This manuscript considers the influence of hydraulic control on exchange between glacial fjords and the ocean. The authors motivate the importance of the topic and demostrate the utility of the model using observations from a handful of major glaciers in North Greenland. The manuscript mainly considers the development of a simple quasi-analytical model that provides a dynamical understanding of when hydraulic control becomes active and its quantifies its influence on basal melt of glaciers inside the fjord. It is convincingly shown using both the observations and model that hydraulic control decreases the temperature of waters reaching the grounding line and thus decreases basal melt rates.

This manuscript is appropriate for the Cryosphere, is well written and I believe the model will provide useful insights for other researchers. I’d suggest minor revisions, with my only main comment being on the straightforward applicability of the model to the situation where subglacial discharge is more prominent. This and other minor comments are detailed below.

Substantial comments

I wonder about the applicability of the model to systems where subglacial discharge is significant (by which I mean that the subglacial discharge flux is comparable to or greater than the basal melt flux). I feel that many (most?) marine-terminating glaciers without ice tongues are likely to fall into this category, at least in summer. This comment has two parts:

(i) On neglecting subglacial discharge in the freshwater budget (L89) - is this appropriate? Based on Table 2 the observational estimates for the basal melt flux, M, are 60 m3/s at Ryder, 300 m3/s at Petermann and 600 m3/s at 79N. Cai et al. (2017) suggest based on RACMO2.3 surface runoff that subglacial discharge at Petermann can reach over 1000 m3/s in summer, which would significantly exceed the M term that is accounted for in the freshwater budget. On the other hand, Schaffer et al. (2020) suggest that only 11% of freshwater leaving the 79N cavity is subglacial discharge, which would support neglecting subglacial discharge. I feel a few more sentences justifying this assumption are needed. Also, for glaciers without ice tongues, it is much more likely that the subglacial discharge will significantly exceed the submarine/basal melt flux, so is the model applicable to fjords with tidewater glaciers, as you say in L53, when the subglacial discharge is neglected in the freshwater budget?

(ii) On the choice of the exponents n1 and n2 (L142 and discussion shortly after). For systems with high subglacial discharge, the buoyancy of the plume can be dominated by the subglacial discharge, so that the plume volume flux (and plume velocity) becomes independent of the thermal forcing and scales only with the subglacial discharge raised to the power 1/3. Some studies that investigate this regime are Jenkins 2011 and Straneo & Cenedese (2015) – see in particular Eqs. 7 & 8 of the latter study. This subglacial discharge-dominated case would have n2=0 and n1 would be close to 1. I think it would be great to mention this possibility when discussing values for n1 and n2. And, if the model is to be widely applicable across Greenland fjords, do the results change much if n2=0? Or is this already a sub-case of what you have presented? I appreciate that this might require a lot extra to look into properly and that is not what I am proposing – maybe just a short consideration of how n2=0 might change things.

Overall, this substantial comment is not really a criticism of the paper and doesn’t require major changes to address, but would be worth considering as I think it has a bearing on how widely applicable the model would be.

Minor comments

L14 – the use of “marine ice” – I worry that this terminology could be a bit confusing. I’d suggest rephrasing using “marine-terminating glaciers”.

L26 – Slater et al., 2022 recently argued that for some regions, the impact of increasing subglacial discharge on submarine melt has been as important as AW temperature – could be worth acknowledging here.

L32 – “can stabilise marine glaciers” – I feel this statement is too certain for this point in the paper. Perhaps “has the potential to stabilise marine glaciers”?

L34 – either here or somewhere else appropriate, I think it would be worth acknowledging that processes other than hydraulic control can also modify AW between the shelf and the glacier – for example vertical mixing due to velocity shear even in the absence of a sill, or icebergs.

Fig. 2, panels b and c – it would be great to have a scale bar for these panels.

Fig. 3 – it would be great to have the locations of these profiles shown on Fig. 2b and 2c

L78 – it would be nice to finish off the introduction with a sentence that bridges into the next section. For example, “We now describe a two-layer model to investigate…”

L122 (and a few other places) – it would be more consistent to refer to “Eq.” instead of “relation”

L129 – is the value of rho0 ever actually used in the model? Or does the density difference always get normalised by rho0 (e.g. Eq. 22), in which case there would be no need to assume a value for rho0.

