Ultra-low frequency waves in the Hermean magnetosphere: On the role of the morphology of the magnetic field and the foreshock

Ultra-low frequency (ULF) waves have been observed in the Mercury's magnetosphere by the Mariner 10 and MErcury Surface, Space ENvironment, GEochemistry and Ranging missions. The observed ∼2 s (∼0.6 Hz) period waves in the magnetic field are proposed to be generated by dynamic processes in the Mercury's magnetosphere. We investigate the Hermean ULF waves with a global hybrid model. We found evidence for ∼2-s circularly polarized right-handed waves in Mercury's magnetosphere at the closest approach of BepiColombo mission's first Mercury flyby in the model. The most intense wave power occurs on the dawn side closed magnetic field lines. These waves were found to be generated on the hemisphere which is magnetically directly connected to the interplanetary magnetic field on the dayside and to the foreshock region. It is therefore possible that the generation mechanism of these waves is associated with the precipitating ion flux or with the wave activity in the foreshock region.


Introduction
Characterization of the properties of the Ultra-low frequency (ULF) waves in planetary magnetospheres provides a way to investigate magnetospheric dynamics and the properties of the magnetosphere, such as the density of magnetospheric plasma. The dynamics of Mercury's magnetosphere are anticipated to have unique properties as it has an intrinsic magnetic field that is weak compared to that of the Earth, but strong compared to the other terrestrial planets and the interplanetary magnetic field (IMF).

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Observations from Mariner 10 flybys, as well as those by the MErcury Surface, Space ENvironment, GEochemistry and Ranging (MESSENGER) (Solomon et al., 2007), mission showed that there are waves around Mercury at various frequencies (Le et al., 2013;Romanelli et al., 2020). In analogy of the Earth's magnetosphere, some of the observed ULF waves were suggested to be Kelvin-Helmholtz waves (Boardsen et al., 2010), associated with field line resonance (FLR) events (James et al., 2019;Russell, 1989) or some kinetic instabilities . However, it has been pointed out that the Mariner 10 ULF wave event has a clear compressional component, which indicates that they are not simple standing waves (Southwood, 1997). The role of the plasma content at high-altitudes and the nature of the reflection of these waves near Mercury's highly conducting core interface must also be considered. Furthermore, the observed ∼1 Hz waves have been proposed to be quasi-trapped (Boardsen et al., 2009). Similar transient phenomena to those found in the Earth foreshock (e.g., Zhang et al., 2022) may exist at Mercury in one way or another.
The first observation of ∼second scale ULF waves was during near closest approach of the first Mariner 10 Mercury encounter (Russell, 1989). These ∼2-s periodic waves were initially attributed to be FLRs, that is, standing Alfvén waves (Russell, 1989). Later, it was suggested that the "ringing" of the Hermean magnetosphere could be caused by small-amplitude compressional ULF waves (Glassmeier et al., 2004). As the frequency of the observed waves was relatively close to the proton gyrofrequency, ion kinetic effects associated with ion cyclotron resonance or Bernstein waves have been suggested to play a role (Boardsen et al., 2009. However, it should be noted that no Na + or other heavy planetary ion cyclotron waves have been observed in the Mariner 10 or MESSENGER magnetic field observations presumably due to their large gyroradii (Boardsen & Slavin, 2007). The BepiColombo mission provides a new opportunity to measure the Hermean solar wind interaction including ULF waves with several Mercury flybys happening prior to the orbit insertion in December 2025 (Mangano et al., 2021;Milillo et al., 2020).
In this paper, the ULF waves at Mercury are investigated with a global hybrid model, in which ions are modeled as particles. The simulation setup is the same as that recently used to investigate ULF waves in the Hermean foreshock (Jarvinen et al., 2020b). We show that circularly polarized waves can exist in Mercury's magnetosphere on closed field lines at the closest approach of BepiColombo's first flyby (MFB1), on the dawn side near the foreshock.
The paper starts with introduction to the basic properties of the hybrid model and the simulation run. Following the analysis of the properties of the observed ULF waves, we suggest a possible origin of the observed waves by examining the global morphology of the magnetic field, especially, near the foreshock. We conclude by comparing the observations with the simulation results.

