Mickael Sigogne

Lukshe Kanagaratnam

Caroline Mora

Malika Pierre

Andreea Petrache

Claude Marcus

Michel Fischbach

Moustapha Dramé

Fatouma Touré

Sae Aizawa

Dominique Delcourt

Naoki Terada

Nicolas André

Identification of the Factors Associated With Intraperitoneal Pressure in ADPKD Patients Treated With Peritoneal Dialysis

Keywords: Mercury's magnetosphere, Kelvin-Helmholtz instability, particle acceleration, transport, planetary ion

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The MErcury Surface, Space ENvironment, GEochemistry, and Ranging (MESSENGER) spacecraft remained in orbit at Mercury from March 11, 2011until April 30, 2015. The many data collected by its instruments have provided key information about the structure of the magnetosphere as well as the dynamics of the plasma environment around Mercury. A number of plasma processes similar to the ones identified at Earth have been reported: magnetic reconnection and Flux Transfer Events (FTE) at the magnetopause [Slavin et al., 2009[Slavin et al., , 2010b[Slavin et al., , 2014;;DiBraccio et al., 2013;Imber et al., 2014], substorm-like dipolarization events in the magnetotail [e.g., Sundberg et al., 2012a, Imber et al., 2017;Dewey et al., 2017Dewey et al., , 2018]], and Kelvin-Helmholtz (KH) vortices in the magnetospheric flanks [Boardsen et al., 2010;Sundberg et al., 2011Sundberg et al., , 2012b;;Liljeblad et al., 2014Liljeblad et al., , 2016;;Gershman et al., 2015].

Moreover, the MESSENGER data have shown evidence in several regions of Mercury's magnetosphere of heavy ions of planetary origin resulting from the ionization of exospheric neutrals by solar UV [Zurbuchen et al., 2011;Schriver et al., 2011a, b;Raines et al., 2011Raines et al., , 2013;;DiBraccio et al., 2013;Gershman et al., 2015]. These planetary ions have enough energy to escape the gravity of Mercury and are trapped by the magnetic field of the planet.

They can either convect inside the magnetosphere of Mercury, escape from its boundaries, or return to the surface of the planet where they are recycled. These planetary ions will be referred to as picked up ions in the rest of the manuscript. These picked up ions can affect the structure and dynamics of the magnetosphere of Mercury by contributing to its mass loading and driving original plasma processes. Since the magnetic field of Mercury is weaker than the magnetic field of the Earth, the size of its magnetosphere is relatively small. As a result, plasma processes at Mercury should occur on shorter timescales, e.g., the Dungey plasma cycle at Mercury lasts for a few minutes only while it lasts for a few hours at the Earth [Siscoe et al., 1975;Slavin et al., 2010a]. In addition, since these picked up ions have a large gyration motion, plasma processes at Mercury should have a scale comparable to their Larmor radius. Under these circumstances, the ion motion and the electromagnetic field variations within the magnetosphere of Mercury are strongly coupled. The interaction between the picked up ions and the variable electromagnetic fields at Mercury has been investigated by various authors. In particular, Delcourt et al. (2003) investigated how magnetic field variations essentially affect the ion motion. They showed that a dipolarization in the magnetotail of Mercury can accelerate ions non-adiabatically by violating the first adiabatic invariant (the magnetic moment ). The violation of the first adiabatic invariant means that the ion may gain a large energy from the field (represented by a large change in ) and its gyration motion may not be circular but become helical or meandering instead [Speiser, 1965;Büchner and Zelenyi, 1989;Burkhart and Chen, 1991;Delcourt and Martin, 1994]. We know that at the Earth the resulting non-adiabatic energization can lead to ion precipitation in the ionosphere of the planet through pitch angle scattering [e.g., Delcourt and Martin, 1999].

Observational evidence at the Earth for such a non-adiabatic acceleration of ions have been reported by several authors [West et al., 1978a, b;Wagner et al., 1979;Lyons and Speiser, 1982;Ashour-Abdalla et al., 1991, 1992, 1993a, 1993b;Keiling et al., 2004]. At Mercury, the signatures of non-adiabatic acceleration in the plasma sheet have been recently reported by Sun et al. (2018). Aizawa et al. (2018) further investigated how sudden electric field variations -so called E burst -occurring during the development of the KH instability can also lead to the non-adiabatic acceleration of ions. They showed that ions can efficiently be accelerated within the KH vortex and discussed how these accelerated ions can effectively sputter the planetary surface and produce secondary neutrals and ions that populate both the exosphere and magnetosphere of Mercury. On the other hand, Gingell et al. (2015) show that the ion motion can also affect the field. They used a simple local two-dimensional hybrid simulation of Mercury's environment in order to reveal that the presence of picked up ions leads to a gyroresonance between growing KH vortices and sodium ion gyration that introduces a strong spectral peak at sodium gyroscales at the dusk magnetopause, modify the growth rate of the KH instability, and increase its dawn-dusk asymmetry, as was previously reported by MESSENGER observations [Gershman et al., 2015]. The KH instability is a macroscopic instability that occurs at the magnetopause of magnetized planets and known to be an important process for mixing plasma of different origin under a particular interplanetary magnetic field (IMF) configuration (e.g., northward for the Earth and Mercury). Indeed, when the instability fully develops, it forms large rolled-up KH vortices where magnetic reconnection can occur.

