the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 02 Jan 2024
Modeling the effect of free convection on permafrost melting-rates in frozen rock-clefts
Abstract. Fully coupled heat transfer modeling during the thawing of frozen rock clefts yields melting rates that differ from those predicted by conventional conduction-based models. This research develops a conceptual model of a karst system subject to mountain permafrost supported by a numerical simulation incorporating free water convection. The numerical simulations rely on the apparent heat capacity method and the Darcy approach for energy and momentum equations. Notably, the anomalous behavior of water between 0 and 4 ℃ causes warmer meltwater to flow downwards, increasing the melting rate by approximately an order of magnitude as compared to conventional models that disregard free convection. The model outcomes are compared qualitatively with field data from Monlesi ice cave (Switzerland) and confirm the close agreement between the proposed model and real-world observations.
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Status: final response (author comments only)
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CC1: 'Comment on egusphere-2023-2349', Wenbing Yu, 04 Jan 2024
Is the icing in the cave in Figure 1 formed by the permafrost melts and seeps out, and then freezes again?
Citation: https://doi.org/10.5194/egusphere-2023-2349-CC1 -
CC2: 'Reply on CC1', Marc Luetscher, 10 Jan 2024
Yes, indeed. The cave depicted on Figure 1 is located in the Swiss Alps at >2300 m a.s.l.. Field evidence suggests this karst system is still prone to mountain permafrost. Water circulations are generally absent and infiltration occurs only sporadically. The figure illustrates the refreezing of water in a cave below 0°C after the cleft shortly thawed.
Citation: https://doi.org/10.5194/egusphere-2023-2349-CC2
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CC2: 'Reply on CC1', Marc Luetscher, 10 Jan 2024
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RC1: 'Comment on egusphere-2023-2349', Anonymous Referee #1, 17 Jan 2024
In their study, Sedaghatkish et al. model ice melt in ice cleft and compare melting rates with or without free convection. They show that free convection is a key process to account for because it increase melt rates by an order of magnitude. The modeling works is based on a commercial software, validated against to modeling experiments from the literature as well as observations. When used in a real world case, their setup shows a good ability to produce a melt rate that matches the water flow observed in a monitored cave.
My overall impression is that the study tackles an interesting topic with a relevant angle and adequate methodology. The model validation is convincing and the results on the real world case are good. The article is well structured and reads well. Some illustrations could be prettier but do the job (see suggestions below). Overall I think it is a good study that maybe won’t catch the attention of everyone because it tackles a very specific question but that TC should be interested in publishing because it is a relevant piece of work on the cryosphere. I don’t see much to change, so I suggest minor revision to the editor.
I made a detailed lists of small comments, I think some significant progresses can be made on the clarity of the explanations. The biggest of my small comments are the following.
- I think the study can be more clear and pedagogic regarding the physics and the equations.
- I think the author should discuss to what extent, fast water flow through the macro-porosity of this kind of massif could compete with free convection by disturbing the small scale density contrast that it requires. Who wins? Would it be nice to have a model that do both… (see below)
- I think the discussion should also try to discuss larger scale implications of the results in terms of consequence for permafrost disappearance at the scale of the massifs/mountains and regarding catchments water balance.
Specific comments
Abstract: I have the feeling that it could be nice to remind in a few words what is free convection to the readers in the abstract as it took me a few minutes to realize what we are talking about. E.g. “free convection (convection driven by density contrasts within the water phase)”. Instinctively I did not think that the water bodies in karstic environments are big enough for free convection to be important, like it would be in a lake for example.
L15: “the anomalous behavior of water between 0 and 4°C”
Do you refer to the density increase of water from 0 to 4°C? If so state it in a less mysterious way.
L22: The first sentence talks about the impact of global climate change on permafrost and the ref is only about the French Alps, maybe add a more large scale ref as well.
