MESSENGER Observations of Standing Whistler Waves Upstream of Mercury's Bow Shock

This paper reports on the standing whistler waves upstream of Mercury's quasi‐perpendicular bow shock. Using MESSENGER's magnetometer data, 36 wave events were identified during interplanetary coronal mass ejections (ICMEs). These elliptic or circular polarized waves were characterized by: (a) a constant phase with respect to the shock, (b) propagation along the normal direction to the shock surface, and (c) rapid damping over a few wave periods. We inferred the speed of Mercury's bow shock as ∼26 km/s and a shock width of 1.87 ion inertial length. These events were observed in 20% of the MESSENGER orbits during ICMEs. We conclude that standing whistler wave generations at Mercury are generic to ICME impacts and the low Alfvén Mach number (MA) collisionless shock, and are not affected by the absolute dimensions of the bow shock. Our results further support the theory that these waves are generated by the current in the shock.

• First survey of standing whistler waves upstream of Mercury's bow shock • Standing whistler waves are common at Mercury during interplanetary coronal mass ejections • Our results support the theory that current in shock generates standing whistler waves

Supporting Information:
Supporting Information may be found in the online version of this article.
Mercury has a miniature and weak bow shock, which is created by the interaction of low Mach number solar wind and a relatively small planetary magnetosphere in the inner heliosphere. The average bow shock subsolar distance has been determined to be only ∼2 R M (radius of Mercury, 1 R M = 2,440 km), which is approximately 1-2 orders smaller than that of the Earth . The "1 Hz" whistler waves have been commonly observed upstream of the bow shock of Mercury (Fairfield & Behannon, 1976;Le et al., 2013), where they propagate along the magnetic field and farther upstream (∼30,000 km). Although phase standing whistler waves have been observed at Mercury, they have not yet been analyzed (Gedalin et al., 2022).
Due to the nature of close-in orbit, there is higher probability for observing low M A shocks at Mercury than at other planets. The typical M A at Mercury orbit is ∼4-6 (Slavin & Holzer, 1981). Especially during ICMEs, the M A can be less than 3 (Liu et al., 2005;Sarantos & Slavin, 2009). The ICME's impact on Mercury's magnetosphere was first analyzed by Slavin et al. (2014). They showed that Mercury's dayside magnetosphere is highly dynamic and greatly compressed by ICME impacts. The bow shock and magnetopause locations during the impact of ICMEs deviate greatly from normal conditions (Slavin et al., 2014;Winslow et al., 2015Winslow et al., , 2017, and the dayside magnetosphere may even occasionally disappear (Slavin et al., 2019;Winslow et al., 2020). MESSEN-GER orbited Mercury during the maximum of solar cycle 24. Over the 4-year mission from February 2011 to April 2015, a total of 69 ICMEs were detected by MESSENGER (Winslow et al., 2015(Winslow et al., , 2017. We use the 69 ICMEs to study standing whistler waves.
Here we report the MESSENGER observations of the standing whistler wave upstream of Mercury's bow shock during ICMEs collected by Winslow et al. (2015Winslow et al. ( , 2017. Among 69 ICMEs, we identified 36 standing whistler wave events, which correspond to at least 20% of the orbits. Our results suggest that Mercury is a natural plasma laboratory to understand the physics of standing whistler waves and low M A collisionless shocks. It is likely that our understanding of such low Mach number shocks will be greatly advanced by the measurements to be collected by the upcoming Bepi-Colombo mission.

