be described via some approximation such as adiabatic, isothermal or polytropic electrons or a fixed ion-to-electron temperature ratio, or by neglecting the small electron pressure gradient term altogether. The standard ion-hybrid Vlasiator code supports isothermal fluid electrons but existing simulations have always set this temperature to zero. This along with assuming charge neutrality (proton number density np=ne) results in the ion-hybrid Vlasiator using the simplified MHD version of Ohm's law with the Hall term included: 6E+V×B=1enpμ0(∇×B)×B.

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 ). Vlasiator propagates distribution functions of particles following the SLICE-3D method and utilizing Strang splitting with advection (also referred to as translation, the second term of Vlasov's Eq. ) and acceleration (the third term of Vlasov's Eq. ) calculated one after the other with a leapfrog offset of 12Δt. In this paper, Δ denotes steps on the full simulation grid and associated time step and δ is used to indicate calculations performed as substepping. For each time step, a Vlasov acceleration is evaluated with time step length Δt which is, amongst other things, limited to a maximal Larmor orbit gyromotion rotation value, which is usually set to 22∘. This value is constrained by the SLICE-3D shear approach, with values much above 22∘ resulting in unphysical deformation of the VDF and smaller values increasing the computational cost of the simulation. For each acceleration step of length Δt, a transformation matrix is initialized as an identity matrix. The transformation matrix is first composed to apply the uniform electric field acceleration and the gyromotion due to the magnetic Lorentz force. Then, the transformation matrix is decomposed into three shear transformations. For a detailed explanation of the approach see chapter 3.5.1 of . The transformation matrix is incrementally built with substepping of δt where each δt corresponds to a 0.1∘ Larmor gyration, with the gyration step derived from convergence tests. Instead of applying linear acceleration by the motional electric field, a method similar to the Boris-push method is applied, where first a transformation is performed to move to a frame in which the electric field vanishes, then the rotation is applied, and then a frame transformation back to the original frame is added. In the standard hybrid formalism, the frame without an electric field is found via the MHD Ohm's law with the Hall term included (Eq. ). This Hall frame estimates the frame of reference of electrons, assuming electrons generate a current density which corresponds to the local magnetic field structure, in accordance with Ampère's law. After substepping is evaluated, the transformation matrix is applied to the gridded velocity distribution function by the SLICE-3D algorithm.

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 ). Both protons and electrons for the eVlasiator simulation are initialized from the read moments as Maxwellian distribution functions, with electron bulk velocity selected so that Ampère's law (Eq. ) is fulfilled. Re-mapping input-run Vlasiator proton VDFs as Maxwellians does not affect the simulation results as eVlasiator only considers the proton number density and bulk velocity for current density calculations and does not propagate the proton distribution functions, instead keeping their characteristics completely constant for the duration of the eVlasiator simulation. For each simulation cell, we use the approach for calculating cell-average volumetric magnetic fields and respective derivatives. The eVlasiator solver uses volumetric field derivatives for calculating ∇×B.

eVlasiator solves the evolution of electron eVDFs similar to how Vlasiator simulates proton VDFs (for a detailed explanation, see , in particular chapter 5.3.1). Solving the Vlasov Eq. () is split into two sections, translation and acceleration, with each of these steps performed in a staggered leapfrog approach. This approach is described in Fig.  with the first row indicating the spatial advection of electrons (translation) and the second row describing the effect of the Lorentz force on electrons through electric field acceleration and gyromotion. At time t (or t0 at the initial state) we have the 5D (2D-3V) or 6D (3D-3V) electron velocity distributions (and, by extension, moments) in the whole simulation domain as well as proton moments and volumetric magnetic fields. Proton and magnetic field data are as read from the Vlasiator simulation and kept constant throughout the eVlasiator simulation. At simulation start, the leapfrog stepping is initialized with a half-length acceleration step (shown in red as step 0. in Fig. ).

During each translation step, as depicted in Fig.  and described by the equation 7∂fs∂ttrans+v⋅∂fs∂x=0, we perform a semi-Lagrangian spatial advection operation using second-order polynomial remapping, in an identical fashion to in a regular Vlasiator. This is evaluated separately for each cell in the gridded electron distribution functions using the velocity for that cell and is evaluated for one Cartesian direction at a time.