Model Description
The ULF waves are investigated with a three-dimensional hybrid simulation where ions are modeled as particles while electrons form a massless charge neutralizing fluid. The model was recently used to investigate ULF waves in the Hermean (Jarvinen et al., 2020b), Venusian (Jarvinen et al., 2020a), and Martian (Jarvinen et al., 2022) foreshocks.
We use a Mercury-centered coordinate system similar to the Mercury Solar Orbital (MSO) system, defined as the x axis pointing toward the Sun, z axis being parallel to the normal vector of Mercury's orbital plane, and y axis completing the right-handed system. All upstream solar wind and IMF parameters remain constant throughout the simulation. The upstream solar wind densities and temperatures were similar to that in the previous foreshock study (Jarvinen et al., 2020b): n(H + ) = 73 cm −3 , n(He ++ ) = 2.92 cm −3 (= 0.04 × 73 cm −3 ), T(H + ) = 1.7 × 10 5 K, T(He ++ ) = 3.5 × T(H + ). Motivated by the solar wind parameters measured during MFB1 (Orsini et al., 2022), the upstream solar wind speed was assumed to be relatively slow and flowing in the −x direction (U(H + ) = U(He + ) = [U x , U y , U z ] = [−320, 0, 0] km/s) that is, the aberration due to the orbital motion was not taken into account. The upstream IMF was set to [−10, 8, 5] nT. The strength of the magnetic dipole was set to 195 nT R M 3 (R M = 2,439.7 km) and the dipole was at [0, 0, 484] km (Anderson et al., 2011). The size of the simulation box was x = [−10, 6] R M , y = [−8, 12] R M , and z = [−8, 8] R M . The size of the Cartesian cell was R M /15 ∼ 163 km, the time step was 10 ms, and each cell contained on average 178 macroparticles. The planet was assumed to have a perfectly conducting core with a radius of 1,800 km and an insulating layer (a mantle) above it (see Jarvinen et al., 2020b, for details).

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The wave analysis was made by adding 144 "virtual detectors" to the simulation, along MFB1 trajectory on 1 October 2021. These detectors recorded particles and macroscopic plasma parameters during the run at every simulation time step (10 ms). Macroscopic plasma parameters ( Figure 1) and fields in the simulation domain were saved at t = 300 s to investigate the morphology of 3D magnetic field lines that provide global context for the particle measurements (see e.g., Milillo et al., 2020;Orsini et al., 2021Orsini et al., , 2022. As the trajectory resembles the orbit of the Mariner 10's first Mercury flyby, when ∼2 s ULF waves were observed, and the MESSENGER's first Mercury flyby (ULF waves observed on 14 January 2008), we can compare our results with those from the previous missions. Figure 1 shows an overview of the Hermean solar wind interaction environment and the MFB1 trajectory in the model. The perpendicular bow shock is clearly identified in the upper hemisphere of the figure by a sudden density enhancement. Density fluctuations are found on the parallel and quasi parallel side of the foreshock region on the dawn side. According to a spectral analysis of the total magnetic field, there is a peak at frequency f ULF = 0.62 Hz, that is, at period T ULF = 1.6 s. Furthermore, these waves contain a wave packet with a modulating period of about 20-30 s, especially, in the B x and B z MSO components (Figures 2a and 2c).

Properties of the ULF Waves
The minimum variance analysis for the time period 310-320 s shows that the waves are quite circularly polarized (Figures 2e and 2f). The eigenvalues of the magnetic field are [l 1 , l 2 , l 3 ] = [65.538, 62.654, 3.3225] nT 2 , that is, variations in the perpendicular direction are more than an order of magnitude stronger than in the parallel (compressional) direction. The angle between the eigenvector e 3 = [−0.095884, −0.98874, 0.11491] and the mean direction of the magnetic field in the analyzed time range, B° (= [−0.41618, −0.86807, 0.27063] nT, |B| = 103.6 nT) was ∼158° implying that the angle between the wave k vector and the ambient magnetic field is ∼22°. The hodogram in Figure 2e, where the magnetic field points out of the plane, shows counterclockwise motion indicating a right-handed wave in the simulation frame.
Detailed determination of the wavelength of the 1.6-s waves is complicated, as the waves propagate in 3D space. However, based on fluctuations of the 3D magnetic field lines, variations of the magnetic field along a line in the direction of e 3 and variations in the time evolution of the magnetic field on 2D planes (see Movie S1), the wavelength (l ULF ) is of the order of 600-800 km. This range corresponds to phase speed (= l ULF /T ULF ) in the range of 375-500 km/s. In the simulation time interval [250,350] s, in turn, the Alfvén velocity is between 568 and 894 km/s with the average velocity being 708 km/s. These Alfvén speeds are, therefore, relatively close to the estimated phase speed of the 1.6-s waves taking into account the possible inaccuracies in the wavelength estimation.

