Unusually Low-frequency Whistler-mode Waves and Their Association with High-energy Protons Observed Upstream of Martian Bow Shock

Whistler-mode waves upstream of planetary bow shock are often referred to as “1-Hz waves” due to the center of their observed frequency range being at ∼1 Hz. A series of whistler-mode waves were observed upstream of the Martian bow shock by MAVEN on 2015 August 14, with unusually low frequencies centered at ∼0.4 Hz. These waves were accompanied (though not synchronized) by the significant flux enhancement of high-energy protons up to ∼10 keV. By analyzing the wave dispersion property and the wave–particle interaction condition, we find that the observed whistler-mode waves have the potential of resonating with protons of ∼1 keV with large pitch angles up to nearly perpendicular to the background magnetic field, thereby providing a feasible means of accounting for proton acceleration. Our results indicate the possible origin of energized protons in the Martian environment through the interaction with whistler-mode waves, and their potential relationship with the unique upstream conditions.


Introduction
High-energy ions up to several tens of keV, or even hundreds of keV, have been observed upstream of the bow shock of Earth (Anderson 1979;Sckopke et al. 1983;Fuselier & Schmidt 1994) and other planets (e.g., Yamauchi et al. 2015).Though limited in amount, with the excessive energy, they can both excite and interact with various kinds of waves in plasma, thus playing a crucial role in the process of energy transfer and redistribution in the upstream region of planets.A variety of possible mechanisms have been proposed for explaining the acceleration of these high-energy ions, such as the shock acceleration (Bale et al. 2005;Eastwood et al. 2005;Desai & Burgess 2008;Burgess et al. 2012), magnetic reconnection (e.g., Che et al. 2021), and wave-particle interaction (e.g., Schreiner & Spanier 2014); however, many questions remain open regarding their nature and origin.
Whistler-mode waves with frequency observed around 1 Hz are common in foreshock regions of planets such as the Earth, Venus, Mars, and Mercury.These waves propagate with a rather small normal angle (θ kB typically less than 45°), and their observed frequency tends to increase when being closer to the Sun (e.g., Russell 2007).When converted into a plasma rest frame, the waves are right-hand polarized, with a frequency typically 20 ∼ 100 times that of the local proton gyrofrequency.The polarization of the wave is also affected by the angle between the wave normal k and the solar-wind velocity v SW (Fairfield 1974;Orlowski & Russell 1991).
The generation mechanism of upstream whistler waves has long been under debate (e.g., Wilson 2016).Discrepancy of arguments focuses on two main problems: (a) Are waves generated by electrons or protons?(b) Are waves generated locally or remotely?While electrons are commonly considered responsible for the generation of whistler-mode waves (e.g., Sentman et al. 1983), Hoppe et al. (1981) reported from the observation that large amplitude waves with frequency at ∼1 Hz are sometimes accompanied by ion beams reflected back at the bow shock, but no clear correlation was found between ion beams and waves.Several theoretical studies suggested that both electrons and protons are capable of exciting whistler waves locally given their velocity distributions (Akimoto et al. 1987;Wong & Goldstein 1987, 1988;Wong & Smith 1994).On the other hand, Orlowski et al. (1995) argued that the damping of waves indicates a propagation of waves originating from somewhere else, most likely at the bow shock; however, the exact mechanism responsible for the generation of waves is yet to be clarified.Brain et al. (2002) from Mars Global Surveyor (MGS) measurements first observed and studied the properties of 1 Hz upstream whistler waves.The authors found the waves have frequencies lying in the range of 0.4 ∼ 2.3 Hz, elliptically polarized, and propagate obliquely to the ambient magnetic field with wave normal angles θ kB typically between 19°and 40°.The waves tend to be right-hand polarized when they propagate at angles to the sunward direction below 66°, otherwise, they appear reversed polarized.The diminishing wave amplitudes with increasing distance from the planetary bow shock (up to ∼10 R M ) led the authors to suggest that waves are generated near the bow shock.A more recent observation was made by MAVEN (Connerney et al. 2015b) in the foreshock region of Mars, while the ambient magnetic field direction was quasi-perpendicular to the shock normal, accompanied by a strong proton cyclotron wave (PCW) signal at ∼0.1 Hz.A statistical analysis by Ruhunusiri et al. (2018) Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. using MAVEN data found the occurrence rate of waves in 0.3 ∼ 1.3 Hz up to 5% or greater; the authors also revealed that the wave frequency tends to be higher for high Mach number and high proton beta value, while ellipticity and wave normal angle hardly change with those two factors.The dependence of wave occurrence rate with the proton beta values was explained as a consequence of Landau damping of waves, which influences the group-standing condition of waves.
So far, the investigations on 1 Hz waves in the Martian space have been restricted to the waves themselves, and not combined with the analysis of the particles' behavior.In this article, we describe an observation of upstream whistler waves at unusually low frequencies, made by MAVEN on 2015 August 14, and a simultaneously enhanced flux of protons at energies greater than the pristine solar wind.Sections 2 and 3 describe the analysis of wave properties and ion behaviors; the conclusions and discussion are summarized in Section 4.

