the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 31 Jan 2025
Ion beam instability model for the Mercury upstream waves
Abstract. An analytic model for the ion beam instability is constructed in view of application to the Mercury upstream waves. Our ion beam instability model determines the frequency and the wavenumber by equating the whistler dispersion relation with the beam resonance condition in favor of planetary foreshock wave excitation. By introducing the Doppler shift in the instability frequency, our model can derive the observer-frame relation of the resonance frequency to the beam velocity and the flow speed. The frequency relation serves as a useful diagnostic tool to the Mercury upstream wave studies in the upcoming BepiColombo observations.
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Status: closed
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RC1: 'Comment on egusphere-2025-60', Anonymous Referee #1, 25 Mar 2025
The authors present a theoretical model of ion beam driven instabilities relevant to Mercury’s foreshock, in anticipation of the upcoming in-situ observations of the BepiColombo mission. They have analytically derived expressions for the resonance condition of whistler-mode waves interacting with beam ions, incorporating Doppler shifts into the spacecraft frame. The model produces practical relations between wave frequency, beam velocity, and solar wind speed, potentially enabling data driven diagnostics using only magnetic field measurements. The manuscript is well written and the conclusions are clear, but a few minor comments and questions come to mind. In addition, there are a few grammatical errors in the manuscript, please correct them.
Q1. Is the Galilean approximation sufficient for all expected velocities?
Q2. The authors assume that the wave propagation is aligned to the magnetic field. What is the expected influence of small angular derivations from this alignment on the resonance condition? Would oblique corrections significantly/slightly alter the model's applicability?
Q3. In paragraph 60, in the main text, tilda is missing for the equation part.
Q4. Threshold value is close to the critical Alfven Mach number. The authors have mentioned it is interesting. What would it indicate scientifically?
Q5. The authors have mentioned that we can estimate the flow velocity if we know the resonance frequency. Have you tried to calculate it from MESSENGER observations? Does it give reasonable flow velocity?
Q6. How do you manage this model with multiple species for pick up ions? Can you distinguish them?
Citation: https://doi.org/10.5194/egusphere-2025-60-RC1 -
AC1: 'Reply on RC1', Yasuhito Narita, 30 Apr 2025
> Referee 1 (RC1)
>
> The authors present a theoretical model of ion beam driven instabilities
> relevant to Mercury’s foreshock, in anticipation of the upcoming
> in-situ observations of the BepiColombo mission. They have analytically
> derived expressions for the resonance condition of whistler-mode
> waves interacting with beam ions, incorporating Doppler shifts
> into the spacecraft frame. The model produces practical relations
> between wave frequency, beam velocity, and solar wind speed,
> potentially enabling data driven diagnostics using only magnetic
> field measurements. The manuscript is well written and the conclusions
> are clear, but a few minor comments and questions come to mind.
> In addition, there are a few grammatical errors in the manuscript,
> please correct them.
>
> Q1. Is the Galilean approximation sufficient for all expected velocities?
R01.01
Yes, for the Mercury upstream waves. The solar wind speed is about 400 km/s at the distance of Mercury orbit. The backstreaming ion beam velocity is in the range from 500 to 1000 km/s. In total, the velocity of up to 1500 km/s is considered in our model. The ratio of the velocity to the speed of light (Lorentz beta) is 0.005, and the Lorentz factor gamma is 1.0000125. So, we think that the Galilean approximation is sufficient.
Our model breaks down at shorter wavelengths because the second-order Taylor expansion (parabolic fitting) of the whistler dispersion relation is no longer valid at shorter wavelengths. Our model is valid at a wavelength (of the low-frequency whistler mode) from MHD scales (1000 km and above) down to about the ion inertial length (say, down to 100 km).
We added a subsection (section 4.1) about the validity of our model with respect to the Galilean approximation, low-frequency approximation, and low-beam-density condition.
See section 4.1 in the revision, from page 8, line 145 to page 9, line 165. See also Appendix B for the relativistic treatment (page 13, lines 268--278).
> Q2. The authors assume that the wave propagation is aligned to
> the magnetic field. What is the expected influence of small angular
> derivations from this alignment on the resonance condition?
> Would oblique corrections significantly/slightly alter the model's applicability?
R01.02
We tested numerically. No significant change. We solved the wave equation for a beam plasma numerically and studied the effect of small propagation angle deviation from the parallel direction (to the mean magnetic field) for a propagation angle of 0 degree, 10 degree, and 20 degree.
One possible effect is that the obliquely-propagating whistler wave has a left-hand polarized component because the wave polarization of elliptic. For highly oblique propagation (at angles of 45 degree or higher), the beam can be in resonance with the left-hand component.
The effect of the oblique magnetic field to the flow (yet the beam is parallel to the magnetic field) is also considered, but the effect is minor in that the flow speed simply needs to be projected to the magnetic field direction using cos(theta).
See section 4.2 in the revision (page 9, lines 166 --176 including Fig. 6) for the propagation angle misalignment and section 4.3 (page 9, lines 177--181) for the oblique magnetic field to the flow.
> Q3. In paragraph 60, in the main text, tilda is missing for the equation part.
R01.03
Equations and symbols are not normalized in Eq. (3). To avoid confusion, the normalization is introduced after Eq. (3) and we do not use the arrow symbol any more but use the equations.
See page 4 in the revision, lines 73--75.
> Q4. Threshold value is close to the critical Alfven Mach number.
> The authors have mentioned it is interesting. What would it indicate scientifically?
