Peer review of: Local heating of radiation belt electrons to ultra-relativistic energies (2020)

Response to reviewers
We thank all of the reviewers for their time. Their comments are addressed below in blue. Unless otherwise stated, line numbers correspond to the model up-to-date version of the track-changes manuscript.
Reviewer #1
Having reviewed the author responses and changes to the manuscript, I unfortunately still have great concerns
about the conclusions of this paper. Overall, I don’t see that the data here supports the conclusions
unambiguously. Nowhere in the manuscript do you say that inward radial diffusion was clearly acting over the day
of Oct 9th on higher energies, although it is very evident in the data (see point 2 below), so it appears that you are
actively avoiding this discussion.
As requested in the previous review, we did add discussion of inward radial diffusion to the revised
manuscript – see lines 102-103 (previously on lines 113 – 114 of the tracked changes document),
quantifying the energy change produced, which is only ~600keV, less than 10% of the energy of the
observed 7.7 MeV electrons. We included a discussion on lines 186 -189 (previously on lines 200 – 201
of the tracked changes document) highlighting that inward radial diffusion is always active alongside
local acceleration because local acceleration introduces radial gradients in phase space density. We
also had a section in the supplementary material discussing inward diffusion (supplementary section
10). Hopefully these discussions that were overlooked by the reviewer in the previous review will
satisfy their questions.
 I think this topic is extremely timely and of great interest to the community but I do not see the evidence in this
 event as presented. It may be possible to find another event which unambiguously shows local acceleration
 directly leading to 7.7 MeV electron enhancements (very clear growing peak signatures – much more clear than
 shown here), and I encourage authors to look for an event like this.
We are glad to see that the reviewer finds this topic interesting, and we take their suggestion that we should continue to analyze the phase space density profiles surrounding other 7.7 MeV enhancement events. We believe that the profiles shown here are conclusive and note that reviewer #2 and reviewer #3 also did not raise concerns regarding the interpretation. Here we have included an annotated version of figure 5 to show where we see growing peaks:
 
Annotated Figure 5
1
 In summary, I do not recommend publication of the manuscript. It is an interesting analysis of the storm event, but
without robust support of the stated conclusion, it is not new enough to constitute publication in this kind of
journal.
We thank the reviewer for their time and review and appreciate the comment that they find the analysis interesting. We are sorry to see that they do not recommend publication. We consider the results presented in the manuscript to be clear and have addressed their points below.
   Major issues still unresolved:
 1. In the first review, I asked that you show evidence that the 7.7 MeV electron channel was above background. It
 turns out it was at the 1 or 2 count level for the beginning of the Oct 8-9 period. The same is true for some of the
 lower energy channels. Channels 5-7 appear to be below background until the pass starting at ~2 UT on 10/9,
 and Channels 6-7 are still below background until the inbound pass starting at ~6 UT on 10/9 and the next
 outbound pass starting at ~11 UT on 10/9, respectively. However, for the Oct 13-14 period, all channels were
 above background levels and that data is viable. The concern that comes with this surrounds the validity of the
 PSD profiles for the Oct 8-9 period. For the first part of this period, the flat PSD curves (dark blue and black in
 your plots) are meaningless. The data used to derive PSD is not robust. The first light blue curve is also not
 representative, since the values at low L will be from those data points below background (you can see the PSD
 values are very close to the dark blue and black curves).
 For the storm time the instruments were above background. Black and dark blue curves may have
 been close to the background level prior to the first event, but these have been included to illustrate
 the upper bound of the phase space density prior to the enhancement. A phrase to this effect has now
been added to the manuscript on lines 108 - 110.
  The second light blue curve is the most valid point to start the analysis – here, all data points are likely above
 background and we can trust the shape of that profile. Except, when I look at the publicly available PSD, I see
 PSD value are not the same as what you show. The magnitudes are very similar, but the trend doesn’t always
 look quite the same. This can be true depending on the different methods used to calculate PSD, but in this case I
 would say it is not clear that there are growing peaks in either storm case.
