The authors wish to thank Reviewer #1 for reading our manuscript and

providing useful and constructive comments. Below we list our point-by-point

response to the Reviewer’s comments. Italics are used to list the reviewer’s

comments and our response is in regular font.

Below the authors list the reviewer’s major comments on the manuscript.

1. Fig. 3 and Fig. 8 show numerical artefacts in XY plots but they do not

show the AMR mesh. It is not clear how these are related exactly. The MS

mentions the AMR mesh can be recovered from these figures (line 143).

No, that is not enough. Please show the AMR mesh explicitly.

We wish to point out that the wording used in Line 143 (“ . . . we demon-

strate results of the boxcar filtering operator in a large magnetospheric

production scale run with four 4 AMR levels as illustrated in Figure 8.”)

is unclear. Figures 8(b,c,e) are quantities on the field solver grid which

is uniform. We have modified the manuscript and now the text reads

(” . . . we demonstrate results of the boxcar filtering operator in a large

magnetospheric production scale run using four AMR levels. Simulation

quantities on the refined AMR mesh are illustrated in Figure 8a and on

the uniform field solver grid in Figures 8(b,c,d).”). To address the raised

point, we have supplemented Figure 1 with an explicit 3D plot of the AMR

Vlasov grid, as the reviewer suggested.

2. Also, why do the authors show smoothed (“fixed”) solutions for Fig.8 but

not Fig.3? This is confusing. If you show numerical issues, it makes sense

to show how you fix them everywhere.

The aliasing effects on the field-solver quantities illustrated in Figure 3

require a lot of simulation time to propagate into the field solver. The

simulations snapshots in Figure 3 are at t=1096.5 simulation seconds.

Vlasiator is a very computationally expensive code and reproducing the

simulation shown in Figure 3 with the filtering operator active would re-

quire large amounts of computational time, for which we do not currently

have a Tier-0 computational grant. The aliasing results can be qualita-

tively compared between different 3D simulations, as shown in Figures

3(d,f) and Figure 8(d,f). As mentioned in the manuscript it is visible

that the profiles in Figures 8(d,f) are not showing unphysical

oscillations unlike their counterparts in Figures 3(d,f) where the filtering operator is

not used.

3. What region exactly do those insets in Fig. 8 represent? Why does one

show coarse cells and the other shows fine cells while they seem to represent

the same AMR mesh with and without filtering?

The spatial coordinates of the insets are exactly the same in Figure 8(a)

and Figure 8(b). Both insets show the mass density at a zoomed in region

close to the Earth’s bow-shock. However, in the revised version of our

manuscript Figure 8(a) illustrates mass density on the AMR Vlasov grid

while Figure 8(b) illustrates mass density on the uniform field solver grid.

The staircase effect is visible at the interfaces of the AMR refinement

levels in Figure 8(a), since the filtering operator is not applied. In Figure

8(b) the filtering operator is applied and the staircase effect is alleviated.

Based on the reviewer’s comment this was not clear enough in the text,

so we have modified the caption of Figure 8 to also read ”. . . The insets

in panels (a,b) show a common zoomed-in region from the bow shock."

4. What sense does it make to show Fig. 8(d,f ) if they cannot be compared

to similar unfiltered profiles?

The authors’ goal in showing Fig8(d,f) is to point that with the filter-

ing operator enabled, the quantities on the uniform field-solver grid do

not suffer from the high frequency oscillations that their counterparts in

Fig3(d,f) suffer due to the staircase effect. We think that even though

the two simulations are indeed different, a qualitative comparison of the

electric and magnetic field profiles in Figure 3(d,f) and Figure 8(d,f) is

enough to gauge whether the staircase effect is alleviated or still induces

artifacts in the field solver.

5.Given all the above, please show unfiltered and filtered plots side by side

for comparison, together with the AMR mesh. Fig 3 and Fig.8(c-f ) do not

carry much information unless you show their counterparts next to them.

We want to clarify that as mentioned in the manuscript the filtering opera-

tor is only applied to the plasma moments and not to the magnetic/electric

fields. Thus we cannot show what the reviewer asks for since unfiltered

electric/magnetic fields do not exist is a simulation where the filtering op-

erator is active. However, Figure 3 is provided as a point of comparison

between a simulation that does not use the filtering operator and one that

does as illustrated in Figure 8. We have also elected not to show the AMR

mesh as it is too fine to allow for the evaluation of the plot and the mesh

at the same time. In the revised version of the manuscript the zoom-ins

in Figure 8(a,b) have the AMR enabled Vlasov grid overlaid on them as

a point of comparison.

6.Overall, my major concern is that it is not clear at this point if the

authors have “cracked” the problem or not. This spatial filtering may be good enough to regularize the bow shock boundary, but this procedure may result in modifying the global solution considerably. The only way to verify

that is to also show a reference solution on a fine uniform mesh. I don’t see those. It looks to me that the MS shows either unfiltered AMR solutions or filtered AMR solutions, without comparing them side by side or showing

uniform mesh solutions next to them.

We thank the reviewer for this comment. We agree that the only way

to verify the filtering mechanism would be to compare the filtered re-

sults with a simulation without the filtering which is also artifact-free.

