Statistical study of magnetic holes in the upstream region of Mercury’s bow shock

1.   Introduction

Magnetic holes with a size of several to tens of ion gyroradii, first reported by Turner et al. (1977), are a kind of magnetic depression structures (Stevens and Kasper, 2007). They exist widely in many astrophysical plasma environments, such as the solar wind (Stevens and Kasper, 2007; Madanian et al., 2020; Potapov, 2020; Volwerk et al., 2021), planetary magnetosheaths (Soucek et al., 2008; Balikhin et al., 2009; Génot et al., 2009; Volwerk et al., 2016; Karlsson et al., 2021; Duanmu XY et al., 2023), and magnetospheres (Wang GQ et al., 2016, 2020a). Magnetic holes can be generated by ion mirror instabilities (Hasegawa, 1969; Tsurutani et al., 2011). In the solar wind, their occurrence rate decreases with an increase in the heliocentric distance from ~0.3 to 8.9 astronomical units (AU), suggesting that they might originate from somewhere within 0.3 AU (Russell et al., 2008).

In the last decade, magnetic holes with a size of ~1 ρ_i (ion gyroradius) or less have been found to be a common phenomenon existing in the terrestrial current sheet (Ge YS et al., 2011; Gershman et al., 2016; Goodrich et al., 2016; Yao ST et al., 2016, 2021; Shustov et al., 2020), planetary magnetosheaths (Huang SY et al., 2017; Liu H et al., 2019; Yao ST et al., 2019a; Goodrich et al., 2021; Wu MY et al., 2021; Chen YJ et al., 2022), and the upstream region of the terrestrial and Martian bow shocks (Yao ST et al., 2019b; Wang GQ et al., 2020b, d, 2021b). These sub-ion-scale structures can trap electrons with an energy up to a few 10⁵ eV (Balikhin et al., 2012; Sun WJ et al., 2012). In addition, electron cyclotron waves, whistler mode waves (or lion roars), and electrostatic solitary waves are able to be generated inside them in the magnetosheath (Maksimovic et al., 2001; Huang SY et al., 2018, 2019; Sun JC et al., 2019; Yao ST et al., 2019a). Thus, they might play an important role in transporting electrons (Balikhin et al., 2012) or dissipating energy (Huang SY et al., 2017). Several generation mechanisms can explain the formation of sub-ion-scale magnetic holes, such as electron mirror instabilities (Sundberg et al., 2015; Yao ST et al., 2019b), tearing mode instabilities (Balikhin et al., 2012), the electron magnetohydrodynamics soliton model (Ji XF et al., 2014; Li ZY et al., 2016), electron vortex magnetic holes (Haynes et al., 2015), and ballooning/interchange instabilities (Shustov et al., 2019).

On the basis of observations by the MAVEN (Mars Atmosphere and Volatile EvolutioN) spacecraft, Wang GQ et al. (2021a) statistically studied the magnetic holes with durations of 0.1–100 s in the upstream region of the Martian bow shock. According to the distribution of their durations, the authors found that these magnetic holes could be divided into two groups with a duration of 0.1–2 s and 2–100 s, respectively. For each group, the durations of the magnetic holes approximately obeyed the log-normal distribution. And the magnetic holes with durations of 0.1–2 s were suggested to be sub-ion scale (Wang GQ et al., 2021a). An abundance of electron and ion dynamics can be created during the interaction between the solar wind and bow shock (Eastman et al., 1981; Hao YF et al., 2016, 2018). The foreshock is a region magnetically connected to the bow shock (Greenstadt et al., 1980; Eastman et al., 1981; Hao YF et al., 2023a, b) and is filled with backstreaming particles, which provide free energy to create waves or excite instabilities (Sentman et al., 1983; Tanaka et al., 1983; Archer et al., 2005; Shan LC et al., 2018; Hao YF et al., 2023c). Both the terrestrial and Martian foreshocks are suggested to be sources of the sub-ion-scale magnetic holes, based on their occurrence rate in the upstream region of the terrestrial and Martian bow shocks (Wang GQ et al., 2020c, 2021a).