In the present study, the non-adiabatic acceleration and related transport of picked up ions during the development of the KH instability in both the dawn and dusk flanks of the magnetosphere of Mercury are investigated in detail. Aizawa et al. (2018) have already revealed that picked up ions can be accelerated by this process, but they have restricted their study to the dawn side of the magnetosphere only. Since MESSENGER observations have shown that a strong dawn-dusk asymmetry for the KH occurrence at Mercury exists [Liljeblad et al., 2014], it appears necessary to extend the study of Aizawa et al. (2018) to the dusk side of the magnetosphere also. This is the aim of the present study that focuses further on the transport of ions due to electromagnetic field variations and the role of the convection electric field. In order to explore the dynamics of picked up ions, we performed the same numerical method that was done by Aizawa et al. (2018), launching single particle trajectory computations in the KH field obtained from MHD simulations. In addition, whereas the previous study considered only sodium ions, we include in the present study additional ions such as solar wind plasma, oxygen, and potassium ions of planetary origin, and discuss based on a statistical analysis the dependence of the acceleration and transport of ions with their Larmor radius. In Section 2, we introduce our modeling approach and describe the initial settings used. We detail the results of our simulations in Section 3 and discuss their interpretation in Section 4. Section 5 summarizes our conclusions.

Modelling approach and initial settings

In order to investigate the dynamics of picked up ions under realistic conditions for Mercury, we adopt the same approach as the one used in Aizawa et al. (2018). We trace the motion of particles in the KH field configurations obtained from 2D local ideal MHD simulations. A detailed description of the numerical scheme and conditions used can be found in Aizawa et al. (2018). In the present study, quantitative and systematic calculations are performed in configurations representative for both the dawn and dusk sides of the magnetosphere of Mercury. All parameters are carefully chosen from observational data when available. (2008). All ions except solar wind species are injected as picked up ions at given times in our simulations. The size of our simulation box in the x-direction, corresponding to the KH wavelength, is set to 1.5 R as estimated in Gershman et al. (2015), where R is the radius of Mercury. The size of our simulation box in the y-direction is set to 12 R with a spatial resolution of 32.2 km (114 x 909 grids). It is necessary to take such a large size in the ydirection in order to avoid that the circular motion of ions does not close in the MHD domain and, hence, to lose them in our simulations. The thickness of the magnetopause (boundary layer) is ~ 515 km (referred to as 2a hereinafter) as estimated in Gershman et al. (2015) who considered orbits with northward-oriented interplanetary magnetic fields and high (i.e., > 1 cm ) Na concentrations. DiBraccio et al. (2013) also estimated the thickness of the dayside magnetopause to be of the order of ~100 km but their study considered mainly the subsolar point where the magnetopause is the most compressed, whereas the KH instability occurs near the terminator.

Various ion species, not only

In our simulations ions are traced in the frame of the average motion of the KH wave, with the periodical boundary condition being set in the x-direction. Since the time scale of the considered MHD simulations is much larger than that of the particle motion such as the ion gyromotion, all ions have to be traced with a time step delta-t much smaller than the gyromotion period. Our time step delta-t is therefore chosen so that it corresponds only to a few degrees of the ion gyromotion. It allows us to capture all the possible motions for the ions as accurately as possible, in particular their helical or meandering motions. All the settings for our parameters are summarized in Table 1. The variations of all physical quantities across the magnetopause follow hyperbolic tangent profiles.

Since MESSENGER observations did not enable us to derive the magnetospheric convection speed, we consider 300 km/s for the anti-sunward flow speed in the magnetosheath and 50 km/s for the sunward flow speed in the magnetosphere as used in previous numerical studies [e.g., Seki et al., 2013;Fujimoto et al., 2007;Jia et al., 2015;Wang et al., 2010]. We consider a purely Bz component across the magnetopause. This configuration corresponds to the most unstable configuration for the development of the KH instability and also allows us to avoid the occurrence of magnetic reconnection in our MHD simulations. In our particle calculation, the electric field is calculated from Ohm's law using the background velocity and magnetic field from the MHD simulation. We both consider the magnetospheric convection speed in the magnetosphere and the anti-sunward flow in the magnetosheath when calculating the electric field. In the rest of the paper the calculated electric field is referred to as the convection electric field.

In order to examine the properties of picked up ions, we take t = 32.2 [s], which is corresponding to the time of the initial stage of the development of the KH instability. This enables us to observe the ion properties with the evolution of scales during the development of the KH instability. Although we also considered other injection times, t = 80.5 [s] for the stage when the KH vortex is forming and t = 128.8 [s] for the saturation stage when the KH vortex starts to break, the results and conclusions of the present paper are unchanged. The results discussed in the rest of the paper will correspond to an ion injection time of t = 32.2 [s].

Figure 1 displays the results of our MHD simulation for those three times in order to illustrate the different stages of the KH development. Although the results in Figure 1 are displayed with only a scale of +/-1.0 R in the y-direction, our calculation domain is large enough so that the gyration motions of all ions are closed within the domain. In addition, although it appears visually that the development of the KH instability seems to be different between the dawn and dusk side configurations studied here, we confirm that the growth rate of the KH instability is almost the same. Due to the periodic boundary condition set up, only the fastest growing mode can be observed in our simulation and this mode is constrained by the thickness of the magnetopause and length of the simulation domain that are identical at dawn and dusk. Although the MHD approximation may not be valid especially at the dawnside magnetopause of Mercury according to previous studies (e.g., Paral and Rankin (2013) who have reported that both the finite Larmor radius of ions and the convection electric field can control the thickness of the magnetopause where the KH instability develops), we consider in the present study both the dawn and dusk sides of the magnetosphere in order to globally understand the effect of a large convection electric field on the ion dynamics and transport. In addition to a configuration with a northward IMF, we also consider a configuration with a southward IMF even if, according to in-situ observations around the Earth, the occurrence of the KH instability significantly decreases under a southward IMF because the magnetic reconnection is the dominant process [e.g., Kavosi and Raeder, 2015].