R01.04
The earlier studies (particularly using the numerical simulations) indicate that the wave-particle interaction can sufficiently scatter the particles in a low Mach number shock up to an Alfven Mach
number of about 2.7 (sub-critical shocks). In a high Mach number shock (Alfven Mach number above 2.7), the dissipation is primarily made by the specular reflection by the cross-shock potential and the wave-particle interactions at the shock transition. Our theory predicts that the critical beam Mach number is about 2.4. There might be a relation between the dissipation mechanism and the ion beam instability in the collisionless shock such that the beam instability potentially contributes to a more efficient shock dissipation.
See page 5, lines 89--94 in the revision.
> Q5. The authors have mentioned that we can estimate the flow
> velocity if we know the resonance frequency. Have you tried
> to calculate it from MESSENGER observations?
> Does it give reasonable flow velocity?
R01.05
No, access to the calibrated flow velocity or beam velocity data in the MESSENGER mission is still limited to the instrument team. In the publications, there are observations of beams in the MESSENGER data (Glass et al., J. Geophys. Res. 2023).
> Q6. How do you manage this model with multiple species for
> pick up ions? Can you distinguish them?
R01.06
The beam species can be other than protons. We added a subsection (sec. 4.4) and discuss the heavier ion species such as the helium alpha particles and exospheric particles. Generalization of our model is also presented for the heavier ions. The heavier species have a lower density and a smaller cyclotron frequency, and thus it is unlikely that the heavy-ion beam instability is relevant in the near-Mercury space. Even if there are, the resonance frequency is far too low compared to the proton beam.
See section 4.4 in the revision, from page 10, lines 182--205.
Citation: https://doi.org/10.5194/egusphere-2025-60-AC1
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AC1: 'Reply on RC1', Yasuhito Narita, 30 Apr 2025
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RC2: 'Comment on egusphere-2025-60', Hongyang Zhou, 01 Apr 2025
This study demonstrates a simple relation between the plasma bulk flow, the observed beam velocity, and the observed wave frequency in the satellite frame that can be used to check whether the observed plasma wave satisfies the ion beam resonance instability. I validated all the equations and reproduced the figures shown in the manuscript and considered them straightforward to be applied to real data. I would recommend this to be published after some minor clarifications and corrections.
1. In the Introduction Section starting at line 15, the authors introduced three kinds of wave modes upstream of Mercury's bow shock. In the second sentence of this paragraph, it is stated that all three waves are driven by right-hand resonant resonant ion beam instability from Earth's foreshock study. However, ion cyclotron waves from pick-up ions are rarely reported at Earth, although mathmetically the difference between ICW and the "whistler-magnetosonic" waves only lies in the initial beam velocity in the satellite frame (Ub vs 0). Do the authors claim that all three modes come from the same generation mechanism? Please clarify.
2. In Figure 2, I suppose the beam cyclotron frequency Omega_b = qi * B / mi = Omega_i. However, in the caption it is stated that Omega_b is negative, which I don't fully understand. Also, in the following text Omega_p is used instead of Omega_i, especially for Equation 3 which is represented as the resonance line in the figure. It is better to be consistent. Besides, what does the dashed line mean in the plot?
3. Starting from Line 62, the normalized quantities are introduced with a tilde symbol. I suggest using \tilde{\omega} and \tilde{k} on the right-hand side of the arrow expressions.
4. Line 70: Consider adding \tilde{U_b} notation to the text.
5. Equation 9: missing tilde for U_+, and subscript b on the LHS.
6. Line 81: As I understand, the parabolic approximation refers to the dispersion curve of R+. However, due to the limited plotting range, the large wavenumber solution is not shown in Figure 2. I think this can be better shown by extending Figure 2 range or simply add an explanation to the text.
7. Line 115: The third condition is the same as Equation 9 (which can be referenced in the text).
8. Line 117: The threshold 0.5 is a rough estimation, as the actual value depends on \omega^\prime. I suggest the usage of approximately less than (i.e. lesssim) or similar notations here.
9. Line 122: Better to be consistent whether or not to use short-hand notations such as Fig.
10. Line 139: The unit of ion inertial length "km/rad" seems wrong.
Citation: https://doi.org/10.5194/egusphere-2025-60-RC2 -
RC3: 'Reply on RC2', Hongyang Zhou, 02 Apr 2025
And one additional question about Equation 1. This equation follows Eq.6.2.5 in Gary 1993, but is different from Eq.2.24 in Hasegawa and Uberoi 1982. Can the authors provide further clarification or derivation here? Thanks.
Citation: https://doi.org/10.5194/egusphere-2025-60-RC3 -
AC3: 'Reply on RC3', Yasuhito Narita, 30 Apr 2025
> Referee 2 (RC3)
>
> And one additional question about Equation 1. This equation follows
> Eq.6.2.5 in Gary 1993, but is different from Eq.2.24 in Hasegawa
> and Uberoi 1982. Can the authors provide further clarification or
> derivation here? Thanks.
R02.11
The two expressions are equivalent when the deviation from the Alfven wave dispersion relation is considered to the first order. Hasegawa-Uberoi's expression uses the frequency-deviation form, and Gary's expression uses the wave number-deivation form. We added Appendix in the manuscript and derived the two dispersion relations from the general from of the parallel-propagating R-mode dispersion equation.
See Appendix A in the revision, page 11, line 233 to page 13, line 266.
Citation: https://doi.org/10.5194/egusphere-2025-60-AC3 -
RC5: 'Reply on AC3', Hongyang Zhou, 30 Apr 2025
Thanks for the further clarification! One small typo though: In Eq. 25, it should be a minus sign on the LHS. Eq. 34 is a repetition of Eq. 26.
