I think this paper is a good contribution towards the understanding of channel evolution. The observations are unique and informative. The modeling complements well the analysis of observations. I like the separation of the paper in two parts, observations and modeling - I think that gives the paper clear structure.

In addition to the comments below, more comments and a few suggestions to improve readability are in the attached pdf.  

General:

I found the paper quite difficult to read. There are many presentational issues that can be easily fixed and this would make a lot of difference to the reader. Even the abstract contains some examples:

"Some ice shelves exhibit channels at the base that are not yet fully understood."

-are all channels not fully understood? or only channels at some ice shelves?

"Time series of melt rate measurements show strong tidally-induced variability in vertical strain-rates."

-How can a melt rate time series show tidal strain rate variability?

-This is a common issue throughout the paper. Especially in section 2.3.2 the authors often confuse the terminology, and with that the reader. I don't think there is an issue with the data analysis, just with the presentation, but I can't be sure.

"The type of melt channel in this study diminishes with distance from the grounding line and are hence not a destabilizing factor for ice shelves".

-This statement is repeated at a few place throughout the manuscript, but it is not clear why it is true. It definitely doesn't seem to be a conclusion drawn from the observational or modeling results presented here. It would need to be elaborated on in a bit more detail and it is unclear whether this sentence is appropriate for the abstract, since this doesn't seem to be a finding of this study.

These are just a few of many examples. I guess the main advice would be, do not assume the reader knows what you mean. Repetition is important.

Representativeness of the measurements:

It seems that primarily the eastern side of the channel was sampled. Where the western flank was sampled (2 locations), the melt rate there was much higher than at the eastern flank of the nearest cross-sections. In the southern hemisphere, Coriolis force deflects flows to the left, which in this case is westward. Therefore, it is not unreasonable to expect relatively higher melt rates on the western flank, and the two western flank measurements seem to go along with this. Can sampling bias explain the apparent discrepancy between observed melt rates and thickness profiles?

Could you get the correct channel geometry assuming asymmetric east west melt rate, higher on the western side?

Basal melting measurements, technique:

One thing I am missing in this paper are some figures of the basal return, how it changed between the two measurements, if at all, and what are the implications for the uncertainty in the basal melt rate.

In particular, I am curious about how the basal melt rate at the steepest channel wall was derived (SW and SE). Is the first basal reflection that you consider from a flat base beneath? Or could it be of nadir? Are there any ambiguities?

How was the basal melt rate time series derived? It seems quite jumpy all together. How robust is the time series? Why is the time series getting noisier with time (Fig 4 - blue high frequency oscillations get gradually higher amplitude)?

Can you exclude the possibility of instrumental artifacts? - Are any of the jumps present in the internal reflector time series too?

Do any of the jumps coincide with the changes in the character of the basal reflector? e.g. splitting/joining peaks as the reflector evolves?

Or is the base just a simple, single peak that doesn't change its shape, in which case a lot of concerns would go away?

I think this is something that should be discussed in the paper if the readers are to believe the presented time series.

I have some more concerns about the melt rate time series now that I am looking at Fig 4 more carefully. The thinning rate doesn't show much seasonality (panel b). But then in panel a and also in Fig A3b it is indicated that the melt rate does have seasonality. Do the strain and melt time series have a seasonal variability that is equal but opposite? Such a result often indicates issues with the derivation of the melt rate time series. Or did you assume that vertical strain rate is constant in time, apart from tidal oscillations?

But as before some of my confusion can be caused by mixing up terminology, specifically whether the time series shown are melt thinning or total thinning. 

Model:

I think it would be useful to present the equations that are solved, as well as the boundary conditions written out mathematically. For those not used to the glaciology jargon, it can be hard to decode from the words (and not always so clear sentences) what system of equations is actually being solved.

There are some viscosity sensitivity experiments briefly mentioned in the end. What I wasn't able to gather from the description is, whether any realistic rheological values could possibly account for the observed geometry and melt rate or not. And if so, at what expense would that be - presumably not a good fit of the vertical displacement profiles in Fig 8?

The authors optimize viscosity to match the channel thickness, but they don't optimize it to match the vertical strain rates. Why not? Could it be that a better fit to the vertical strain rates (especially in the middle of the channel) could yield a melt rate solution closer to that what was actually observed? It could be that an answer to this is already in the paper/experiments, but I wasn't able to find it.