In the main manuscript we used the
 TS07 field model for the K and L* invariants and the local measurement of the magnetic field for the calculation of mu. The profiles corresponding to the TS07
field model do not load in the gateway. The gateway returns the following figure:
 So here we have included the phase space density profiles from the publicly available gateway with the
 TS04 field model instead. Note that growing peaks are still observed.
 2

 The gateway does not provide documentation on how the phase space density was calculated, so it is difficult to comment on whether there is a differing approach to how the data was processed. Note that we have made our script for interpolating the phase space density to a target mu and K pair available to the reviewers, as well as the PSD values, so that you can see our processing steps.
In
During the first storm, as shown in Figure 5 (and the annotated figure 5 at the start of this document):
• negative phase space density gradients are observed between L* = 4.6 and L* = 4.9 throughout the enhancement, at both mu = 10,000 MeV/G and mu = 8,000 MeV/G.
• During this time, the peak phase space density increases from levels near the background threshold of the instrument to 4 ́109 c3cm-3MeV-3 for mu = 8,000 MeV/G and to 8 ́1010 c3cm- 3MeV-3 for mu = 10,000 MeV/G.
This is a growing peak.
During the second storm, the initial phase space density is higher than for the first storm event, as it has not been fully depleted following the previous enhancement.
Figure 5 (and the annotated figure 5 at the top of this document) shows that:
• The final profile has a peak in phase space density at L* =4.2 with a phase space density value greater than the pre-storm levels
• Profiles show negative phase space density gradients between L* = 4.6 and L* = 4.9 throughout the enhancement.
• The light blue inbound pass of Van Allen Probe A (solid line) is the exception, which shows a positive gradient in this L* range. However, the phase space density at the largest L* during this pass is still lower than the value of the final peak, and so inward diffusion from this point cannot explain the resulting enhancement. Also, the pass immediately after (dotted outbound light blue pass from Van Allen Probe B) again shows negative phase space density gradients
  fact, from your Oct 13-14 plots, I don’t see growing peaks at all. Green and blue curves look like positive or flat
 gradients – the peak is only evident in the red (maybe orange for 8000 MeV/G) traces, at which time the
 maximum magnitude has stopped increasing.
  3

between L* = 4.6 and L* = 4.9 (as do all subsequent passes). The change in phase space density between the solid light blue and dashed light blue passes is small for mu = 8,000 MeV/G and not seen for mu = 10,000 MeV/G.
The negative phase space density gradients between L* = 4.6 and L* = 4.9 and the increase in the level of phase space density constitutes a growing peak.
Figure S1 shows the phase space density for all of the different values of mu on the same color scale. We presented the data in this way for this supplementary figure to enable a comparison of the phase space density magnitude between the mu values. As a result, for the larger values of mu, where the phase space densities are much smaller, it is difficult to assess the scale of the changes.
The line plots of phase space densities shown in figure 5, and throughout the supplementary material, enables the change in phase space density to be quantified. These line plots show growing
peaks.
For the benefit of the reviewer, we include here the mu = 8,000 MeV/G and mu = 10,000
MeV/G panels of Figure S1 (now Figure S2) on a different color scale. An increase in color at some mid-L value can be observed.
 You can see this in the Supplemental information as well. Look at your Fig S1, 8000 and above panels. You see
 no growing peak – or else you would see a step by step increase in color at some mid-L value. You can check the
 public PSD data on the RBSP Gateway for comparison: http://rbspgway.jhuapl.edu/PSD
    2. This next issue is in line with the previous one, but directly relates to my original second major issue (called
 “Line 88-89” in the first review). I still see clear evidence of inward radial diffusion from Figure 4. It’s hard to see all
 the Mu values clearly, but at least for the 10000 MeV/G and 8000 MeV/G values (the two red layers on top), there
 is very clearly an inward movement of the peak before a flattening and/or outward motion. I’m including an
 annotated figure with my review to show the period I’m referring to.
 In the previous revision of the manuscript, we included lines 113 -114 (line numbers correspond to the
 previous track-changes version of the manuscript) to address this comment, acknowledging the slight
 inward motion of the peaks and quantifying the increase in energy range that this corresponds to.