L198 – I don’t quite follow why the exchange flow increases with deltaT when n1-n2>1. From Eq. 21, don’t we require n2/(n1-n2)>0? Which would give 2*n2-n1>0, but perhaps I am mistaken.

L200 – I think somewhere in this paragraph it would be appropriate to cite Zhao et al. (2021), which similarly looked at parameterising hydraulically-controlled transport (e.g. Eq. 17 in that paper).

Fig. 4 – could you say how the axes are non-dimensionalised?

L315 – suggest adding “in the case n1=2 and n2=1” at the end of the first sentence

Figs. 6 & 7 – it could be better not to use the jet colorscale

Fig. 12 caption – “See the text for details” – did this mean to look at the text for details on the refreezing, or for details on the figure more generally? I took it to mean details on refreezing, and I think I didn’t see those, so perhaps revise.

L492 – suggest adding “unmodified” before “AW”

L506 – on the two idealised scenarios – can you speculate which might be more realistic? Scenario 2 feels more realistic to me because there is more a gradual transition from no entrainment to some entrainment, but perhaps we are not able to say yet.

L536 – on Ryder and 79N having “basal melt processes that are less sensitive to thermal forcing than Petermann” – surely according to your model, the basal melt processes at all of the glaciers are equally sensitive to thermal forcing (because M varies as T^n1)? So is the higher melt rate at Petermann likely due to factors beyond thermal forcing (i.e. gamma1 in your equations), such as subglacial discharge or grounding line depth or basal slope?

L538 – suggest adding “at 79N” after Schaffer et al. (2020)

Typos etc

L12 – suggest changing to “…it may contribute up to…”

L21 – raise > rises

L22 – transport > transports

Fig. 2 caption – where Petermann > into which Peterman

Fig. 2 caption – has two sill > has two sills

Fig. 3 caption – observations where > observations were

Fig. 3 caption – has two sill > has two sills

L69 – atmospheric condition > atmospheric conditions

L70 – bathymetry is > bathymetry are

L77 – ice-free condition > ice-free conditions

L83 – basal melting on > basal melting of

Last line in Table 1 – I guess the Gade temperature should have units of Celsius?

L116 – add comma after “very cold”

L130 – coefficient > coefficients

L194 – determines > determine

L223 – Eq. 8 > Eq. 9

L232 – suggest removing comma after “Essentially”

L258 – h is fixed also TL is fixed > h is fixed then TL is also fixed

L264 – has > have

L292 – The reasoning above and suggest that > The reasoning above suggests that

L294 – introduce additional > introduce an additional

L309 – simplifies > simplify

L316 – and opposite applies > and the opposite applies

L331 – constrains > constraints

L356 – adjust > adjusts

Fig. 8 caption – than one the > than one in the

L422 – no need for comma after “show”

L429 – qualitative > qualitatively

L447 – temperaturs > temperatures

Fig. 11 caption – function the AW > function of the AW

L479 – Currently, AW > Currently, the AW

Fig. 12 caption – function the AW > function of the AW. Also Ryders  Ryder’s

L525 – suggest adding comment after “contrast”

L545 – half > halve

L552 – on non-dimensional > in non-dimensional. Also variabels > variables

L560 – on a closed > in a closed. Also allow us examine > allows us to examine

L568 – (A6) on > (A6) in

L569 – forms > form

References that are not in the paper

Slater et al., 2022. Submarine melting of glaciers in Greenland amplified by atmospheric warming. Nature Geoscience, doi:10.1038/s41561-022-01035-9.

Zhao et al., 2021. Geometric Constraints on Glacier Fjord-Shelf Exchange. Journal of Physical Oceanography, doi:10.1175/JPO-D-20-0091.1.

