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 that, in turn, often trigger secondary instabilities at smaller scales. As an example, secondary instabilities such as magnetic reconnection, Kelvin–Helmholtz and/or Rayleigh–Taylor instability can efficiently develop on the shoulder of the primary instability; the Kelvin–Helmholtz instability at the low-latitude magnetopause is a prime example of such a case (see and references therein).

All of these phenomena can cause significant entry of magnetosheath plasma mass , momentum and energy into the magnetosphere. The study of the magnetopause is of particular interest, as this system offers the unique opportunity to study a two-plasma large-scale interaction in conditions not achievable in the laboratory. The magnetopause physics is also of basic importance in studies addressing the Sun–Earth interaction, in particular concerning the impact of solar wind disturbances on the terrestrial environment and space-weather forecasting efforts (see for instance ). Modeling space plasmas using data provided by multi-spacecraft missions has been much developed during (and in preparation of) the “cluster era” . Concerning the magnetopause data, one of the key points relates to the mixing between magnetospheric and magnetosheath plasmas and the resulting non-Maxwellian shape of the distribution functions (hereafter referred to as “dfs”) observed in these regions (). These dfs are often reminiscent of those observed in reconnection kinetic simulations (). Some of these dfs can be compared using simple analytic models, as in the pioneering work by . As the populations originating from the two different sides of the magnetopause differ with respect to density and temperature, modeling the mixing requires the use of at least a multi-population model. In the perspective of investigating the dynamics of the magnetopause mixing layer using a three-fluid numerical simulation, the main target of this paper is to build up a three-fluid equilibrium that is as realistic as possible to initialize the simulation.

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, Bn=un=0), the kinetic models are by definition the most complete ones as they rely on a Vlasov equilibrium. Nevertheless, as these models are considerably more complicated than the fluid models, using the latter can be considered to be a necessary first step for modeling the magnetopause as closely to the observational data as possible. Note also that all of the equilibria built via dfs that are only single-valued functions of the particle invariants of motion () cannot really be considered to be “multi-population” models: they ignore the issue of accessibility () and, therefore, cannot distinguish between particles of magnetospheric or magnetosheath origin. Some models () have partly accounted for the accessibility problem by considering functions of the invariants that are bi-valued. In recent years, new models have been developed by and that account for accessibility by considering more general multivalued dfs. Nevertheless, it must be kept in mind that all of these models involve many free parameters. In , the authors even emphasize that there is no general constraint for fixing the normal electric field profile in their kinetic model; thus, the normal electric field profile can be chosen in a fully arbitrary manner. Furthermore, even if a few profiles can be fixed in a more or less realistic way, all of the others depend on simple mathematical assumptions, which are largely arbitrary; therefore, the latter profiles generally still remain far from realistic.

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 (), although the question of the spatial resolution is then difficult to solve for investigating the internal structure of the magnetopause boundary. It will be interesting to compare the results obtained using the abovementioned methods with those obtained using the methods discussed in the present paper where the model instead relies on spacecraft data as the initial condition for the simulation.

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 , although for a different purpose from that in this study. Using a particle-in-cell (PIC) code, they investigated the influence of cold magnetospheric ions on the development of magnetic reconnection. In this paper, the magnetospheric plasma includes two populations with different temperatures in order to account for the presence of the cold ions that are observed in the magnetosphere close to the magnetopause.

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 (), the solar chromosphere and basic plasma physics problems (e.g., drift turbulence in ).

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 (Bn=0) at the magnetopause. The magnetic and velocity shear are both taken into account in a realistic way. Profiles are chosen that best fit the data from the Magnetospheric MultiScale (MMS) mission , for which the time-to-space conversion has been performed using recent techniques presented in . As will be shown in Sect. , the method provides a cold and a hot contribution that are in qualitative agreement with observations, even if the model only uses the global ion macroscopic moments as inputs.

We use MMS data from 16 October 2015 at 13:05:34 + 40 s UT that occurred at [8.3, 8.5, -0.7] RE in GSE frame, which embeds a magnetopause crossing. In Fig. , we plot the experimental data that the equilibrium model attempts to reproduce. This interval shows the standard signatures of the region where the magnetospheric and magnetosheath plasmas meet (magnetopause crossing): a reversal of the magnetic field and a change in the energy distributions.

