Modeling saline fluid flow through subglacial ice-walled channels and the impact of density on fluid flux

Jenson, Amy; Skidmore, Mark; Beem, Lucas; Truffer, Martin; McCalla, Scott

doi:https://doi.org/10.5194/egusphere-2023-792

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https://doi.org/10.5194/egusphere-2023-792

Preprints

28 Apr 2023

| 28 Apr 2023

Modeling saline fluid flow through subglacial ice-walled channels and the impact of density on fluid flux

Amy Jenson, Mark Skidmore, Lucas Beem, Martin Truffer, and Scott McCalla

Abstract. Subglacial hydrological systems have impacts on ice dynamics, as well as, nutrient and sediment transport. There has been extensive effort to understand the dynamics of subglacial drainage through numerical modeling. These models, however, have focused on freshwater in warm ice and neglected the consideration of fluid chemistry such as salts. Saline fluid can exist in cold-based glacier systems where freshwater cannot and understanding the routing of saline fluid is important for understanding geochemical and microbiological processes in these saline cryospheric habitats. A better characterization of such terrestrial environments may provide insight to analogous systems on other planetary bodies. We present a model of channelized drainage from a hypersaline subglacial lake and highlight the impact of salinity on melt rates in an ice-walled channel. The model results show that channel walls grow more quickly when fluid contains higher salt concentrations which lead to higher discharge rates. We show this is due to a higher density fluid moving through a gravitational potential. This model provides a framework to assess the impact of fluid chemistry and properties on the spatial and temporal variation of fluid flux.

Received: 20 Apr 2023 – Discussion started: 28 Apr 2023

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Journal article(s) based on this preprint

25 Nov 2024

Modeling saline-fluid flow through subglacial channels

Amy Jenson, Mark Skidmore, Lucas Beem, Martin Truffer, and Scott McCalla

The Cryosphere, 18, 5451–5464, https://doi.org/10.5194/tc-18-5451-2024,https://doi.org/10.5194/tc-18-5451-2024, 2024

Short summary

Amy Jenson, Mark Skidmore, Lucas Beem, Martin Truffer, and Scott McCalla

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2023-792', Anonymous Referee #1, 01 Jun 2023

see attached referee report.

Citation: https://doi.org/10.5194/egusphere-2023-792-RC1
- AC1: 'Reply on RC1', Amy Jenson, 24 Jul 2023
  
  Please see the attached pdf to view our response to RC1.
  
  Citation: https://doi.org/10.5194/egusphere-2023-792-AC1
RC2:
'Comment on egusphere-2023-792', Anonymous Referee #2, 05 Jun 2023

This paper describes a jökulhlaup model, based on R-channel drainage, taking into account effects of saline water on the dynamics. It finds that the impact of salinity on melt-rates is limited and the biggest impact is due to the increased density of the fluid on hydraulics.
This short paper has interesting findings and is novel as no-one has looked into impacts of salinity on R-channel dynamics. It has however a few shortcomings which need to be rectified before publication.
Major comments

==============
The somewhat surprising take-home (that the main difference compared to freshwater discharge is due to the higher density and not due to higher melt rates) should be elaborated a bit more. I would state the following, possibly in the Abstract & Conclusion:

- salinity makes liquid water possible at sub-zero temperatures: fresh water would be frozen. So this is really the biggest difference: liquid water versus no liquid water.

- once the equations are solved at the sub-zero temperature given by the melting point of the saline solution, it is actually pretty obvious that impact on melt is minimal: dilution of the water as it traverses the R-channel is minimal as melt is really small compared to discharge. Even for a setting where there is much more potential energy available, this statement holds. Thus it makes sense that density has the biggest impact.
Provide a sketch (as fig. 1) of the setup for an overview, that way also the coordinate system and orientation are defined.
With many of the equations in section 2 I struggle:

- Eq6 and the following three unnumbered eqs have some issues, see line-by-line comments

- my understanding is that the authors assume that water-temp equal to the local melting point as stated on l72 (i.e. temperature is dictated by pressure and salinity). Thus \theta, \theta_i and \theta_b are equal and hence the last term in Eq7 is zero. If temperature was treated as a free variable, as in Fowler 1999, then this term would be needed.

--> this whole temperature treatment is confusing. I had a quick look at the code and I think that temperature is indeed not a free variable.