During each acceleration step, as depicted in Fig.  and described by the equation 8∂fs∂tacc+qsmsE+v×B⋅∂fs∂v=0, we perform a semi-Lagrangian velocity space SLICE-3D update of the whole local distribution function, separately for each spatial cell. This method evaluates the acceleration due to electric fields (uniform movement in velocity space) and the rotation due to the Lorentz force magnetic component (a rotation in velocity space). The uniform movement and the rotation are composed into a transformation matrix. To apply the transformation with the SLICE-3D scheme, the matrix is then decomposed into three shear motions, one along each Cartesian velocity coordinate axis, and performed using semi-Lagrangian fourth-order polynomial remapping, similar to how the regular Vlasiator Vlasov solver works. This approach is applicable as long as velocities are non-relativistic. For a detailed description, see chapter 5.3.1 of .

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. ), allowing for the interaction of electron distribution functions with electron–oscillation electric fields. Whistler mode propagation is not included in this study. We do not include any electric field due to Gauss's law. We will consider each term of the eVlasiator field solver separately:

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

For electron dynamics to be modelled, electron gyration and plasma oscillation must both be considered. We choose to limit the acceleration time step Δt to a maximum of 22∘ of Larmor rotation or 22/360 of a single plasma oscillation. The value of 22∘ is used to ensure our VDF remapping algorithm SLICE-3D remains stable and the value 22/360 was chosen for equal resolution of both oscillations as a result of convergence tests. Much larger values will result in neighbouring simulation cells with different plasma characteristics diverging into an unstable state, and much lower values will needlessly cause an increase in computational cost. Due to the computational cost of SLICE-3D remapping, a substepping approach is used in order to more accurately model the electron gyromotion and plasma oscillation. Whilst the 22∘ step models eVDF evolution to a high accuracy, the accurate and stable simulation of feedback between electron velocity, plasma oscillation, and the electric field due to the electron inertia term in Ohm's law requires substepping and places strict requirements on the length of the substep δt. This substepping is performed in tandem with the SLICE-3D transformation matrix generation. The electron gyroperiod is τce=2πωce-1 and the plasma oscillation time is τpe=2πωpe-1, where the electron plasma frequency is 9ωpe=nee2ε0me and the electron gyrofrequency is 10ωce=eBme. In transformation matrix generation, substepping is constrained to a maximum of δt≤min⁡(τpe,τce)/3600. This value was defined as a result of convergence tests, and its dependence on the relationship between τpe and τce is discussed more in Sect. .

The electron oscillation and electric field feedback loop is handled in parallel with gyration by tracking a cell-volume-averaged electric field EJe which is itself derived from the small-scale electron oscillation. For each acceleration substep, we update electron motion V and the electric field EJe by performing two parallel fourth-order Runge–Kutta propagations. The RK4 algorithm was chosen instead of a Runge–Kutta–Nyström method as it provides a good balance between general applicability, stability, and computational performance. The first one is 11δVe=δtemeEJe, tracking electron bulk velocity response δVe to the EJe field. This simple acceleration term is in fact equal to evaluating current changes via the electron inertia term in Ohm's law with the EJe field included in the left-hand-side electric field. The second Runge–Kutta propagation tracks the evolution of the EJe field due to changing current density, according to the displacement current on the right-hand side of Ampère's law (Eq. ) with the ∇×B term in Ampère's law fixed to the static input magnetic fields. Thus, for each Runge–Kutta step, the electric field EJe is updated with 12δEJe=δt∇×Bε0μ0-Jε013=δt∇×Bε0μ0-eVpnp-eVeneε014=δtc2∇×B+μ0eneVe-npVp, where c is the speed of light, and B, np and the proton bulk velocity Vp are assumed to be constant throughout the substep. Each of the four δVe Runge–Kutta coefficients are updated with the latest estimate for δEJe, and vice versa. Values for EJe are stored between acceleration steps to ensure continuity of the oscillation. The change δVe calculated via each Runge–Kutta step is then applied to the transformation matrix, allowing the solver to proceed to perform gyration in the electron frame of reference. The substepping procedure is visualized in the third row of Fig. . Further details of the solver and advection methods in Vlasiator can be found in .