L33-34: “have shown that heat advected by water and air fluxes may significantly disturb the geothermal field, challenging classical models of heat propagation based on conductive fluxes”
Here you want to talk about the general case, not specifically karstic environment, so it would be good to come up with a reference that demonstrates that in non-karstic environments. Since you then talk about well-developed conduits right after, it would be also nice to add a line to explain the difference between convection in a porous media and convection in a conduit.
L36: “this permafrost” which permafrost? Maybe better “permafrost in karstic environments” or something similar. But the “this” refers to something undefined I believe.
L42: If rock-glaciers are relevant to this list, add it.
L43: lower than what? Than 2 or 3D models?
L47: “This distributed model is efficient for large domains (catchment scale).”
In what regard? What does that mean?
L49-50: Tubini et al., (2021)
With which processes? Conduction only or also convection?
L53-54: “The effect of water flow inside the clefts is noticeable because of creating thermal shortcut between atmosphere and subsurface.”
This is a general statement that does not really fit the list of modelling studies you are going through. It probably fits better earlier when you talk about medias with conduits.
L59-60: Maybe one more sentence to be more specific. You imply that atmospheric warming could warm melt water located close to the upper boundary close to 4°C, which would later sink right?
Figure 1. As I am still at the point where I try to understand what you did, I am surprised that you show on the picture a volume which order of magnitude is tens of meters and your modeled domain is in the order of 80 cm. I missed earlier on some explanations on why 80 cm is a relevant size and how something that small can have a larger scale relevance (i.e. small features but very frequent I suppose).
Also Figure 1b can be improved. Remind what are the x and y axis (I am surprised you did not use z for the vertical axe by the way, I have the feeling it is what the general reader would expect to understand your work more smoothly). As it is I am still unsure which one is the vertical one. If you put dT/dx on the side I am tempted to believe it is a boundary condition for the side, but with the proximity of the other dt/dx to the H, I would be tempted to think x is the vertical axis. Add arrows as well maybe. The red arrows look like they were made with MS Paint. I think in general you can give more love in Fig 1B.
L86: The sentence misses a point at the end.
L93: I feel there is maybe a lack of pedagogy regarding the A x… term that is introduced. I don’t know what is a Darcy like pressure drop (a bit of physics “with the hand” to help intuition maybe) and I don’t see why theta=0 will nullify the velocity. What I see is that you will add the term A v / epsilon to the Navier Stokes equation (or A u / epsilon horizontally). How does that nullifies the velocity?
Similarly, I think you should explicit what is the Boussinesq approximation. I don’t suspect many readers of TC will know that and it can be disorienting to see rho0 in your equations whereas you intend to work with density contrasts in your study (and your actual rho variable is hidden in beta, so at first glance, rho seems to be fixed).
L109: “domain”, to me at this stage, it is not clear if your domain is just the water/ice or also the surrounding rock. I see a mesh on the rock on Figure 1 but the rock is impermeable. So I think you should find a way to be clear on this, talking about the “water and ice domain only” or the “whole domain” or anything that would reach the same goal. So for Temperature, your domain is water/ice + rock? Because the cp you describe looks like it is for water only, it is not a freeze curve that would account for suction effect in the rock, that would spread the phase change below 0°C.
L131: “no-slip” same, explicit quickly. Also what about heat fluxes between the water and the wall? Is it just conduction or also convection?
Section 2.1. Finite elements? Finite volumes?
L151-152 : Here you just mention comparison with simulations even though Virag also has observation if I understood correctly. Reproducing observations is an even better validation, so make it more clear.
L159: « with (Kahraman et al., 1998). »
Fix parenthesis
L172-174 « The isothermal line corresponding to T=4 ℃ is plotted inside the temperature contours implicating well the interface of the two counter rotating convection cells with two temperature ranges. »
I do not understand this sentence. Please reformulate.
Sect 3-2
Here be more precise whether you replicate the simulations from Virag or their observations.
L165: “The density of water increases between 0 and 4°C, and decreases above 4°C by increasing temperature.”
If not already the case, this should appear earlier. Not in the experiment explanation.
L196-197 “thorough”
Through
L200 “for two scenarios”
They are the same 2 scenarios as before right?