Mercury, the closest planet to the Sun, has a global intrinsic magnetic field (Diego et al., 2020; Jarvinen et al., 2020). The magnetic holes with durations of >1 s have been statistically studied based on observations by the MESSENGER (MErcury Surface, Space ENvironment, GEochemistry, and Ranging) spacecraft (Volwerk et al., 2020; Karlsson et al., 2021). Their sizes follow a log-normal distribution (Karlsson et al., 2021), and their occurrence rate is ~3.4 per day (Volwerk et al., 2020). Karlsson et al. (2021) further found that their ambient magnetic field strength is generally lower than the average interplanetary magnetic field (IMF) strength, indicating that they might be more likely to occur in the high-β region. The authors also found that the properties of the magnetic holes in Mercury’s magnetosheath are similar to those in the solar wind; thus, they argued that the magnetic holes in the magnetosheath have a solar wind origin.

In this study, we statistically investigate the magnetic holes in the solar wind at Mercury’s orbit based on observations by the MESSENGER spacecraft. We show that a new group of magnetic holes with durations of 0.1–0.6 s exist in the upstream region of Mercury’s bow shock.

2.   Observation

2.1   Selection of Magnetic Holes

The MESSENGER spacecraft, launched on August 3, 2004, was inserted into a high-eccentricity and ~80° inclination orbit around Mercury in March 2011 (Solomon et al., 2007). In the present study, we use the magnetic field data sampled at 20 Hz (i.e., with a time resolution of 0.05 s) measured by the magnetometer instrument (Anderson et al., 2007) from January 1 to December 31, 2012, to investigate the magnetic holes in the solar wind at Mercury’s orbit. Philpott et al. (2020) have provided Mercury’s magnetopause crossings and bow shock crossings for the MESSENGER spacecraft based on the procedure outlined by Winslow et al. (2013). We obtain the time intervals in the solar wind from Mercury’s bow shock crossings listed in the Supplement of Philpott et al. (2020). In total, 221.6 days of data in the solar wind are available. Figure 1 shows the orbit coverage of the MESSENGER spacecraft in the X–Y and X–Z planes in the Mercury solar orbital (MSO) coordinate system. The color denotes the dwell time of MESSENGER in each grid with a grid size of 0.2 R_M (Mercury radius). Figure 1a shows that the data at Y_MSO < 0 is more than that at Y_MSO > 0. Figure 1b shows that the orbit coverage of the MESSENGER spacecraft is mainly at Z_MSO < 0.

Figure  1.  Orbit coverage of the MESSENGER spacecraft in the solar wind in the X_MSO–Y_MSO (a) and X_MSO–Z_MSO (b) planes from January 1 to December 31, 2012. The grid size is 0.2 R_M.

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The selection of magnetic holes generally requires consideration of the ambient magnetic field (Zhang TL et al., 2008; Sun WJ et al., 2012). Because the duration of the magnetic hole in the solar wind has a wide range of values from <1 s to tens of seconds (Zhang TL et al., 2008; Wang GQ et al., 2021c) and the IMF is time variant, the magnetic field data low-pass filtered with the same cut-off frequency is no longer reasonable as the ambient magnetic field. In other words, it is difficult to identify both the small and large durations of the magnetic holes in the solar wind when using the same temporal window. Therefore, we use different cut-off frequencies to obtain the ambient magnetic field to select the magnetic holes with different ranges of durations.

We set temporal windows of 2, 10, 60, and 300 s to select the magnetic holes with a duration of 0–1 s, 1–5 s, 5–30 s, and 30–150 s, respectively. We first search out the local minima of the magnetic field strength |B|. For each minimum |B| at T₀, the starting and ending times of the temporal window are set to be T₀ – 0.5T_win and T₀ + 0.5T_win, respectively, where T_win is 2, 10, 60, or 300 s. We then determine the time of the boundary for each event by using the method of Wang GQ et al. (2021c). For a given temporal window T_win, we can obtain the ambient magnetic field strength B_sm and the standard deviation σ of |B| – B_sm, where B_sm is the |B| smoothed with a temporal window of T_win. We select the time with |B| = B_sm + 0.5σ closest to T₀ from the left (right) as the time of the left (right) boundary of the potential magnetic hole. If the times of the left and right boundaries are not found simultaneously, we then select the time with |B| = B_sm and |B| = B_sm − 0.5σ as the second and third choices to find the times of the left and right boundaries, respectively. Figure 2 shows an example of the selection of the magnetic holes with a minimum |B| at 08:58:55.6 universal time (UT) on March 2, 2012. The use of the |B| smoothed with a 2-s boxcar filter as the ambient magnetic field is suitable for this event. And the times where |B| = B_sm + 0.5σ closest to 08:58:55.6 UT are selected as the boundary times of this magnetic hole.

