the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 04 Mar 2024
Modelling active layer thickness in mountain permafrost based on an analytical solution of the heat transport equation, Kitzsteinhorn, Hohe Tauern Range, Austria
Abstract. The active layer thickness (ALT) refers to the seasonal thaw depth of a permafrost body and is an essential parameter for natural hazard analysis, construction, land-use planning and the estimation of greenhouse gas emissions in periglacial regions. The aim of this study is to model the annual maximum thaw depth for determining ALT based on temperature data measured in four shallow boreholes (SBs, 0.1 m deep) in the summit region of the Kitzsteinhorn (Hohe Tauern Range, Austria, Europe). We set up our heat flow model with temperature data (2016–21) from a 30 m deep borehole (DB) drilled into bedrock at the Kitzsteinhorn north-face. For modeling purposes, we assume 1D conductive heat flow and present an analytical solution of the heat transport equation through sinusoidal temperature waves resulting from seasonal temperature oscillations (damping depth method). The model approach is considered successful: In the validation period (2019–21), modeled and measured ALT differed by only 0.1±0.1 m. We then applied the DB-calibrated model to four SBs and found that the modeled seasonal ALT maximum ranged between 2.5 m (SB 2) and 10.6 m (SB 1) in the observation period (2013–2021). Due to small differences in altitude (~ 200 m) within the study area, slope aspect had the strongest impact on ALT. To project future ALT deepening due to global warming, we integrated IPCC climate scenarios SSP1-2.6 and SSP5-8.5 into our model. By mid-century (~ 2050), ALT is expected to increase by 48 % at SB 2 and by 62 % at DB under scenario SSP1-2.6 (56 % and 128 % under scenario SSP5-8.5), while permafrost will no longer be present at SB 1, SB 3 and SB 4. By the end of the century (~ 2100), permafrost will only remain under scenario SSP1-2.6 with an ALT increase of 51 % at SB 2 and of 69 % at DB.
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RC1: 'Comment on egusphere-2023-3006', Anonymous Referee #1, 18 Jun 2024
General comments.
This study is on active layer thickness in an alpine rock permafrost environment with borehole temperature measurements and analytical modeling of the heat diffusion equation. The measurements and data set used is very interesting and the underlying problem of using shallow ground temperature measurements (here at 0.1 m) for making predictions of ALT and permafrost warming is very important especially in the context of climate change and its severe impacts for alpine regions.
However, the thermal modeling approach used is not new or novel, and has very limited applicability for active layer modeling. The equations described are well-known analytic solutions to the heat conduction (diffusion) equation and can be found in textbooks on the subject, see e.g. Carslaw and Jaeger (Conduction of heat in solids, 1959), Williams (The Frozen Earth, 1989), Woo (Permafrost Hydrology, 2012), and others. It is severely limited for active layer and permafrost modeling because it does not represent many of the critical freezing and thawing processes which control heat propagation in the subsurface.
Many of the assumptions imposed are mentioned in various parts of the manuscript and include neglecting latent heat exchange, advective heat carried by water migration (both lateral and vertical), snow insulation and snowmelt, and heterogeneity in subsurface thermal and hydraulic properties. Critically, even for homogenous ground, latent heat, thermal conductivity and heat capacity change due to variations in liquid-ice-air phase saturation in the active layer, and these changes need to be accounted for in order to be able to calculate heat propagation through the active layer and to the permafrost table and below. A possible exception could be if the ground is extremely dry. Or if the interest is only long-term and deep temperature responses, say below the depth of zero annual amplitude. Even if the intact/unfractured bedrock contains little pore space and hence may be suitably represented by heat conduction, there are evidently fractures feeding the subsurface with unfrozen water. This is noted in the discussion in Section 5.1 and is clearly seen by the prolongation of temperature near the freezing point (the zero-degree curtain) for the thermal sensors located at 3 m depth in Fig 3a.
