the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 19 Jun 2025
Stochastic Modelling of Thermokarst Lakes: Size Distributions and Dynamic Regimes
Abstract. Thermokarst lakes are among the most common and dynamic landscape features in ice-rich permafrost lowland regions. They influence carbon, water and energy fluxes between atmosphere and land surface and are an important component of Arctic lowland hydrology. Despite their significant role in the climate system, thermokarst lakes are only rudimentarily or not at all represented in Earth system models (ESMs). Attempts at stand-alone modelling of their dynamics have mostly been limited to the scale of individual lakes. Because lake formation, expansion, and drainage depend on small-scale surface and subsurface heterogeneities that are difficult to measure, a deterministic modelling-approach would be a challenge at the regional or pan-Arctic scale. We therefore treat these processes as probabilistic across a landscape and create a model of thermokarst lake dynamics using stochastic approaches. With the inclusion of stochasticity and volatility, our method allows us to account for the diversity of individual lake behaviour that results from the small-scale differences in environmental conditions. We present idealized simulations and, additionally, test novel high-resolution remote sensing data products that capture annual lake areas for model initialization and the calibration of inherent or climate-induced lake dynamics. Our model is able to capture three plausible regimes by incorporating the main processes behind thermokarst lake dynamics and represents a new step towards stochastic representation of permafrost landscapes in ESMs. Furthermore, our findings emphasize the importance of continued remote sensing data retrieval and additional data products containing information on past thermokarst lake behaviour for model parameterization.
Competing interests: At least one of the (co-)authors is a member of the editorial board of The Cryosphere.
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 paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.-
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Preprint
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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- Final revised paper
Journal article(s) based on this preprint
Interactive discussion
Status: closed
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CC1: 'Comment on egusphere-2025-1817', Elchin Jafarov, 17 Jul 2025
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AC1: 'Reply on CC1', Constanze Reinken, 19 Dec 2025
Dear Dr. Jafarov,
we sincerely thank you for your comments and will make sure to incorporate your suggestions:
- We will add one or two paragraphs outlining technical details of how the model could be incorporated or used for ESMs, which would mainly work through providing the ESMs with water area fractions for grid cells in thermokarst-affected regions. We will mainly use the example of the MPI-ESMs land surface component JSBACH / ICON-Land, which some co-authors are experts on.
- The main differences are that Nitzbon et al. 2020 is not stochastic and consists of tiles that each contain a one-dimensional vertical representation of the subsurface that needs to be initialized and parameterized. With our approach, on the other hand, we try to bypass the necessity of such sub-surface initializations and parameterizations by using stochastic processes and only looking at the changes of the land surface. We realize now that the placement of the mention of Nitzbon et al. 2020 within the text is confusing and might suggest that this model is also stochastic. We will rearrange this part of the text and try to make the differences between our approach and Nitzbon et al. 2020 more clear.
- In the “Introduction” and “Conclusions” we will add more insight into how the model can potentially contribute to the estimation of permafrost carbon emissions by drawing connections to previous work and knowledge on carbon emissions from high-latitude lakes. For instance, we will add details on how the model’s projections of changes in water area fraction and lake size distribution can help estimate methane vs. carbon dioxide emissions by giving an indicator for surface water area and lake coastline lengths and depths, which influence carbon emission pathways and processes of microbial decomposition (e.g. Rehder et al. 2023, Turetsky et al. 2020).
Citation: https://doi.org/10.5194/egusphere-2025-1817-AC1
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AC1: 'Reply on CC1', Constanze Reinken, 19 Dec 2025
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RC1: 'Comment on egusphere-2025-1817', Anonymous Referee #1, 16 Sep 2025
This is a well-structured paper that addresses a critical gap in Earth System Modeling (ESM) by proposing a novel stochastic approach to represent thermokarst lake dynamics at a regional scale. The model is conceptually sound, clearly explained, and the exploration of different dynamic regimes is a particular strength. The main limitations are openly acknowledged and are largely tied to the current state of observational data, which is a field-wide challenge, not a flaw specific to this study. Regarding the specific content of this manuscript, I have the following comments and suggestions.
1. The application of Geometric Brownian Motion (GBM) and Poisson processes to model lake area change and formation/drainage events is innovative and well-justified within the context of landscape-scale heterogeneity and stochasticity. This is a significant step beyond previous deterministic or analytical models.
2. The three idealized regimes (Complete Drainage, Oscillation, Stabilization) effectively demonstrate the model's behavioral range and provide a useful conceptual framework for understanding possible long-term trajectories of thermokarst landscapes. The links to real-world examples for each regime are well-made.
3. The current merging algorithm is identified correctly as a significant weakness. The assumption that merged lake area is the sum of the two original areas and that the new lake is perfectly circular is physically unrealistic and computationally expensive. It likely leads to a drastic overestimation of lake sizes and an underestimation of lake numbers.
4. The high percentage of unusable data points (30-33%) in the remote sensing dataset severely impacts the robustness of the parameter estimation. This uncertainty propagates through the observation-based simulations and limits the confidence in the derived parameters (λf, λd, μ, σ).
5. A crucial result is that the stochastic component (σ) dominates the deterministic drift (μ) in the observed period. This implies that, for the 2000-2020 period in this region, random environmental variability was a stronger driver of annual lake area changes than any clear climate-driven trend, explaining the lack of strong correlations.