- the melt rate $m$ can be solved for in Eq7. However, I do not understand what is going on here. As stated above, the last term should ==0. Also there should be a pressure dependent term to capture the pressure melting point effects. At least when a linear relation between temperature and pressure is assumed then it takes the form $rho_w c_t c_t dP_w/ds*Q$ (e.g. Röthlisberger 1972, Eq 17); maybe the authors try to capture this with their last term of Eq7, but I don't think that can be done like this. A term in similar spirit would need to be added for the salinity dependent effects, presumably featuring a $d\beta/ds$. In fact, this is the quintessential equation to be stated as that is the novel one, all others are known.

- How Eq15, also featuring the melt rate, then ties in with this is also not clear to me.
The lake has only ever a simple geometry, thus I would recommend to just keep it simple and shorten that part. A simple "refer to Kingslake 2015 for a treatment of more complicated lake hypsometries."
Line by line

============

12: nice Introduction
13: (any many others): I prefer incomplete lists of references to be prefaced by a "e.g.". Many instances in the Introduction (l.15, 18, 24, 26, etc.)
30: I would prefer to use an SI units throughout instead of "psu". At least give a translation (1psu=1kg/m3).
70: Cite the original too, Röthlisberger (1972). I would write something like: "We construct a lake-drainage model in which the water flows through a subglacial channel (Röthlisberger, 1972; Nye, 1976). We follow the implementation and notation of Fowler (1999) and Kingslake (2015). We assume..." I also like the term "R-channel" and would use that, but no strong feelings here.
74: I disagree with Fowler (1999) here and thus with this manuscript: the given definition is the "negative basic hydraulic potential". I think it would be good to state the "negative" somewhere. Similarly Eq2 gives the negative potential gradient.
95: the "(instead of Darcy-Weisbach)" I find confusing as D-W is also for freshwater. Just delete?
97: I think 0.6m^{-1/3}s should be 0.06 (according to the code). Also cite where these values are from (maybe Clarke 2003?) as Fowler 1999 does not provide those values.
Eq6 and following: I don't think those formulas should be stated with so many digits. At most two significant digits are needed. Also state that for the $\rho_b$ and $\Delta\theta_P$ the 3rd and 2nd order terms are not important. Also, Eq6 is written with \beta as an argument, the others are without argument: should be consistent.
Eq6: is this correct? I evaluate $\rho_b(0)=0.1$ but fresh water has a density of 1000kg/m3.
Eq6 / l109: I don't understand how the melting point depression can have a constant term. That would mean that it is always depressed! Similarly for the change in melting-point due to pressure. Also, I guess these formula depend on temperature being given in deg-Celsius, that should be stated.
123: cite Röthlisberger here too, drop Kingslake.
132: why suddenly salinity in kgm^-3 instead of "psu"?
139-143: this can be abbreviated to just state Eq 8. The reader does not need to see a derivation of a conservation equation.
115: doesn't line 72 state that these temperatures are always equal?
167: I wouldn't call this a "no flux" boundary condition as there is water flux through the boundary. Just call it a Neumann BC. Also, what does it mean? What is meant with "We do not require the channel to exit at a terminus or end subaerially"? Where does it end? Or at least where does the simulation end and how do we know that there dN/ds=0?
174: reference the suggested Fig.1 (the sketch) here.
Fig.1: could the temperature as a function of distance also be plotted? Maybe even the different components, i.e. press-melt-point term and salinity-melt-point term.
230: I think this is vice versa.
242: what are "hyperconcentrated sediment concentrations"? What is this "threshold"?
253-259: I recommend to delete these lines as they don't add much
288: I concur with Reviewer 1: these N values do not make sense to me. Is this caused by the Neumann BC on N? Note that a, in my opinion, more natural and standard BC N=p_i (or p_w=0) leads to dN/ds not equal zero. See for instance in Röthlisberger (1972): Fig.5d shows a setup similar to here and shows p_w -> 0 at the outlet.
312: Excellent that the code is provided, thanks! Ideally the authors should also provide a third script that, when run, will reproduce and write to disc all figures used in the publication.

Citation: https://doi.org/10.5194/egusphere-2023-792-RC2
- AC2: 'Reply on RC2', Amy Jenson, 24 Jul 2023
  
  Please see the attached pdf to view our response to RC2.
  
  Citation: https://doi.org/10.5194/egusphere-2023-792-AC2

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2023-792', Anonymous Referee #1, 01 Jun 2023

see attached referee report.

Citation: https://doi.org/10.5194/egusphere-2023-792-RC1
- AC1: 'Reply on RC1', Amy Jenson, 24 Jul 2023
  
  Please see the attached pdf to view our response to RC1.
  