Figure  2.  An example of the selection of magnetic holes. B_sm is the |B| smoothed with a 2-s boxcar filter, and σ is the standard deviation of |B| – B_sm between 08:58:54.6 and 08:58:56.6 UT on March 2, 2012. The black curve denotes |B|. The blue, green, and red dashed lines denote B_sm + 0.5σ, B_sm, and B_sm − 0.5σ, respectively.

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After obtaining the time of the boundaries, the potential magnetic hole is required to meet the following criteria: (1) B_min/B_edge < 0.85, where B_min is the minimum |B| between the two boundaries of the event, and B_edge is the average |B| at the two boundaries; (2) B_edge − B_min > 0.5 nT; (3) ω < 15°, where ω is the angle between the magnetic field vectors at the two boundaries. The value 0.5 nT of the criterion B_edge − B_min is an empirical value to reduce the effect of the magnetic field noise, such as the variable spacecraft field (Anderson et al., 2007). We visually check all the automatically selected events to select the unambiguous magnetic holes. Finally, 2656 magnetic holes are selected for 1 year of MESSENGER observations.

2.2   Durations and $B_{{\mathrm{min}}}$/$B_{{\mathrm{edge}}}$ of Magnetic Holes

Figure 3 shows histograms of the durations of our selected magnetic holes, which can be clearly divided into two groups: 0.1–0.6 s (orange) and 0.6–100 s (blue). We classify the 757 magnetic holes with durations of <0.6 s and 1899 magnetic holes with durations of >0.6 s as the small and large durations of the magnetic holes (abbreviated as SDMHs and LDMHs), respectively. The durations of the LDMHs approximately obey a log-normal distribution, expressed by

$f\left(\mathrm{l}\mathrm{g}\left(x\right)\right)=\dfrac{1}{\sqrt{2\mathrm{\pi }}\sigma }\mathrm{e}\mathrm{x}\mathrm{p}\left(-\dfrac{{(\mathrm{l}\mathrm{g}(x)-\mu )}^{2}}{2{\sigma }^{2}}\right)$,

where x denotes the duration of the magnetic hole, σ is the standard deviation, and μ is the median of the log-normal distribution. For the SDMHs, their durations roughly obey a log-normal distribution, but not convincingly. One possible reason is that the resolution of the magnetic field data at 20 Hz is not high enough, leading to a gap between some adjacent orange histograms, as shown in Figure 3. For the SDMHs and LDMHs, μ is −0.61 and 0.48, respectively. Thus, the median durations of the SDMHs and LDMHs are expected to be ~0.25 and 3 s, respectively.

Figure  3.  Histograms of the durations of the SDMHs (orange) and LDMHs (blue). The red and blue curves denote the fitted log-normal distributions for the orange and blue histograms, respectively. The corresponding median μ and standard deviation σ of the fitted log-normal distribution are also given. The median duration of the magnetic holes for the orange distribution is approximately 0.25 s and that for the blue distribution is approximately 3 s.

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Figure 4a shows the histograms of B_min/B_edge for the SDMHs (orange) and LDMHs (blue). The B_min/B_edge is >0.5 for 98.3% of the SDMHs, and the proportion of the magnetic holes tends to be larger when B_min/B_edge increases from 0.5 to 0.85. For the LDMHs, 32.6% of the magnetic holes have a B_min/B_edge value of <0.5, and their proportion tends to gradually increase with the increase in B_min/B_edge from 0 to 0.85. Figure 4b shows that the values of B_edge – B_min for 256 (425) out of 757 SDMHs are in the range of 1–2 (2–4) nT. The values of B_edge – B_min are much larger for the LDMHs than the SDMHs overall.

Figure  4.  Histograms of B_min/B_edge (a) and B_edge – B_min (b) for the SDMHs (orange) and LDMHs (blue). B_edge is the average |B| at the two boundaries of the magnetic hole, and B_min is the minimum |B| inside the magnetic hole.