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

Apply the acceleration δVe derived from RK4-substepped EJe acceleration.

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 , , , , , , and . It has solar wind values of β=0.7, magnetosonic Mach number Mms=5.6, Alfvén Mach number MA=6.9, proton number density np=1cm-3, and solar wind speed usw along the e^x (Earth–Sun) direction with usw,x=-750kms-1, simulating fast solar wind conditions and ensuring efficient simulation initialization. The simulation input interplanetary magnetic field is purely southward with Bz=-5nT and the Earth's magnetic dipole is a e^z-aligned line dipole scaled to result in a realistic magnetopause standoff distance . The simulation has an inner boundary at 3×106m≈4.7 Earth radii, modelled as a perfectly conducting sphere. The spatial resolution is Δx=300km.

For this eVlasiator sample run, we choose a region from the magnetotail with 70×1×40 simulation cells in the X, Y, and Z directions, respectively. The subregion extent is from x-=-75.6×106m to x+=-54.6×106m, from y-=-0.15×106m to y+=+0.15×106m, and from z-=-6×106m to z+=+6×106m. Within this domain, visualized with a small rectangle in Fig. a, the electron plasma period τpe ranges from ∼0.7ms in the magnetotail plasma sheet up to ∼2.5ms in the near-plasmasphere lobes. The electron gyroperiod τce ranges from ∼14ms in most of the lobes up to ∼770ms at a tail current sheet X-line site.

The electron distributions are discretized onto eVlasiator velocity meshes, with the electron velocity mesh consisting of 4003 cells, extending from -4.2×107 to +4.2×107ms-1 in each direction, resulting in an electron velocity space resolution of 210kms-1. The eVDF sparsity threshold was set to 10-21m-6s3, ensuring good representation of the main structure of the eVDF. Discretizing a hot and dense electron distribution onto a Cartesian grid is numerically challenging without using vast amounts of memory. As portions of our simulation domain have proton temperature up to 108K, we use an empirical estimate of Ti/Te∼4 as magnetosheath temperature ratios are usually around 4 to 12 . , , , and show similar proton–electron temperature ratios in the magnetotail. In order to constrain the extent of our velocity space and numerical requirements of our solver, we implement our electrons with a mass of 10 times the true electron mass, resulting in an ion-to-electron mass ratio of mi/me=183.6. As mentioned above, we calculate the required electron bulk velocity for each cell using the local volumetric (cell-average) derivatives so that the ion and electron fluxes in each cell correspond with the current density J required for fulfilling Ampère's law (Eq. ) (with the displacement current neglected at initialization). This is equal to performing a transformation to the Hall frame of reference. Proton densities, magnetic field lines, proton temperatures, proton bulk velocities and electron bulk velocities calculated for simulation initialization are shown in Fig.  along with an overview of the input Vlasiator simulation and the selected electron subdomain.

To validate the performance of our electron solver, we performed single-cell tests, with resultant electron bulk velocities Ve and plasma oscillation electric fields EJe shown in Fig. . These single-cell tests did not have magnetic field curvature or an ion population present, resulting in the electron motion oscillating around a stability point at Ve=0 and EJe=0. We set the electron number density to ne=0.1cm-3 and the magnetic field to Bx=20nT (panels a through d) or Bx=200nT (panels e and f). We set an initial velocity perturbation of Ve,0=(-100,-150,200)kms-1, close to but below our electron velocity resolution of Δv=210kms-1. As can be seen from Fig. , the electron oscillatory motion is well resolved and remains stable over an extended period. In panels (e) and (f) where the magnetic field strength was artificially increased in order to set the plasma and gyroperiods to values closer to each other (1.11 and 1.79 ms, respectively), we see a gradual evolution of oscillation amplitude and, thus, EJe field magnitude as the two types of electron motion interact. Over longer periods of time this growth becomes unstable, but it can be counteracted by using a smaller substep. Also, this instability occurs only when τce≈τpe, which does not occur in our full simulation domain.