I love the newly included discussion section which clearly points out the approximation and extension of the analytical model.
Citation: https://doi.org/10.5194/egusphere-2025-60-RC5 -
AC6: 'Reply on RC5', Yasuhito Narita, 30 Apr 2025
Eq. (25) was erroneous. It is a sum of the squared plasma frequency divided by the cyclotron frequency over species, which reduces into a sum of charge times number density per species. Equation (25) reads correct with this version (including the explanation of the symbols used in Eq. 25 after the equation).
Redundancy of Eq. (26) was removed by referring to Eq. (26). The equation number of 34 is moved to Appendix B for the continuation of numbering.
Citation: https://doi.org/10.5194/egusphere-2025-60-AC6
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AC6: 'Reply on RC5', Yasuhito Narita, 30 Apr 2025
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RC5: 'Reply on AC3', Hongyang Zhou, 30 Apr 2025
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AC3: 'Reply on RC3', Yasuhito Narita, 30 Apr 2025
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AC2: 'Reply on RC2', Yasuhito Narita, 30 Apr 2025
> Referee 2 (RC2)
>
> This study demonstrates a simple relation between the plasma bulk flow,
> the observed beam velocity, and the observed wave frequency in
> the satellite frame that can be used to check whether the observed
> plasma wave satisfies the ion beam resonance instability. I validated
> all the equations and reproduced the figures shown in the manuscript
> and considered them straightforward to be applied to real data.
> I would recommend this to be published after some minor clarifications
> and corrections.
>
> 1. In the Introduction Section starting at line 15, the authors introduced
> three kinds of wave modes upstream of Mercury's bow shock.
> In the second sentence of this paragraph, it is stated that all three waves
> are driven by right-hand resonant resonant ion beam instability from
> Earth's foreshock study. However, ion cyclotron waves from pick-up ions
> are rarely reported at Earth, although mathematically the difference
> between ICW and the "whistler-magnetosonic" waves only lies in the
> initial beam velocity in the satellite frame (Ub vs 0). Do the authors
> claim that all three modes come from the same generation mechanism? Please clarify.
R02.01
We are constructing a model of the whistler wave excitation just below the ion cyclotron frequency in the spacecraft frame (the waves at 0.3 Hz in Le et al., 2013) and the the pickup ion cyclotron wave (Schmid et al., 2022). The wave excitation at higher frequency such as at 2 Hz in the spacecraft frame needs a different excitation mechanism such as parametric instability, an anti-sunward streaming ion beam, or a sunward streaming electron beam.
See page 1, lines 17--22 in the revision.
> 2. In Figure 2, I suppose the beam cyclotron frequency
> Omega_b = qi * B / mi = Omega_i. However, in the caption it is
> stated that Omega_b is negative, which I don't fully understand.
> Also, in the following text Omega_p is used instead of Omega_i,
> especially for Equation 3 which is represented as the resonance
> line in the figure. It is better to be consistent. Besides, what does the
> dashed line mean in the plot?
R02.02
Both figure 2 and figure caption are improved. Figure 2 shows an extended curve for the R+ mode so that the two roots of the resonance equation are visible.
Figure caption is changed as follows:
"Dispersion relations and the beam resonance condition. Protons are assumed for the bulk ions and the beam ions with the cyclotron frequency Omega_i . The wave modes are the whistler wave propagating forward to the mean magnetic field (R+ mode) and backward (L- mode) and the ion cyclotron wave propagating forward (L+ mode) and backward (R- mode). The symbols R and L refer to the polarization, i.e., dielectric response in the Stix notation, and the plus and minus signs refer to the propagation sense with respect to the mean magnetic field. The beam resonance condition is indicated by the thin line, and intersects the y-axis (long wavelength limit) the ion cyclotron frequency in the negative frequency domain. The beam resonance with the R+ mode is considered here. The resonance with the R- mode is unlikely because of the opposite group velocity direction to the beam (no sufficient time for the energy exchange)."
We used Omega_i not Omega_b in the caption. The dashed line (indication of the ion cyclotron frequency) was deleted. Also, we use Omega_i coherently in the revision (both main text and figures), and not Omega_p anymore.
See figure 2 and figure-2 caption in the revision, page 4 top.
> 3. Starting from Line 62, the normalized quantities are introduced
> with a tilde symbol. I suggest using \tilde{\omega} and \tilde{k}
> on the right-hand side of the arrow expressions.
R02.03
Arrow was replaced by the equation in the definition of normalization (paragraph after Eq. 3).
See page 4, lines 73--75.
> 4. Line 70: Consider adding \tilde{U_b} notation to the text.
R02.04
Done. See page 4, line 80.
> 5. Equation 9: missing tilde for U_+, and subscript b on the LHS.
R02.05
Done. See page 5, Eq. (9).
> 6. Line 81: As I understand, the parabolic approximation refers to the
> dispersion curve of R+. However, due to the limited plotting range,
> the large wavenumber solution is not shown in Figure 2.
> I think this can be better shown by extending Figure 2 range or
> simply add an explanation to the text.
R02.06
Done. See Figure 2 (page 4 top).
> 7. Line 115: The third condition is the same as Equation 9
> (which can be referenced in the text).
R02.07
Done. See page 7, lines 133--134.
> 8. Line 117: The threshold 0.5 is a rough estimation, as the
> actual value depends on \omega^\prime. I suggest the usage of
> approximately less than (i.e. lesssim) or similar notations here.
R02.08
Done. See page 7, lines 136--137.
> 9. Line 122: Better to be consistent whether or not to use
> short-hand notations such as Fig.