In Fig. 1a–c, data are plotted as functions of a spatial coordinate Xn, which is the projection of the spacecraft path along the direction normal to the magnetopause (units of di,MSh with di,MSh≃70 km). Once the function Xn(t) is known, it allows for the recovery of the spatial profiles of all quantities of interest from the corresponding temporal series of the spacecraft magnetosheath–magnetosphere crossing, even with a variable velocity. For the sake of completeness, we also give the time corresponding to each given value of Xn in the abscissa of Fig. 1c. The spatial coordinate X(t) is determined by the integration of the velocity of the spacecraft with respect to the magnetic structure. In order to determine this spatial velocity, we use a new method from , which is an extension of the spatiotemporal difference (STD) method first proposed by . The main difference with respect to the Shi model consists of taking the possible weak nonstationarities of the magnetic structure into account , i.e., assuming ∂t,scB=∂t,0X⋅∇B+∂t,0B and not just ∂t,scB=∂t,0X⋅∇B (the indexes “sc” and “0” indicate the spacecraft frame and the frame where the intrinsic variation of the magnetopause structure is the smallest, respectively). This allows for some singularities in the results to be avoided.

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 Xn∼3di.

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 ∼0.75 s long interval (equivalent to 0.5di). The radius of the distribution functions is 103 km s-1 and the purple full circle drawn at their centers determines the bottom limits of the energy of the FPI instrument (10 eV ∼ 53 km s-1 for ions). The direction of the local magnetic field is indicated by a white arrow. In Fig. 1e (mixing region), one can observe that the idf contains two peaks that are emphasized by the blue and red dashed circles. These two circles have a diameter equal to the magnetosheath and magnetospheric thermal velocities, respectively. The same circles are shown for the magnetosheath and magnetospheric idfs in Fig. 1f and d. In these two asymptotic media, we see that there is only one single peak. Note that the idf shown in Fig. 1d was recorded a little earlier (10:20:00 UT + 2 s) during a “clear” observation of the magnetosphere and, therefore, allows the presence of magnetosheath particles when the spacecraft is too close to the magnetopause to be avoided. Conversely, Fig. 1e shows a mixture of the magnetosheath and magnetospheric populations at the same time. However, as the two peaks are close to each other and the distributions of the two populations are partly superposed, it is not possible to clearly separate the hot/cold contributions, which is a necessary input for the multi-population model to be built by a direct fit of this region. In the next section, we will explain a new method capable of separating the two particle components – even in such complex situations. Note that the electron spectrogram of Fig. 1c shows energy maxima that lay just at the bottom limit of the instrument (≃10 eV) in the magnetospheric region (Xn≤2di). This indicates the presence of cold electrons in the magnetosphere. The role of this poorly measured cold electron population is not relevant for the magnetopause pressure equilibrium, but it could be significant in the electron bulk velocity. However the electron population parameters will be not determined by a direct fit of the data nor will those of the two ion populations (cold/hot). Instead, they will be determined by another method based on the equilibrium equations, which we will describe in the next section.

We apply the procedure to the case study introduced in Sect. . In Fig. , we compare the model field profiles with those obtained with the MMS data. The model profiles for the magnetic field, the ion temperature and density are obtained by a fit procedure (Fig. 3a, b, and c, respectively), whereas the others are calculated from the equilibrium equations. The fits are obtained by means of analytic functions. For a given quantity Q, the fitting functions have the following form: 8Q=∑jaQ,j+bQ,jtanhXn-cQ,jdQ,j, where Xn is the coordinate along the direction normal to the magnetopause (as discussed in Sect. ). The parameters aQ,j, bQ,j, cQ,j and dQ,j are the free parameters shaping the analytic profiles, and j is the component index. The maximum value of j depends on the fitted quantity: the analytic profiles are considered to be good fits of the data if they correctly shape the large-scale configuration as well as the position and the length scale of the gradients within the magnetopause layer. An example of a “good fit” is given in Fig. a–c. It is worth noticing that the particle boundary, observed on density and temperature, has a length scale smaller than the magnetic boundary (by a ratio ≃0.25) and that its position is considerably shifted toward the magnetosphere with respect to the center of the magnetic jump. This may indicate the presence of a boundary layer, possibly made of magnetosheath plasma, that is observed on the magnetospheric side of the magnetopause . Such features cannot be reproduced in the framework of a magnetohydrodynamic (MHD) equilibrium model. To the best of our knowledge, these features have not even been introduced in the context of a kinetic model.