 From Figure 2, the motion from L* = 4.5 to L* = 4.2 is ~600keV, less than 10% of the energy of a 7.7
 MeV electron. We have now reworded lines 102 -103 (in current version of the manuscript) for
 clarification.
4

 As for the radial diffusion modeling, this can be tuned according to what diffusion rates are used and we know that
 many events do not follow the Kp parameterized values but have event-specific rates that can differ greatly from storm to storm. Showing one run of a 1-d diffusion model does not support the conclusion that local acceleration
 was the main driver here. Other groups have successfully modeled all Van Allen Probes observations using radial
 diffusion alone (see Mann and Ozeke, 2016, JGR) – it just depends on the rates you decide to use.
While we agree that radial diffusion coefficients can be tuned to give particular results, we stress that
  for the runs included in the supplementary material we have used the standard parameterized radial
 diffusion coefficients from Brautigam and Albert, 2000, which have not been tuned, and are driven by
 the observed values of the Kp index for the period in question. These coefficients have been used in
 many publications (Glauert et al., 2018; Zhao et al., 2018; Drozdov et al., 2017, etc) and generally
 reproduce observed dynamics well. Of course, the model runs in the supplementary material are not
 the main focus of the paper, which instead concentrates on observations, and are included only as an
 illustration, and do not affect the conclusions. Note that Drozdov et al., 2017 provides a comparison
 between using both the Brautigam and Albert, 2000 diffusion rates and Ozeke et al., 2014 diffusion
 rates in a numerical model, and finds that they give similar results.
 References
Glauert, S. A., Horne, R. B., & Meredith, N. P. (2018). A 30-year simulation of the outer electron radiation belt. Space Weather, 16, 1498– 1522. https://doi.org/10.1029/2018SW001981
Zhao, H., Baker, D. N., Li, X., Jaynes, A. N., & Kanekal, S. G. (2018). The acceleration of ultrarelativistic electrons during a small to moderate storm of 21 April 2017. Geophysical Research
Letters, 45, 5818– 5825. https://doi.org/10.1029/2018GL078582
Drozdov, A. Y., Shprits, Y. Y., Aseev, N. A., Kellerman, A. C., and Reeves, G. D. ( 2017), Dependence of radiation belt simulations to assumed radial diffusion rates tested for two empirical models of radial transport, Space Weather, 15, 150– 162, doi:10.1002/2016SW001426.
Ozeke, L. G., I. R. Mann, K. R. Murphy, I. Jonathan Rae, and D. K. Milling (2014), Analytic expressions for ULF wave radiation belt radial diffusion coefficients, J. Geophys. Res. Sapce Phys., 119, 1587–1605, doi:10.1002/2013JA019204.
Each pitch angle and energy bin is converted to phase space density by dividing by the particle momentum squared. We then average the phase space density within each energy and pitch angle channel into 5 minute intervals. The resulting PSD grid is then interpolated to the targeted mu and K values. The PSD does not vary much over the 5 minute time frame.
Related to that, in your method you say that you interpolate PSD to a target value of K, but don’t mention a model
                                               3. I suggest plotting RBSP-B PSD profiles as well, since A shows some complicated dynamics during both of
 these storms, while B seems to catch the belts during the more stable intervals (overall). I don’t think this will
 strengthen the argument you are putting forth, but it may be worth looking at more closely.
 There is a misunderstanding here, we do show data from probe A and B as stated in the text and
 figure captions.
 4. In your newly added Supplemental material, you say you used 5 minute averaged REPT data (Line 33-34).
 How are you using averaged data to get individual sector information for pitch angle direction? Or do you mean
 that you are averaging after calculating PSD? Or are you averaging within each pitch angle bin by 5 minutes?
  pitch angle distribution. Is it a liner interpolation from one K-value to the next to ascertain the point in space that
 matches your target K?
 As discussed on line 43-45 of the supplementary material, we linearly interpolate the log of the phase
 space density to the target value of K.
