Observation of the solar wind–magnetosphere boundary provides a unique opportunity to investigate the physics underlying the interaction between two collisionless magnetized plasmas with different temperature, density and magnetic field topology. Their mixing across the interface as well as the boundary dynamics are affected by the development of fluid (and kinetic) instabilities driven by large-scale inhomogeneities in particle and electromagnetic fields. Building up a realistic initial equilibrium state of the magnetopause according to observations is still a challenge nowadays. In this paper, we address the modeling of the particles and electromagnetic field configuration across the Earth's magnetopause by means of a three-fluid analytic model. The model relies on one hot and one cold ion population as well as a neutralizing electron population. The goal is to create an analytic model that is able to reproduce the observations as closely as possible. Some parameters of the model are set using a fitting procedure that aims to minimize their difference with respect to experimental data provided by the Magnetospheric MultiScale (MMS) mission. All of the other profiles, concerning the electron pressure and the relative densities of the cold and hot ion populations, are calculated in order to satisfy the fluid equilibrium equations. Finally, using a new tri-fluid code, we check the stability of the large-scale equilibrium model for a given experimental case and provide proof that the system is unstable to reconnection. This model could be of interest for the interpretation of satellite results and for the study of the dynamics at the magnetosphere–solar wind boundary.

The solar wind–magnetosphere boundary, known as the magnetopause, is characterized by the presence of magnetic and velocity shears as well as jumps in magnetic and velocity magnitudes in addition to jumps in plasma density and temperature. These inhomogeneities are the sources of many plasma instabilities at different spatiotemporal scales

All of these phenomena can cause significant entry of magnetosheath plasma mass

The three-fluid model that we propose in the present paper aims to distinguish between ions of magnetospheric and magnetosheath origin. It must be considered as a first step towards a four-population model, which would also allow one to distinguish between the electrons from both origins. This simplification allows nonessential details related to the presence of two electron species, which are not likely to have a major role in structuring the large-scale equilibrium, to be avoided. In addition, note that for a correct modeling of the electrons in the magnetopause vicinity, one should also split the magnetospheric electrons into at least two subpopulations: one “cold” poorly measured population, carrying the density, and one “hot” population, carrying the pressure. Finally, starting from a previously existing two-fluid code, it is relatively straightforward to create a multi-ion one-electron code where the electrons only provide an “Ohm's law”. However, a fully multi-population code (i.e., including several electron populations) requires a radically different approach that is presently under investigation and is a matter for future work.

In the past, several multi-population models trying to simulate the plasma interaction between the magnetosheath and magnetosphere have been developed. In particular, for the modeling of tangential layers (i.e., without normal magnetic field and without normal velocity,

A different approach to creating a large-scale configuration at the boundary between the magnetosheath and the magnetosphere is to run a global simulation that eventually reaches a steady state. This has been done recently utilizing hybrid and kinetic codes (

In summary, the lack of realistic equilibria in the literature generally makes the initialization of the magnetopause studies difficult with respect to kinetic simulations. Recently the multi-population character of the medium has been taken into account by

Multi-fluid models have been developed in various domains, although generally not for magnetopause studies. These studies address multispecies evolution involving chemical processes and collisions. They have been used to investigate planetary atmospheres (

In this paper, we present a new technique to create a three-fluid equilibrium that derives directly from satellite observations. The model assumes 1-D gradients in the normal direction and a tangential boundary (

We use MMS data from 16 October 2015 at 13:05:34

MMS data for the event on 16 October 2015 at 13:05:34 UT

In Fig. 1a–c, data are plotted as functions of a spatial coordinate

Black points have been superimposed on the two spectrograms (Fig. 1b, c) to indicate their maxima. This allows one to more easily distinguish where the magnetosheath and the magnetospheric plasma interact, as indicated by discontinuities in the curve joining the maxima. The region where the two plasmas partially overlap in space is emphasized by a blue rectangle in Fig. 1b, which is centered at

In Fig. 1d–f, we plot the 2-D ion distribution functions (idfs) in the plane tangential to the magnetopause. They are obtained by integration over the out-of-plane (normal) component of the velocity. Each plot is the average of five single idfs recorded within a

We present a three-fluid collisionless model here that includes two proton populations (one cold and one hot) and one electron population. The cold ion population models the ions of magnetosheath origin and disappears more and more on the magnetospheric side. Conversely, the hot population models the ions of magnetosphere origin and disappears on the magnetosheath side.

The continuity and ion momentum equations are derived from the first two moments of the Vlasov equation.
We impose charge neutrality, and the displacement current is neglected.
We assume isotropic pressures and adopt a polytropic closure for all populations. These equations are coupled to the electromagnetic field
via the Faraday equation, and we use an Ohm's law that takes the electron pressure gradient into account but neglects electron inertia effects.
The three-fluid system of equations reads as follows:

We aim to establish a tangential 1-D equilibrium
to mimic the magnetopause observations previously presented as closely as possible. Assuming

In step 1, we impose the magnetic field

In step 2, we deduce the electron density

As far as

Comparison between

In step 3, we now split the global proton population into two different populations, cold and hot (hereafter referred to as “ic” and “ih”, respectively) to distinguish the magnetospheric and magnetosheath populations. The densities (

The temperatures of the cold and hot ion populations,

The temperature ratio between the two populations is set by the value of the dimensionless parameter as follows:

In order to implement this model into a numerical simulation, a compromise is necessary because the multi-fluid code cannot deal with a population having a zero density somewhere in the domain. To avoid this problem, we introduce the parameters

We apply the procedure to the case study introduced in Sect.