An overview of the properties of Na ions for the northward IMF case is shown in significantly different between the dawn and dusk sides. In particular, a large ion density gap can be observed near the magnetopause on the dawnside, while such a gap is not seen on the duskside. Moreover, ions of magnetospheric origin (yellow dots) can enter the magnetosheath region at t = 96.6 [s] due to the development of the KH vortices on the dawnside, whereas those on the duskside could not cross the magnetopause. Still on the dawnside, once ions penetrate inside the magnetosheath region, they seem to move independently from the filamentary structure of the KH vortex. On the contrary, planetary ions picked up in the magnetosheath (blue dots) did not penetrate into the magnetosphere. The situation is opposite for the duskside configuration. Ions in the magnetosheath (blue dots) can enter inside the magnetosphere region, whereas ions in the magnetosphere (yellow dots) penetrate less inside the magnetosheath region compared to those in the case of the dawnside configuration.

Results

Transport from the magnetosphere to the magnetosheath

Northward IMF case

The initial energy of injected picked up ions depends on the energy calculated from the ExB drift speed in each region. In this section, we focus on ions picked up in the magnetosphere, which are represented by yellow dots in Figure 2. Figure 3 shows the properties of Na ions for the northward IMF case, at given times. Although all the colored dots represent Na ions picked up in the magnetosphere, the different colors correspond to different distances from the magnetopause. Each color has an initial thickness corresponding to the thickness of the magnetopause, 2a (the magnetopause here corresponds to the region within +/-0.1 R of the center of the boundary layer). The magenta dots represent ions initially picked up in the region located at distances between 0.1 -0.3 R from the magnetopause, the green dots represent ions picked up in the region located at distances between 0.3 -0.5 R from the magnetopause, the cyan dots represents ions picked up in the region located at distances between 0.5 -0.7 R from the magnetopause, and the yellow dots represent ions picked up in the region located at distances between 0.7 -0.9 R from the magnetopause. The Na ions were initially injected with a thermal energy of 1 eV at t = 32.2

[s], which corresponds to the time when the KH instability begins to develop. For simplicity, we choose all ions with 90 degrees of pitch angle. At t = 128.8 [s], the magenta and green dots on the dawnside seem to be distributed randomly and the corresponding ions are transported from the magnetosphere to the magnetosheath. On the other hand, no ions are observed in the magnetosheath on the duskside. This is because of the orientation of the convection electric field. At t = 193.2 [s], ions in the dawnside are distributed throughout the entire simulation box, which indicates that they are well mixed in all regions. In contrast, only little ions are observed in the magnetosheath on the duskside, whereas they are well mixed inside the magnetosphere. The three right panels of Figure 3 show, from top to bottom, the growth of the KH instability, the number of energized ions, and the number of transported ions as a function of time. The 2nd and 3rd panels indicate the occurrence rate, obtained by counting the number of energized/transported ions and dividing it by the total number of ions in each region. When the magnetic moment of an ion undergoes a large change, it is no longer conserved and this is the signature of non-adiabatic acceleration. In the rest of the paper, energized ion will refer to non-adiabatically energized ions. For example, in the second panel of Figure 3, when the magnetic moment of an ion in magenta reaches a value ⁄ > 10, it is counted as an energized ion. In the second panel the total number of energized ions is furthermore divided by the total number of ions initially injected in the magenta region. A similar counting method is applied for the third panel, where net transport means that an ion crosses the magnetopause and is located up to a distance of +2a from the magnetopause (see white dashed lines at t = 64.4 [s] in Figure 3(a)). For example, we count an ion picked up on the dawnside in the 0.3 -0.5 R region (see the colored dots in green in Figure 3(a)) when it has crossed the magnetopause and moved to a distance farther than -0.3 R from the boundary. In the third panel those counts are furthermore divided by the total number of ions initially injected in the green region. The magenta and orange profiles in the 2nd and 3rd panels of Figure 3(b) correspond to the color dots represented in magenta and yellow, which are respectively, the closest region to and the farthest region from the magnetopause. Profiles for the properties of ions colored in green and cyan in Figure 3(a) are omitted here for readability.

Looking at Figure 3(b), we first find that non-adiabatic acceleration occurs on both the dawn and dusk sides, with ions more significantly accelerated on the dawnside. The solid and dashed lines refer to the dawn and dusk configurations. For example, 85% of the ions are accelerated in the region closest to the magnetopause at dawn, while only 35% of them are accelerated in the same region at dusk. The time when the acceleration starts to be observed closely matches the time of the growth of the KH instability, suggesting that the formation of KH vortices efficiently accelerate ions. It is again stressed that ion transport here means that ions are observed in another region after having crossed the magnetopause. On the dawnside, we observe ions having moved into another region only under their gyration motion. We also observe ions with a large Larmor radius near the magnetopause after having crossed the boundary during one of their cyclotrons motions. This can lead to an oscillatory motion as illustrated in the bottom last panel in Figure 3(b). The fact that there is no dashed line in the 3rd panel of Figure 3(b) indicates that no transport from the magnetosphere to the magnetosheath occurs on the duskside even if the ions are energized. In the present analysis, we count all energized ions with ⁄ > 10 as non-adiabatic energized ions, thus some of them may be even more energized. Ions with ⁄ ~ 10 may not cross the boundary but ions with ⁄ ~ 100 may do so because of their large Larmor radius. In addition these nonadiabatic energized ions may show the oscillation signature described previously. In order to investigate this effect in more detail, we look at changes in the magnetic moment of picked up ions when accelerated and energized in the closest region to the magnetopause (delineated by A and A' in Figure 3). Figure 4 shows the distribution of their energization and transport for different values of magnetic moment as a function of time. First of all, the main difference between the dusk and dawn sides is observed for their energization profile. The left panel shows that energized ions have their magnetic moment increasing from 10 to 50 whereas the right panel shows more dispersed changes in magnetic moment. On the duskside, 78 % of energized ions represented in magenta have their magnetic moment increasing from 10 to 25, while those represented in blue have their magnetic moment increasing from 25 to 50. Figure 4 also clearly shows that the ions on the dawnside are efficiently accelerated, and the profile in the 2nd panel shows that ion transport occurs even for the lower value of magnetic moment change. The level of ion energization does not correspond to the level of ion transport. A different behavior is observed in the duskside where we observe no transport at all together with little ions significantly energized. In order to investigate further this dawn-dusk asymmetry we select model trajectories for ions and detail their properties.