I am wondering why the authors do not find a strain rate structure similar to the modeling of Vaughan 2012, which promotes formation of basal crevasses. Is that because the ice here is so much thicker and the channel relatively shallow compared to the ice thickness? Or is this a fundamental difference between viscous and viscous-elastic rheology?

Is there any evidence of basal crevasses on this channel top?

Discussion:

There is some comparison with a study from Ross Ice Shelf, but it is not clear what the purpose of the discussion is. The authors are citing high melt rate values observed elsewhere but those were  measured much closer to the grounding line than in the current study. So what is the purpose of this comparison? To state that elsewhere people measure higher melt rates in channels? Or that it is possible that had you measured closer to the grounding line, you might have found higher values of melting?

I think the first paragraph of the discussion should be rewritten.

Another point regarding the main conclusion of the paper, which is that melt rates must have been higher in the past. Have you consider whether melt rate on side walls, which can be 10 times higher than on the base (Dutrieux et al 2014), could explain the maintenance of the channel without having to evoke a major change over the past 250 years? 

Naughten et al 2021 does idealized abrupt change experiments. I don't think that paper can be used as a supporting evidence for the hypothesis that melt rates in the channel decreased over the past 250 years.

Some more comments (page/line):

Again, need to differentiate between melting and total thickness change. On 11/225 you claim 1 cm tidal melt amplitude, but on 12/235 you say you were unable to get tidal melt rate.

6/130 How does the tidal bending at daily or so timescales (so presumably mainly elastic response) translate into long term vertical strain rate that is depth dependent (primarily viscous response)? Is that just coincidental, or is that expected from theory? 

I would draw different conclusions than the authors from some of the provided figures, e.g. Fig. 2c:

8/200

My interpretation of this figure would be that except for a few outliers (very high melt rates) the melt rate pretty much lies on a line, increasing with increasing draft. And it doesn't matter whether you measure melt inside or outside the channel.

13/265

The authors you Poisson’s ratio of 0.325. Jenkins 2006 found this to be 0.5 near Rutford Ice Stream, which is relatively nearby. How sensitive are the simulations to this parameter?

Related to that, there has been some discussion recently on n in glen's flow law being closer to 4 than to 3 (e.g. Milstein 2021), in which why would you expect that to influence the result, if at all? - I am just curious about that one, I am not really asking you to run more simulations.

Manuscript Review: Cryosphere Discussions

On the evolution of an ice shelf melt channel at the base of Filchner Ice Shelf, from observations and viscoelastic modeling

A. Humbert et al.

Summary:

This study presents an interesting combination of geophysical observations (pRES, GPS, seismics) and numerical modelling to gain insight into melting at an ice shelf basal channel within the Filchner Ice Shelf. The channel is found within a flow unit originating from Support Force Glacier.

Using a collection of phase-sensitive radar observations (pRES) that follow the basal channel and span its width, ice-shelf basal melting is calculated, accounting for vertical strain and surface accumulation. At the two upstream sites, melting is higher in channel than outside. Melting in the channel decreases in the along flow direction, turning to freeze-on downstream. Outside the channel melting persists, leading to a reduction in channel height. These present-day observations of melting and ice-shelf flow cannot be used to explain the observed geometry of the downstream basal channel. As such, it is assumed that the melt rate must have been higher melting in the past. There is a suggestion that this lower present-day melt may be linked to a reduction in subglacial runoff. 

Complementary to these observations, a viscoelastic model is used to simulate the evolution of the basal channel geometry. A 2D domain is considered transverse to ice flow that evolves in time and along-flow in the Lagrangian sense. This modelling confirms that higher-than-present melt rates are required to reproduce the height of the observed channel.

Overall comments:

The manuscript is well organised and fairly well written. I have included some suggested edits in the attached PDF to improve the clarity of the text and the ease of reading.

It would be informative to show more results from the viscoelastic modelling. At the moment I don’t feel the reader can appreciate the modelling without seeing more results. The point is made that it is important to consider elastic effects, but this is not demonstrated in any of the results. It would be good to explicitly state the Maxwell Time. Normally it is assumed to be on the order of hours to days (Gudmundsson 2011, Ultee et al., 2020), so on short time periods, such as the tidal cycle or large hourly fluctuations in the melt rate, elastic effects will be important, but here you are considering approximately steady melt rates over long time periods (decades to centuries).