Morphology of the Magnetic Field and the Foreshock Region
The relation of the waves in Figure 2 to the Hermean global magnetosphere is then further examined in Figure 3. The morphology of the magnetic field is assessed by starting field line tracing along the flyby trajectory at the virtual detector points. Figure     The properties of the field lines on the dawn side around point #96 are investigated in more detail in Figure 3b. The color coding after point #99([0.03, −1.43, −0.63] R M ) shows how these magnetic field lines go through the high density magnetosheath while the plasma density is low on the closed magnetic field lines. Note also a small "twisting" on the closed magnetic field line on the northern hemisphere. The magnetic field lines on the dawn side connected to the northern hemisphere are in the region where the IMF is relatively parallel to the bow shock normal, that is, on the foreshock region. Figure 4 shows the global scale morphology of the magnetic field near Mercury in more detail. The figure shows the magnetic field B y component on two planes and on a spherical shell 360 km above the Hermean surface; this altitude is still clearly within the magnetosphere. Magnetic field lines connecting to the planetary surface are also shown. As the dipole field B y value in the y = 0 plane is zero, the B y component on this plane is caused by the IMF and is a measure of the planet's interaction with the solar wind.
Figures 4a and 4b demonstrate clear wave activity on the Northern hemisphere around the north magnetic cusp. No clear wave activity can be detected on the southern hemisphere. When the positions of the magnetic cusps are determined by calculating the angle (α) on the planet's surface between the surface normal and the magnetic field, the north magnetic cusp (α ~ 180°) is located near z-axis at high latitudes, within the bounds set by in MESSENGER observations (Raines et al., 2022;Winslow et al., 2012). The south magnetic cusp (α ~ 0°) instead is shifted toward dawn to lower latitude (~40°S) in the analyzed solar wind parameters. The shifting and twisting of the position of the cusp from the y = 0 plane is caused partly by the IMF B y component (see e.g., Kallio et al., 2008, Figures 2 and 3) and partly by the off-centered magnetic dipole.
Note that clear wave activity can also be seen near the dawn equator both on the surface of the shell and as a fluctuations of the magnetic field (Figure 4a). While the snapshot shown in Figure 4 cannot be used to identify the propagation of the waves, we can draw conclusions from the time series of B y on the y = 0 plane. These data suggests that the waves originate from near the planet on the northern dayside hemisphere, from where they propagate tailward, then toward the center of the tail and continue to the southern hemisphere (see Movie S1).

Discussion and Conclusions
In this paper, ULF waves in the Hermean magnetosphere with period ∼2 s were studied with a hybrid model. The analyzed region was chosen to be along the orbit of MFB1, which provides insight of the plasma regions sampled by the spacecraft. We find that ULF waves are formed naturally in a global hybrid simulation even when stationary upstream parameters are used as boundary conditions.
Although this study showed some basic properties of the ULF waves in the simulation, the study leaves open the question of their generation mechanism. It has been suggested that the ion-ion instability could be a potential origin for the observed ULF waves at Mercury (see e.g., Boardsen et al., 2012Boardsen et al., , 2015, and references therein). As the hybrid simulation does not include planetary ions, planetary ions cannot be the source of the ULF waves seen in the simulation. In the analyzed region, the average magnetic field was ∼106 nT corresponding to gyro periods of 0.6 and 1.2 s for H + and He ++ ions, respectively. Both time scales are shorter than the observed 1.6 s waves in the closed field line region.