Figure 5 shows two model trajectories for picked up sodium ions in the magnetosphere under both (a) a dusk and (b) a dawn configuration. In this figure, we consider that an ion with ⁄ < 10 at dusk undergoes a limited energization whereas an ion with ⁄ > 100 at dawn undergoes an efficient energization. From top to bottom we show the ion position in the y-direction with arrows showing the electric field orientation, the electric field intensity that the ion experiences, its magnetic moment which is normalized to its initial value , and the ion energy. The electric field orientation shown on the 1st panel corresponds to the direction of the Lorentz force that the ion experiences. For example, in Figure 5(a) at t = 50

[s], the green arrows point towards the magnetosheath indicating that the ions are accelerated towards this direction. On the contrary, at t = 250 [s] in the same figure, the green arrows point in the opposite direction indicating that the ions are essentially accelerated towards the magnetosphere. The two selected ions were initially injected at the same distance from the magnetopause. In Figure 5(b), the change in their magnetic moments is observed around the time when the KH instability grows (indicated by the red dashed line), and occurred at t ~ 115

[s]. On the contrary, such clear change in their magnetic moment is not observed in Figure 5(a). We look at snapshots of the ion and electric field properties in order to build the following scenario. At dawn, when the ion is just picked up in the magnetosphere, it is first following its normal gyration motion. It may experience a small electric field variation with the development of the KH instability and the resulting polarization drift gradually accelerates it adiabatically (equivalent to an increase in its Larmor radius). When the ion enters the initial boundary layer region, here at t ~ 115 [s], the ion suddenly experiences a large electric field directed towards the magnetosheath where the change in its magnetic moment occurs. After that, the ion follows again its gyration motion in the magnetosheath. The model trajectory displayed in Figure 5(a) shows that although E burst-like signatures can be seen on the 2nd panel, the ion does not gain a large amount of energy on the duskside. Due to the direction of the electric field (given by the green arrows in the 1st panel) in this region, the E burst occurs when the ion moves against the electric field, and thus deceleration occurs. Some ions are able to gain large energy, for example up to ⁄ ~ 30, due to the stronger electric field inside the rotating KH vortex, which is directed towards the magnetosphere, during the later stage of the development of the KH instability. We conclude that the transport of ions is controlled by the electric field configuration in the magnetosheath under a northward IMF.

Southward IMF case

Magnetic reconnection at Mercury generally occurs when the IMF is southward, leading to a more complex configuration for the magnetopause. Despite the fact that the KH instability less likely occurs under this configuration we performed our MHD simulation with the magnetic field varying across the magnetopause from -Bz in the magnetosheath to + Bz in the magnetosphere, with the same magnitude in the entire simulation domain. Because we considered ideal MHD, no magnetic reconnection was observed, whereas the KH instability was growing. In the dawn magnetosheath we have a large electric field directed towards the magnetopause, whereas in the magnetosphere we have a small electric field pointing away from the magnetopause. Conversely, in the dusk magnetosheath we have a large electric field pointing away from the magnetopause, whereas in the magnetosphere we have a small electric field directed towards the magnetopause. Figure 6 shows the properties of sodium ions in the case of a southward IMF. Contrary to the case of a northward IMF, there is no dawn-dusk asymmetry for the energization and transport of ions. The energized ratio for ions picked up in the closest region adjacent to the magnetopause is much higher in the case of a southward IMF than in the case of a northward IMF, and this is true for all three heavy ion species (O , Na , and K ) as well as for both the dawn and dusk sides. This is due to the net electric field change across the magnetopause, with the electric fields in the magnetosphere and magnetosheath cancelling each other in the case of a northward IMF but adding each other in the case of a southward IMF, leading to an efficient energization in the latter case. We also investigated the distribution of ion energization in the same manner as done previously for the case of a northward IMF (see Figure 7). We found that for highly energized ions, ⁄ > 100, the energization increases in time due to the rotating electric field component. Interestingly, although the magnetosheath electric field is directed toward the magnetopause at dawn in the case of a southward IMF, the ions are nevertheless transported to the magnetosheath. This result totally differs with the case of a northward IMF. Indeed, at dawn when the IMF is southward a large rotating electric field inside the KH vortex directed towards the magnetosheath is observed, leading to a significant ion energization and transport in the magnetosheath. In summary when the IMF is southward, signatures of energization and transport of ions picked up in the magnetosphere are observed on both the dawn and dusk sides. The levels of ion transport on both sides are almost identical during the later stage of the KH development. Even if a large convection electric field affects the transport of ions, these ions cannot cross the magnetopause without the development of the KH instability. In more realistic configurations, however, additional components of the magnetic field, Bx and By, are observed in the magnetospheric flanks, which may lead to a more effective transport of magnetospheric ions.

Transport from the magnetosheath to the magnetosphere

Northward IMF case

Here we study the properties of ions initially picked up in the magnetosheath. A noticeable difference is the electric field that the ions initially experience: ions picked up in the magnetosheath might not experience a large E burst because any additional electric field is small compared to the electric field in the magnetosheath. The results for the case of a northward IMF are shown in Figure 8(a) and (b). We can first notice that non-adiabatic acceleration does not occur on both the dawn and dusk sides of the magnetosphere (see the 2nd panel of Figure 8(b)). On the dawnside, ions cannot be transported to the magnetosphere.