L202: “and the extending the meltwater depth”
Problem with the sentence.
Figure 8. Make a more explicit legend than W/WO.
L221: “Irregular water circulations”
What is that?
L238: “completely are removed.”
L240: You left an exclamation mark.
L242-L245: not completely clear to me, can you reformulate? What is a aperture size threshold? So you back calculate Ra based on the empirical relationship and check for threshold values? With the last sentence do you mean that the empirical relationship is not valid?
L257: What is the soil temperature? Do you have a real pedological soil? To make things clear maybe you can talk about the ground surface (I guess it’s the ground surface outside right, not in the cave?).
L260-261: “The daily temperature variation induces a water…”
Temperature of what? A cave is a complex setup compared to an homogenous media, be more specific to help the reader get a clear picture of what you describe.
L264: “we allow” you allow but you are talking about the natural case no? Because you said “In contrast to our model” just before?
L273: “(red dash-dotted in Fig. 11-b)”
If I understand correctly, should be “(red solid line and red dash-dotted line…” because the good news is that the dash-dotted line follows the moving average of the red solid line right? I feel you should make the 3 red lines thicker or anyhow more visible. They convey the key message of your paper right?
Discussion : the 2 first paragraphs feel redundant with things you already said in the introduction and methods. I would start at line 285.
L293-295: “Whether this water results from the melting of ice in the cleft or recharges from the surface (storm events or snowmelt) does not matter.”
I don’t understand what you mean with this sentence. Additionally, if these are the sources of water, it is unlikely to be warm, so I miss the connection with the previous sentence.
L302-306: I don’t see where this paragraph goes. In soils things work differently than in karstic cavities right? The freezing will spread below 0 because of the suction in the soil. I don’t see what perspective that gives on your work.
L312 :”… is in the same order of magnitude as the measured water flow rate (Fig. 11) in Monlesi cave.” Here would be a nice place to discuss why the water flow is oscillating and your melting rate is smooth. If the average values are similar, what create the contrast in timing?
L313: “The effect of free convection is not limited to hourly or daily oscillations and can be studied over much longer timescales, including centennial to millennial fluctuations.”
When melting very big ice clefts? If the weather warms up from one year to another, once you start melting a cleft of one or 2 meters long, isn’t it going to melt in a few years? Or are you discussing the melt at the scale of a massif?
L326-328: “In karst systems and fractured aquifers, where secondary porosity is exceptionally well developed, frozen conduits/fractures may all of a sudden drain water into depth and change the local hydrological regime leading a thermal anomaly within the surrounding permafrost (Phillips et al., 2016).”
That’s something you could discuss further to think against yourself. How much fast flow in karstic system is likely to actually advect temperature quickly over long distances and disrupt the peace of free convection? It gives an opportunity to discuss the representativity of the process you pinpoint in the perspective of the general functioning of a karstic/fractured massif exposed to seasonal/long term cold weather. It Is also an opportunity to compare your work with Hasler et al. (2011). Is there a process more important than the other? Would we gain something trying to represent both at the same time?...
L333-335 “But also at shallower depth, acknowledging the potential role of convective heat fluxes in ice-rich permafrost degradation may help predicting the rate of greenhouse gas releases, mainly carbon dioxide and methane, due to the decomposition of formerly frozen organic matter (Schaefer et al., 2014; (Schuur et al., 2015).”
This sounds a bit far-fetched. What carbon pools are you talking about? There is not much carbon in the fractures/karstic cavities of a rock massif right? And you do not expect much free convection in an organic peat soil right? I have the feeling that free convection is relatively low in the list of missing processes to accurately represent permafrost thaw where you find a lot of organic carbon, but I am happy to be proved wrong. You can check Kane et al. (2001, GPC, 10.1016/S0921-8181(01)00095-9).
L335-338: Free convection is everywhere so it is beautiful, ok but not super relevant for your study. Funny that you did not explain the TC reader what is Boussinesq approximation but you do explain what is an iceberg 😉.