R02.09
Done. See page 7, lines 141--143.
> 10. Line 139: The unit of ion inertial length "km/rad" seems wrong.
R02.10
It is not wrong. The Alfven speed is given in units of km/s and the
ion cyclotron frequency in units of rad/s. The combination V_A/Omega_i
is [(km/s) (s/rad)] = [km/rad].
Citation: https://doi.org/10.5194/egusphere-2025-60-AC2
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RC3: 'Reply on RC2', Hongyang Zhou, 02 Apr 2025
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RC4: 'Comment on egusphere-2025-60', Anonymous Referee #3, 03 Apr 2025
This paper presented an analytic model for the ion beam instability with application to the Mercury upstream waves. The ion beam instability model determines the frequency and the wavenumber by equating the whistler dispersion relation with the beam resonance condition in favor of planetary foreshock wave excitation. By introducing the Doppler shift in the instability frequency, their model derived the observer-frame relation of the resonance frequency to the beam velocity and the flow speed. The frequency relation serves as a useful diagnostic tool to the Mercury upstream wave studies in the upcoming BepiColombo observations.
Comments:
The authors gave the example of pickup ions at Mercury as part of the low-frequency EM waves observed upstream of Mercury's bow shock in the introduction section. However it is difficult to see the relevance of this whistler wave model to the pickup ions at Mercury. Furthermore paragraph 95 stated that "the ion cyclotron frequency is expected for the beam instability for the pickup ions by substituting the sign-reversed flow speed into beam velocity as U˜b = −U˜f (pickup ion cyclotron waves) ". However, it is unclear how the authors made that "jump" from equation 12 to pickup ions when the equations were derived for whistler mode waves. Further clarification should be provided.
It is also unclear how this technique is useful to future Mercury's studies. The idea to "back-calculate" the resonance frequency in the spacecraft (or observer) frame if there is information on the beam velocity and flow speed is interesting. The flow speed can be easily assumed to be the solar wind flow speed. However, the ion beam is challenging to observe even with high-resolution plasma data at Earth, let alone accurately calculate its velocity. Further clarification is needed to address how such information can be obtained from future Mercury studies in the context of Bepi-Colombo's observational capability.
Citation: https://doi.org/10.5194/egusphere-2025-60-RC4 -
AC4: 'Reply on RC4', Yasuhito Narita, 30 Apr 2025
> Referee 3 (RC4)
>
> This paper presented an analytic model for the ion beam instability
> with application to the Mercury upstream waves. The ion beam instability
> model determines the frequency and the wavenumber by equating
> the whistler dispersion relation with the beam resonance condition
> in favor of planetary foreshock wave excitation. By introducing the
> Doppler shift in the instability frequency, their model derived the
> observer-frame relation of the resonance frequency to the beam velocity
> and the flow speed. The frequency relation serves as a useful diagnostic
> tool to the Mercury upstream wave studies in the upcoming
> BepiColombo observations.
>
> Comments:
> The authors gave the example of pickup ions at Mercury as part of the
> low-frequency EM waves observed upstream of Mercury's bow shock
> in the introduction section. However it is difficult to see the relevance
> of this whistler wave model to the pickup ions at Mercury.
R03.01
Our model can be applied to the pickup ion cyclotron wave in the 1-D setup for parallel propagation. Figure 3 has been improved to the foreshock ion scenario (left column) and the pickup ion scenario (right column). The observed wave appears as left-hand polarized wave at the ion cyclotron frequency, but in our model, it is the Doppler shifted whistler wave with the reversal of frequency.
See Figure 3 and caption text (page 6 top) in the revision.
> Furthermore paragraph 95 stated that "the ion cyclotron frequency is
> expected for the beam instability for the pickup ions by substituting
> the sign-reversed flow speed into beam velocity as U˜b = −U˜f
> (pickup ion cyclotron waves) ". However, it is unclear how the
> authors made that "jump" from equation 12 to pickup ions when
> the equations were derived for whistler mode waves. Further clarification
> should be provided.
R03.02
A paragraph was added to explain the scenario of pickup ion cyclotron wave.
See page 5, lines 111--113.
> It is also unclear how this technique is useful to future Mercury's studies.
> The idea to "back-calculate" the resonance frequency in the spacecraft
> (or observer) frame if there is information on the beam velocity
> and flow speed is interesting. The flow speed can be easily assumed
> to be the solar wind flow speed. However, the ion beam is challenging
> to observe even with high-resolution plasma data at Earth, let
> alone accurately calculate its velocity. Further clarification is needed
> to address how such information can be obtained from future
> Mercury studies in the context of Bepi-Colombo's observational capability.
R03.03
The backstreaming ions are observed by MESSENGER (Glass et al., 2023). The pickup ion cyclotron waves are also observed by MESSENGER (Schmid et al., 2022). Combination of Mio MGF and Mio MPPE/MIA instrument is ideally suited to test for our instability model against the spacecraft data.
See page 11, lines 222--225 in the revision.
Citation: https://doi.org/10.5194/egusphere-2025-60-AC4
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AC4: 'Reply on RC4', Yasuhito Narita, 30 Apr 2025
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AC5: 'Comment on egusphere-2025-60', Yasuhito Narita, 30 Apr 2025
Here are more detailed explanations and reference as a supporting material to my reply.
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AC7: 'Reply on AC5', Yasuhito Narita, 30 Apr 2025
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AC8: 'Reply on AC7', Yasuhito Narita, 30 Apr 2025
The first term on r.h.s. in Equation (24) has to be corrected accordingly to the change in Eq. (25). Here is the correct one. The first term vanishes in Eq. (24), and the second term is rewritten using c and V_A, and the third term the Hall-term correction (which makes the waves dispersive into whistler sense).