  Citation: https://doi.org/10.5194/egusphere-2023-792-AC1
RC2:
'Comment on egusphere-2023-792', Anonymous Referee #2, 05 Jun 2023

This paper describes a jökulhlaup model, based on R-channel drainage, taking into account effects of saline water on the dynamics. It finds that the impact of salinity on melt-rates is limited and the biggest impact is due to the increased density of the fluid on hydraulics.
This short paper has interesting findings and is novel as no-one has looked into impacts of salinity on R-channel dynamics. It has however a few shortcomings which need to be rectified before publication.
Major comments

==============
The somewhat surprising take-home (that the main difference compared to freshwater discharge is due to the higher density and not due to higher melt rates) should be elaborated a bit more. I would state the following, possibly in the Abstract & Conclusion:

- salinity makes liquid water possible at sub-zero temperatures: fresh water would be frozen. So this is really the biggest difference: liquid water versus no liquid water.

- once the equations are solved at the sub-zero temperature given by the melting point of the saline solution, it is actually pretty obvious that impact on melt is minimal: dilution of the water as it traverses the R-channel is minimal as melt is really small compared to discharge. Even for a setting where there is much more potential energy available, this statement holds. Thus it makes sense that density has the biggest impact.
Provide a sketch (as fig. 1) of the setup for an overview, that way also the coordinate system and orientation are defined.
With many of the equations in section 2 I struggle:

- Eq6 and the following three unnumbered eqs have some issues, see line-by-line comments

- my understanding is that the authors assume that water-temp equal to the local melting point as stated on l72 (i.e. temperature is dictated by pressure and salinity). Thus \theta, \theta_i and \theta_b are equal and hence the last term in Eq7 is zero. If temperature was treated as a free variable, as in Fowler 1999, then this term would be needed.

--> this whole temperature treatment is confusing. I had a quick look at the code and I think that temperature is indeed not a free variable.

- the melt rate $m$ can be solved for in Eq7. However, I do not understand what is going on here. As stated above, the last term should ==0. Also there should be a pressure dependent term to capture the pressure melting point effects. At least when a linear relation between temperature and pressure is assumed then it takes the form $rho_w c_t c_t dP_w/ds*Q$ (e.g. Röthlisberger 1972, Eq 17); maybe the authors try to capture this with their last term of Eq7, but I don't think that can be done like this. A term in similar spirit would need to be added for the salinity dependent effects, presumably featuring a $d\beta/ds$. In fact, this is the quintessential equation to be stated as that is the novel one, all others are known.

- How Eq15, also featuring the melt rate, then ties in with this is also not clear to me.
The lake has only ever a simple geometry, thus I would recommend to just keep it simple and shorten that part. A simple "refer to Kingslake 2015 for a treatment of more complicated lake hypsometries."
Line by line

============

12: nice Introduction
13: (any many others): I prefer incomplete lists of references to be prefaced by a "e.g.". Many instances in the Introduction (l.15, 18, 24, 26, etc.)
30: I would prefer to use an SI units throughout instead of "psu". At least give a translation (1psu=1kg/m3).
70: Cite the original too, Röthlisberger (1972). I would write something like: "We construct a lake-drainage model in which the water flows through a subglacial channel (Röthlisberger, 1972; Nye, 1976). We follow the implementation and notation of Fowler (1999) and Kingslake (2015). We assume..." I also like the term "R-channel" and would use that, but no strong feelings here.
74: I disagree with Fowler (1999) here and thus with this manuscript: the given definition is the "negative basic hydraulic potential". I think it would be good to state the "negative" somewhere. Similarly Eq2 gives the negative potential gradient.
95: the "(instead of Darcy-Weisbach)" I find confusing as D-W is also for freshwater. Just delete?
97: I think 0.6m^{-1/3}s should be 0.06 (according to the code). Also cite where these values are from (maybe Clarke 2003?) as Fowler 1999 does not provide those values.
Eq6 and following: I don't think those formulas should be stated with so many digits. At most two significant digits are needed. Also state that for the $\rho_b$ and $\Delta\theta_P$ the 3rd and 2nd order terms are not important. Also, Eq6 is written with \beta as an argument, the others are without argument: should be consistent.
Eq6: is this correct? I evaluate $\rho_b(0)=0.1$ but fresh water has a density of 1000kg/m3.
Eq6 / l109: I don't understand how the melting point depression can have a constant term. That would mean that it is always depressed! Similarly for the change in melting-point due to pressure. Also, I guess these formula depend on temperature being given in deg-Celsius, that should be stated.
123: cite Röthlisberger here too, drop Kingslake.
132: why suddenly salinity in kgm^-3 instead of "psu"?
139-143: this can be abbreviated to just state Eq 8. The reader does not need to see a derivation of a conservation equation.
115: doesn't line 72 state that these temperatures are always equal?
167: I wouldn't call this a "no flux" boundary condition as there is water flux through the boundary. Just call it a Neumann BC. Also, what does it mean? What is meant with "We do not require the channel to exit at a terminus or end subaerially"? Where does it end? Or at least where does the simulation end and how do we know that there dN/ds=0?
174: reference the suggested Fig.1 (the sketch) here.
Fig.1: could the temperature as a function of distance also be plotted? Maybe even the different components, i.e. press-melt-point term and salinity-melt-point term.
230: I think this is vice versa.
242: what are "hyperconcentrated sediment concentrations"? What is this "threshold"?
253-259: I recommend to delete these lines as they don't add much
288: I concur with Reviewer 1: these N values do not make sense to me. Is this caused by the Neumann BC on N? Note that a, in my opinion, more natural and standard BC N=p_i (or p_w=0) leads to dN/ds not equal zero. See for instance in Röthlisberger (1972): Fig.5d shows a setup similar to here and shows p_w -> 0 at the outlet.
312: Excellent that the code is provided, thanks! Ideally the authors should also provide a third script that, when run, will reproduce and write to disc all figures used in the publication.