Several analytical and semi-analytical approaches have been developed to address the limitations of the analytical solution to the heat conduction equation, other than full numerical solutions of the coupled system of differential equations for heat transport with water flow. For some background and an overview, please see the excellent papers by Riseborough et al. (2008) and Kurylyk et al. (2014). Perhaps some of those more robust methods could be employed to make predictions on ALT based on the shallow boreholes, or even better, numerical models which incorporate even fewer limitations.
Riseborough, D., Shiklomanov, N., Etzelmüller, B., Gruber, S., Marchenko, S., 2008. Recent advances in permafrost modeling. Permafrost and Periglacial Processes 19, 137–156. https://doi.org/10.1002/ppp.615
Kurylyk, B.L., McKenzie, J.M., MacQuarrie, K.T.B., Voss, C.I., 2014. Analytical solutions for benchmarking cold regions subsurface water flow and energy transport models: One-dimensional soil thaw with conduction and advection. Advances in Water Resources 70, 172–184. https://doi.org/10.1016/j.advwatres.2014.05.005
Another problem is the biased implementation of the NSE as an error metric, which seems to be based on modeled values after discarding values which perform poorly (Fig 5a). This is a biased filtering and is probably the reason high values of the NSE metric are obtained. Clearly, if all the modeled data are used, including the freeze-up period, and compared against the measurements, the NSE metric will be much lower.
Finally, the interpretation of the ALT projections for the shallow boreholes SB 2, 3 and 4 in Section 4.3 and 5.3 is questionable. These boreholes seem to be grouped near a peak and the spatial distance between them only seems to be a few 10s of meters. Despite their proximity, the prediction on the ALT/PF table are very different (Table 5), from no PF (“infinite” ALT) in SB3 to relatively shallow ALT of 2-3 meters in SB2. It is unrealistic to infer that the permafrost table varies as extremely as this. The result is probably an artifact of the method which assumes 1D conduction only, and that the ALT prediction is based solely on a single near-surface temperature sensor. In reality of course, lateral subsurface heat transport distributes the thermal response and the permafrost extent will be smoother even if differential warming occurs on different faces of the mountain surface.
Detailed comments.
L125-130. Has the thermal effect of the concrete annulus been evaluated? It could be that the thermal properties of concrete have an effect on heat conduction between the rock and the brass segments and potentially bias sensor measurements. Also, there could be effects of vertical heat conduction in the concrete annulus which may differ from that of the bedrock.
L130-135. Is the surface smooth bedrock or is there overburden and unconsolidated material? It is difficult to judge from the photograph in Fig 2b, and not entirely clear how the depth 0.1 m is obtained, because it seems surface roughness could easily vary within this range of a few dm.
L250-255. Water saturation generally increases with depth and, more critically, changes over time with variable infiltration from snowmelt, rainfall, and seasonal freezing/thawing.
L195-200. The problem with the damping depth calibration approach used is that it applies to the thaw depth of the time period considered. With the climate warming scenarios imposed later, the thaw depth changes, yielding the calibration obsolete. The calibration-validation exercise simply indicates no significant changes occur within the time period.
Fig 3a shows the zero-degree isotherm, indicating presence of water and showing that latent heat exchange is playing a significant role. Thus, the assumptions used for the model are questionable.
Eqn 7. It is not clear how temperature comes in to play for the equation for d_phase?
L195-200. The NSE reference is not included in the bibliography. Also, values can be negative, indicating predictions are worse than the mean of the measured data, i.e., not capturing the variability of measurements.
Fig 4b. Unclear why only two years are shown. Please make the markers more visible, currently they are difficult to see. The colors for 2017 and 2018 are inconsistent with Fig 4c. Is it correct there are only 3 measurements in the interval 0-2 m; if so, then a linear regression is of course expected to fit well. Same goes for Fig 4c.