Case Analysis of Standing Whistler Wave
The dynamics of the dayside magnetosphere and magnetotail response to an ICME observed by MESSENGER on 23 November 2011, have been analyzed in detail by Slavin et al. (2014) and Zhong et al. (2020), respectively. This study analyzes its effects on the bow shock. Figures 1a-1d show an overview of MESSENGER's bow shock crossings during this ICME. High-resolution magnetic field data (20 vectors s −1 ) obtained from the magnetometer (MAG; Anderson et al., 2007) were used and displayed in the aberrated Mercury solar magnetic (MSM) coordinates. The MSM coordinate system was centered on the offset internal dipole of Mercury (Anderson et al., 2011), wherein the X-axis was pointed toward the Sun, the Y-axis was pointed in the opposite direction of the orbit motion, and the Z-axis completed the right-handed system. The average radial solar wind speed of 700 km s −1 during the ICME was applied to correct for the aberration. The spacecraft crossed the bow shock thrice; the crossings are denoted as inbound crossing 1, outbound crossing, and inbound crossing 2 in Figure 1. The multiple crossings were caused by the forward and backward motion of the bow shock, possibly due to temporal variations in upstream IMF conditions.
The shock normal was determined using the magnetic coplanarity method (Lepping & Argentiero, 1971) that substitutes the average magnetic field upstream and downstream of the shock; it is expressed as = The shock normals for the inbound crossing 1, outbound crossing, and inbound crossing 2 were observed to be very close at ( The accompanying upstream waves are considered key features of these bow shock crossings. The polarizations and k vectors of these waves were obtained from the results of the minimum variance analysis (MVA) of the magnetic field within an interval between two dotted lines upstream (Sonnerup & Cahill, 1967). Using these, the direction of propagation for an assumed planar wave can be estimated. For inbound crossing 1, the small ratio of the maximum to intermediate eigenvalues λ 1 /λ 2 = 1.9 and the large ratio of the intermediate to minimum eigenvalues λ 2 /λ 3 = 7.8 suggest that the waves had relatively stable elliptic polarizations. The wave vector k corresponds to the minimum variance eigenvector e 3 (0.70, −0.11, −0.69), whereas the corresponding mean magnetic field (B 0 ) is directed out of the maximum-intermediate plane. The hodograms of the magnetic field for several wavelengths in the MVA coordinates are shown in Figure 1f. The gyration of the magnetic field with respect to B 0 indicates that the wave polarization was right-handed in the spacecraft coordinate frame (SCF). The angles between k and n (θ kn ) and k and B up (θ kB ) were 11.85° and 55.49°, respectively, wherein the small θ kn and large θ kB suggest that the wave propagated approximately along the shock normal direction rather than the magnetic field. The waves observed during the outbound crossing and inbound crossing 2 were also elliptically polarized (Figures 1f2 and 1f3), with θ kn = 23.18°, 2.82° and θ kB = 64.17°, 58.14°, respectively. Moreover, the polarization direction of the outbound crossing was opposite to that of the inbound crossing, wherein it was left-handed, which is consistent with the characteristics of standing whistler waves (Fairfield & Feldman, 1975;Mellott & Greenstadt, 1984).
Wavelet analysis was used to calculate the total power at each moment. Figure 1e shows the variations of the total power along n in the SCF. The function = 0 − ∕ 0 was fitted to the total power. For inbound crossing 1, the damping time (T 0 ) was 11.17 s, which was 5.36 times the wave period (T wave ), indicating rapid damping. The damping distance was 1.83 km along k and the normalized wave amplitude (δB wave /B up ) was 0.40. This rapid damping of waves was also observed during the outbound crossing and inbound crossing 2.
The power spectral density shown in Figure 1e demonstrates that these waves were mainly restricted to the plane perpendicular to e 3 , as indicated by the PSD 1 , and PSD 2 ≫ PSD 3 around the wave frequency (f sc ) in the SCF. The f sc for inbound crossing 1, outbound crossing, and inbound crossing 2 were ∼0.48, 1.01, and 0.19 Hz, respectively; the different values indicate the change in the relative velocity between the spacecraft and bow shock in the normal direction.