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AC8: 'Reply on AC7', Yasuhito Narita, 30 Apr 2025
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AC7: 'Reply on AC5', Yasuhito Narita, 30 Apr 2025
Status: closed
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RC1: 'Comment on egusphere-2025-60', Anonymous Referee #1, 25 Mar 2025
The authors present a theoretical model of ion beam driven instabilities relevant to Mercury’s foreshock, in anticipation of the upcoming in-situ observations of the BepiColombo mission. They have analytically derived expressions for the resonance condition of whistler-mode waves interacting with beam ions, incorporating Doppler shifts into the spacecraft frame. The model produces practical relations between wave frequency, beam velocity, and solar wind speed, potentially enabling data driven diagnostics using only magnetic field measurements. The manuscript is well written and the conclusions are clear, but a few minor comments and questions come to mind. In addition, there are a few grammatical errors in the manuscript, please correct them.
Q1. Is the Galilean approximation sufficient for all expected velocities?
Q2. The authors assume that the wave propagation is aligned to the magnetic field. What is the expected influence of small angular derivations from this alignment on the resonance condition? Would oblique corrections significantly/slightly alter the model's applicability?
Q3. In paragraph 60, in the main text, tilda is missing for the equation part.
Q4. Threshold value is close to the critical Alfven Mach number. The authors have mentioned it is interesting. What would it indicate scientifically?
Q5. The authors have mentioned that we can estimate the flow velocity if we know the resonance frequency. Have you tried to calculate it from MESSENGER observations? Does it give reasonable flow velocity?
Q6. How do you manage this model with multiple species for pick up ions? Can you distinguish them?
Citation: https://doi.org/10.5194/egusphere-2025-60-RC1 -
AC1: 'Reply on RC1', Yasuhito Narita, 30 Apr 2025
> Referee 1 (RC1)
>
> The authors present a theoretical model of ion beam driven instabilities
> relevant to Mercury’s foreshock, in anticipation of the upcoming
> in-situ observations of the BepiColombo mission. They have analytically
> derived expressions for the resonance condition of whistler-mode
> waves interacting with beam ions, incorporating Doppler shifts
> into the spacecraft frame. The model produces practical relations
> between wave frequency, beam velocity, and solar wind speed,
> potentially enabling data driven diagnostics using only magnetic
> field measurements. The manuscript is well written and the conclusions
> are clear, but a few minor comments and questions come to mind.
> In addition, there are a few grammatical errors in the manuscript,
> please correct them.
>
> Q1. Is the Galilean approximation sufficient for all expected velocities?
R01.01
Yes, for the Mercury upstream waves. The solar wind speed is about 400 km/s at the distance of Mercury orbit. The backstreaming ion beam velocity is in the range from 500 to 1000 km/s. In total, the velocity of up to 1500 km/s is considered in our model. The ratio of the velocity to the speed of light (Lorentz beta) is 0.005, and the Lorentz factor gamma is 1.0000125. So, we think that the Galilean approximation is sufficient.
Our model breaks down at shorter wavelengths because the second-order Taylor expansion (parabolic fitting) of the whistler dispersion relation is no longer valid at shorter wavelengths. Our model is valid at a wavelength (of the low-frequency whistler mode) from MHD scales (1000 km and above) down to about the ion inertial length (say, down to 100 km).
We added a subsection (section 4.1) about the validity of our model with respect to the Galilean approximation, low-frequency approximation, and low-beam-density condition.
See section 4.1 in the revision, from page 8, line 145 to page 9, line 165. See also Appendix B for the relativistic treatment (page 13, lines 268--278).
> Q2. The authors assume that the wave propagation is aligned to
> the magnetic field. What is the expected influence of small angular
> derivations from this alignment on the resonance condition?
> Would oblique corrections significantly/slightly alter the model's applicability?
R01.02
We tested numerically. No significant change. We solved the wave equation for a beam plasma numerically and studied the effect of small propagation angle deviation from the parallel direction (to the mean magnetic field) for a propagation angle of 0 degree, 10 degree, and 20 degree.
One possible effect is that the obliquely-propagating whistler wave has a left-hand polarized component because the wave polarization of elliptic. For highly oblique propagation (at angles of 45 degree or higher), the beam can be in resonance with the left-hand component.
The effect of the oblique magnetic field to the flow (yet the beam is parallel to the magnetic field) is also considered, but the effect is minor in that the flow speed simply needs to be projected to the magnetic field direction using cos(theta).
See section 4.2 in the revision (page 9, lines 166 --176 including Fig. 6) for the propagation angle misalignment and section 4.3 (page 9, lines 177--181) for the oblique magnetic field to the flow.
> Q3. In paragraph 60, in the main text, tilda is missing for the equation part.
R01.03
Equations and symbols are not normalized in Eq. (3). To avoid confusion, the normalization is introduced after Eq. (3) and we do not use the arrow symbol any more but use the equations.
See page 4 in the revision, lines 73--75.
> Q4. Threshold value is close to the critical Alfven Mach number.
> The authors have mentioned it is interesting. What would it indicate scientifically?
R01.04
The earlier studies (particularly using the numerical simulations) indicate that the wave-particle interaction can sufficiently scatter the particles in a low Mach number shock up to an Alfven Mach
number of about 2.7 (sub-critical shocks). In a high Mach number shock (Alfven Mach number above 2.7), the dissipation is primarily made by the specular reflection by the cross-shock potential and the wave-particle interactions at the shock transition. Our theory predicts that the critical beam Mach number is about 2.4. There might be a relation between the dissipation mechanism and the ion beam instability in the collisionless shock such that the beam instability potentially contributes to a more efficient shock dissipation.