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2.3   Polarization

The minimum variance analysis (MVA; Sonnerup and Scheible, 1998) can be used to investigate the polarization of waves or magnetic holes (e.g., Sonnerup and Cahill, 1967; Balikhin et al., 2012). We transform the magnetic field from the MSO into the LMN coordinate system by MVA using the data during each magnetic hole, where the L, M, and N axes denote the maximum, intermediate, and minimum variance directions, respectively; the three eigenvalues are λ₁, λ₂, and λ₃. If λ₂ and λ₃ are small in comparison to λ₁ and are close to each other, the polarization is linear (e.g., Balikhin et al., 2012).

Figure 5a shows a SDMH between 14:35:59.4 and 14:35:59.65 UT on September 18, 2012. During this structure, B_L shows a decrease of ~2 nT, whereas B_M and B_N show little change. For this magnetic hole, λ₁, λ₂, and λ₃ are 5.54, 0.0588, and 0.0459 nT², respectively, and λ₁/λ₂ is ~94, much larger than 1. The bottom panels in Figure 5a show the plots of B_M versus B_L, B_N versus B_L, and B_N versus B_M during the magnetic hole, which illustrates that this structure has linear polarization. Figure 5b shows another SDMH between 20:52:49.02 and 20:52:49.38 UT on March 20, 2012. Its three eigenvalues are 0.604, 0.241, and 0.00358 nT², respectively. In contrast to the SDMH in Figure 5a, this SDMH has a bipolar variation of B_M. The plot of B_M versus B_L indicates that this SDMH has elliptical polarization.

Figure  5.  The magnetic field in LMN and the plots of B_M versus B_L, B_N versus B_L, and B_N versus B_M during the magnetic hole between 14:35:59.4 and 14:35:59.65 UT on September 18, 2012 (a) and between 20:52:49.02 and 20:52:49.38 UT on March 20, 2012 (b).

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Figure 6a shows a LDMH with a duration of ~14 s observed at ~11:04:15 UT on August 7, 2012. Its duration is much larger than that of the magnetic hole (~0.3 s) in Figure 5. The magnetic field depression is mainly in B_L during the magnetic hole, whereas B_M and B_N have high-frequency fluctuations with an amplitude of <1 nT. The eigenvalues λ₁, λ₂, and λ₃ are 5.55, 0.059, and 0.046 nT², respectively. And the plots of B_M versus B_L, B_N versus B_L, and B_N versus B_M during this magnetic hole indicate that this structure has linear or quasi-linear polarization, although λ₂ is not small enough in comparison to λ₁. Figure 6b shows another LDMH with a duration of ~2.7 s, for which λ₁/λ₂ is ~6.76. In contrast to the LDMH in Figure 6a, B_M shows a significant change, with a temporal scale of 10 s. The plot of B_M versus B_L indicates that this structure has elliptical polarization.

Figure  6.  The magnetic field in LMN and the plots of B_M versus B_L, B_N versus B_L, and B_N versus B_M during the magnetic hole between 11:04:09 and 11:04:23 UT on August 7, 2012 (a) and between 00:01:01.61 and 00:01:04.27 UT on May 27, 2012 (b).

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Figures 7a and 7b show the distributions of the magnetic holes with different values of λ₁/λ₂ and λ₂/λ₃ for the SDMHs and LDMHs. For each magnetic hole, the eigenvalues are determined by MVA using the magnetic field data for the entire interval of this magnetic hole. For the SDMHs, 38.3% of the magnetic holes have λ₂/λ₃ < 10. In contrast, Figure 7b shows that 79.8% of the LDMHs have λ₂/λ₃ < 10. The medians of λ₂/λ₃ are ~14.7 and 4.3 for the SDMHs and LDMHs, respectively. Approximately 8.9% (3.4%) of λ₁/λ₂ is in the range of 1–5 for the SDMHs (LDMHs), and ~17.8% (12.4%) of λ₁/λ₂ is in the range of 5–10. This indicates that the polarization of both SDMHs and LDMHs can be elliptical, quasi-linear, or linear. The distribution of λ₂/λ₃ for the SDMHs is clearly different from that of the LDMHs, as shown in Figures 7c and 7d. Such a large difference may have been overestimated because the sampling points can affect the accuracy of MVA (Khrabrov and Sonnerup, 1998).