Wave Observations
Before describing the event, we briefly introduce the data set used in this study.The magnetometer MAG onboard MAVEN (Connerney et al. 2015a) provided the magnetic field data; the 32 Hz high-resolution data are adopted.The STATIC measurements, given their ability of ion distinguishment, are utilized for all ion analyses.We selected the d1 data package, providing measurements of eight mass channels with full angular resolution in every 8 s, for ion moments and pitch-angle calculations.
On 2015 August 14, MAVEN followed an orbit with its apoapsis located north to Mars (see Figure 1).At ∼22:04 UTC, the spacecraft crossed the bow shock at (0.19, −0.30, 2.73) R M (Martian radius, 1 R M = 3389.5km) and entered the solar wind, observing the interplanetary magnetic field (IMF) at ∼5 nT (corresponds to proton gyrofrequency ∼0.1 Hz), dominated by the B x component.With the distance to the bow shock increasing, the cone angle of the IMF decreased from ∼70°to ∼45°.After reaching the apoapsis at ∼22:45 (Figure 1  Figure 3 presents the wavelet analysis of waves during 23:00 ∼ 00:00.Analysis of the magnetic field under the MFA frame revealed the dominancy of transverse fluctuation in the power spectrum.The ellipticity of waves is close to −1, indicating a left-hand circular polarization (also seen from the minimal variance analysis (MVA) of bandpass filtered magnetic field projection, shown in Figure 4); the wave normal angle remains less than 20°for most of the time, never exceeding 30°.From the analysis above, we concluded that these are whistler-mode waves.The last panel shows the integrated amplitude of the wave event, which is calculated from the selected spectrum with the following criteria: frequency between 0.3 and 0.5 Hz, ellipticity less than −0.5, and wave normal angle less than 30°.From Figure 3(f) one sees the amplitude of the wave is less than 2 nT for most of the period, never exceeding 4 nT.
To examine the Doppler effect on the frequency and polarization of the waves, the wavevector k is evaluated using the MVA method.Given the limitation of the single-point measurement, only the direction of k can be determined, with ±180°ambiguity.Here we assume that the wave propagated into the solar wind, which means that the x-component of vector k should be positive.A 1 minute nonoverlapping window was adopted for MVA to give a series of estimations of k largely parallel to the x-axis of the Mars-Solar orbital (MSO) frame (the angles 〈k, v〉 lie within the range 150°∼ 170°), meaning that the waves were traveling almost against the solar wind.The wavenumber |k| was estimated from the observed wavelength, calculated by multiplying the wave period (∼2.4 s) and the phase velocity of waves (∼200 km s −1 ; from Russell 2007).With all the above, the wave frequency in the plasma rest frame was obtained to be 1.14 ∼ 1.34 Hz (the ambiguity coming from the spread of wave frequencies).Note that the local proton gyrofrequency Ω i is estimated to be ∼0.1 Hz, implying that the wave in the plasma rest frame has a frequency of ∼10 Ω i .The estimated group velocity of the wave was less than that of the solar wind, implying that the wave was intrinsically right-hand polarized in the plasma rest frame and had its polarization inversed by the solar wind when observed by the spacecraft.The frequency and polarization of the wave in the plasma rest frame further confirm our deduction that the observed event was a series of whistler waves.