 If so, this may introduce errors since the pitch angle bins can be very under-sampled (even somewhat close to 90
 degrees depending on angle of spacecraft to Bfield). Usually a model pitch angle distribution is used to account
 for this (which I know has its own problems).
 As discussed on lines 46 – 49 of the supplementary material, we do not use extrapolation and do not
 use a model pitch angle distribution. Although a model pitch angle distribution was not used, as
 shown below, using one has a very limited effect on the output. This analysis has been added to the
 supplementary material as Figure S1 and lines 49 - 51.
5

  The plot shows the phase space density profile at mu = 8,000 MeV/G and K = 0.11 REG1/2, from Van Allen Probe A on an example pass (15th October 2012 02:25:00 to 05:20:00). The linear interpolation of the logged phase space density is used as is in the paper (red line) and we also used a psd(a) = A sin(a) + C pitch angle distribution between the pitch angles surrounding the target value of K (blue curve). The two curves are mostly indistinguishable. This Figure has been added to the supplementary information.
 As an additional test, we again compute the phase space density for the above example pass using all
 of the pitch angle bins (black line) and also using a reduced dataset where we had removed every
 other pitch angle bin (blue line). This reduces the number of available pitch angles from 9 to 5. As you
 can see from the below figure, reducing the number of pitch angles has a limited effect on the phase
 space density output.
  In the paper, the same method for determining the phase space density at a particular mu
The plot shows the phase space density profile at mu = 8,000 MeV/G and K = 0.11 REG1/2, from Van Allen Probe A on an example pass (15th October 2012 02:25:00 to 05:20:00). As in the figure above. The phase space density is calculated using all pitch angle bins (black line) and also calculated using a subset of the data comprising every other pitch angle bin (blue line).
  and K has been used as was used in the Reeves et al., 2013 study, even down to the 5 minute
 averaging. See supplementary information of that reference for details. Reference:
 Reeves, G. D., H. E. Spence, M. G. Henderson, S. K. Morley, R. H. W. Friedel, H. O. Funsten, D. N. Baker, S. G. Kanekal, J. B.
Blake, J. F. Fennell, S. G. Claudepierre, R. M. Thorne, D. L. Turner, C. A. Kletzing, W. S. Kurth, B. A. Larsen, and J. T. Niehof (2013), Electron acceleration in the heart of the Van Allen radiation belts, Science, 341 (6149), 991{994, doi:10.1126/science.1237743.
 What happens if you instead target a certain Mu value and base your interpolation on that? Does it change your
 results much? The reason I asked is based on the discrepancy between your PSD profiles and the ones I have
 access to on the Gateway, which may have been derived in a different way.
 We do interpolate between energy measurements to a specific mu. In the analysis the phase space
 density shown is always a specific mu value and a specific K value. It is unclear what reviewer means
 here.
6

 The mu values we present are calculated with the locally measured magnetic field model as
 described in lines 35-38 of the supplementary material. Note that we made the phase space density
 data available to the reviewers, along with the Matlab script for interpolating the data to the target
 mu and K pair.
 Reviewer #2
 The authors have addressed some of my comments. However, I still have some concerns regarding the
 responses to my third major comment.
 We thank the reviewer for his/her second review. We really appreciate their time and useful
comments.
  The authors have added some discussions (lines 180 – 190) and also figures (Figs S8 and S9) to the manuscript
 to address the spatiotemporal ambiguity in the phase space density radial profile. Specifically, it was stated that
 “Furthermore, the last closed drift shell location is at L* ~5.5 - 6.5 for the first storm and L* ~6 - 6.5 for the second
 (Fig S8). The minimum energy of the source population that would be required for the observed 7 MeV
 enhancements is >3 MeV (Fig 2) and it is unclear where this would originate.” However, Fig 2 does not give the
 electron energy at L*>~5.6, making it hard for readers to know what the corresponding energy of the potential
 source population around GEO would be.