Figure  7.  Distributions of the number of magnetic holes with different values of λ₁/λ₂ and λ₂/λ₃ for the SDMHs (a) and LDMHs (b), respectively. Histograms of λ₁/λ₂ (red) and λ₂/λ₃ (black) for the SDMHs (c) and LDMHs (d), respectively.

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2.4   Occurrence Rate

Figure 8a shows the orbit coverage of the MESSENGER spacecraft in the solar wind during the time intervals we used in the X–R_YZ plane, where R_YZ = $\sqrt{{Y}_{\mathrm{M}\mathrm{S}\mathrm{O}}^{2}+{Z}_{\mathrm{M}\mathrm{S}\mathrm{O}}^{2}}$. The grid size is 0.5 R_M. We require the amount of data in each grid to be at least 5 h. The average bow shock position under a solar wind ramp pressure of 14.3 nPa (Winslow et al., 2013) is also given in Figure 8. The bow shock boundary changes with the variation in the solar wind ramp pressure as well as the Alfvén Mach number. For example, its subsolar standoff distance varies from 2.29 to 1.89 R_M when the Alfvén Mach number changes from 4.12 to 11.8 (Winslow et al., 2013). Nevertheless, the bow shock shape remains unchanged when the solar wind ramp pressure or the Alfvén Mach number changes (Winslow et al., 2013).

Figure  8.  Orbit coverage of the MESSENGER spacecraft (a) and the occurrence rates of the SDMH (b) and LDMH (c) events in the upstream region of Mercury’s bow shock in the X–R_YZ plane. The gray curve denotes the average bow shock position under a solar wind ramp pressure of 14.3 nPa, as determined by the model of Winslow et al. (2013).

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Magnetic holes are often observed successively within several minutes, and some are observed in isolation during a long period (see Zhang TL et al., 2008). Here, two adjacent magnetic holes in each group are considered to belong to the same train of magnetic holes if they are observed within 300 s (see Zhang TL et al., 2008). A single magnetic hole or a train of magnetic holes is regarded as a magnetic hole event, and we focus on the occurrence rate of the magnetic hole event instead of each single magnetic hole. In total, we obtain 160 SDMH and 862 LDMH events, respectively. The average position during the entire event is considered the position of this event. Figure 8b shows the occurrence rates of the SDMH events in the X–R_YZ plane. The black grid indicates that no events are observed in the corresponding region. The occurrence rates are mainly in the range of 0.3–5.5 events per day, and they tend to be larger when closer to the bow shock. Figure 8c shows the occurrence rates of the LDMH events in the X–R_YZ plane, which are mainly in the range of 0.6–16.6 events per day. Several local maxima of the occurrence rates generally occur at the bins with <10 h of data. Except for these local maxima, the distribution of the occurrence rates is approximately uniform.

To check whether Mercury’s foreshock can affect the occurrence of the magnetic hole events, we attempt to determine the relation between the direction of the magnetic field line and the occurrence of the magnetic hole events. For the same magnetic field line, the direction of the magnetic field vector B has two possibilities. Thus, we set B₁ (= [B_X1, B_Y1, B_Z1]) = B if the X_MSO component of B is positive, otherwise B₁ = 0 – B. An abundance of waves are in the frequency range of 0.1–4 Hz in the foreshock (Jarvinen et al., 2020; Romanelli et al., 2020). To reduce the effect of these high-frequency waves, the magnetic field data have been smoothed with a 20-s boxcar filter before determining the IMF direction.

Figure 9a shows the amount of data in the magnetic field we used at different θ and ϕ for the data measured by MESSENGER at Y_MSO < −1 R_M, where

θ = $\dfrac{\pi }{2}$ – arctan(B_Y1/B_X1),

and

ϕ = arctan(B_Z1/$\sqrt{{B}_{X1}^{2}+{B}_{Y1}^{2}}$).

