This paper describes a modular Eulerian-Langrangian approach solving the coupled non-linear processes. The authors report the shortcomings of the majority of snow models that do not give an explicit numerical solution of the ice mass continuity equation but are built around. The model formulation of the snow scheme presented explicitly accounts for the ice mass balance and couples mechanical settling, heat transport and vapour transport whereby these processes can also be analysed individually due to the modular structure of the model. This modular design of the new approach allows for high degree of flexibility, which the authors use to analyse different isolated and coupled scenarios involving of heat transport, vapour transport and mechanical settling of a dry two-layer snowpack.

The work provides a good basis for future work and discussions on the future generation of snow models. The work is very exciting and gives new approaches that will be useful for the snow modelling community. However, I have some suggestions for changes in restructuring of the work and some visualisations.

Major Comments:

The discussion section lacks comparisons to other studies and observations from which one can see the progress. This would provide an important context for the presented results and an evaluation of the presented results of the different cases and model behaviour. More discussion and comparison would for example be desirable in section 4.2: are there other studies that show the same? Also section 4.3: How about a comparison to observations and other work e.g. for lines 408 to 410?

The authors also use references sparingly in the method sections. For example, section 2.3 get along without using a single reference. If this is a result of the author’s work, I recommend emphasising this to the reader.

I recommend restructuring section 4 and using a separate method section for the difference steps you took to apply the model. I suggest moving line 363 – line 379 to a subsection of the methods. I also recommend moving section 4.1 to the methods. For me, the results section starts with line 392.

Section 4.6 and Fig. 10: Here you write that you compare cases 5 and 8 and use Fig. 10 for visualisation. However, Fig. 10 only visualise for case 8. I can only find the visualisation for case 5 in Fig. 6 and 7 but only for different variables or different visualisation. This makes it difficult for the reader to compare the two cases. I suggest adding the visualisation for case 5 in Fig. 10. Also, I cannot find a comparison to case 5 in the text. You describe Fig. 10, briefly compare your results to section 4.2, which deals with case 1, and mention the sublimation peak as observed for other cases. However, I cannot find a comparison to case 8.

Detailed Comments:

Line 20: I suggest to add “e.g. Snowpack, Crocus” behind “when not even detailed snowpack models”.

Line 29: Please name the snowpack models.

Line 46: The vapour transport is very important and interesting. I encourage the authors to explore further the importance and the differences in e.g. alpine and arctic snow of the importance.

Line 50-52: See comment above: Please describe in more detail why it is important.

Line 57: Please also add the details for Crocus.

Line 61: You mention that a finer spatial resolution is needed. Please mention the spatial resolution of Snowpack and Crocus.

Line 74: Reference is needed for the current treatment of densification in snowpack models.

Line 80: Define “σ-coordinates”.

Line 111: Please add a reference where this common starting point is used in snow models.

Line 113: I recommend deleting “without explicitly mentioning every time”. You already write “if not stated otherwise”.

Line 115: “snow density” did you mean “ice density” ? Please clarify.

Line 114-117: Reference is needed here.

Line 119: Would there be a lot change if wet snow were used? Please clarify in 2-3 sentences the differences that dry snow and wet snow would make (more detailed description of what you have already started in line 121)

Line 121: I recommend adding “compared to wet snow situation” after “as the more challenging (yet less investigated) one”

Line 122: I suggest starting a new paragraph for “Note, that water transport …”

Table 1: Is there a value missing for density? I suggest ordering the variables within the headings according to their occurrence in the equations. So ice volume fraction first, followed by vertical velocity, etc.

Line 129: “source term c” do you mean ice deposition rate?

Line 132 – 134: You write that vertical motion results either from mechanical settling changes in ice volume from sublimation/deposition. What about melting and snow redistribution?

Line 139: see comment for line 129

Line 141: “Hence we will refer to c as the deposition rate” I suggest to introduce this term earlier maybe line 129 (see comments line 129, line 139)

Line 152: Please add references for snowpacks where the approach typically chosen in the snowpack models is apparent.