Citation: https://doi.org/10.5194/egusphere-2023-792-RC2
- AC2: 'Reply on RC2', Amy Jenson, 24 Jul 2023
  
  Please see the attached pdf to view our response to RC2.
  
  Citation: https://doi.org/10.5194/egusphere-2023-792-AC2

Peer review completion

AR: Author's response | RR: Referee report | ED: Editor decision | EF: Editorial file upload

ED: Reconsider after major revisions (further review by editor and referees) (09 Aug 2023) by Elisa Mantelli

AR by Amy Jenson on behalf of the Authors (15 Aug 2023) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (16 Aug 2023) by Elisa Mantelli

RR by Anonymous Referee #2 (30 Aug 2023)

RR by Anonymous Referee #1 (07 Sep 2023)

ED: Reconsider after major revisions (further review by editor and referees) (27 Oct 2023) by Elisa Mantelli

Dear Authors,

first of all my apologies for getting back to you with delay, due to a combination of delayed summer holidays and illness.

I have considered the manuscript and the referees' comments carefully, and I have to admit that I have been very torn regarding the best decision for this manuscript. While on one hand the novelty of the work is commendable (I'm here referring to the treatment of salinity in subglacial hydrology), the mathematical treatment of the model presented remains, in my view, shallow, which gives rise to major concerns regarding the model itself (see referees comments), as well as the conclusions drawn.

In spite of the latter point, and motivated primarily by novelty considerations, I would like to provide here one further opportunity for revision, so my decision is to reconsider this manuscript after major revisions.

In your review I would ask you to:

- consider carefully the point raised by ref 1 regarding energy conservation, and the dependence of the melting point on the along-channel coordinate. I trust that this point bears substantial consequences to the model itself and potentially the results/ conclusions drawn. I would expect to see careful recalculations and reconsideration of the stated conclusions.

- Discuss simplifications/ model choices and their implications with respect to results/ conclusions extensively. And ideally present a full (non-simplified) model that gets progressively reduced on the basis of physical considerations. At the moment, section 2 results extremely confusing in that it collates formulas/ equations from other authors with apparently limited coherence. Rather, I would expect to see a full model building on conservation laws that gets progressively and consistently simplified, starting from the statement of all model variables (and their dependence on independent variables), and including all initial and boundary conditions as each governing equation is presented. Then the choices made to simplify such model should be discussed and justified first in the context of the physical system that is being described (presumably in sec. 2) , and later on again in the discussion section so as to put the results obtained/conclusions drawn into proper context.

I look forward to seeing a substantially revised version of this manuscript. Should you have any questions regarding the review, or require extra time to perform a thorough revision of the paper, please do not hesitate to get in touch.

With best regards,
Elisa Mantelli

Hide

AR by Amy Jenson on behalf of the Authors (05 Mar 2024) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (26 Jun 2024) by Nanna Bjørnholt Karlsson

RR by Anonymous Referee #2 (27 Aug 2024)

Suggestions for revision or reasons for rejection

[This paper describes an R-channel model which takes into account the effects of saline water. Its main finding is that an R-channel conducting saline water at the local pressure and salinity melting point will open less fast that an R-channel conducting fresh water. This is because as the channel walls melt, salinity drops and with that the melting point increases. Thus some energy will be needed to heat up the water to the new melting point and thus is not available to melt the channel wall. Note that this is similar to the effect of the pressure dependence of the melting point of ice on R-channel dynamics for fresh water R-channels.