That would mean running a 3D simulation with no AMR, using the

maximum spatial resolution (highest refinement level in the AMR runs

demonstrated in the manuscript) in the whole simulation domain. We

can extrapolate the computational requirements of such a reference so-

lution. Based on the computational resources used for the simulation

in Figure 8, the best case scenario without taking into account scaling

performance, the same simulation with no AMR would require upwards

of 350 MCPUh for a total of 500 simulation seconds. That would make

the simulation about 22 times more expensive than the ones presented

in the manuscript. To put this into perspective, Prace’s EuroHPC JU

Call for Proposals grants at most 306 MCPUh (https://prace-ri.eu/

hpc-access/eurohpc-access/eurohpc-ju-call-for-proposals-for-regular-access-mode/)

on LUMI which ranks 3rd at TOP500 as of June 2022(https://www.

top500.org/lists/top500/2022/06/). Further, the authors wish to point

out that the filtering mechanism is not applied to the highest resolution

regions in the simulation which are the regions with high scientific impor-

tance.

Below the authors list the reviewer’s minor comments on the manuscript:

1). Lines 100-105: How is Eq.6 related to numerical filters actually used in

the MS? Either strike this equation out or explain how it is supposed to

be used. Is ‘j’ the imaginary unit? Is this formula supposed to be used in

Fourier transforms? Show that exactly in Eq. 7.

We agree with the reviewer’s comment that this equation is not aiding in

the scientific quality of the manuscript and we have removed it.

2. Line 5: strike out ‘resolution induced’

We have removed ”resolution induced” from the text as per the

reviewer’s suggestion.

3. Line 14: rephrase with ‘3 dimensions . . . . space’.

Based on the reviewer’s comment this sentence has been modified to:

”. . . hybrid-Vlasov plasma simulation code that models collisionless plasma

by solving the Vlasov-Maxwell system of equations for ion particle distri-

bution functions on a six dimensional Cartesian mesh, representing three

spatial and three velocity dimensions.

4. Line 19: strike out ‘which in practice’

Done, thank you.

5. Line 21: there’s no Gauss law here (divE= 4 pi rho)

We have modified the text accordingly and thank the reviewer

for pointing this out.

6. Lines 30-31: Lapenta (2012) is not appropriate for referencing the PIC

method in general.

We have modified the citation and are now citing Nishikawa et

al. 2021 as more appropriate reference to the Particle-in-Cell method.

7. Line 134: replace ‘artificial step’ by ‘density step’.

We have modified the manuscript accordingly.

Comment on the Semi-Lagrangian solver

Lines 39-40: I am actually surprised that Vlasiator uses a semi-Lagrangian

scheme in configuration and velocity space. These schemes are known to enhance

numerical diffusion due to their inherent need to map Lagrangian particles back

to the mesh. Please comment on diffusion effects they cause, compared to high-

order Eulerian approaches. I understand that the Lagrangian step preserves

positivity and is conservative. However, it is highly diffusive too, which is not

discussed here.

As mentioned in the manuscript Vlasiator uses the SLICE-3D algorithm to

solve the Vlasov equation. Since this is a semi-Lagrangian solver it does not

need to abide by the CFL limit which in essence means that less total steps

are required to propagate the plasma compared to a Eulerian approach. Since

diffusion is accumulated over many simulations steps this aids in keeping the

total diffusion low. Moreover, Vlasiator is using slope limited polynomials for

the mapping instead of tracking Lagrangian quasi-particles and that does not

exactly compare to common SL approaches. Finally, Vlasiator uses a 5th order

polynomial reconstruction to perform the Lagrangian mapping which is accurate

enough to ensure low diffusivity.

This paper designs some kernels to filter the staircase effects arising from

AMR, which is innovative. The authors carefully examine the mass conservation

and computational overhead. Great work!

The authors wish to thank the reviewer for his comments on our manuscript.

We encourage the authors to check WarpX’s ( https: // warpx. readthedocs.

io/ en/ 21. 02/ theory/ amr. html work to see if any techniques related to the

absorbing layers can be utilized. Also, moving the codes to the GPU architecture

is another trend.

The authors have modified the introduction to cite WarpX as related work.

However, as also mentioned in the revised version of our manuscript, WarpX’s

methods are not compatible with Vlasiator since Vlasiator does not use a particle

approach to modeling plasmas and also because Vlasiator’s field solver operates

on a uniform mesh.

In the ”Code and Data Availability section” of your manuscript, you state

that the code for your work is archived on GitHub. Admittedly, later in the

references, a ZENODO repository is cited, as you addressed the previous com-

ments by the Topical Editor. However, the mentioned section must contain this

information about the permanent repository. Therefore, please, in a potential

reviewed version of your manuscript modify the ’Code and Data Availability’

section, including the DOI of the Zenodo repository.

We wish to thank the executive editor for reading our manuscript and point-

ing out that our permanent repository is not included in the ’Code and Data

Availability’ section. We have revised that section in our udpate version of the

manuscript to include our Zenodo reference DOI as well.