In Fig. 3b, we show the temperature profiles as obtained with our model equilibrium. The total ion population temperature

One observes that the global temperature is well fitted by the model outside the mixing region, but the fit is less accurate in the

In Fig. 3c, we show the density profiles. As explained in the previous section, the hot and cold ion contributions to the total density

The electron density and velocity profiles are obtained from the equilibrium equations. However, these quantities are not plotted here because
their experimental counterparts are likely to be biased in the magnetosphere by the cold electron population, which is below the bottom energy threshold of the FPI instrument (as mentioned in Sect.

Comparison between the magnetopause profiles as observed by MMS on 16 October 2015 at 13:05:34

The parallel components of the cold and hot ion currents are set by

Between the two limits above, a reasonable choice for the

In Fig.

Finally, in Fig.

Here we give an example of a three-fluid numerical simulation with the aim of demonstrating the possibility of numerically studying the above system by starting from an equilibrium not far from a real one (not only qualitatively but also quantitatively). For the sake of simplicity, we limit the geometry (to 2-D) and the numerical code (to 3-D). A detailed numerical study relying on such an approach will be the focus of future work.

The three-fluid model introduced in this paper has been used to initialize a 2-D three-fluid numerical simulation of the interaction between the solar wind and the Earth's magnetopause.
The numerical simulation is intended to mimic the MMS crossing on 16 October 2015 at 13:05:34

To initialize the simulation presented in this paper, we take the model profiles represented in Fig.

The simulation box dimensions are given by

The large-scale equilibrium configuration used to initialize the simulation is unstable with respect to the reconnection mode. At

Development of the reconnection instability.

The simulation is run for about 1500 ion cyclotron times. Very rapidly, the initial perturbation reorganizes and sets up the reconnection eigenmodes that are identified by their wave number in the

Shaded iso-contours of the cold ion fluctuations,

Same as in Fig.

The huge number of spacecraft data available today offers a lot of information about the magnetopause, especially those from the MMS mission due to their high-resolution particle data. Thus, magnetopause modeling can now be improved in view of these observations, which show that this boundary is never the simple textbook boundary generally considered. Beyond the natural asymmetry in temperature and density between the magnetosphere and magnetosheath plasmas, the first important ingredient to consider is the strong velocity shear that arises at the boundary as well as the magnetic shear which is a defining property of the magnetopause. Furthermore, the gradients concerning the particles and those concerning the magnetic field generally have different locations and show different scale lengths. Therefore, the model also has to be able to take these characteristics into account.

In this paper, for the first time, we present a three-fluid equilibrium directly derived from data using a magnetopause crossing by MMS. The derivation of the model is based on a fit of the experimental data to the most reliable data, which is completed by a “realistic” solution of the equilibrium fluid equations for the others.
The relative densities of the hot and cold ion populations calculated using the equilibrium equations provide an a posteriori check of our three-fluid model. In particular, this information helps to understand the different bulk quantities observed in the ion distribution functions (see Fig.

Furthermore, a preliminary study shows that the model can be implemented in a three-fluid numerical simulation, validating the correctness of the equilibrium solution. The detailed study of the long time evolution of the magnetopause instability will be the subject of future work.

It may seem contradictory to consider the data as characteristic of some magnetopause equilibrium and observe afterward that this equilibrium is not stable and should not last for long (even if the reconnection phenomenon is never “immediate”). To justify this point, one must understand that the main characteristics that are taken into account are the asymptotic values on each side and, in particular, the velocity shear between the magnetosheath and magnetosphere. These conditions are not changed by the instability. On the contrary, the positions and the scale of the different gradients can indeed be partly modified by the instability. We think that this is one of the interesting issues that can be investigated by the time evolution observed in the simulation. However the following questions are issues that will be resolved in future work: how is the system stability impacted? (A parametric study is needed.) How does the system change in time due to nonlinear effects? Will the simulation converge toward a new more stable equilibrium state representative of the real system?

Investigating the magnetopause stability and trying to understand topics such as when and where reconnection phenomena can be triggered and how the plasmas from both sides can be mixed are still currently challenging issues to attack using numerical simulations. However, knowing that the stability of a physical system is given by the specific initial equilibrium state, it must be kept in mind that the resulting nonlinear dynamics, in particular the mixing properties, also strongly depend on the choice of the initial equilibrium. As a consequence, it is very important to initialize a simulation with a configuration that is as realistic as possible. In most of the published literature, simulations have been initialized with relatively simple configurations, such as Harris sheets or modified Harris sheets, with little relationship to the real magnetopause. Therefore, the realistic three-fluid equilibrium presented in this paper should allow for this work to be taken a step further, and the same method could be applied to other experimental cases in the future.

All the data used are available from the MMS Science Data Center:

RM, LR and GB performed the experimental work. RM and FC developed the numerical code and carried out the simulation. All authors contributed to the final paper.

The authors declare that they have no conflict of interest.

We acknowledge ISCRA for awarding us access to the supercomputer facilities at CINECA, Italy, where the calculations were performed. The French involvement in MMS is supported by CNES and CNRS.

This project (Francesco Califano) has received funding from the European Unions Horizon 2020 research and innovation programme under grant agreement no. 776262 (AIDA,

This paper was edited by Minna Palmroth and reviewed by Johan De Keyser and one anonymous referee.