With remarkable dexterity and elegance in the use of the equations and assumptions, the authors of this study propose a simplified two-layer model where the properties of the in- and out- flows in a sill fjord depend on the thermal forcing at the grounding line and the relative-to-the-sill height where the upper limit of the AW layer is located. Directly or indirectly, both variables influence the basal melt rate at the ice tongue. Depending on the fraction of glacially modified waters exported to the other side of the sill, the authors distinguish two regimes: melt-controlled or hydraulically-controlled exchange flows. In the first regime, the exported flow is a function of the basal melting, which depends on the AW temperatures, but is independent of the geometry of the fjord, the sill and the AW height. With an increase in AW temperature, a decrease in AW height and/or an intensification of the subglacial discharge, a transition from a melt-controlled to a hydraulically-controlled flow regime may occur. Under the hydraulically-controlled regime, the AW temperatures reaching the GL have been R-decreased, and both basal melt and plume flux depend on AW height over the sill. The sensitivity of the hydraulic regime dynamics to different parameters of the model is analyzed in two different scenarios: with and without reflux of the GMW (i.e. part of the outflowing plume flux is entrained into the inflowing AW flux). Finally, the model is applied to three glacier-fjord systems in northern Greenland (Peterman, Ryder and 79ºN glaciers), from which values ​​of the different model parameters are obtained and the type of regime that controls flow exchanges is determined: Hydraulic control in Ryder and 79ºN glaciers, and melt control in Peterman Glacier. Thus, by a different path, the authors reach the same conclusion as Jakobson et al., (2020), who used observations together with the well-known 1D plume model.

In general, I found this to be a well-reasoned and well-organized paper. The methodology is appropriate, the line of reasoning is clear, and the limitations of the study are acknowledged. I believe the key point made in this paper is of relevance to many aspects related to glacial fjords , and should therefore be of interest to readers of Egusphere Journal. I recommend the publication of this manuscript with some minor revisions, which I have outlined below.

GENERAL COMMENTS

From my point of view, the manuscript is well written and the study is well prepared and with many of the limitations of the model considered. However, there are two general aspects that I would like to comment on.

1) Regarding the content, I have missed several details about the physical settings of Peterman and Ryder glaciers (see specific comments below), as well as the entire description of 79ºN Glacier system. In order to understand the transition range to which each system is subject, it would also be convenient to report approximately how deep the SPW, the front of the ice tongue, the sill and the AW reach in each system. In the limitations of the model, I believe that the implications of not including subglacial discharge in the melting flow should be further developed. It would also be convenient to take into account that melt buoyant plumes can reach neutral buoyancy at different depths. If either of these depths is close to hL, it could encourage a transition to the hydraulically-controlled regime. Even, if the plume NBD is reached deeper than the sill, the plume flux would be totally trapped in the ice cavity; there would be no exchange flux to the oceanward side of the sill (entrainment ratio of 1) and the waters from the ice cavity would turn colder and fresher without input of the AW.

2) Regarding the organization of the content, I would consider it appropriate to make some adjustments to facilitate the understanding of the study, although I also understand that each person must have their own style and I propose my comment as a suggestion. I think the most appropriate structure would be:

1. Introduction

    The state of the art, motivation and objectives as they are (I would not include here the subsection about the Physicalvsettings of the glacier-fjord systems).

    2. A two-layer model

    As it is, although I am not sure about whether subsection 2.2.1 should be at the same level than Sections 2.2 and 2.3.

    3. The dynamics in the hydraulic regime

    4. Application of the hydraulic regime to glacial sill fjods (as a 'Case of study')

    Independent from the previous section, and including here the glacier-fjords' description of Peterman, Ryder and 79ºN.

    5. Conclusions

SPECIFIC COMMENTS & TYPOS

L.18 - Basal melt relates to the melting occurring at the base of an ice body (either grounded or floating). However, basal melt here only refers to the floating part of the ice (ice shelves and ice tongues), but it doens't to tidewater glaciers with vertical ice front, which are also marine-terminating glaciers. 

L.19 - The definition of grounding line given here is only valid for ice shelves and ice tongues. The grounding line feature is also present in tidewater glaciers with a nearly-vertical ice front, where no permanent floating ice exists.

L.20 - The effects of water column stratification on submarine melting was also reported on De Andrés et al. (2020).

L.46 - 'effects due Earth's rotation' -> 'effects due to Earth's rotation'.