See page 5, lines 89--94 in the revision.
> Q5. The authors have mentioned that we can estimate the flow
> velocity if we know the resonance frequency. Have you tried
> to calculate it from MESSENGER observations?
> Does it give reasonable flow velocity?
R01.05
No, access to the calibrated flow velocity or beam velocity data in the MESSENGER mission is still limited to the instrument team. In the publications, there are observations of beams in the MESSENGER data (Glass et al., J. Geophys. Res. 2023).
> Q6. How do you manage this model with multiple species for
> pick up ions? Can you distinguish them?
R01.06
The beam species can be other than protons. We added a subsection (sec. 4.4) and discuss the heavier ion species such as the helium alpha particles and exospheric particles. Generalization of our model is also presented for the heavier ions. The heavier species have a lower density and a smaller cyclotron frequency, and thus it is unlikely that the heavy-ion beam instability is relevant in the near-Mercury space. Even if there are, the resonance frequency is far too low compared to the proton beam.
See section 4.4 in the revision, from page 10, lines 182--205.
Citation: https://doi.org/10.5194/egusphere-2025-60-AC1
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AC1: 'Reply on RC1', Yasuhito Narita, 30 Apr 2025
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RC2: 'Comment on egusphere-2025-60', Hongyang Zhou, 01 Apr 2025
This study demonstrates a simple relation between the plasma bulk flow, the observed beam velocity, and the observed wave frequency in the satellite frame that can be used to check whether the observed plasma wave satisfies the ion beam resonance instability. I validated all the equations and reproduced the figures shown in the manuscript and considered them straightforward to be applied to real data. I would recommend this to be published after some minor clarifications and corrections.
1. In the Introduction Section starting at line 15, the authors introduced three kinds of wave modes upstream of Mercury's bow shock. In the second sentence of this paragraph, it is stated that all three waves are driven by right-hand resonant resonant ion beam instability from Earth's foreshock study. However, ion cyclotron waves from pick-up ions are rarely reported at Earth, although mathmetically the difference between ICW and the "whistler-magnetosonic" waves only lies in the initial beam velocity in the satellite frame (Ub vs 0). Do the authors claim that all three modes come from the same generation mechanism? Please clarify.
2. In Figure 2, I suppose the beam cyclotron frequency Omega_b = qi * B / mi = Omega_i. However, in the caption it is stated that Omega_b is negative, which I don't fully understand. Also, in the following text Omega_p is used instead of Omega_i, especially for Equation 3 which is represented as the resonance line in the figure. It is better to be consistent. Besides, what does the dashed line mean in the plot?
3. Starting from Line 62, the normalized quantities are introduced with a tilde symbol. I suggest using \tilde{\omega} and \tilde{k} on the right-hand side of the arrow expressions.
4. Line 70: Consider adding \tilde{U_b} notation to the text.
5. Equation 9: missing tilde for U_+, and subscript b on the LHS.
6. Line 81: As I understand, the parabolic approximation refers to the dispersion curve of R+. However, due to the limited plotting range, the large wavenumber solution is not shown in Figure 2. I think this can be better shown by extending Figure 2 range or simply add an explanation to the text.
7. Line 115: The third condition is the same as Equation 9 (which can be referenced in the text).
8. Line 117: The threshold 0.5 is a rough estimation, as the actual value depends on \omega^\prime. I suggest the usage of approximately less than (i.e. lesssim) or similar notations here.
9. Line 122: Better to be consistent whether or not to use short-hand notations such as Fig.
10. Line 139: The unit of ion inertial length "km/rad" seems wrong.
Citation: https://doi.org/10.5194/egusphere-2025-60-RC2 -
RC3: 'Reply on RC2', Hongyang Zhou, 02 Apr 2025
And one additional question about Equation 1. This equation follows Eq.6.2.5 in Gary 1993, but is different from Eq.2.24 in Hasegawa and Uberoi 1982. Can the authors provide further clarification or derivation here? Thanks.
Citation: https://doi.org/10.5194/egusphere-2025-60-RC3 -
AC3: 'Reply on RC3', Yasuhito Narita, 30 Apr 2025
> Referee 2 (RC3)
>
> And one additional question about Equation 1. This equation follows
> Eq.6.2.5 in Gary 1993, but is different from Eq.2.24 in Hasegawa
> and Uberoi 1982. Can the authors provide further clarification or
> derivation here? Thanks.
R02.11
The two expressions are equivalent when the deviation from the Alfven wave dispersion relation is considered to the first order. Hasegawa-Uberoi's expression uses the frequency-deviation form, and Gary's expression uses the wave number-deivation form. We added Appendix in the manuscript and derived the two dispersion relations from the general from of the parallel-propagating R-mode dispersion equation.
See Appendix A in the revision, page 11, line 233 to page 13, line 266.
Citation: https://doi.org/10.5194/egusphere-2025-60-AC3 -
RC5: 'Reply on AC3', Hongyang Zhou, 30 Apr 2025
Thanks for the further clarification! One small typo though: In Eq. 25, it should be a minus sign on the LHS. Eq. 34 is a repetition of Eq. 26.
I love the newly included discussion section which clearly points out the approximation and extension of the analytical model.
Citation: https://doi.org/10.5194/egusphere-2025-60-RC5 -
AC6: 'Reply on RC5', Yasuhito Narita, 30 Apr 2025
Eq. (25) was erroneous. It is a sum of the squared plasma frequency divided by the cyclotron frequency over species, which reduces into a sum of charge times number density per species. Equation (25) reads correct with this version (including the explanation of the symbols used in Eq. 25 after the equation).
