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: open (until 02 May 2025)
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RC1: 'Comment on egusphere-2025-60', Anonymous Referee #1, 25 Mar 2025
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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 -
RC2: 'Comment on egusphere-2025-60', Hongyang Zhou, 01 Apr 2025
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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
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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
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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
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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
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