This is the third review for this paper I am doing and I am happy in how this progressed. I think the model is now correct, however, I am not 100% sure about the results. In particular the ones presented in Fig2d seem odd to me: a difference of 10^-5 C between brine and freshwater runs, as shown in Fig2d right axis, is a tiny difference compared to the difference in channel radius or discharge between those runs (Fig2a,c). To get some more context for the amount of melt and thus change in salinity consider the following back of the envelope calculation.

The total potential energy available is approximately 150m*rho_w*g (100m of ice thickness and about 50m from the 3-degree slope). Taking a typical discharge of 1m3/s the total melt along the channel is approximately (ignoring pressure and salinity effects, i.e. an upper estimate)

Q * H * rho_w * g / (rho_w L) ~ 1 * 150 * 1000 * 10 / (1000 * 3.3e5) ~ 0.004 m3/s

thus the relative salinity change should be about 0.004/Q ~ 0.004. This is about 3 orders of magnitude larger than what is plotted in Fig2d. Maybe my above estimate is wrong or I do not understand what is plotted in Fig2d. Therefore this needs to be either explained better, in particular how such minuscule changes could have such a big effect, or the calculation needs to be revisited.

Similarly, the change in temperature along the channel is at most 10^-5C according to Fig2d. However, I would expect a change in temperature due to pressure melt point effects alone of c_n * 100m * rho_w * g ~ 0.07C as the water de-pressurizes from the ice overburden pressure at the lake to zero at the outlet. Again many orders of magnitude difference between this estimate and Fig2d.

Line by Line comments
-------------------------------

34: state the salinity of sea water for comparison for those of us who do not deal with brine regularly

43: not sure what is meant by "edge dynamics"

Table 1: for some variable their value is given (e.g. rho_i) but not for others (e.g. A). This should be consistent.

99-101: Likely the biggest uncertainty in Manning's n is due to the great variability in its value (e.g. Pohle et al., 2022 find order of magnitude difference for one R-channel as it evolves). Maybe state something in this regard.

Eq 7: I guess in this equation only the constant and linear term could be retained, the effects of the higher order terms are minimal for the salinities considered here.

143: use `\times` to typeset the "x" in scientific notation numbers

Eq 9: it should be \sigma_w, the heat capacity of water (sorry, I miss-stated this in my last review); also in the summary of model equations

190: repetition

255: "evolving" -> "increasing"

259: I'd rather use "reduce" than "limit" as the latter suggest a hard limit but it only reduces the effects. Also other occurrences, e.g. l295.

287-289: state which assumption is more likely correct. Potentially cite, e.g. Röthlisberger here about his choice for the pressure melting point.

Fig 4: plot 0psu also.

Eq 15: Pretty sure this is wrong. In both denominators should be rho_s. This then leads to, for instance, a density of 1056kg/m3 for a SSC of 91g/l. Probably easiest to just correct the vector of SSCs in line 315 to give the stated rho-vector.

Hide

ED: Publish subject to minor revisions (review by editor) (10 Sep 2024) by Nanna Bjørnholt Karlsson

AR by Amy Jenson on behalf of the Authors (26 Sep 2024) Author's response Author's tracked changes Manuscript

ED: Publish subject to technical corrections (02 Oct 2024) by Nanna Bjørnholt Karlsson

AR by Amy Jenson on behalf of the Authors (03 Oct 2024) Author's response Manuscript

Journal article(s) based on this preprint

25 Nov 2024

Modeling saline-fluid flow through subglacial channels

Amy Jenson, Mark Skidmore, Lucas Beem, Martin Truffer, and Scott McCalla

The Cryosphere, 18, 5451–5464, https://doi.org/10.5194/tc-18-5451-2024,https://doi.org/10.5194/tc-18-5451-2024, 2024

Short summary

Amy Jenson, Mark Skidmore, Lucas Beem, Martin Truffer, and Scott McCalla

Model code and software

Subglacial brine flow Amy Jenson, Mark Skidmore, Lucas Beem, Martin Truffer, Scott McCalla https://doi.org/10.5281/zenodo.7829316

Amy Jenson, Mark Skidmore, Lucas Beem, Martin Truffer, and Scott McCalla

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Short summary

Water in some glacier environments contains salt which increases the density of the fluid and decreases the freezing point of the fluid. As a result, hypersaline lakes can exist in places where freshwater cannot and can contain unique microbiological communities. We model the flow of saline fluid from a subglacial lake through a channel at the glacier bed. The results suggest that fluid with higher salinity reach higher discharge rates compared to fresh water due to increased fluid density.


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