Although our method is geared towards solving electron motion at coarse spatial resolutions, to further validate the solver, a wave dispersion test was run . As waves are a collective, emergent phenomenon of the kinetic simulation approach, a correct reproduction of wave dispersion behaviour is a good indicator of correct physical behaviour of the simulation system.

Two 1D-simulation set-ups with a spatial grid resolution of Δx=300m (= 0.01 de) and Nx=1000 cells were initialized with an electron number density of ne=0.4×106m-3, an electron temperature of Te=2.5MK, and a magnetic field magnitude of 50 nT. In one simulation, the magnetic field direction was chosen to coincide with the extended simulation direction (resulting in parallel plasma wave modes to be resolved), in the other one, the magnetic field was set up perpendicular to the long dimension, resulting in perpendicular mode resolution. The plasma had zero bulk velocity in the simulation frame, with an added white noise velocity fluctuation of ṽ=1000ms-1. The simulation was run for 0.037 s (433 ωpe-1).

Figure shows the dispersion data resulting from a spatial and temporal Fourier transform (using a von Hann window). Overlaid are analytic dispersion curves for the Langmuir wave (black dashed curve) and electron Bernstein modes (black solid curves). The wave behaviour in the simulation shows good agreement in both parallel and perpendicular directions. One noteworthy additional feature visible in the parallel direction (Fig. a) is the presence of an entropy wave feature at low wave number k and angular frequency ω that shows a quantization consistent with the electron velocity space resolution.

We also evaluate the stability of our solver over the larger simulated domain described in Sect. , with initialization values derived from the Vlasiator hybrid-Vlasov simulation. These graphs are shown in Fig. . Panels (a) through (e) show the evolution of electron temperature values over a simulation of 1.0 s, covering hundreds of electron plasma periods and, for the most part, tens of gyroperiods. We evaluate minimum, maximum, mean, and median values for total, B-parallel, and B-perpendicular electron temperatures. The system is seen to relax somewhat towards a final state, though some evolution is still apparent at the end of the simulation, possibly due to boundary effects. The maximum temperature plot in panel (b) is of particular interest as the hottest plasma cells appear to diffuse into their surroundings until t∼0.4 s when dynamic gyration processes overtake this temperature diffusion with perpendicular heating.

Panel (f) shows the agyrotropy measure calculated from the electron pressure tensor, indicating that in the majority of the simulation domain electrons remain gyrotropic and even peak values do not grow past 10-3. Panel (g) shows statistics for the electron number density deviation from the initialization value, indicating loss of plasma neutrality due to the motion of electrons. The minimum value oscillating between approximately 10-9 and 10-6cm-3 indicates the level of numerical fluctuations, and the maximum, mean and median values show how charge imbalance does grow initially but stabilizes within about 0.1s and does not grow beyond 10-1cm-3.

In panels (h) through (k) of Fig.  we show how the instantaneous plasma oscillation electric field EJe is well-behaved throughout the simulation box, converging towards stable values. We note that as the EJe field oscillates around zero, the averages are indeed zero throughout (not shown) and the values used for inferring minimum, maximum, mean and median values are instantaneous values from an arbitrary phase of the oscillation. In panel (l) we show the normalized current density J departure from the balance current JB=∇×Bμ0 which would be required to maintain the magnetic field structure according to Ampère's law (Eq. ). This metric is seen to also stabilize, mostly at values well below unity. We expect the maximum-value outliers to be due to locally small values of JB. Panels (m) and (n) show statistics for the parallel and perpendicular components of the electric field caused by electron pressure gradients, that is, the -∇⋅Penee term. As expected due to the ability of electrons to propagate along field lines, perpendicular components are much larger than parallel components. All components remain stable at roughly their initial values. A minimum value is not shown, as the use of a numerical slope limiter in the calculation of pressure gradients gives identically zero field components at local extrema of pressure.

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, EJe, as could be expected (not shown).