They mainly move outwards from the magnetopause due to the orientation of the electric field in the magnetosheath (see top panels of Figure 8(a) and the 3rd panel of Figure 8(b)). On the duskside, some ions can cross the magnetopause and move towards the magnetosphere where they are well mixed. However they are not energized due to the fact that they are initially experiencing a large electric field that does not increase during their gyration motions. The transport of ions on the duskside starts at the time when the KH instability grows linearly and its depth of penetration seems to be of the same order as the fluid structure represented in black and gray, suggesting that the formation of the KH vortex plays a role in the transport of ions even if they initially have a large Larmor radius.

The exosphere of Mercury extends to a few R [McClintok et al., 2008] and so many ions of exospheric origin can be picked up in the magnetosheath. Since these ions have a large Larmor radius (approximately of the same order or more than the thickness of the magnetopause), they can penetrate inside the magnetosphere. However, according to our results, without the x and y components of the magnetic field, it is difficult to cross the magnetopause on the dawnside. This is also true for ions picked up on the duskside with little ions transported to the magnetosphere in comparison with ions transported in the magnetosheath, as described in the previous section.

Southward IMF case

When the IMF is southward a small number of ions can be energized and some of them transported on both the dawn and dusk sides (see Figure 8(c) and (d)). However, the ion transport on the dawnside suddenly starts when the ions are injected. This is not due to the development of the KH instability itself, but to the presence of a large electric field instead.

We restrict ourselves in this study to the only effect of the electric field variations due to the development of the KH instability and so we do not take into consideration the ion transport without the growth of the KH instability. In that case only a small (negligible) ion energization is observed on both sides due to the development of the KH instability (see 2nd panel of Figure 8(d)), as well as ion transport on the duskside (see 3rd panel of Figure 8(d)).

The small ion energization observed can be explained due to the orientation of the electric field which points in the same direction across the magnetopause.

Properties of lighter ion species

Compared to the properties of heavy ions of planetary origin, lighter ion species that were injected with initial energies of 0.047 eV (= 540 K) have different properties. Figure 9(a) and (b) shows the properties of lighter ions (dihydrogen ions) initially picked up in the magnetosphere for a northward IMF. These lighter species have a Larmor radius small enough compared to the magnetopause thickness so that they can be treated as fluid. Although some ion energization can be seen after the linear growth of the KH instability, no transport occurs. Our results indicate that the fine structures arising from the late phases of the development of the KH instability are responsible for the acceleration. Acceleration of dihydrogen ions is observed in the region closest to the magnetopause (in magenta) at t ~ 120

[s], whereas it is observed at t ~ 140 [s] for protons with a smaller Larmor radius. Lighter ion transport does not occur either at dawn or at dusk: these ions only follow the expansion of the magnetopause due to the development of the KH instability unlike the case of heavy ions.

Moreover, the lighter ion energization occurs at similar level on both sides contrary to the properties of heavy ions. When the IMF is southward we found that the ion transport can be observed at both dawn and dusk (see Figure 9(c) and(d)). The lighter ion energization on both sides is significant, reaching more than 90% (see purple lines in the 2nd panel of Figure 9(d)). So, in conclusion, the rotating components of the electric field work efficiently on the energization and transport of lighter ions.

Lighter ions picked up in the magnetosheath region when the IMF is northward are not energized and only a small transport (~ 10%) can be observed on the duskside similarly to the case of heavy ions. On the other hand, when the IMF is southward, lighter ions injected in the magnetosheath are energized and transported both at dawn and dusk. The main difference compared to the properties of heavy ions of planetary origin is the fact that lighter ions picked up in the magnetosheath region are more significantly energized and transported. Indeed, because of their lighter masses, it is easier for them to be trapped inside KH vortices and be affected by small variations of the rotating electric field variations inside the structures.

Discussion

Table 2 summarizes the energization and transport of planetary ions for both a northward and a southward IMF. Three symbols are used to describe the properties of ion energization and transport. Overall, the level of energization is higher in the case of a southward IMF as well as in the dawnside due to the orientation of the electric field. For example, we can look at the double circles used for the energization of heavy ions picked up in magnetosphere on the dawnside under both a northward and a southward IMF. The energization ratio in the case of a northward IMF reaches 80% while the ratio in the case of a southward IMF reaches 100%. The energization and transport of heavy ions of planetary origin picked up in the magnetosphere are illustrated in Figure 10. When the IMF is northward, we found that no acceleration of heavy planetary ions picked up in the magnetosheath is observed and that ion transport from the magnetosphere to the magnetosheath occurs only at dawn, while a small amount of heavy ion transport in the opposite direction occurs at dusk because of the orientation of the electric field. Because of their smaller Larmor radius, lighter ions behave magnetohydrodynamically. Acceleration of lighter planetary ions picked up in the magnetosphere occurs at both dawn and dusk but no transport towards the magnetosheath is observed.