Statistical Results
We used 69 ICMEs (94 orbits) collected by Winslow et al. (2015Winslow et al. ( , 2017 to find bow shock crossings during ICMEs. As standing whistler waves typically occur upstream of the quasi-perpendicular bow shock, the θ Bn was calculated, wherein 486 quasi-perpendicular bow shock crossings (θ Bn > 45°) were identified to select the events. Multiple bow shock crossings are common during inbound or outbound crossings in each orbit owing to the up-and-down displacement of the shocks. MVA was performed on the magnetic field data upstream for each quasi-perpendicular shock crossing under the assumption that the eigenvalues conform to λ 1 /λ 2 < 2 and λ 2 / λ 3 > 7, which indicate that the waves are elliptically or circularly polarized. In all elliptically polarized waves, 36 perpendicular bow shock crossings with rapid damping were identified, including 20 inbound and 16 outbound crossings. They occurred during 19 orbits, with an orbital occurrence rate of ∼20%.
The characteristics of the wave during each event were observed (Supporting Information S1). A statistical analysis indicated the following: Wave polarization. Right-handed polarization was observed in all 20 upstream to downstream traversals, whereas left-handed polarization was observed in all 16 downstream to upstream traversals. These polarizations were consistent with the previous theory and observation of standing whistler waves presented by Perez and Northrop (1970) and Fairfield and Feldman (1975).
Propagation direction. The calculated θ kn ranged from ∼0° to 45°, while the θ kB ranged from ∼45° to 90° (Figure 2a). The mean θ kn and θ kB were 17.31° and 69.55°, respectively. These results suggest that the waves were propagating along the shock normal instead of the magnetic field. These results are also consistent with the observations at Earth (Mellott & Greenstadt, 1984); however, they are different from the propagating direction of "1 Hz" waves observed at Mercury (Fairfield & Behannon, 1976;Le et al., 2013).
Wave damping. Figure 2b shows the distance from the wave damping at e −1 to the ramp along n in the SCF, where the amplitudes of most waves damp at e −1 within 10 km. There is one outlier with the damping distance of ∼40 km. This was significantly less than the damping distance of the "1 Hz" wave within ∼30,000 km at Mercury (Le et al., 2013). The ratio of T 0 to T wave was nearly <10 (Figure 2c). The standing whistler waves on Earth also exhibit this rapid damping, which is <10 times the wave periods (Mellott & Greenstadt, 1984); hence, the damping mechanism can be considered Landau damping (Gary & Mellott, 1985).
Wave frequency. The mean frequency of standing whistler waves was 1.67 Hz in the SCF. Here, 70% of the events had a frequency <2 Hz (Figure 2d), suggesting that the standing whistler waves have a lower frequency than the "1 Hz" waves at Mercury (2-3 Hz, Russell, 2007).
Based on the statistical results of these events, it was discovered that their characteristics were similar to those of the standing whistler wave (Mellott & Greenstadt, 1984). Other properties (Figures 2e and 2f) of the waves are described in detail in Section 3.2.

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Considering that the solar wind beam observation of MESSENGER's fast imaging plasma spectrometer was obstructed by the sunshade, the upstream solar wind M A could not be derived directly. The M A was estimated using the following formula: M A − 1 = (B down /B up − 1) sin 2 θ Bn , which is suitable for low Mach number and low-β shocks (Balikhin et al., 2008). The magnetic field compression ratio (B down /B up : magnetic field intensity (average value of ∼1 min) ratio of downstream to upstream) of all perpendicular bow shock crossings during ICMEs are plotted in Figure 3a, wherein it can be observed that most of the identified wave events (blue) had lower magnetic field compression ratios than the mean ratio during ICMEs. The distributions of the calculated M A and wave event occurrence rates are plotted in Figure 3b. Eighty-nine percent of the wave events have lower M A values than the average value during ICMEs. The occurrence rate was 11.6% when the M A was below average and 1.9% when the M A was above average. Therefore, these waves have a high likelihood of occurrence under a relatively lower M A .