In Fig. 3b, we show the temperature profiles as obtained with our model equilibrium. The total ion population temperature Ti has been obtained by a fit, and it is superposed onto the cold ion population temperature Tic (blue curve) and the hot ion population temperature Tih (red curve). The figure has been drawn using ϵ(h)=0.35 and ϵ(c)=0.05, which determine the values of Tic and Tih via Eqs. () and (), respectively.

One observes that the global temperature is well fitted by the model outside the mixing region, but the fit is less accurate in the ∼1.0di≤Xn≤∼2.5di interval. In this interval, the real total ion temperature actually becomes larger than its magnetospheric asymptotic limit. Unfortunately this feature can not be reproduced by the present three-fluid model with hot and cold constant temperatures, as the Γ≥0 constraint forces the Ti profile to be lower than Tih everywhere (see Eq. ). This little deviation is acceptable as the model mainly aims to reproduce the asymptotic trends, with the observed inner region most likely being out of equilibrium.

In Fig. 3c, we show the density profiles. As explained in the previous section, the hot and cold ion contributions to the total density ni are computed by means of the Γ function, which is fixed once the global Ti profile and the temperature ratio Υ are fixed (Eq. ). In Fig. a–g, the two vertical (dashed black) lines indicate the limits of the region where 1/4≤Γ≤3/4. Note that the cold ion density rapidly falls to very low values in more or less ∼2 di (where di≃70 km is the ions inertial length measured in the magnetosheath), whereas the hot population density remains at nearly the same value over a longer interval (between 0 and 8 di).

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. ). Conversely, in Fig. 3d, we plot the normal component of the electric field, which is obtained from the three-fluid model using Eq. (). We see that the electric field calculated by the model agrees quite well with the experimental field. As the electric field reflects the electron dynamics, this shows (independently of the electron measurements) that the electron dynamics is correctly accounted for in the model.

The parallel components of the cold and hot ion currents are set by ϕ (Eq. ). As long as there are no cold ions on the magnetospheric side and no hot ions on the magnetosheath side, the asymptotic constraints on ϕ would be limx→MSphϕ=0;limx→MShϕ=1. Nevertheless, owing to the compromises necessary to implement the model in the multi-population numerical simulation, the cold and hot densities actually take small but not strictly null values on both sides. To determine the corresponding parallel currents, corrections similar to Eqs. () and () are applied with the assumption that, on each side, the parallel velocities of the cold and hot populations, in the electron frame, are equal to each other and, therefore, equal to the global one. Under this assumption, it can be easily shown that the asymptotic values of Φ are equal to those of Γ: limx→MSphϕ=ϵ(h);limx→MShϕ=1-ϵ(c). Note that for the particular MMS event considered, the global ion parallel velocities are observed to be quasi-null on each side, so that the same asymptotic property holds for the velocities of the two populations.

Between the two limits above, a reasonable choice for the ϕ profile is that the length of its gradients is of the same order as the scale length of the density and temperature gradients, i.e., ∼ 1–2 di. The position of the main gradient of ϕ is set in order to separate the magnetopause thickness into two parts, each representing the length proportional to the gyro-radii of the two populations (their ratio is ≃2).

In Fig. e, we show the model profiles for Γ, Π and Φ. Because of the differences in temperature between the two components, the profile in Π (concerning the pressures) noticeably differs from the profile in Γ (concerning the densities).

Finally, in Fig. f and g, we show the results concerning the parallel and perpendicular components of the ion currents. Once more, one observes that the global ion current is well fitted with the exception of the perpendicular current in the mixing region, which is less accurate. This is due to the previously mentioned small inaccuracy in the modeled ion temperature in this region.

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. d).

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.