We thank the Referee for carefully reading our paper  and for appreciation of our work.

Reviewer (R)

The manuscript deals with observations by the MMS spacecraft from Earth's quasi-perpendicular bow shock. The paper covers a very wide range of topics like: the fine structure of the shock, oscillatory shock motion, and both ion and electron heating/energization. The paper builds mainly on work done by the same author(s), which does not give the impression that this work is of very wide interest. Furthermore, to my assessment, much of the reasoning and many of the conclusions are poorly supported or even wrong. I therefore can't recommend this paper to be published in ANGEO and instead recommend to reject the paper in its present form. Please see more detailed comments below. I don't think any revisions to this manuscript can sway me to recommend publication. I can see that the other reviewer has a very different opinion to my own. Perhaps a third reviewer can brought in to asses this paper?

Authors (A)

We thank the Reviewer for carefully reading our paper and providing many comments. Unfortunately, some negative comments and the conclusion focus on imponderables or refer to some misconceptions in shock physics. We hope that we have now clarified these issues in our reply and in the revision of the manuscript.

(R1) Lines: 86-87 "The decomposition suggests that the oscillatory behaviour of the shock and wave steepening process are related to the ∼1 mHz wave at the bottom, which triggers cascades of compressional waves extending to 1 Hz and above". It is not clear to me what the authors mean with this statement. The 1 mHz wave in the bottom of Figure 1d is the result of the spacecraft crossing the bow shock several times and filtering the time series. The peaks are the magnetosheath intervals and the dips are the solar wind intervals. So this is not a wave at all. The higher-frequency waves appear when the spacecraft are near the shock ramp/foot. If the authors wish to argue that a 1 mHz compressional wave modulates the position of the bow shock, they would have to identify this wave in the upstream solar wind where the measurement is not affected by observing the compressed magnetosheath.

(A1) The Reviewer claims that the observed wavy motion of the shock front in Figure 1 is not a wave. The data clearly show that shock oscillates with frequency of ~1 mHz. Periodic oscillations are generally regarded as waves. However, to avoid confusion, we have changed the sentence in question to:

“The decomposition suggests that the oscillatory behaviour of the shock and wave steepening process are related to the large-scale ∼1 mHz oscillation seen at the bottom of Fig. 1, which causes the spacecraft to exit and re-enter the shock.  The oscillation triggers cascades of compressional waves extending to 1 Hz and above."

(R2) Lines 141-149: The authors claim that the well-described gyrating ion beam observed in the foot of quasi-perpendicular shocks is not due to ion reflection. Instead, they claim that these are accelerated by lower hybrid waves near the shock. 

(A2) Yes, this is a correct statement, part of our conclusions supported by test-particle simulations. We are not familiar with any other plasma process which could create coherent secondary beam by scattering the solar wind beam on magnetic field turbulence in shocks with δB/B~1. Furthermore, the secondary beam is not "gyrating" as the Reviewer claims, but displaced perpendicularly from the ExB drift direction. Our model of SRA (stochastic resonant acceleration) provides for the first time physically justified explanation of this phenomenon. 

(R3) The idea that supercritical collisionless shock waves reflect a portion of the incoming ions is fundamental to how energy is understood to be dissipated at shocks, see (e.g. Kennel 1987). Any alternative theory to ion reflection needs to address this central question to collisionless shock physics.

(A3) Protons accelerated to some 5 keV by the ExB (or SRA) mechanism described in our paper would have gyroradius of 600 km in the magnetic field of 10 nT. The gyroradius of these ions exceeds by factor 6 the thickness of the shock ramp, so they could remove excess heat from the shock needed by the process described by Kennel (1987) and based on resistive MHD equations. Many brilliant physicists working on shocks some 30 years ago had no access to multipoint measurements of high resolution now available, and were unaware of the stochastic heating mechanism that operates in shocks due to large amplitude waves generated by plasma instabilities. The stochastic heating does not need "anomalous resistivity" and is clearly outside the scope of fluid models. 

(R4) The authors show that a proton can be accelerated by waves but it is not clear to me how these calculations correspond to the observations or if they are able to quantitatively reproduce the observed ion distributions.

(A4) We show indeed with test-particle simulations that secondary beams in the perpendicular plane can be created by SRA mechanism. The secondary beam can appear in any direction, related to the phase velocity of waves. Observations are consistent with our model. A dedicated simulation of the whole particle distribution will be the subject of a separate paper.

(R5) I think the lower hybrid wave model is unlikely since shock reflected ions are also observed in hybrid simulations where these waves are not resolved (e.g. Leroy+, 1983; Lowe+, 2003; Hellinger+ 2007, Caprioli+, 2015).