Redundancy of Eq. (26) was removed by referring to Eq. (26). The equation number of 34 is moved to Appendix B for the continuation of numbering.
Citation: https://doi.org/10.5194/egusphere-2025-60-AC6
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AC6: 'Reply on RC5', Yasuhito Narita, 30 Apr 2025
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RC5: 'Reply on AC3', Hongyang Zhou, 30 Apr 2025
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AC3: 'Reply on RC3', Yasuhito Narita, 30 Apr 2025
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AC2: 'Reply on RC2', Yasuhito Narita, 30 Apr 2025
> Referee 2 (RC2)
>
> This study demonstrates a simple relation between the plasma bulk flow,
> the observed beam velocity, and the observed wave frequency in
> the satellite frame that can be used to check whether the observed
> plasma wave satisfies the ion beam resonance instability. I validated
> all the equations and reproduced the figures shown in the manuscript
> and considered them straightforward to be applied to real data.
> I would recommend this to be published after some minor clarifications
> and corrections.
>
> 1. In the Introduction Section starting at line 15, the authors introduced
> three kinds of wave modes upstream of Mercury's bow shock.
> In the second sentence of this paragraph, it is stated that all three waves
> are driven by right-hand resonant resonant ion beam instability from
> Earth's foreshock study. However, ion cyclotron waves from pick-up ions
> are rarely reported at Earth, although mathematically the difference
> between ICW and the "whistler-magnetosonic" waves only lies in the
> initial beam velocity in the satellite frame (Ub vs 0). Do the authors
> claim that all three modes come from the same generation mechanism? Please clarify.
R02.01
We are constructing a model of the whistler wave excitation just below the ion cyclotron frequency in the spacecraft frame (the waves at 0.3 Hz in Le et al., 2013) and the the pickup ion cyclotron wave (Schmid et al., 2022). The wave excitation at higher frequency such as at 2 Hz in the spacecraft frame needs a different excitation mechanism such as parametric instability, an anti-sunward streaming ion beam, or a sunward streaming electron beam.
See page 1, lines 17--22 in the revision.
> 2. In Figure 2, I suppose the beam cyclotron frequency
> Omega_b = qi * B / mi = Omega_i. However, in the caption it is
> stated that Omega_b is negative, which I don't fully understand.
> Also, in the following text Omega_p is used instead of Omega_i,
> especially for Equation 3 which is represented as the resonance
> line in the figure. It is better to be consistent. Besides, what does the
> dashed line mean in the plot?
R02.02
Both figure 2 and figure caption are improved. Figure 2 shows an extended curve for the R+ mode so that the two roots of the resonance equation are visible.
Figure caption is changed as follows:
"Dispersion relations and the beam resonance condition. Protons are assumed for the bulk ions and the beam ions with the cyclotron frequency Omega_i . The wave modes are the whistler wave propagating forward to the mean magnetic field (R+ mode) and backward (L- mode) and the ion cyclotron wave propagating forward (L+ mode) and backward (R- mode). The symbols R and L refer to the polarization, i.e., dielectric response in the Stix notation, and the plus and minus signs refer to the propagation sense with respect to the mean magnetic field. The beam resonance condition is indicated by the thin line, and intersects the y-axis (long wavelength limit) the ion cyclotron frequency in the negative frequency domain. The beam resonance with the R+ mode is considered here. The resonance with the R- mode is unlikely because of the opposite group velocity direction to the beam (no sufficient time for the energy exchange)."
We used Omega_i not Omega_b in the caption. The dashed line (indication of the ion cyclotron frequency) was deleted. Also, we use Omega_i coherently in the revision (both main text and figures), and not Omega_p anymore.
See figure 2 and figure-2 caption in the revision, page 4 top.
> 3. Starting from Line 62, the normalized quantities are introduced
> with a tilde symbol. I suggest using \tilde{\omega} and \tilde{k}
> on the right-hand side of the arrow expressions.
R02.03
Arrow was replaced by the equation in the definition of normalization (paragraph after Eq. 3).
See page 4, lines 73--75.
> 4. Line 70: Consider adding \tilde{U_b} notation to the text.
R02.04
Done. See page 4, line 80.
> 5. Equation 9: missing tilde for U_+, and subscript b on the LHS.
R02.05
Done. See page 5, Eq. (9).
> 6. Line 81: As I understand, the parabolic approximation refers to the
> dispersion curve of R+. However, due to the limited plotting range,
> the large wavenumber solution is not shown in Figure 2.
> I think this can be better shown by extending Figure 2 range or
> simply add an explanation to the text.
R02.06
Done. See Figure 2 (page 4 top).
> 7. Line 115: The third condition is the same as Equation 9
> (which can be referenced in the text).
R02.07
Done. See page 7, lines 133--134.
> 8. Line 117: The threshold 0.5 is a rough estimation, as the
> actual value depends on \omega^\prime. I suggest the usage of
> approximately less than (i.e. lesssim) or similar notations here.
R02.08
Done. See page 7, lines 136--137.
> 9. Line 122: Better to be consistent whether or not to use
> short-hand notations such as Fig.
R02.09
Done. See page 7, lines 141--143.
> 10. Line 139: The unit of ion inertial length "km/rad" seems wrong.
R02.10
It is not wrong. The Alfven speed is given in units of km/s and the
ion cyclotron frequency in units of rad/s. The combination V_A/Omega_i
is [(km/s) (s/rad)] = [km/rad].