In this discussion, since your main conclusion is that we need to be careful about not underestimating the melt rates in rock massifs with ice cleft, I missed a bit of large scale discussion on the implication for:
- catchments water balance. If you try to upscale your results, how can this impact runoff in mountain catchments, river flow, lake levels, at catchment scale and a global scale? Where should we start worrying more about this question?
- Permafrost disappearance at the scale of the massif. Does it change what we forecast for the Alps, by much?
So that we can grasp how significant these results could be at broader scales.
L345: “only impact on the temperature” I suspect the “on” should be removed.
Citation: https://doi.org/10.5194/egusphere-2023-2349-RC1 - AC1: 'Reply on RC1', Amir Sedaghatkish, 18 Apr 2024
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RC2: 'Comment on egusphere-2023-2349', Anonymous Referee #2, 24 Jan 2024
GENERAL COMMENTS
The manuscript presents numerical analysis of the thawing of ice-saturated cleft in rocks, demonstrating the relative importance of gravity-driven convection of liquid water due to the subtle increase in density as the temperature rises from 0 to 4 degrees. While it is not difficult to imagine the significance of free convection based on observations in other environments, it is important to advance the quantitative understanding of the effects of free convection in ice-filled clefts. Therefore, this study has the potential to make a significant new contribution to cryospheric sciences. The manuscript is reasonably well organized and written in a clear language. However, it is missing some essential information on the methodology and as such, it is difficult for the reviewer to evaluate the rigor of numerical and experimental methods in some places. Theoretical interpretation of numerical modelling results are sound except for some missing details (see above), but the comparison between numerical results and field observation is much weaker. To strengthen the comparison, I suggest that the authors consider the following: (1) enhance the description of field methods, (2) acknowledge the magnitude of uncertainty in flow measurements more specifically, and (3) use independent evidence to support the match between model results and field observation. I will elaborate more on these in my specific comments below.
SPECIFIC COMMENTS
Line 18-19. The agreement the model and real-world observations is qualitative at best (see my comment on Line 273). ‘The close agreement’ is an overstatement. Please rephrase the sentence.
Line 91. Eq. 1 assumes no volume change in water, implying that the change in volume associated with ice-to-liquid transition is neglected in the analysis. If this is the case, then the model domain will have void space, presumably at the top. How does the model take this into account? Please present a clear explanation.
Line 92. Reduced pressure. This term is not familiar to most readers of the journal. Please define it.
Line 94. A kind of Darcy-like pressure drop. I do not understand what this means. Please explain it more clearly.
Line 96. What does the variable ‘A’ physically represent? What is the unit?
Line 109. The heat transfer in the rock phase is limited to conduction, implying negligible rock porosity. This is contrary to numerous field-based studies of karst hydrogeology, where the fracture network in karstified rocks can play a major role in water transfer. What is the expected porosity of the system the authors are intending to model? Please add a sentence or two on porosity and fractures.
Line 122. The authors assume -0.5 to +0.5 C as the temperature rage of ice-liquid transition. While ice and liquid water can co-exist under negative temperature due to freezing-point depression, there is no known mechanism to sustain the ice-liquid mixture under positive temperature. Please justify the choice of temperature range. If it is not justifiable, please re-run the simulations using a physically feasible temperature range.
Line 127. Impermeable solid rock. Please see my comment on Line 109.
Line 132. How is the depth of the cleft defined with respect to the actual clefts in the field. In natural systems, water will drain from the bottom of the cleft as shown in Figure 1a.
Line 132. 80 and 10 cm. How are these values chosen? Please provide an explanation.
Table 1. Density 2320 kg/m3. This is much smaller than a typical density of solid carbonate rocks, and implies substantial porosity (14%?). Is this consistent with the model assumption? Please explain.
Table 1. Thermal conductivity 1.656 W/m/K. This is much smaller than that of solid carbonate rocks, implying substantial porosity. Is this consistent with the model assumption?
Line 141. Please spell out 14k, 24k, etc.