(A5) Shock reflected, or rather magnetic mirror reflected ions streaming along the magnetic field are indeed observed, mainly in quasi-parallel shocks. Interacting with the solar wind beam they produce counter-streaming ion-ion resonant right-handed instability responsible for the creation of quasi-parallel shocks, which is addressed in separate paper (Stasiewicz & Klos, submitted to MNRAS). 

Different situation is observed in quasi-perpendicular shocks discussed in this paper. Distribution functions measured by MMS and shown in Figure 2a,b,c are inconsistent with the concept of reflection which should produce reflected ion beam in panel 2c (parallel direction) and possibly in panel 2a (ExB direction). Instead, the secondary beam is in panel 2b (E direction), which can be explained by the SRA mechanism discussed extensively in our paper. Ions heated and accelerated by the ExB or SRA mechanism would have large gyroradius ~600 km (see A3 above) and should be seen in front of the shock in kinetic simulations. These accelerated ions could appear also in front of the shock in simulations cited in R5. However, we are not in a position to compare results of these simulations with the measured distributions functions shown in Figures 2 and 3. Such comparison should be made and publicised by involved authors. 

(R6) I do not understand the authors' claim that the solar wind and reflected ions "are in the same electric field so they should have the same V_perp". In the solar wind frame (where the electric field vanishes), the reflected ions gyrate around the center of mass. This leads to perpendicular acceleration of the reflected ions in the shock frame (but not of the solar wind ions).

(A6) We explain: the electric field measured by a satellite acts on all particles measured at the concurrent position. If this electric field corresponds to the primary beam marked with magenta circle in Figure 3C1,D1,E1, then the displaced secondary beam does not obey ExB motion, despite the fact of being in the same electric field as the primary beam. To our knowledge, this discrepancy can be explained only by the SRA mechanism as demonstrated in our paper. 

(R8) Of course, it's welcome to see new ideas that challenge old truths about the field of shock physics. But in the end, I don't think that the current manuscript does this convincingly. 

(A8) We hope that our answers A1-A11 clarify the physics of the quasi-perpendicular shock based on careful interpretation of MMS data, and in particular the role of large amplitude electrostatic waves for particle acceleration. 

(R9)  Timing analysis and shock thickness: The inter-spacecraft separation at this event was roughly 20 km. The small separation, together with the strong wave activity at the shock, can reasonably make the timing analysis uncertain. The authors claim the uncertainty is roughly the orbital speed of the spacecraft without any explanation why. In my opinion, this casts doubt on the statement on line 248: "Using exceptional quality, multipoint measurements of MMS we have made exact determinations of the shock ramp thickness". 

(A9) The timing analysis (Schwartz, 1998) was made by finding the time lags Δt between two signals within the ramp using the least squares method. Strong wave activity in the magnetic signal sampled at 64 Hz introduces indeed some uncertainty into the results. We have used multiresolution wavelet decomposition to remove high frequencies which produce jitter. Wavelet decomposition was chosen instead of low-pass filtering to avoid introducing phase distortions. The least squares values were minimised for signals at frequencies f=0-2 Hz, which were used to determine the shock velocity. The neighbouring frequency level f< 4 Hz gave a velocity difference ~2 km/s, which we assumed corresponds to the error of the analysis. 

(R10)  Line 115: "Lower hybrid drift waves, can be identified in the frequency range fcp-flh." It is not clear to this reviewer how these waves are identified as lower hybrid waves. Frequency is generally not a good tool to identify waves in the fast-flowing solar wind due to the unknown doppler shift. Identifying lower-hybrid waves at shocks require careful analysis of the observed dispersion relation, see (e.g. Walker+, 2008).

(A10) Observations of lower hybrid waves in the dayside magnetosphere have been reported by many authors (Bale et al.,2002, Vaivads et al.,2004, Walker et al.,2008, Norgren et al.,2012). We have discussed identification of LHD and other wave modes in previous articles, so we feel it is not necessary to duplicate it again in this paper. The measured waves have large amplitudes 10-100 mV/m in the frequency range 1-1000 Hz, so the linear dispersion equations in broadband, fully developed turbulence are not valid. There are also other wave modes, like whistlers in this frequency range. Separation of wave modes in strong turbulence is difficult, if at all possible. These waves are observed in regions of strong density gradients with spatial gradient scales L ~ 50-200 km, as demonstrated in earlier papers. Since the density gradients destabilise first LHD waves in the frequency range f_cp-f_lh, the observed waves in this frequency range have been labeled as 'LHD' waves.  These waves would accelerate and heat ions irrespectively on the labels 'LHD' or  'f_cp-f_lh' which we may put on them.