Line 170: Which properties of the snow microstructure?

Line 185f: Define g (only defined in line 299).

Line 186: Please define the term “snowpack’s effective density”

Line 189: Please define ζ.

Line 203 – 210: First, you write that you are extending the model for mechanics but a few lines later you write “and thus it is neglected in the following.” I cannot follow this thought. Please clarify.

Line 214 – 215: “Instead of following […] closure for the source term” repetition of line 202. I suggest to add the references mentioned in line 215 to line 202 and delete 214-215.

Line 244: What do you mean with “necessary accuracy”? Please add a sentence what the necessary accuracy for the scenarios is.

Line 245: I suggest to add “using a 1d snow column” at the end of the sentence “[…] scenarios considered in the paper.”

Line 247:”In that situation”. Which situation do you mean with “that”? What is a complex geometry in this context? 2d/3d?

Line 252: For the interested reader, it would be helpful to add an example reference for the use of a second order Strang splitting.

Line 275: You only explain the usage of a Lagrangian approach in this line but already use the term before in line 237. I recommend moving this explanation to an earlier line.

Line 301: What about Crocus? Please add reference here.

Line 313: “within”

Line 323: Define α.

Figure 2: Nice and helpful overview. I suggest inserting the names of Calonne, and Hansen and Folien in the boxes of the approaches and the equation numbers used to give the reader a quick overview.

Line 347: Figure appears in manuscript before it is mentioned in the paper.

Line 349: What is the minimum time step for the output? Does it also vary?

Line 351 – 356: Already in Figure 2. Please bring both information together in Figure 2 (see also comment about Figure 2).

Line 360: I recommend deleting “(Sect. 4)”

Line 383: “The densities are in the range of […]” For which regions/ type of snow is this the case?

Line 384: Correct “over over”.

Figure 3: Why do you choose these densities? Why does your layer have equal thickness? Representative for what region? Why do you use the values for ice volume, representative for what? Any reference for the used values?

Line 392: I recommend changing “As the first step” to “first”

Line 394: “Furthermore, the vertical velocity varies less in the upper layer than it does in the lower layer.” Misleading formulation, difficult to understand on first reading. Please clarify: in comparison of the same time step or between/within time steps?

Line 397: Does it increase at all in the upper layer? I can only see the thinning of the snowpack but no colour changes in the upper layer.

Line 398: “the extent of the upper layer decreased only slightly with time”, how many cm at the end?

Figure 4: It might help the reader if you mark the upper and lower layers at least on the y-axis, as you write about the upper layer and the lower layer e.g. line 394, line 397.

Line 420: You mention that this is explained in detail in Schürholt et al. but a short summary would be helpful. 1 to 2 sentences e.g. why, also in reality? Where observed? Why peak, why peak between layers?

Line 421 – 426: Very interesting. It would be valuable if you could discuss this further and in relation to reality. What is expected in reality? Why is the peak 4 times higher?

Line 430: In equation 23 und 24, Ec and Et are used, but only one number is given here. The meaning of this is unclear. Shouldn’t there be one number for Ec and one for Et?

Figure 6: It would be valuable for the work if you also add a plot c) for the temperature gradient for case 5. Perhaps a plot d) for vapour density gradient for case 5 over time would also be beneficial. You write about both in your text.

Figure 7: (a) Where is the dashed line? If it coincides with the solid line, please change the visualisation so that both lines are visible.

Figure 8: Not mentioned in the text.

Line 441: I recommend adding at the end of the sentence “for case 7 of Table 2, which refers to the fully coupled process in combination with dynamically varying viscosity”.

Line 443 – 444: Please add a discussion why this is the case.

Line 449: “As discussed before” please add section number.

Line 452: I suggest adding “Figure 10 (a) […] for case 8”.

Line 458: “decreases in height”. I recommend providing numbers from what height to what height or by how many cm?

Line 456: You use alternately “paper, study and article” in your manuscript. I recommend deciding on one term for the whole manuscript.