Pitch Angle and Velocity Distribution of Particles
The behaviors of particles (H + and O + ) were analyzed through their pitch-angle distributions calculated from STATIC measurements.The differential fluxes of H + and O + during the event are plotted in Figure 5.Given nearly radial IMF, the pitch angles of H + in <4 keV were concentrated near ∼180°(not shown here), indicating an antiparallel propagating solar wind.For H + with energy greater than 5 keV, no concentration of pitch angles was seen near the antiparallel direction.The enhancement and perturbation in the flux of H + at high energy (>4 keV; Figure 5(b)) is obvious after the crossing of the HFA structure, sustained for about 1 hr until ∼23:53, the pitch angle of which is concentrated at 90°∼ 150°.No obvious signals were seen in energy channels greater than 10 keV (Figure 5(a)).All enhancement in H + flux was observed post-HFA (∼22:53), which is quite the opposite to the O + signal, which occurred only prior to HFA, concentrating near 90°(see Figure 5(c)).
From the given differential flux, the velocity distribution of H + is calculated using converted phase space density (PSD) during the time interval 23:16 ∼ 23:19, when significant ion flux in 90°∼ 150°was detected to form a reflect population.Figure 7 shows the measurement of PSD with fitted velocity distribution overplotted using three ion populations, all written in the form of the following bi-Maxwellian distribution function (Equation ( 1)): .
The values of the parameters fitted are given in Table 1.From the figure above, one can see that the solar wind and the energetic ions are well-fitted, peaks of which can be clearly seen at around (110, −420) km s −1 and (580, −200) km s −1 , respectively.Another peak around (125, −20) km s −1 with a relatively large temperature presumably represents the hydrogen corona.

Dispersion Relation and Resonance Energy Range
Having obtained the velocity distribution of ions, the growth rate of possible instability is calculated using the PDRK program, a numerical dispersion relation solver developed by Xie (Xie & Xiao 2016;Xie 2019).This program is now integrated into a larger general solver named "BO", supporting not only kinetic mode but also fluid mode.Here we selected kinetic mode for proper consideration of nonzero drift velocities.One solution for our observed whistler-mode wave was found from the resulting ω-k curve, as is seen in Figure 6(c).While the imaginary part (orange curve, right axis of the panel) of the frequency is negative, the real part (blue curve, left axis of the panel) reaches nearly 4 at a normalized k around 25.
The real part of the solution is then used as input for solving the resonant energy range of protons after being transferred into the plasma rest frame.The result is shown in Figure 7 above.The x-and y-axis represent the pitch angle and energy of ions, respectively, with color indicating the frequencies of the waves.The wave observed has a frequency of 1.88 ∼ 3.14 rad s −1 in the spacecraft frame.From the results in Figure 7, one sees that the wave observed is capable of resonating with the proton of ∼1 keV with large pitch angles up to nearly perpendicular to the background magnetic field.The resonance order of N = ±2 ∼ 5 is also calculated (not shown here), which resembles largely the result of N = ±1.Added together, the resonant range of ∼1 keV protons from the order of N = −5 to N = 5 covers pitch angles from 87°∼ 89°, indicating the potential of energizing the solar-wind protons through the interactions with the observed whistler-mode waves.