Section 1.1 - As stated in L.52-53: 'the model results are discussed in relation to observations from... and 79ºN glacier'. However, no information of the physical settings of 79ºN glacier is provided within the whole manuscript. Is there any reason for this lack of information?

- In order to contextualize the rate of frontal advancing/retreating and get a frame of reference for the sill influence on hydraulic control, could the author specify what the lengths and widths of these two glaciers and fjords are? and the depth range of the grounding lines and ice-tongue fronts? I can only found GL and front details for Ryder glacier in Fig.2 caption.

 

L.57 - It would be better to quantify 'a relatively deep and wide sill'.

Fig.2 - In panel a), it would be nice to have the coordinates frame and the North arrow.

 - In panels b) and c), a scale bar (and a more precise bathymetric colorbar) would help with the fjord and sill dimensions. It seems that the coordinate 63º0'W appearing on the left y-axis of pannel b) is a mistake. It would also be helpful to have in these pannels (b and c) the location of the CTD casts used in Fig. 3.

L.80 & 86 - Based on CTD observations, it would be helpful to give a thickness range of the two layers considered in the model, as well as the thickness of the surface-polar-waters layer (it could also be highlighted on Fig.3). What are the limitations on the study (if any) of avoiding mixing between glacially modified and surface polar waters? 

L.82-83 - What are the limitations of neglecting subglacial disharge as a mechanism enhancing basal ice-tongue melting?

Table 1 - Last row, in Gade temperature relation, change the 'equal symbol' by the 'almost-equal symbol' (as it appears in L.111 and L.116). Also, I am a bit confused, since it seems to be inconsistencies with the magnitude and units of this Gade temperature. A value of 80 K is given in Table 1 (which is consistent applying the proposed relation therein), but a value of 80 ºC is given in L.111. From other studies (e.g. Jenkins, 1999; Mankoff et al., 2016), this Gade temperature values are about -90 ºC. Could you, please, unravel this question?

L.111 - Modify L/c value/units according to my previous comment.

L.198 - 'This show the' -> 'This shows that the'

L.204-205 - Could also a significant subglacial discharge flux motivate this subcritical-to-critical transition? Answered in L.272-273.

L.206-207 - See also Hager et al. (2022), where a simple model is used to estimate the proportion of refluxed freshwater in a silled fjord and the potential impacts on submarine melting are discussed.

Foot note 1 - The word 'than' is repeated twice in the second line.

L.245 - Shouldn't it be R < 1 in the hydraulic regime, since R = 1 is reserved for the melt-controlled? regime.

Fig.4 - What are the h and deltaT used to make axes non-dimensional?

L.289-290 - Increased stratification generated by strong surface melting has also been observed to dampen submarine melting in tidewater glacier-fjord systems (De Andrés et al., 2020).

L.292 - 'The reasoning above and suggest that' -> 'The reasoning above suggests that'.

Fig.8 - in L.3, 'is smaller (greater) than one the hydraulic' -> 'is smaller (greater) than one in the hydraulic'.

L.446 - I understand the near-bottom temperatures for the ice cavity, to get better estimates of those temperatures near the grounding line, but, shouldn't outside-ford temperatures be those at the near-sill depth, where the flow exchanges are taken place?.

L.460 - Please, quantify 'with large error bars'.

Fig.11 - in L.2, 'and 79º' -> 'and 79ºN'.

Fig. 12 - To get a more comprehensive understanding, tt would be nice to have the squares for the three glaciers, not only for the Ryder glacier. 

L.490 - 'Our results suggests' -> 'Our results suggest'.

L.521 - 'longer that today' -> 'longer than today'.

REFERENCES

De Andrés, E., Slater, D. A., Straneo, F., Otero, J., Das, S., and Navarro, F.: Surface emergence of glacial plumes determined by fjord stratification, The Cryosphere, 14, 1951–1969, https://doi.org/10.5194/tc-14-1951-2020, 2020. 

Hager, A. O., Sutherland, D. A., Amundson, J. M., Jackson, R. H., Kienholz, C., Motyka, R. J., & Nash, J. D.: Subglacial discharge reflux and buoyancy forcing drive seasonality in a silled glacial fjord, JGR: Oceans, 127(5), https://doi.org/10.1029/2021jc018355, 2022.