(R11) The FPI-DIS instrument onboard MMS was not designed to measure the cold solar wind beam and tends to overestimate the ion temperature in the solar wind. The values of ion beta and gyroradius in the manuscript are likely overestimates.

(A11) Yes, we agree with the Reviewer and we have added a sentence that the values are likely overestimated.

In summary, we have carefully addressed all points raised by the Reviewer and amended the text in the revision, where appropriate. We hope that the points raised have now been clarified.

Thank you very much for this opinion. Indeed, the stochastic resonant acceleration (SRA) presented in our paper is the first physically grounded mechanism that explains the origin of secondary perpendicular beams observed at the Earth’s bow shock.

Reviewer (R)

The manuscript presents observations of the Earth’s bow shock made in a series of crossings by Magnetospheric Multiscale on 3 January 2020, and examines the relationship these observations have with stochastic resonant acceleration (SRA), a relatively novel particle acceleration mechanism developed by the authors. The SRA mechanism is based around evaluation of the divergence of the electric field, whereby gradients in the electric field can cause non-adiabatic particle behavior and therefore heating. The mechanism further recognises that the relevant electric field is not simply the large-scale convection electric field, but must also consider the impact of the electric fields of different waves, at a variety of spatial and temporal scales. Through so-called acceleration lanes, it is possible for particles to be preferentially and therefore efficiently accelerated. I find that the basic idea proposed by the authors is certainly interesting and thought-provoking, and in particular it is exciting that the experimental capabilities of MMS are at a point where it can be sensibly tested. Overall the manuscript is well written and presented and so here I restrict my input to only major comments, which I offer to the authors with the aim of helping further improve the manuscript.

Authors (A)

We thank the Reviewer for this insightful opinion and for detailed comments below. 

R1

Novelty. The authors have published a lot of papers in a short period of time on the same data interval and concept. Please note this is not a criticism! The SRA mechanism has been discussed in Stasiewicz MNRAS 2020 (original presentation); Stasiewicz and Eliasson ApJ 2020a, 2020b (further development); Stasiewicz and Eliasson MNRAS 2021, Stasiewicz et al. JGR 2021 (ion acceleration to 100 keV). These papers use test particle simulations to examine stochastic behavior and also compare with MMS observations from 3 January 2020 and 6 January 2018. I had to look carefully to really understand what was new here, and the same would apply for the average reader. Please expand the paragraph starting on line 49 to explain to the reader what the essential new developments are in this manuscript. From my perspective, this would include the examination of the ion distribution function but of course there may be other things that should be highlighted.

A1

The novel results of this paper include:

(a) the first identification/introduction of the SRA mechanism in literature. The mechanism was implicit, but not identified and named in previous publications.

(b) demonstration that the "shock reflected ions" discussed in numerous publications are in fact ions accelerated by the SRA mechanism.

(c) new interpretation of double peaks in perpendicular ion distributions observed at shocks.

(d) demonstration that high ion temperature in the foot of the shock is an artefact of  double beams in ion distribution (Figure 2b,d).

(e) demonstration that Equation (3) provides a good estimate of ion acceleration in quasi-perpendicular shocks (Figure 1a).

(f) high-accuracy determinations of thickness and velocity of the bow shock.

In our opinion, each of these findings alone is sufficient to warrant publication in any journal. 

We believe that a priori listing of these findings in the Introduction, before data presentation and numerical analysis would be inappropriate and not understood by readers. The new results are listed in Conclusions.

We regard explanation of double peaked distributions in the perpendicular plane by the SRA mechanism introduced in this paper as one of the most important achievements in shock physics since the discovery of the bow shock in 1964. These secondary beams have been described in literature as "shock reflected" ions, without explaining the physical mechanism that produces these distributions.

R2

Shock motion. Lines 79-83 describe the decomposition of the measured magnetic field. The residual dc field shows an oscillation, but I don’t understand what the physical meaning of this is, at least in the context of the original data where the upstream field strength is fairly uniform. The shock crossings are some sort of square wave in the time series which would give this decomposition, but is it then the case that this oscillation can be related to surface waves on the shock itself? Apologies if I am misunderstanding things here. I think I also do not follow line 87, on how this low frequency wave ‘triggers cascades’ to higher frequency. Please clarify this discussion.

A2

Indeed, the decomposed magnetic field in Figure 1d can be interpreted in different ways. To remove the ambiguity we have re-phrased this paragraph to the form: 

"The decomposition shows cascade of compressional waves with the lowest frequency of oscillation at ~1 mHz  seen at the bottom. The oscillatory movement of the shock causes the spacecraft to exit and re-enter the shock.  The  compressional waves extend from 1 mHz to 1 Hz and above with maximum amplitude collocated with the strongest gradients of B and N." 