It would be useful to show a trumpet diagram of the temperature measurements with depth (similar to Fig 4a) for the deep borehole with systematically selected times, e.g., once a month. This serves to present data without interpolations as in Fig 3b and A1, and will help visualize the active layer depth, the depth of zero annual amplitude, if there is an inflection point, and if the temperature trend with depth indicates a transition to the permafrost base. This can readily be combined with annual averages for each depth.
Fig 5a. The modeled values can of course be calculated at all depths but the measurements are only made at specific depths. Therefore, it would be reasonable to plot lines for the simulations and markers for the measurements at their corresponding depths. This would help clarify the comparison between modeled results and measurements. Also, it would be helpful to clarify that the values in brackets indicate the ALT.
Fig 5a and Section 4.2. It seems by ignoring the data which does not fit the model is the reason high NSE and RMSE values are obtained – but this is by biasing the measurements and as such highlights the limitations of the model approach.
Fig 5a. The main reason the freeze-up is not represented well by the model is the absence of latent heat.
Citation: https://doi.org/10.5194/egusphere-2023-3006-RC1 -
AC1: 'Reply on RC1', Wolfgang Aumer, 10 Jul 2024
Many thanks for your comprehensive review and insightful comments. We fully agree that the model approach used is neither new nor novel and has significant limitations in its application for modeling Active Layer Thickness (ALT) due to the neglection of latent heat and convection. We will emphasize this limitation more clearly in the manuscript.
However, despite these limitations, we firmly believe that our study provides valuable evidence for the first time that ALT in conduction-dominated bedrock permafrost can be effectively modeled using this cost-efficient and simple approach.
We also believe that the applicability of the presented approach is not limited to the studied rock-faces (Kitzsteinhorn) but much rather can be transferred to other active layers in conduction-driven systems in bedrock. We do not see indications that the studied site has unique site-specific conditions, which would significantly set it apart from other high-alpine bedrock sites. We can also rule out the suggested possibility that the studied site is “extremely dry”. Abundant meltwater availability during the melt season (~ May to August) leads to full subsurface saturation, which has been demonstrated based on electrical resistivity measurements, piezometric borehole measurements and borehole temperature jumps induced by water flow in fractures (the latter is mentioned in the manuscript). We will integrate complementary data and references into the manuscript to further prove our point. The fact that the presented simplistic analytical solution provides excellent results in an environment that is – at least periodically (spring/summer season) – highly impacted by subsurface water flow indicates, in our view, that our approach should have wide applicability (in frozen, steep bedrock environments).
We are fully aware that an analytical solution is less sophisticated than a numerical solution and is not capable of integrating water flows and latent heat. However, in alpine terrain, missing information on water balance and subsurface flow is the norm (with only a few exceptions), which can also lead to inaccuracies in model calibration. This means that in conduction-driven active layers, an analytical solution does not necessarily have to be less robust than a numerical solution, especially when the goal is determining annual thaw depths rather than precise modeling of ground temperatures. This is the case for the present manuscript, which explicitly focuses on the simulation of the maximum ALT in high-alpine environments due to its significant impact on natural hazard occurrence and infrastructure management.
The analytical solution of the heat conduction equation eliminates computational constraints that can limit the number of measurement points and simulation runs. Simulations with different boundary conditions can be carried out very easily compared to numerical solutions. For example, using the idealized boundary conditions of the analytical solution, we determined that a warmer initial thermal situation also causes a greater increase in ALT in relation to a colder initial thermal situation with the same temperature increase. Estimating ALT based on near-surface temperature data in combination with analytical modeling can therefore be a simple yet very help- and powerful tool, for example, to implement in an expert system for bedrock permafrost regions.
A detailed response to the review will be provided after the second reviewer has posted his/her comment. We very much appreciate your valuable feedback and look forward to further enhancing our study based on the forthcoming evaluations and discussion.