 We agree, the number given was inferred from the trends shown in the figure. We have now
rephrased this line in the manuscript so that it reads “The trends shown in Figure 2 then suggest that
  the minimum energy of the source population required for the observed 7 MeV enhancements is likely
 to be several MeV” as it is difficult to provide the exact energy value. See line 164-166. This is included
 to justify why we look at the >2 MeV channel of GOES.
 Also, the statement that “it is unclear where this would originate” is hard to understand – is it saying that there are
  not so many high
energy electrons at the high L region to act as the source population for >7 MeV electrons in the
 center of the outer belt?
 Here we were commenting that near the last closed drift shell, particles can be lost very quickly and
 there is a limited time for local acceleration to act. Due to limited observations of the region it is
 difficult to understand the dynamics there. This phrase has now been removed as it is not directly
 needed for the conclusions of our study. Instead, we lead into the discussion of the GOES
 observations.
 In addition, it was stated that “These additional observations suggest an absence of a rapidly appearing source
 population for radial diffusion during both events.” However, GOES data in Fig S9 does show some flux
 enhancements of >2 MeV electrons especially during the first storm; and because of GOES’s integral channel
 and often steep-falling electron energy spectrum in the radiation belts, GOES measurements may not show the
 actual corresponding source populations of >7 MeV electrons in the center of the outer belt. I would suggest to
 change the wording and point out this limitation of current in situ measurements, which are not sufficient to infer
 the source populations of ultrarelativistic electrons at the high L region.
 Because of the large pitch angle range sampled by the GOES satellites, and the fact that L* is pitch
 angle dependent, there is some ambiguity as to the exact L* of the populations measured by GOES.
 To account for this, we showed the L* values for a number of different local pitch angles at the GOES
 location alongside the GOES flux plots. The L* range covered by a single GOES measurement can
 extend over ~1 L*. Note that the changes in the flux correspond to transitions in this sampled L*
 range and so are likely a reflection of changes in the population that is being measured. Note also
7

that the flux recorded by GOES is higher in the days following the events than it was during the
acceleration window at L* ~4. For this period, that is consistent with locally accelerated source that
diffuses outwards.
We have now changed the wording here, as requested, and include a discussion regarding
the limitation of current in situ measurements as well as the integral energy channels. See lines 168 –
177 which now read: “
When considered alongside the changes in the L* measured, these additional observations do not suggest a rapidly appearing source population for radial diffusion during either event. Furthermore, the GOES flux is higher in the days following the acceleration occurring at L* ~ 4 than it was during the enhancement period, consistent with a locally accelerated source that diffuses outwards. However, we note that these are again in situ measurements and therefore also subject to certain spatial limitations. The GOES spacecraft may not have been appropriately situated to measure a source population during the period. Additionally, because radiation belt electrons often show steep falling energy spectra, the integral energy channels may not necessarily show the source populations of >7 MeV electrons in the outer belt.”
Another minor comment: a manuscript recently published by Zhao et al. suggested the dominant role of inward
radial diffusion in the acceleration of ultrarelativistic electrons through statistical analysis on 19 electron flux
enhancement events during the Van Allen Probe era. While their results may not necessarily contradict this study as they focused on the general statistics while this study focuses on two specific storms, it is beneficial to discuss
their results in the context of this study. Reference: Zhao, H., Baker, D. N., Li, X., Malaspina, D. M., Jaynes, A. N.,
&Kanekal, S. G. (2019). On the acceleration mechanism of ultrarelativistic electrons in the center of the outer
radiation belt: A statistical study. Journal of Geophysical Research: Space Physics,124, 8590–8599.
https://doi.org/10.1029/2019JA027111.
We agree that this is beneficial to highlight that our results do not conflict with this previous work
and have now included a discussion of the results from the above reference in lines 190-196 of the
manuscript. The above reference has been added to the reference list.
 Reviewer #3
The authors revisions have adequately all of my concerns and I am happy to recommend publication.
We thank the reviewer for their time and review of the manuscript and are happy to see that they recommend publication.
  This paper addresses the acceleration of ultra-relativistic
electrons using phase space density observations from
 the Van Allen Probes. This is an important and relevant result and should be an impactful contribution to the
 community.