I’m not an expert on the role the tidal signal plays on controlling ice-shelf melt rates and deformation. It would be helpful to provide more details on these processes and how they can be linked to the findings here. At present this section (2.3.2) is confusing. 

Imposed melt rate in numerical modelling: looking at Figures B2 and B3 the width of the channel seems to grow considerably downstream between seismic line IV and V, but this is no reproduced in the simulation, particularly on the right flank. This feature isn’t addressed. Would a different spatial distribution of the imposed melt be required to reproduce it? And if so, what does this imply? 

The current set of figures are well presented and informative. It would be good to include some more figures showing results from the viscoelastic modelling.

The work is of interest to many in the scientific community: oceanography, glaciology, geophysics, numerical modelling.

Individual Comments:

Line 11-12: “The type of melt channel in this study diminishes with distance from the grounding line and are hence not a destabilizing factor for ice shelves.” I agree that channels that close towards the calving front are not destabilizing in the way that they may breakthrough the ice shelf, but unless they close completely, these areas of thinner ice may be the initiation sites for fracturing due to extension perpendicular to the ice front, as in Dow et al., (2018).

Line 33: “The channel increases in height close to the grounding line and widens afterwards.” Have you thought about the processes that lead to the widening of the channel?

Line 39-40: “In lateral direction, the melt rate is only 0.82ma−1 demonstrating enhanced melt inside the channel.” I'm not sure what you mean here. Initially I thought you were suggesting that the horizontal melt rate is 0.82 m/a (i.e. channel is growing wider). But when you mention enhanced melt inside the channel it suggests you mean that outside of the channel the melt rate is 0.82 m/a?

Line 43-44: “At some point, it becomes super-cooled due to the falling pressure. Thus, the melt rate decreases and could even change to refreezing.” I think these sentences need rewriting. If the water is super-cooled it can no-longer melt?

Line 49-50: “M2 tidal constituent” Maybe add detail of the related time period here for those unfairly with tides. (i.e. approximately 12.5 hr or semi-diurnal).

Lines 61-66: Is it possible to calculate the corresponding Maxwell Time?

Line 81-82: “A GPS station was also in operation at this point from December 24, 2015 to May 5, 2016.” This is prior to the time period in which the pRES was deployed. It would be good to acknowledge this and note what the purpose of the GPS data is here to avoid confusion. 

Lines 132-137: This is an interesting observation. Has this been observed elsewhere? It would be good to include some references if so. Also why would e_zz decrease to 0 at the base? Is there a physical process behind this idea?

Line 146: “interpolated the ab, ∆Hε and ∆Hs along the distance of the channel” What method of interpolation did you use?

Line 155-156: Good to remind the reader here that ApRES measures are taken every 2 hours.

Line 170-174: How do you expect tides to impact melt rates? It would be good to give the reader an insight into the physical process you think might be impacting melt rates.

Line 203-204: “inside the channel (L) the melt rates decrease with reduced draft.” Here we observe a big jump in melt rate at around 750m – what could be causing this?

Line 205: “The distribution of ∆Hε shows a significant thickening of more than 1ma−1 at the most upstream cross-section at L and OE” It would be good to see the along-flow and transverse-to-flow components of strain in this region – is thickening due to along-flow compression or lateral compression?

Line 228-229: “cumulative melt shows a tidal signal with amplitudes of ~ 1cm within 12h around the low-pass filtered cumulative melt.” Can you distinguish how the melt rate is related to the tidal signal? When is the largest melt rate?

Line 233-234: “We found evidence for a clear accordance of the strain in the upper ice column with the tidal signal as recorded by GPS measurements.” How does the tide impact strain within the ice column? These datasets are from different time periods – how are they compared?

Line 238-239: “Consequently, we infer that strain in the lower part compensates the one in the upper part and there is only a small variation of basal melt on tidal time scales.”  I don't understand this. What is happening within the ice column? What is the sign of the strain in the upper and lower portions? What physical processes lead to these values?

Line 260-261: “there is no justification to expect a priori the deformation to be small for simulation times of more than 200a.” The deformation is induced by melt rate and accumulation only? Therefore this is limited by the magnitude of these terms?