Here, θ = 0 (180°) and ϕ = 0 indicate that the magnetic field vector points to the positive (negative) Y axis. The step lengths of both θ and ϕ are 45°. The distribution of the amount of data is symmetrical with respect to ϕ = 0. The maximum value is at 90° < θ < 135° and −45° < ϕ < 45°, which meets the expectation of the Parker spiral configuration (Meyer-Vernet, 2007; Chang et al., 2019). Figure 9b shows the distribution of the magnetic field data in the θ–ϕ plane for the data measured at Y_MSO > 1 R_M, whose characteristics are the same as those in Figure 9a except that the overall amount of data in each grid in Figure 9b is relatively smaller.

Figure  9.  The distribution of the magnetic field data at Y_MSO < −1 R_M (a) and Y_MSO > 1 R_M (b) in the θ–ϕ plane.

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Forty-eight and 17 SDMH events are observed at Y_MSO < −1 R_M and Y_MSO > 1 R_M, respectively. Figure 10a shows the occurrence rate of the SDMH events at Y_MSO < −1 R_M in the θ–ϕ plane. The occurrence rate is ~1.92 events per day in the bin with 90° < θ < 135° and −45° < ϕ < 0°, much larger than that of other bins with a value of ~0.26–1.1 events per day. In contrast, Figure 10b shows that the maximum occurrence rate (~2.03 events per day) is in the bin with 45° < θ < 90° and −45° < ϕ < 0°, and it is much larger than that in other bins, with a value of 0–0.58 events per day. At Y_MSO < −1 (Y_MSO > 1) R_M, the spacecraft has a higher probability of being located in the foreshock when ϕ < 0 and 90° < θ < 135° (45° < θ < 90°) than that when ϕ < 0 and other values of θ. This result suggests that the SDMH events are more likely to be observed in Mercury’s foreshock. Contrary to the SDMH events, Figure 10c show that the occurrence rate is the minimum at 90° < θ < 135° and −45° < ϕ < 0°, suggesting that the LDMH events have a smaller occurrence rate in Mercury’s foreshock.

Figure  10.  (a, b) Occurrence rates of the SDMH events observed at Y_MSO < −1 R_M and Y_MSO > 1 R_M in the θ–ϕ plane. (c, d) Occurrence rates of the LDMH events observed at Y_MSO < −1 R_M and Y_MSO > 1 R_M in the θ–ϕ plane.

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In the solar wind at Mercury’s orbit, the magnetic holes with durations of >1 s are more likely to be observed when the IMF |B| is lower than the average strength (Karlsson et al., 2021), indicating that the IMF |B| might affect the occurrence of these magnetic holes. Figure 11a shows the amount of magnetic field data at different IMF |B| with a step length of ∆B = 2 nT. We require the amount of data in each bin to be at least 1 d. If the duration of an event is >6 min, the average |B| during this event is considered the ambient magnetic field strength. Otherwise, the average |B| during the interval 3 min before and after the central moment of this event is considered the ambient magnetic field strength of the magnetic hole event. Figure 11b shows the occurrence rates of the SDMH (blue) and LDMH (orange) events. For both groups, the occurrence rates tend to decrease with an increase in |B| until it is ~16 nT, and they are small when |B| > 16 nT, suggesting that the magnetic hole events in both groups are more likely to be observed when the IMF |B| is weak compared with the average IMF |B|.

Figure  11.  (a) Amount of magnetic field data in the solar wind we used at different IMF |B| with a step length of ∆B = 2 nT. (b) Occurrence rates of the SDMH (blue) and LDMH (orange) events at different IMF |B|.

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3.   Discussion

An abundance of magnetic holes are in the solar wind at Mercury’s orbit (Russell et al., 2008; Volwerk et al., 2020; Karlsson et al., 2021). On the basis of MESSENGER data, Volwerk et al. (2020) and Karlsson et al. (2021) statistically studied the properties of these magnetic holes with durations of >1 s. However, the short-duration (<1 s) magnetic holes have not attracted much attention. We analyze the properties of the magnetic holes with durations of 0.1–100 s in the upstream region of Mercury’s bow shock. We find it interesting that these magnetic holes can be divided into two groups according to the distribution of their durations (see Figure 3). The SDMHs and LDMHs have a duration of 0.1–0.6 s and 0.6–100 s, respectively. One must note that there is no clear boundary between the durations of the SDMHs and LDMHs, as shown in Figure 3. Such a distribution of the magnetic holes is similar to that upstream of the Martian bow shock (Wang GQ et al., 2021c). Wang GQ et al. (2021c) found that the group of magnetic holes with small durations are sub-ion scale. We speculate that the SDMHs are also sub-ion scale, and the observational duration of 0.6 s might be the boundary between typical sub-ion- and ion-scale magnetic holes.