On the other hand, when the IMF is southward, the energization of both heavy and lighter ions initially picked up in the magnetosphere and their transport from the magnetosphere to the magnetosheath occur both at dawn and at dusk, whereas the energization of heavy ions initially picked up in the magnetosheath appears negligible and there are only little ions transported from the magnetosheath to the magnetosphere, However, in this case, the heavy ion transport from the magnetosheath to the magnetosphere at dawn can occur without the development of the KH instability. On the contrary, lighter planetary ions picked up in the magnetosheath can be accelerated and transported due to the initial orientation of the large electric field across the magnetopause. Since they are light, they have a small Larmor radius and thus, they stay inside the KH vortices. Those lighter ions are affected by small electric field variations within the KH vortex, leading to efficient acceleration and transport. In the present paper we have mainly focused our illustrations and discussion on two ion species, the sodium and dihydrogen ions. We now discuss the dependence on Larmor radius for the ion acceleration and transport. The different properties of heavy ions compared to the properties of lighter ions indicate that there is a relationship between the size of the KH vortex (equivalent to the magnetopause thickness) and the Larmor radius of the ion. Figure 11 shows the dependence of Larmor radius for the energization of ions picked up in the magnetosphere, especially in the region closest to the magnetopause (in magenta in the previous figures), for the case of a northward IMF. The x-and y-axes are scaled by the ion Larmor radius normalized by the initial half thickness of the magnetopause, and the time when the energization reaches 30% is normalized by the linear growth time of the KH instability (t ~ 90 [s]). The red dashed line represents x = 1 and y = 1. The circle and triangle symbols are used for the duskside and dawnside cases, respectively. We performed additional calculations for helium, carbon, and iron ions. According to this figure, ions are less energized at dusk compared to dawn. The acceleration of lighter species take place during a later stage of the growth of the KH instability, whereas ions with a larger Larmor radius, such as Fe and K , are accelerated just after the formation of the KH vortex. Although the oxygen ions seem to be accelerated efficiently, the Larmor radius dependence of the ion energization seems to vary gradually (see Figure 11). A similar feature is observed for the case of a southward IMF. In our study, when the ions move farther than a distance of 0.3 R from the magnetopause, we assume that they are transported to another region. However, since the lighter planetary ions follow the background fluid structure (MHD-like behavior) and since the background structure does not expand more than 0.3 R this time, they are not considered as transported. To first order, the number of planetary ions transported increases with increasing mass. For example, for the case of a northward IMF, 42% of carbon ions, doubled-charged iron ions and potassium ions, 44% of oxygen ions, and 41% of sodium ions are transported from the magnetosphere to the magnetosheath at dawn, whereas none of the protons and dihydrogen ions, and only 10% of helium ions are. For the case of a southward IMF case, the corresponding values are 25% for protons, 31% for dihydrogen ions, 30% for oxygen and sodium ions, and 32% for potassium ions. Some threshold for the MHD-like behavior of ions could be set when looking at Figure 11 for the properties of helium and dihydrogen ions. Our results show that the ions picked up in the magnetosphere near the magnetopause are non-adiabatically energized during the development of the KH instability, and half of them are transported to the magnetosheath and escape from the magnetosphere of Mercury. The other half of those non-adiabatically energized ions stays inside the magnetosphere and they can sputter the surface of Mercury in order to produce secondary neutrals at the terminator or in the dayside magnetosphere.

All the cases shown in Section 3 correspond to the case of an injection time of t = 32.2 s. We have performed additional runs with injection times for picked up ions of t = 80.5 s and t = 128.8 s as mentioned in Section 2. Our conclusions for the ion energization and transport in both the case of a northward and a southward IMF do not change qualitatively. However, the energization ratio decreases when enhancing the injection time due to the relation between the scale of the ion gyromotion and the structure resulting from the development of the KH instability, indicating that heavy ions are less energized when they are picked up during the saturation phase of the development of the KH instability.

After having discussed the ion acceleration and transport within KH vortices, we now discuss the possibility of solar wind plasma entry inside the magnetosphere. Two different ion species, H and He are included in our modeled solar wind. These ions were injected with initial energies of 10 eV on the ExB frame at the start time of our simulation. Although H can originate both from the solar wind and from planetary ions, He cannot be produced in the Hermean environment, and thus the detection of He can be used as a proxy for the entry of solar wind plasmas inside the magnetopause. Previous results suggested that lighter ions cannot be accelerated under a northward IMF and that the small amount of solar wind penetrating inside the magnetosphere is expected because those lighter ions follow the background structure of the KH instability. In addition, because we assume that our background plasma is dominated by protons in our MHD simulation, the violation of the 1st adiabatic invariant for H leads to the violation of the MHD approximation. Previous studies on the KH instability did not discuss the limitation of the MHD approximation on the nonlinear stage of the development of the KH for the sake of simplicity. However, when the 1st adiabatic invariant of H is not conserved, it is a clear signature of the MHD approximation violation. We discuss our results for both a northward and a southward IMF. In both cases, we consider solar wind protons in the magnetosheath region closest to the magnetopause. Their Larmor radius normalized to the magnetopause thickness is 0.018, whereas the Larmor radius of He is ~ 0.026. In the case of a northward IMF, similarly to the properties of lighter ions of planetary origin, solar wind plasmas can also be energized within KH waves. The ratio of energized proton is almost the same on both the dawn and dusk sides and corresponds to 20% of the injected ions in the region closest to the magnetopause (see purple lines in Figure 12(a)). Identically, no ion transport was found on both the dawn and the dusk sides. Considering our ion injection time here at t = 0 [s] and the fact that the ion acceleration occurs at t ~ 145 [s] which is the same time on both the dawn and dusk sides, we conclude that the fine structures created when the KH vortex breaks are responsible for the acceleration of H ions. A similar behavior is observed for He , with the acceleration starting at t ~ 122 [s] (see pink lines in Figure 12(a)). These observations explain the evolution of fine structures in MHD fields by considering the fact that protons have a smaller Larmor radius (by a factor 2) compared to the Larmor radius of He . The transport of these light species in the solar wind can be considered MHD-like since their motion seem to follow the motion of the fluid background protons. The scale of large structures of the KH vortex is larger enough than that of proton gyromotion in the linear growth phase of the KH development, however, once the fine structures, which have a scale comparable to the proton gyromotion, appear in the saturation phase of the KH development, solar wind protons start to be energized. On the contrary, the Larmor radius of He is larger than the Larmor radius of protons, hence, they can be energized by larger fine structures compared to protons. On the other hand, in the case of a southward IMF, both He and H show qualitatively similar results (see Figure 12(b)). Ion energization and transport can be observed both on the dawnside and the duskside. The ion energization starts at t ~ 110 [s] for H and t ~ 108 [s] for He . Moreover, the ion transport starts at t ~ 120 [s] for H and at t ~ 120 [s] for He , contrasting with the case of a northward IMF. The ion acceleration occurs on both the dawn and dusk sides with almost the same ratio, but with the ion transport being more effective on the duskside. In addition, the ion transport in the case of a southward IMF starts at the same time as the ion energization starts. This suggests that ions can be transported due to the acceleration arising from the rotational components of the electric field inside the KH vortex in the same manner as in the case of ions of planetary origin. Those results are summarized in Table 3. At the same time, the 1st adiabatic moment of a proton at t = 145 [s], for the case of a northward IMF, indicates that the MHD approximation is not valid anymore. Although our injected proton had different properties compared to those of the background protons, it implies that we need to consider ion kinetic effects during the later stage when the KH instability becomes non-linear. In the present study, these kinetic effects have been neglected and they should be studied in the future. Is it possible to have the entry of solar wind plasmas through a KH vortex? Our results suggest that the ions can be transported magnetohydrodynamically, meaning that the size of the solar wind plasma entry region is determined by the size of the KH vortex. In particular, even in absence of magnetic reconnection, solar wind plasmas are subject to enter into the magnetosphere region when the IMF is southward due to the convection electric field.