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To examine the 1.6 s wave association with FLR, we calculate the time it takes for an Alfvén wave to propagate along the field line between its footpoints on the planet's surface. The length of the magnetic field line #96 is ∼6,400 km. We estimate the propagation time by integrating over the field line in distance steps dl i and summing the propagation time, dt i , over each time step using , where the local Alfvén velocity at the step i is v Alfvén , i . This gives a surface-to-surface propagation time of ∼12 s, which is over seven times longer than the period of the 1.6 s wave. Moreover, the bounce time is larger than this because the waves are bounced from the surface of the iron core. Thus, if the observed wave would result from FLR, it would be at the seventh harmonic frequency.
One possible cause for the hemispherical asymmetry is that the intrinsic magnetic field is larger on the surface on the northern hemisphere than on the southern hemisphere, because the magnetic dipole is centered 484 km ∼0.2 R M from the origin toward the northern hemisphere in the simulation. The asymmetric intrinsic magnetic field is not, however, a necessary requirement for all ULF waves in the Hermean magnetosphere, as clear ULF wave activity was found also in the hybrid model run (Jarvinen et al., 2020b) in which the dipole was placed at the center of the planet.
The hemispherical asymmetry may also be related to the direction of the IMF B x component. In the analyzed solar wind condition with negative B x , the northern hemisphere was magnetically connected to the IMF. If, instead, there would have been a strong positive IMF B x component, the southern hemisphere would have been magnetically connected to the IMF . Earlier hybrid and analytical models have shown that the hemisphere magnetically connected to upstream is also the region of intense precipitation of solar wind particles to Mercury's surface (see e.g., Kallio & Janhunen, 2003;Massetti et al., 2003). Moreover, the IMF connected on the surface on the dawn side passed through the quasi-parallel bow shock (e.g., Figures 1 and 3), where ULF waves have been observed (see e.g., Romanelli et al., 2020) and identified from hybrid simulations (Jarvinen et al., 2020b).
One should note that also the IMF B z plays an important role in magnetosphere's dynamics. When the IMF points away from the Sun (B x < 0), reconnection will shift to occurring just tailward of the cusps when IMF is northward (B z > 0). This adds additional mass from the solar wind to the Dungey cycle. This IMF configuration was the case in the analyzed simulation. However, in the southward B z case (B z < 0) the site for magnetopause reconnection will tend to be around the sub-solar region. In such way southward B z adds energy, momentum, magnetic flux, and mass to Mercury's Dungey cycle.
Finally, we note that the simulation contains also longer period 10-20 s variations or modulations of the magnetic field, as can be seen at point #96 (cf. in Figure 2d) and, for example, at point #97 closer to the magnetopause. The ∼16 s ULF waves, which have been suggested to be associated with Kelvin Helmholtz instabilities, have frequently been identified in MESSENGER's magnetic field measurements (Boardsen et al., 2010;Sundberg et al., 2012). For example, MESSENGER observed a close correlation between Kelvin-Helmholtz waves along the afternoon local time magnetopause (exactly where BepiColombo entered the magnetosphere), the appearance of bursts of Na + ions, and ULF waves (see Gershman et al., 2015;Sundberg et al., 2012, Figures 2-6).
An especially interesting question is the possible role of the precipitating particles and/or foreshock waves for the generation of the ULF waves in Mercury's magnetosphere. Ion measurements during MFB1 have revealed rapid flux fluctuations within just a few seconds near the closest approach (Harada et al., 2022). However, until now, no ULF wave observations have been published from MFB1. This is not unexpected, as BepiColombo's magnetic field measurements during the flyby were limited by the cruise configuration of the mission (Baumjohann et al., 2020;Heyner et al., 2021). BepiColombo flybys and orbit phase beginning in 2026 provide unique new possibilities to investigate ULF waves at Mercury (Mangano et al., 2021;Milillo et al., 2020).
In summary, we show hybrid simulation results on ULF waves generated in the Hermean magnetosphere with ∼2-s period consistent with observations by the Mariner 10 and MESSENGER missions. We show that the waves were generated on the hemisphere with direct connection to the IMF through the foreshock region, pointing to the foreshock region as a potential source for the wave activity.

Data Availability Statement
Hybrid simulations were performed using the RHybrid simulation platform, which is available under an open-source license by the Finnish Meteorological Institute https://doi.org/10.5281/zenodo.7391464. The work was supported by the Academy of Finland (Decision 310444). J. A. Slavin's contributions were supported by NASA Grant 80NSSC21K005. E. Kallio and R. Jarvinen thank the ISSI (International Space Science Institute) and ISSI-BJ (International Space Science Institute Beijing) team "Dayside Transient Phenomena and Their Impact on the Magnetosphere-Ionosphere" for discussions and suggestions. Figures 1, 3, and 4 were created using the ParaView open-source visualization tool. We acknowledge the computational resources provided by the Aalto Science-IT project.