Citation: https://doi.org/10.5194/egusphere-2023-3006-AC1
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AC1: 'Reply on RC1', Wolfgang Aumer, 10 Jul 2024
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RC2: 'Comment on egusphere-2023-3006', Anonymous Referee #2, 29 Oct 2024
The paper, "Modelling Active Layer Thickness in Mountain Permafrost Based on an Analytical Solution of the Heat Transport Equation, Kitzsteinhorn, Hohe Tauern Range, Austria" by Wolfgang Aumer, Ingo Hartmeyer, Carolyn-Monika Göres, Daniel Uteau, and Stephan Peth, introduces a pioneering approach to modelling active layer thickness (ALT) in mountain permafrost. It is the first study to use analytical solutions of the heat transport equation for this purpose. The field protocol presented is highly valuable to the permafrost research community, as its methods have the potential to provide an enhanced understanding of permafrost characteristics in similar environments. The study meticulously analyzed methodological parameters and validated its results by comparing them with borehole data, achieving strong agreement. The analytical approach effectively identified the maximum ALT during the observation period. Additionally, the model incorporated IPCC climate projections to simulate ALT changes expected by the middle and end of this century.
The paper is well-structured, thoroughly illustrated, and includes critical reflections that lead to well-supported conclusions. The methodology is rigorous and consistent and the results support the findings. I therefore recommend acceptance of the manuscript, pending minor revisions as outlined below:
- In the Introduction, you mention that this approach is novel for modelling active layer thickness. However, I feel that the potential of this technique for addressing existing limitations is not sufficiently discussed. I suggest adding several sentences in Section 5 (Discussion) to clarify how this analytical approach can help overcome current limitations and how it might be integrated into a more complex model to incorporate additional variables (e.g., snow cover).
- Lines 34–35: You mention large-scale rock avalanches at Piz Cengalo and Fluchthorn, but only specify the country for Fluchthorn. To maintain consistency, please also consider adding the country for Piz Cengalo.
- Line 37: "Direct ground temperature measurements have shown, in some areas very rapid, warming in the European Alps over recent decades (Harris, 2003; Etzelmüller et al., 2020)." Harris (2003) is now 20 years old, so this phrasing feels outdated. I suggest rephrasing this sentence and reconsidering the use of this citation.
- Table 2: The slope aspect for SB3 and SB4 seems unusual. Could you clarify the terms "ONO" and "SO"?
- Figure 1: Could you overlay permafrost distribution on this map? It’s unclear if all sites are situated in permafrost conditions.
- I'm also curious about the primary reasoning behind the sampling strategy. Why were three SBs placed close together, while one was set farther away? Was this influenced only by the slope aspect?
- For model development, you calculate mean daily values using 12 measurements per day, but for model application, you used 8 measurements per day for the mean. Why not use a consistent approach for both?
- Line 191: You cite Dobiński (2011), but it is missing from the reference list.
- Line 382: It would be interesting to compare your findings with existing permafrost distribution models.
- Lines 378–389: You state that the slope aspect is the dominant topographic factor influencing ALT in this study. However, you also mention that the south-facing borehole is slightly colder and has a higher permafrost probability, likely due to local snow cover effects. This suggests it would be beneficial to discuss the interaction between snow cover and ground surface temperature in more detail. While slope aspect is a key factor in the evolution of ALT, its influence is straightforward only when the snow cover regime is uniform across all sites; otherwise, interpretations become more complex.
- The IPCC scenario simulations consider only temperature, but you previously showed that snow cover also plays a key role. Please add a sentence addressing the uncertainties associated with excluding other controlling factors such as snow cover.
- Several references cited in the text are missing or contain inconsistencies. For example, Dobiński (2011) is missing from the reference list, line 573 shows inconsistent author name formatting (Schrott et al.), and line 585 lacks author initials. Please carefully review and revise the reference list for accuracy and consistency.
Citation: https://doi.org/10.5194/egusphere-2023-3006-RC2
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