Line 275: y-direction – what does this correspond to? Along-flow? Vertical?

Line 306: “After the spin-up, the width W(t0) of the simulated geometry is 10km.”  Is the initial width something you vary as part of the spin up process? Assuming that the width is not fixed, how does this change during the simulation? Is it similar to the width of the embayment?

Line 309-310: “Short-term forces like the time-varying climate forcing as well as the lateral extension or compression demand the usage of a viscoelastic instead of a viscous model to simulate the temporal evolution of the basal channel.”  In your model are these short-term forcings? The change in the imposed melt rate is fairly slow (maximum melt from 3 m/yr to 0 m/yr in 250 yr). You’re not using the annual variability from your time series of ApRES measurements as forcing?

Line 322-323: “the ice thickness OE increases due to the prescribed displacement at the lateral boundaries.” Is this due to the fact that the flow regime is compressive here and that the lateral boundaries are moving towards the centre of the channel? If so, it would be clearer to say this explicitly. 

Line 330-331: “This match confirms that present day melt rates would not lead to the observed channel evolution over 250a.” How different is this result using the viscoelastic model to that using the simple advection assumption, eq (5)?

Line 343-344: “Above the channel, the surface elevation is first overestimated by 4m at the end of the spin-up.” On first reading I thought; you can't really say that this is an over estimation as your spin up produced this initial geometry, and therefore couldn’t you change your spin up to more closely match this initial geometry? On closer reading I realised you have made a distinction between surface elevation and ice thickness. It would be good to make this clear. i.e. "While ice thickness is in good agreement, surface elevation..."

Line 356-357: “At the position of the furthest upstream pRES observations we can see from the seismic IV profile that the influence of the grounding line has not completely vanished.”  What exactly do you mean by this? What features are you referring to? Could this also be a result of incorrect assumption for density?

Line 359-360: “Hence, simulations carried out using a higher SMB within the channel would result in better agreement with the observed values of hTDX.” Because higher SMB implies a larger firn column and reduced mean density?

Line 367-368: “The generally good agreement of the simulated displacements outside the channel comes from tuning ux at the lateral boundary to match uz from the pRES measurements at OE.” Is it possible to investigate the influence of gradients in both ux and uy, (i.e. exx and eyy) seeing as compressive exx would increase closure of the channel, while extensive/compressive eyy would thin/thicken whole shelf.

Line 397-398: “We thus do not find any evidence that such channels are a cause for instabilities of ice shelves as suggested by Dow et al. (2018).” Dow et al., (2018) suggests thinner ice, at along-flow channels, may act as initiation sites for fractures perpendicular to flow, which would still be the case unless the channel completely closes.

Line 399 – 402: I agree. It would be good to note somewhere that generally ice shelves comprise of mainly extensive ice flow regimes. The compressive regime within this study area seems to stem from the fact that the ice from Support Force Glacier runs into the main body of the Filchner-Ronne Ice Shelf directly opposite to the pinning point of Berkner Island.

Line 410-411: “However, our model results suggest that the mismatch between the past melt rates needed to explain the observed channel geometry and those that were observed applies only to the channel, and not to the ambient ice.” Ocean conditions at the grounding line may trigger melting and formation of plume/focused melting in the channel. What is the along-flow profile in ice thickness from the grounding line? Seismic 1? Is the grounding line considerably deeper? Could we expect variability there to impact just the channel and not the ambient ice downstream?

Line 439-441: “The simulated geometry change is mainly due to the elastic response to thinning by basal melt and ice accumulation. Any purely viscous simulation would overrate the deformation.” It would be good to highlight this result with a figure that demonstrates this. Can this difference be quantified?

Line 444-445: “The elevation difference is most likely caused by the constant density that we used for the simulations, as the ice thickness matches well.” Does this geometry mismatch lead to a difference in stresses?

Line 452-453: “first few cross-sections still being influenced by the hinge zone.” This is also where highest channel melt rates are. Would this have an impact?

Line 456-461: Also necessary to know spatial variation in ice density to infer thickness from surface elevation. 

Line 466-467: “The channel diminishes because the reduced melt rate is unable to maintain the channel geometry against viscoelastic deformation.” It would be good to include a figure that demonstrates this.