In the solar wind at Mars’ orbit, the group of magnetic holes with large (small) durations is suggested to be ion (sub-ion) scale (Wang GQ et al., 2021c). Unfortunately, we are unable to determine the sizes of our selected SDMHs and LDMHs because of the lack of plasma data. The SDMHs and LDMHs show some different properties, such as the distributions of B_min/B_edge and λ₂/λ₃. In contrast to the previous observations in the solar wind at Mercury’s orbit, our results suggest that the SDMHs are a new group of magnetic holes.

We use 221.6 d of the data in the solar wind to select 160 SDMH events and 862 LDMH events, respectively. The average occurrence rates are ~0.72 and 3.89 events per day for the SDMH and LDMH events, respectively. This result suggests that the occurrence rate of the LDMH events is much larger than that of the SDMH events. For the SDMH events, the maximum occurrence rate occurs when 90° < θ < 135° and −45° < ϕ < 0° at Y_MSO < −1 R_M, and it occurs when 45° < θ < 90° and −45° < ϕ < 0° at Y_MSO > 1 R_M. As shown in Figures 10a and 10b, both maximum occurrence rates are much larger than the values in other bins. Considering the orbit of the MESSENGER spacecraft, the IMF with 90° < θ < 135° (45° < θ < 90) and −45° < ϕ < 0° might have a higher probability of being connected to the bow shock than other bins when the spacecraft is at Y_MSO < −1 (Y_MSO > 1) R_M. This suggests that the SDMH events are more likely to be observed in Mercury’s foreshock. In contrast, the occurrence rate in Figures 10c is minimum at the bins where the occurrence rate is maximum for the SDMH events. We speculate that the LDMHs originate from the upstream solar wind and that the foreshock can destroy some of these magnetic holes, leading to the decrease in their occurrence rate in the foreshock.

The occurrence rates of the magnetic hole events in both groups are significantly affected by the IMF strength (see Figure 11). The lower the magnetic field intensity, the higher the occurrence rate of the magnetic hole event. This finding indicates that the magnetic holes in both groups are more likely to occur in the high-β plasma, which is consistent with the findings of Karlsson et al. (2021). Thus, mirror instabilities are a potential generation mechanism of these magnetic holes, or the mirror unstable condition contributes to their formation. The foreshock has different impacts on the occurrence rate of the SDMH and LDMH events. If the foreshock is one source of the SDMHs, the plasma in Mercury’s foreshock might provide some clues that would reveal their generation mechanism, which needs study in the future.

4.   Summary

We investigate magnetic holes with durations of 0.1–100 s in the upstream region of Mercury’s bow shock by using the magnetic field data from the MESSENGER spacecraft from January 1 to December 31, 2012. According to the distributions of their durations, the magnetic holes can be divided into two groups: smaller and larger than 0.6 s. The durations of both groups approximately obey a log-normal distribution, with a median of ~0.25 s and 3 s, respectively. Both groups have linear, quasi-linear, or elliptical polarization. Approximately 1.7% and 32.6% of the magnetic holes have a value of B_min/B_edge < 0.5 for the SDMHs and LDMHs, respectively. In addition, these two groups have different distributions of λ₂/λ₃, with a median of ~14.7 and 4.3 for the SDMHs and LDMHs, respectively.

For both groups, their occurrence rates tend to be smaller when the IMF strength is stronger until the IMF strength is up to 16 nT, and they are small when the IMF strength is >16 nT, indicating that they prefer to occur in the high-β plasma. The SDMH events have a much larger occurrence rate when the IMF has a higher probability of being connected to the bow shock, suggesting that Mercury’s foreshock might be a source of these magnetic holes. In contrast, the LDMH events have a smaller occurrence rate when the IMF has a higher probability of being connected to the bow shock, suggesting that some LDMHs could be destroyed in Mercury’s foreshock.