Discussion
On Earth, the theoretical wavelengths of standing whistler waves are consistent with those observed by Mellott and Greenstadt (1984). Single spacecraft observations are normally used to infer the bow shock speed (Fairfield & Feldman, 1975) based on theoretical predictions of the wavelength (Tidman & Krall, 1971): = 2 ccos Bn pi (MA 2 − 1) 1∕2 , where pi is the proton plasma frequency. By applying the typical values from the ICME model at 0.38 AU (Liu et al., 2005), pi = 10034rad∕s, mean MA = 1.55 , and mean Bn = 68 • of all wave events, the theoretically predicted wavelength of = ∼59 km was calculated. The average T wave was ∼2.26 s; hence, the shock speed (λ/T wave ) can be inferred as ∼26 km/s. Notably, this was slightly less than the shock speed of ∼40 km/s estimated through overshoot observations under normal conditions (Masters et al., 2013).
The shock ramp scale was also estimated using the scale relationship between the standing whistler waves and the shock ramps. A shock ramp scale of 56 km was obtained using the formula × T ramp /T wave (59 km × 0.95). Considering an ion inertial length ( ∕ pi ) of 30 km, the width of the ramp was 1.87 ∕ pi . Based on the results of Hobara et al. (2010), the scale can be larger than 1 ∕ pi when M A is low. Previous theories have suggested that standing whistler waves are generated by a stable current in the shock ramp, from which the formula for the wave amplitude can be derived (Tidman & Krall, 1971). This suggests that δB wave / B up has a positive correlation with cosθ Bn , and this relationship is demonstrated in Figure 2f. The best linear fit produced Y = [0.65 ± 0.15]X − [0.01 ± 0.06], which was also consistent with this theory. Based on the fitted values, the maximum amplitude of the standing whistler wave was approximately 0.8 times the intensity of the background magnetic field.
The shock is hypothesized to be the largest-amplitude cycle of the upstream standing whistler wave, wherein its width is half of the wavelength (Farris et al., 1993;Goncharov et al., 2014;Mellott & Greenstadt, 1984). However, this results in conflicting ratios of the standing whistler wave wavelength to the shock thickness at Earth and interplanetary shock, as some researchers have estimated this ratio to be two (Goncharov et al., 2014) while others have estimated it to be closer to one (Farris et al., 1993;Mellott & Greenstadt, 1984). In the SCF, the ratio between the period (T wave ) of the upstream whistler waves and the shock ramp crossing time (T ramp ) can be a good approximation of the shock width to wavelength ratio. Figure 2e shows the T wave /T ramp in the spacecraft frame of the upstream standing whistler waves of Mercury. In cases where almost all T wave was less than 2 × T ramp , the average ratio of the two was 0.95, indicating that the initial hypothesis must be reexamined to further determine the scale of the relationship between standing whistler waves and shock ramps.

Conclusions
In this study, we reported and statistically analyzed the standing whistler waves upstream of the bow shock of Mercury during ICMEs. These waves occur at lower M A and propagate along the normal of the bow shock. However, in a few cases, the propagate angle θ kn can reach 45°. At this stage, we cannot determine whether this is a natural θ kn distribution characteristic or from the data processing methods. It was observed that, similar to the waves at Earth, these waves were rapidly damping with a few wave periods; however, the damping distance in SCF was significantly shorter, only a few kilometers upstream of the bow shock of Mercury. Our results support the theory that these waves are generated by the current in the shock and that the shock is not the largest-amplitude cycle of the waves. Hence, the generation of standing whistler waves was determined to be generic to the low Mach number collisionless shock. Additionally, a high occurrence rate of the standing whistler waves observed during ICMEs suggests that the bow shock of Mercury can be a natural plasma laboratory that can be used to further study low M A planetary shocks. Considering that BepiColombo will arrive at Mercury in 2025 during the ascending and maximum phases of solar cycle 25, it is expected to encounter a large number of ICMEs. This study provides an understanding of standing whistler wave generation and its underlying physics, which can be used for the upcoming high-resolution BepiColombo observations.

Data Availability Statement
The MESSENGER MAG data used in this study are available at NASA's Planetary Data System: https:// pds-ppi.igpp.ucla.edu/search/view/?f=yes&id=pds://PPI/mess-mag-calibrated/data/mso; The list of the identified standing whistler wave events is publicly available via NSSDC Space Science Article Data Repository (https://doi.org/10.57760/sciencedb.space.00591).