Line 531: “in the respective section” Please add section number.

Line 537: (e.g. Vionnet et al. (2012)

Section 5: In your summary and conclusion section, you use about one page to write about future work and challenges. I recommend moving this part to a separate section called e.g “Future work and challenges” before the summary and conclusion section.

Line 561: “Audet and Fowler, 192 please remove brackets.

Line 607: Define ρa.

Figure D1: Why are different times shown here than in the other plots (15h, 32 h, 48h vs. 0 h, 16 h, 48h)? I suggest making the graphic square.

Line 729: 2021.

Simson et al. present rigorous development of a numerical model for coupled heat transport, vapour diffusion and settling in snow. I expected this to be a difficult paper to read, but I was pleasantly surprised by how readable and understandable it was. Because this is presented as a contribution towards snow model development rather than a full snow model, the test cases that can be considered are necessarily limited, but it is still disappointing that the paper contains no comparisons with observations at all.

Specific comments listed by line number:

2

The majority of models use non-deforming layers with ice and water moving between them and enforced conservation of mass.

18

Doesn’t the focus on snow water equivalent suggest that mass is the most important prognostic variable?

23

Bartelt and Lehning (2002) do use a Lagrangian coordinate system that moves with the ice matrix, but “Lagrangian coordinate system that moves with the ice matrix” is not actually a quote from that paper.

57

Considering this motivation from Domine et al., demonstrating whether modelled vapour transport can produce the sort of density stratification observed in shallow Arctic snowpacks subject to high temperature gradients would be an important test case.

Table 1

c and v are not state variables in the thermodynamic system sense; they can be derived from known temperature, ice volume fraction and vapour density.

Excessive precision for latent heat of sublimation

129

Could note that the settling velocity was neglected in Part 1, equation 7

216

Better to write vapour density with the eq subscript hereafter.

220

Equation 10 differs from the corresponding equation 5 in Part 1. Checking units, the error is actually in Part 1.

Figure 2 caption

The last three sentences don’t really fit in a figure caption.

Table 2 caption

Case 8 is also fully coupled.

Figure 3

Why do the ice volume fraction and temperature axes run right to left?

Figure 4

An additional plot with vertical profiles of density at 0, 16 and 48 hours could be interesting. Compaction from 150 to 420 kg/m3 over 2 days at the base of a 50 cm snowpack seems high if there were to be a comparison with observations (but not as implausible as Figure 10).

Levels on plot (b) appear quantized but the colour bar is not.

Figure 7

Deposition rate would be better shown with a diverging colour scale centred on 0.

Reference to Figure 8 is missing in the text.

470

A layer-based snowpack model could be viewed as having computational nodes in the centres of the layers. Overburden on the top layer being half the layer’s weight then makes more sense.

If using Vionnet et al. (2012) as an example, this two-layer, 50 cm snowpack is something that would not arise in Crocus; thin layers are maintained at the top and bottom of the snowpack for heat conduction calculations.

594

a0, a1, a2 and f all have units, which should be given. f is the gas constant for water vapour. This same formula was attributed to Mason (1971) in Part 1.

Minor corrections:

25

“has been well established”

56

“Both require”

57

“uses time steps on the order of 15 minutes or longer”

74

“do not take full advantage”

157

“depends on both the physical regime and computational feasibility”

161

“challenging to determine from experiments”

197

“In the remainder of this paper”

244

“results in the necessary accuracy”

253

“implemented in Python”

254

“but also allows”

305

“avoids numerically approximating”

309

The lengthy parenthetical clause “see for instance Sect. 3.4. in Bartelt and Lehning (2002) or its recent extension Jafari et al. (2020)” would be better inside parentheses than between commas.

317

“allows inferring the most plausible process model … given certain data”

319

“results in a mesh”

530

“without conceptual difficulty”

568

“While including potential phase changes”

598

Part 1 used capital Theta for the Heaviside function.

Appendix C

Matrix elements should be enclosed in brackets.