Conclusions and Discussions
A series of low-frequency waves were observed by MAVEN in the Martian foreshock region.The waves lasted for nearly 1 hr, with frequencies centered at ∼0.4 Hz.Given their features including left-handed polarization and rather small wave   normal angles, the waves are likely to be whistler-mode waves.
During the observation period, H + with energies higher than the pristine solar wind were observed to have their flux enhanced in the pitch-angle range of 90°∼ 150°.The velocity distribution of ions is fitted using bi-Maxwellian functions, based on which the dispersion relation of the wave, as well as the resonant energy range of protons, are calculated.The results showed that the wave observed can interact with protons of ∼1 keV with large pitch angles up to nearly perpendicular to the background magnetic field, which thus serves as a potential mechanism of accelerating solar-wind protons through the process of resonant wave-particle interaction.
Here we would like to briefly discuss the possible contribution of observed ultra-low-frequency waves and HFA structure to the generation of the whistler-mode wave and the energized protons.The observed whistler-mode waves, compared with previous observations and their counterparts in terrestrial foreshock, seem to have unusually low frequencies.Brain et al. (2002) using MGS observation gave a frequency range of whistler waves as 0.4 ∼ 2.3 Hz; Ruhunusiri et al. (2018) using MAVEN observations revealed that the frequency of waves can be extended to a lower value of ∼0.3 Hz.They also found with the variation of frequency with the plasma beta value and the magnetosonic Mach number of the solar wind, that the frequency of whistler waves tends to be lower for a lower Mach number (<6.1) as well as a lower plasma beta value (<0.6).Given the averaged plasma parameters measured after the HFA structure was observed, we further calculated the Mach numbers and plasma beta values to be ∼2.45 and ∼7.2, respectively; prior to the structure, the values were ∼2.97 and ∼4.8, respectively.While the Mach numbers lie in the lower range stated in their study, thus favoring a lower wave frequency, the plasma beta values are much higher, implying a possibly much more complicated relation between the plasma environment and the properties of whistler-mode waves.
Although the exact generation mechanism of upstream whistler waves is yet unknown, the event studied in this article does have its uniqueness viewed from its surrounding environment.From Figures 2(c) and 3 one sees a fairly strong signal occurred near ∼0.04 Hz during 23:00 ∼ 23:20 and 23:30 ∼ 23:50.This signal shows moderately left-handed polarization (∼−0.5) and also a moderate wave normal angle (∼30°), with transverse power stronger than a compressional one.The frequency range resembles that of the magnetosonic wave, but its transverse feature indicates otherwise.Nevertheless, such an ultra-low-frequency wave might be a source of either the observed whistler-mode wave or the high-energy protons.Connerney et al. (2015b) in their first results of MAVEN magnetic field measurement reported a case in which a 1 Hz wave coexisted with a PCW; the authors also reported the observation of a simultaneous enhancement in flux of >1 keV electrons.Both electrons and protons were reported to be enhanced in early observations of 1 Hz waves in the terrestrial foreshock region.Although the damping nature of wave amplitudes propagating away from the shock surface indicates the bow shock as the more likely origin of these waves, their generation mechanism due to ion or electron distributions has never been actually discarded.
On the other hand, the HFA structure observed prior to the whistler waves is another possible candidate for the generation of waves and energetic protons.HFA structures possess the feature of having a motion electric field on at least one side pointing at the structure itself, thus concentrating ions on the discontinuity plane.A more detailed analysis of this observed HFA structure is provided in the Appendix.It is worth noting that only the temperature of protons was moderately enhanced (from ∼100 to ∼150 eV) after crossing the HFA structure, but leaving all other parameters (magnetic field strength, proton density, and bulk velocity) unchanged.The increase in proton temperature might be a consequence of the observed HFA structure; an alternative explanation would be the reflected ions from the bow shock.Given the almost radial IMF during the event, MAVEN was traveling through the foreshock region of Mars, in which accelerated ions are commonly seen, forming diffused signatures on the velocity map, as was seen in Figure 6.However, the velocity calculation of energetic H + (Figure 8) showed that these high-energy protons were flowing  nearly toward the shock surface instead of coming from it (note that v x indicated by the deep blue line for most of the time remained negative).The possibility remains that the trajectories of these protons were modified by electric fields in waves or the solar wind.Further studies will be dedicated to deeply analyzing the behavior of these ions, aiming at finding the possible relations between these features and their contributions to the observed events.calculated.Compared with the normal vector of the plane, it is obvious that E 1 points to the plane, while E 3 does not, fulfilling another criterion of the formation of HFA, which is that at least one side of the structure has its convection electric field pointing to the discontinuity.Given all the results above, we concluded with confidence that MAVEN crossed an HFA structure.

Figure 1 .
Figure 1.MAVEN trajectory in MSO frame.(a) 3D plot; (b) projection on x-r plane, in which = + r x y ; 2 2 (c) projection on x-y plane; (d) spacecraft altitude.The red and blue lines (curves) in (a)-(c) indicate the magnetic pileup boundary and bow shock of Mars, respectively.

Figure 2 .
Figure 2. Event overview.(a) IMF measurement; (b) cone and clock angle of IMF; (c) wavelet spectrum of |B|; (d) density of H + and O + (on left axis), with temperature of H + (on right axis); (e) solar-wind bulk velocity; (f) energy spectrum of H + ; (g) energy spectrum of O + .

Figure 3 .
Figure 3. Wave properties obtained from wavelet analysis.Power spectrum of (a) transverse component of B in MFA frame; (b) compressional component of B in MFA frame; (c) ratio of transverse component over compressional component; (d) ellipticity; (e) wave normal angle; (f) integrated wave amplitude.

Figure 4 .
Figure 4. MVA analysis on the band-passed magnetic field during 23:02 ∼ 23:03, in the mean-field frame.Projection on (a) min-mid plane; (b) min-max plane; (c) mid-max plane.The green dot and red cross mark in each plot indicate the starting and ending point of the vector series.

Figure 6 .
Figure 6.(a) Distribution of measured PSD; (b) fitted velocity distribution; (c) dispersion relation solved for observed whistler-mode wave.Note that both (a) and (b) are in a field-aligned frame and share the same color bar located east of (a).

Table 1
Fitting Parameters of Initial Ion Distribution for Calculating Dispersion Relation