Citation: https://doi.org/10.5194/egusphere-2025-60-AC2
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RC3: 'Reply on RC2', Hongyang Zhou, 02 Apr 2025
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RC4: 'Comment on egusphere-2025-60', Anonymous Referee #3, 03 Apr 2025
This paper presented an analytic model for the ion beam instability with application to the Mercury upstream waves. The ion beam instability model determines the frequency and the wavenumber by equating the whistler dispersion relation with the beam resonance condition in favor of planetary foreshock wave excitation. By introducing the Doppler shift in the instability frequency, their model derived the observer-frame relation of the resonance frequency to the beam velocity and the flow speed. The frequency relation serves as a useful diagnostic tool to the Mercury upstream wave studies in the upcoming BepiColombo observations.
Comments:
The authors gave the example of pickup ions at Mercury as part of the low-frequency EM waves observed upstream of Mercury's bow shock in the introduction section. However it is difficult to see the relevance of this whistler wave model to the pickup ions at Mercury. Furthermore paragraph 95 stated that "the ion cyclotron frequency is expected for the beam instability for the pickup ions by substituting the sign-reversed flow speed into beam velocity as U˜b = −U˜f (pickup ion cyclotron waves) ". However, it is unclear how the authors made that "jump" from equation 12 to pickup ions when the equations were derived for whistler mode waves. Further clarification should be provided.
It is also unclear how this technique is useful to future Mercury's studies. The idea to "back-calculate" the resonance frequency in the spacecraft (or observer) frame if there is information on the beam velocity and flow speed is interesting. The flow speed can be easily assumed to be the solar wind flow speed. However, the ion beam is challenging to observe even with high-resolution plasma data at Earth, let alone accurately calculate its velocity. Further clarification is needed to address how such information can be obtained from future Mercury studies in the context of Bepi-Colombo's observational capability.
Citation: https://doi.org/10.5194/egusphere-2025-60-RC4 -
AC4: 'Reply on RC4', Yasuhito Narita, 30 Apr 2025
> Referee 3 (RC4)
>
> This paper presented an analytic model for the ion beam instability
> with application to the Mercury upstream waves. The ion beam instability
> model determines the frequency and the wavenumber by equating
> the whistler dispersion relation with the beam resonance condition
> in favor of planetary foreshock wave excitation. By introducing the
> Doppler shift in the instability frequency, their model derived the
> observer-frame relation of the resonance frequency to the beam velocity
> and the flow speed. The frequency relation serves as a useful diagnostic
> tool to the Mercury upstream wave studies in the upcoming
> BepiColombo observations.
>
> Comments:
> The authors gave the example of pickup ions at Mercury as part of the
> low-frequency EM waves observed upstream of Mercury's bow shock
> in the introduction section. However it is difficult to see the relevance
> of this whistler wave model to the pickup ions at Mercury.
R03.01
Our model can be applied to the pickup ion cyclotron wave in the 1-D setup for parallel propagation. Figure 3 has been improved to the foreshock ion scenario (left column) and the pickup ion scenario (right column). The observed wave appears as left-hand polarized wave at the ion cyclotron frequency, but in our model, it is the Doppler shifted whistler wave with the reversal of frequency.
See Figure 3 and caption text (page 6 top) in the revision.
> Furthermore paragraph 95 stated that "the ion cyclotron frequency is
> expected for the beam instability for the pickup ions by substituting
> the sign-reversed flow speed into beam velocity as U˜b = −U˜f
> (pickup ion cyclotron waves) ". However, it is unclear how the
> authors made that "jump" from equation 12 to pickup ions when
> the equations were derived for whistler mode waves. Further clarification
> should be provided.
R03.02
A paragraph was added to explain the scenario of pickup ion cyclotron wave.
See page 5, lines 111--113.
> It is also unclear how this technique is useful to future Mercury's studies.
> The idea to "back-calculate" the resonance frequency in the spacecraft
> (or observer) frame if there is information on the beam velocity
> and flow speed is interesting. The flow speed can be easily assumed
> to be the solar wind flow speed. However, the ion beam is challenging
> to observe even with high-resolution plasma data at Earth, let
> alone accurately calculate its velocity. Further clarification is needed
> to address how such information can be obtained from future
> Mercury studies in the context of Bepi-Colombo's observational capability.
R03.03
The backstreaming ions are observed by MESSENGER (Glass et al., 2023). The pickup ion cyclotron waves are also observed by MESSENGER (Schmid et al., 2022). Combination of Mio MGF and Mio MPPE/MIA instrument is ideally suited to test for our instability model against the spacecraft data.
See page 11, lines 222--225 in the revision.
Citation: https://doi.org/10.5194/egusphere-2025-60-AC4
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AC4: 'Reply on RC4', Yasuhito Narita, 30 Apr 2025
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AC5: 'Comment on egusphere-2025-60', Yasuhito Narita, 30 Apr 2025
Here are more detailed explanations and reference as a supporting material to my reply.
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AC7: 'Reply on AC5', Yasuhito Narita, 30 Apr 2025
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AC8: 'Reply on AC7', Yasuhito Narita, 30 Apr 2025
The first term on r.h.s. in Equation (24) has to be corrected accordingly to the change in Eq. (25). Here is the correct one. The first term vanishes in Eq. (24), and the second term is rewritten using c and V_A, and the third term the Hall-term correction (which makes the waves dispersive into whistler sense).
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AC8: 'Reply on AC7', Yasuhito Narita, 30 Apr 2025
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AC7: 'Reply on AC5', Yasuhito Narita, 30 Apr 2025
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