When combining all these results, we observe that the ions on the dawnside are less dense than the ions on the duskside when the IMF is northward. Indeed, in that case, on the duskside, ions picked up in the magnetosphere cannot cross the magnetopause, whereas in addition those in the magnetosheath can penetrate inside the magnetosphere. In order to be accelerated by the variations of the electric field, the ions should be inside the magnetosphere where the initial electric field is smaller than the electric field in the magnetosheath. Moreover, the ratio of energized and transported ions on both sides is reaching a plateau as observed in Figure 3 for instance, 85% (35%) of the energized ratio at dawn (dusk) in the closest region.

This seems to indicate that the heavy ions are not affected by the small-scale structures (socalled fine structures) arising when the KH vortices break during the non-linear growth (saturation phase) of the instability but we note that these fine structures have an effect on the acceleration of lighter ions that have smaller Larmor radius.

Conclusions

Particle trajectory computations have been performed in a time-varying MHD model of KH vortices in order to understand the properties of ions within KH vortices. First of all, we revealed that the large-scale convection electric field plays an important role in the ion acceleration and transport. Our results for both a northward and a southward IMF showed that the ion transport is well controlled by the orientation of the electric field in the magnetosheath.

Ions picked up in the magnetosphere are energized non-adiabatically on both the dawn and dusk sides for both a northward and a southward IMF. In particular, we found when the IMF is southward that ions are efficiently accelerated within KH vortices due to the orientation of the developed electric field inside the vortex. In the particular case of lighter ions, including the solar wind plasmas, their Larmor radius is small enough compared to the magnetopause thickness and thus small enough not to be accelerated during the linear growth phase of the KH instability, resulting in an MHD-like behavior. In their non-linear stage, KH vortices start to break and make fine structures, whose scale becomes comparable to the Larmor radius.

Ions can be energized non-adiabatically with these fine structures. In the later stage of the nonlinear phase, the ion kinetics should be considered. In this study, although we did not consider any component of the magnetic field parallel to the wavenumber of the KH instability and the existence of ions already moving non-adiabatically, some general features should be detected by instruments aboard a spacecraft. Observations of non-adiabatically energized ions in the magnetosphere on both the dawn and dusk sides together with the development of the KH instability should be detected. If we have an ion spectrometer with a wide energy range from a few eV to a few keV, the ion acceleration should be observed and thus, the energy distribution would be changed between the KH and non-KH events. The FIPS (Fast Imaging Plasma Spectrometer) instrument onboard MESSENGER had an energy range from 100 eV/e to 13 keV/e and, therefore, was too limited to identify the acceleration process for picked up ions [e.g., Raines et al., 2011;2013]. The BepiColombo mission now en route to Mercury includes a full set of more sophisticated charged particle instruments. In particular, the Mass Spectrum Analyzer (MSA) instrument onboard the Mio spacecraft has a wider energy range that can cover the picked up ion acceleration theoretically found in this study, and with its wider field of view it will be used for a better understanding of this nonadiabatic energization and transport within KH vortices at Mercury [Delcourt et al., 2016]. White arrows indicate the electric field orientation. The upper and lower color-coded panels correspond to electric field maps at distinct simulation times during the evolution of the KH instability on the dawn and dusk sides, respectively: at t = 0.0 s for the initialization of MHD simulation, for the linear growth phase of the KH instability at t = 80.5 s, and when the KH vortex is fully formed at t = 96.6 s. Then when t = 128.8 s, the KH instability reaches saturation. Here we assumed that the solar wind is flowing tailward with a velocity of 300 km/s and that the magnetospheric convection flows from the nightside to the dayside with a velocity of 50 km/s. Initial variations of physical quantities across the magnetopause follow hyperbolic tangent profiles. The thickness of the magnetopause is 515 km and the wavelength of the KH instability is set to 1.5 R (in Mercury radius), which is consistent with MESSENGER observations. From top to bottom, the ratio of energized ions and the ratio of transported ions are shown versus time. Each colored line corresponds to a change in the magnetic moment of the ions. The red line shows ions energized with ⁄ > 100, the orange line is for ions with magnetic moment in the range of 75 < ⁄ ≤ 100, the green line is for ions with magnetic moment in the range of 50 < ⁄ ≤75, the blue line is for ions with magnetic moment in the range of 25 < ⁄ ≤ 50, and the magenta line is for ions with magnetic moment in the range of 10 ≤ ⁄ ≤ 25. Red and blue dashed vertical lines show the linear growth time of the KH instability and the injection time of particles, respectively. 2. Summary table for the energization and transport of planetary ions (h for heavy ions (O , Na , and K ), l for light ions (H and H )) in the closest region to the magnetopause. +Bz and -Bz show the northward and southward IMF cases, respectively. Each symbol, double circle, circle, and triangle indicates the level of energization and transport. A double circle is used when the level of energization (transport) is higher than 80%, whereas a triangle is used when those are lower than 20%. A circle represents levels between 20 -80 %. A cross is used when those are zero.

