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

Preprints

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

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this preprint. The responsibility to include appropriate place names lies with the authors.

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

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

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

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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