Line 179. At the bottom of the cavity. The model also under-simulates the advance of thawing front in the upper part by 2300 sec. Please point this out.
Line 170. The overall performance of our model is sufficient. This is a subjective statement. Please explain the basis for this statement.
Line 188-189. The authors state that the model conceptualization (Figure 1b) is similar to the physical setting depicted in Figurer 1a. However, I do not see a clear similarity. Please improve the description. A schematic diagram depicting typical clefts observed in the field will be useful to bridge the gap between Figures 1a and 1b.
Line 193. Total volume of liquid water. The total volume of liquid water should be much smaller than the volume of ice before melting. Therefore, there should be some void spaces if the model obeys the mass-conservation law. Violation of mass conservation is considered a major deficiency of any mass and water transfer models. Please explain how the water mass is conserved in the model.
Line 223. Rayleigh number. Please report the Rayleigh numbers computed for the numerical experiments presented in the manuscript.
Line 230. Melting rate. Does it refer to the melting rate (kg/hr/m2) or cumulate amount of melt (kg/m2)? Linear plots in Figure 10 seem to suggest the latter. Please clarify.
Line 244. 8 m meltwater depth. Is this 8 m or 0.8 m? The model has 0.8 m, not 8 m. Please clarify.
Line 253. Seasonal freezing seals them periodically. This implies that the clefts (or chimneys) are not always saturated. How can this be adequately represented in the saturated model? Please explain. By the way, are clefts and chimneys the same thing? If so, please use a consistent term.
Line 254. The distance. Does this refer to vertical distance? Does the external surface indicate the ground surface? Please clarify. A schematic diagram will be useful (see my comment on Line 188-189).
Line 255. Clefts of different sizes. Please indicate a rage of sizes observed at the site.
Line 256. A few centimeters. Please report the actual depth, even if it is approximate.
Line 257. 4.5 days. Please report the actual dates.
Line 259. Cave temperature. Where in the cave (in relation to clefts) and how was it measured? Please explain.
Line 260. How was the water flow monitored? This information is critical. Please explain it carefully with sufficient details.
Line 264-265. I do not exactly understand this sentence. ‘We allow for the meltwater to drain deeper’ implies that a new model is set up to simulate drainage, but ‘in contrast to our model’ implies that the same model without drainage is still used. Please clarify. Also, if drainage is allowed, please explain how it is done in the model.
Line 271. 3 meters depth. Is this based on the measured depth in the field? If not, how is it selected? The same applies to 10 cm aperture.
Line 271. In general, how long are the clefts observed at this site? Please indicate it in the sentence. I am referring to the third dimension in addition to the depth and the aperture.
Line 273. The same order of magnitude. This is not a meaningful comparison because the measured flow rate may have a large degree of uncertainty depending on how it was measured (see my comment on Line 260). It is not uncommon for this kind of measurements to have uncertainties greater than an order of magnitude. This is a major weakness of the manuscript. Independent evidence demonstrating the qualitative match between the model and field observation will be useful (see my comment on Line 311-312).
Line 290. Sufficiently high. Please indicate the number and compare it with the critical Rayleigh numbers reported in previous studies of free convection (not necessarily in water-ice systems).
Line 311-312. The modeled melting rate alone is not sufficient to support the model performance due to the large degree of uncertainty in flow measurements. Independent evidence will be useful. For example, how long does it usually take to thaw a cleft completely from the top to the bottom? Will it be possible to estimate the thawing rate and compare it with the modelled thawing rates? Please explore this and other approaches further to provide independent evidence.
Line 447. At 8 m meltwater depth. Where does 8 m come from? The numerical model was 0.8 m deep, not 8 m.
Citation: https://doi.org/10.5194/egusphere-2023-2349-RC2 - AC2: 'Reply on RC2', Amir Sedaghatkish, 18 Apr 2024
Video supplement
Modeling the effect of free convection on permafrost melting-rates in frozen rock-clefts Amir Sedaghatkish and Marc Luetscher https://doi.org/10.5281/zenodo.8435167
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