The modified text describes only observations without any interpretation, which we hope will remove the controversy.

R3

Universality of results. Line 94 notes that the analysis pertains to shock 4. This provides a good illustration, but please explain why this shock was chosen. Do the other shocks show similar behavior and properties? This is related to a separate point below.

A3

All shocks in the studied case have similar wave content and heating/acceleration capacity, which can be seen in Figure 1. Any shock could be used in this paper with the same results/conclusions. 

R4

Operation of SRA. It would be valuable to also show the stochastic heating threshold as per equation 2 and/or the energisation capacity, to help the reader get a sense of where the SRA mechanism could be important. Please add this to Figure 2.

A4

We include a new figure that shows the energisation capacity given by Eq (1) and frequency decomposition of parameter χ.

R5

Choice of time for distributions in Figure 3. Obviously it is not possible to show every distribution function, but please (check and) confirm that these are representative of the different regions in Figure 2 and note this in the text.

A5

Yes, the distributions shown in Figure 3 are representative for the locations where they are measured. However, all 1D  distribution functions measured across the shock have been displayed in  Figures 2a,b,c with 0.15 s resolution.

R6

Theoretical development. The analysis in section 2.3 shows plausibility, but the manuscript would be more convincing if it was possible to say more about (a) if the mechanism uniquely increases the energy of particles (can you construct counter-examples where the particle will be decelerated?) (b) if a population of particles will tend to be accelerated (c) if there is an efficiency that one can calculate? On this last point, I can see the argument that a particle may indeed be accelerated in some sort of stochastic sense that depends on its interaction with separate wavepackets of different location, duration, and frequency, but does this mechanism act on sufficient numbers of particles? I am not sure that the MMS data here is sufficient, i.e. the simple existence of energised particles is not proof in and of itself. Is there a simple theoretical calculation based on the amplitude of the waves/electric field properties etc. that can be used to predict the fraction of the population that is accelerated? This could be an important theoretical development to understand where and when SRA is important. As an alternative approach, do you find that there are different numbers of accelerated particles in the different shock crossings, and are these correlated to the presence or otherwise of electric field fluctuations that would enable SRA?

A6

Despite the simplicity of the Lorentz equation used to compute particle trajectories, in the stochastic regime particle trajectories are widely different for small differences in initial conditions. There are initial conditions that lead to reduced energy of particles. However the process is generally in one direction in a statistical sense. Accelerated ions acquire higher velocities which puts them outside resonant interactions with waves. They cannot be decelerated by a reverse process.  The efficiency of heating in a statistical sense has been addressed  using a large number of particles with initial Maxwellian distribution in papers http://dx.doi.org/10.3847/1538-4357/abb825, http://dx.doi.org/10.3847/1538-4357/abbffa, http://dx.doi.org/10.1093/mnras/stab2739. The results are in the form of heating maps which are parametrised by 3 parameters: normalised frequency Ω, initial temperature or normalised gyroradius, and the value of chi. The maps make it possible to identify parameter range where heating is most efficient. In paper by Stasiewicz et al JGR 2021 we have shown that a collection of waves is capable to reproduce the measured particle distribution in energy range 10 eV - 200 keV, which speaks in favour of this mechanism. 

R7

On line 214-215 it states that “The stochastic condition in Equation (2) is necessary for energisation of particles. When χw < 2 no significant acceleration can be produced by Equations (5)-(9), irrespectively of the values of other parameters.” I realise that this is a somewhat imperfect approach, but it would seem to me that there should be some macroscopic trends that can be found from the data, for example the duration of observed intervals where χw > 2 compared to the number density of energised particles that could provide further evidence for the underlying ideas. A valuable addition to the manuscript would be characterising/quantifying in all the crossings the stochastic condition and the occurrence or otherwise of acceleration. This could be summarised in a table or scatterplot; comparing and contrasting two different shock crossings could also be useful. From my perspective, adding more information about the other shock crossings, and making an attempt to quantify more clearly the presence or absence of the proposed mechanism would strengthen the case for publishing the work contained in the manuscript.

A7

As mentioned earlier all shocks have similar wave content and similar energies of accelerated particles. In all cases the stochasticity parameter is much larger than the threshold of 1. Strong turbulence and strong stochasticity that characterises all shocks excludes the possibility of obtaining predictive semi-analytical results. However, there is a  dramatic difference between quasi-perpendicular shocks, where acceleration is up to 10 keV and quasi-parallel shocks where acceleration is up to 200 keV. Quasi-parallel shocks are analysed in a separate paper, now under consideration by MNRAS.  

We hope that our paper would inspire other researchers to continue the investigation of this mechanism. There is certainly need for kinetic simulation of the described processes by independent researchers.