  Na like in Aizawa et al. (2018) but also O , K (heavy ions of planetary origin), H , H , (lighter ions of planetary origin), and H , He (solar wind plasmas) are included in the present study. Their properties are taken from Zurbuchen et al.

Figure 2 .

 2 Figure 2. Here 10,000 ions were injected in three different regions as picked up ions. Blue

Fig 1 .

 1 Fig 1. Illustration of MHD simulation conditions for the case of a northward IMF. The left panel shows the location of the bow shock (BS) and magnetopause (MP) from models by Slavin et al. (2009) for BS and Shue et al. (1997) for MP in MSO coordinates, respectively. The parameters for those paraboloids are taken from Winslow et al. (2013). The two red boxes delineate the regions where the KH instability occurs on both flanks of the magnetosphere of Mercury and, therefore, our simulation domain. Initial profiles of normalized physical quantities (velocity Vx, mass density , magnetic field Bz) and colorcoded electric field intensity obtained from MHD modeling are shown on the right panels.White arrows indicate the electric field orientation. The upper and lower color-coded panels correspond to electric field maps at distinct simulation times during the evolution of the KH instability on the dawn and dusk sides, respectively: at t = 0.0 s for the initialization of MHD simulation, for the linear growth phase of the KH instability at t = 80.5 s, and when the KH vortex is fully formed at t = 96.6 s. Then when t = 128.8 s, the KH instability reaches saturation. Here we assumed that the solar wind is flowing tailward with a velocity of 300 km/s and that the magnetospheric convection flows from the nightside to the dayside with a velocity of 50 km/s. Initial variations of physical quantities across the magnetopause follow hyperbolic tangent profiles. The thickness of the magnetopause is 515 km and the wavelength of the KH instability is set to 1.5 R (in Mercury radius), which is consistent with MESSENGER observations.

Fig 4 .

 4 Fig 4. Energization and transport of ions initially picked up in the magnetosphere in the case of a northward IMF. The left (right) panel shows the results for the duskside (dawnside).From top to bottom, the ratio of energized ions and the ratio of transported ions are shown versus time. Each colored line corresponds to a change in the magnetic moment of the ions. The red line shows ions energized with ⁄ > 100, the orange line is for ions with magnetic moment in the range of 75 < ⁄ ≤ 100, the green line is for ions with magnetic moment in the range of 50 < ⁄ ≤75, the blue line is for ions with magnetic moment in the range of 25 < ⁄ ≤ 50, and the magenta line is for ions with magnetic moment in the range of 10 ≤ ⁄ ≤ 25. Red and blue dashed vertical lines show the linear growth time of the KH instability and the injection time of particles, respectively.

Fig 5 .

 5 Fig 5. Modelled ion trajectories at (a) dusk and (b) dawn in the case of a northward IMF. The red dashed line represents the linear growth time of the KH instability. From top to bottom, each panel shows the ion position in the y-direction with the electric field orientation (green arrows), the electric field intensity, the magnetic moment (normalized to the initial value), and the energy as a function of time. The purple and green lines in the bottom panel show the total energy of particle and energy calculated from the ExB velocity.

Fig 8 .

 8 Fig 8. Energization and transport of sodium ions picked up in the magnetosheath under a northward IMF ((a) and (b)) and southward IMF ((c) and (d)). The format is the same as the one used in Figure 3 except that transported ions move now in the opposite direction.

Fig 9 .

 9 Fig 9. Energization and transport of dihydrogen ions picked up in the magnetosphere under a northward IMF ((a) and (b)) and southward IMF ((c) and (d)). The format is the same as the one used in Figure 8. The initial Larmor radius is not shown because it is too small to be visualized in the same format.

Fig 10 .

 10 Fig 10. Illustration of general features of ion energization and transport of heavy ions of planetary origin (O , Na , and K in this study) for the northward and southward IMF cases. From top to bottom, each panel shows the initial large electric field configuration, ion acceleration, and ion transport in the x-y plane. The grey area in the ion acceleration panel correspond to regions with no energization.

Figure 1 :Figure 2 :

 12 Figure 1: Inclusion criteria for ADPKD patients IPP = Intraperitoneal pressure

Figure 3 :

 3 Figure 3: Positive correlation between intraperitoneal pressure (IPP) and body mass index (BMI) in ADPKD patients.

Table 1 .

 1 Summary of parameter settings

	region	Magnetosheath (MSH)	Magnetosphere (MSP)
	Background H number density		100 ions/cc	10 ions/cc
	Flow velocity (MSO)		-300 km/s	+ 50 km/s
	Injected Planetary ion species (m/q)		H (1), H (2), O (16), Na (23), K (39)
	Solar wind plasma in ExB frame (m/q)	H (1), He (2)	-
			H (1), H (2) : 0.047 eV (540K) in Mercury frame
	Initial particle energy	O (16), Na (23), K (39) : 1 eV in Mercury frame H (1), He (2) : 10 eV in ExB frame -
	Injection time		32.2 s
	Magnetic field (Bz)		+/-48.7 nT	+ 48.7 nT
	Cases
		IMF (Bz)	Magnetopause thickness 2	KH wavelength
	Northward IMF	+ 48.7 nT	515 km	1.5 R
	Southward IMF	-48.7 nT	515 km	1.5 R

Acknowledgments. The work of S. Aizawa was supported by the International Joint Graduate Program in Earth and Environmental Sciences (GP-EES). French co-authors acknowledge the support of CNES for the BepiColombo mission. The method used in this study is referred to properly in the reference (e.g., Aizawa et al., 2018). Simulation data used in this study is available in the link: http://rosina1.irap.omp.eu/documents/PUBLICATIONS/AIZAWA/AIZAWA2020_PSS/.