6. The inability to find significant correlations between model parameters and climate variables (TDD, P) is a notable negative result. While honestly reported, it highlights the current impossibility of confidently projecting lake dynamics under climate change scenarios with this model, as intended in the abstract. This is a major constraint on its immediate application in ESMs.
7. The observation-based simulations project water area fractions increasing to over 50%, which is acknowledged as rare. This, combined with the high volatility, suggests the model parameters derived from 20 years of data may not be stable or representative of centennial-scale dynamics, potentially overestimating expansion.
8. The suggestion that the idealized simulations could be interpreted as spanning 10 ka with a 10-year time step is helpful for context, but the parameters would then be "per decade." This should be stated explicitly in the text to avoid confusion (e.g., in Table 1, add a note: "Parameters are per year; for a 10-year time step interpretation, values would be per decade").
9. The comparison with the van Huissteden et al. (2011) model is good. The explanation for the differing results (their reliance on a prescribed river network vs. your data-driven drainage rates) is plausible and highlights the advantage of your approach, but also its current data dependency.
10. The definition of "abrupt drainage" as a complete (>90% loss) and rapid event is clear. However, the discussion of results from other studies (Jones et al., 2011, 2020) that use different thresholds (e.g., >25% loss) is slightly confusing. A small table summarizing different study's definitions and converting their rates to a common framework would be helpful.
11. The axis titles of many figures are obscured, or the figures are not very clear, for example, in Figures 5, 6 and B1.
12. There are some spelling errors in the manuscript. For instance, the first reference should read "thermokarst lake" instead. The authors should carefully proofread the manuscript to avoid such errors.
Overall, this work is a valuable and necessary contribution to the field. It moves the conversation forward from deterministic single-lake models and analytical equilibrium solutions to a dynamic, stochastic, landscape-scale framework. While current data limitations restrict its predictive power, the model provides a powerful tool for hypothesis testing and a clear roadmap for what data is needed to eventually integrate these processes into climate projections.Citation: https://doi.org/10.5194/egusphere-2025-1817-RC1 -
AC2: 'Reply on RC1', Constanze Reinken, 19 Dec 2025
Thank you for your comments. We sincerely appreciate your positive feedback. We are happy to address and implement the suggestions for improvement, mainly mentioned in comments 10. to 12.
For detailed point-by-point replies please refer to the supplementary document.
-
AC2: 'Reply on RC1', Constanze Reinken, 19 Dec 2025
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RC2: 'Comment on egusphere-2025-1817', Anonymous Referee #2, 26 Sep 2025
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AC3: 'Reply on RC2', Constanze Reinken, 19 Dec 2025
We sincerely thank you for your constructive feedback. We strongly appreciate your extensive recommendations and suggestions for the improvement of both the model as well as analysis. We are happy to incorporate these suggestions as far as we can, and, as recommended, will prioritize adding basic analytical results with Fokker-Planck or partial moment equations (iii) as well as replacing the merging algorithm with a stochastic and geometrically-consistent alternative (iv). The GBM noise scaling (i) does not need to be corrected in the model as this was only an error in our description. We will correct the text accordingly. We do not yet see a way to formalise the event rates differently (ii) without adding additional degrees of freedom to the model, but we are happy to put further thought into this.
Implementing the suggestions from comments 2. to 5. (incl. recommendations (ii) and (iv)) requires substantial changes to our model. While we are in principal open to work on each suggestion/recommendation, it is difficult to judge in advance how successful and feasible each of these changes will be, especially considering our capacities and time. We argue that, even without these changes to the model, a publication of the paper would still be beneficial for the permafrost modelling community and invite further model development led by other researchers with more expertise in fields relevant to the respective approaches.
For detailed point-by-point replies to the comments please refer to the supplementary document.
-
AC3: 'Reply on RC2', Constanze Reinken, 19 Dec 2025
Interactive discussion
Status: closed
-
CC1: 'Comment on egusphere-2025-1817', Elchin Jafarov, 17 Jul 2025
The study presents a new stochastic model to simulate the formation, expansion, and drainage of thermokarst lakes in Arctic permafrost regions. These lake processes are significant for carbon and energy fluxes, and are often underrepresented in Earth System Models (ESMs). The authors employ a probabilistic framework, where lake formation and abrupt drainage are modeled as Poisson processes, while size variations follow Geometric Brownian Motion. The model is calibrated using high-resolution satellite data and tested through both idealized simulations and observation-based scenarios from Siberia's Yana-Indigirka Lowland. Three dynamic regimes are demonstrated: complete drainage, oscillation, and stabilization of lake areas. The model’s simplicity and flexibility make it potentially integrable into ESMs, which could lead to more accurate projections of permafrost landscape changes and associated climate feedbacks. The study underscores the importance of remote sensing for parameterization and future model refinement. Overall, the study is well-conceived and clearly written. I have a few comments that could help clarify the model’s potential applications and provide more detail on how the authors envision its integration into Earth System Models (ESMs).
The authors briefly mention the challenges associated with incorporating phase change processes into their model. Expanding on this point would improve the manuscript, particularly by discussing how this stochastic model could be integrated into ESMs. For example, assuming a grid cell resolution of 0.5° or coarser, would the model estimate the ratio of land to water and apply distinct terrestrial and aquatic parameterizations based on that ratio? Providing more technical insight into how the model could be coupled with ESMs, specifically regarding the integration of thermal and hydrological processes, would be valuable.
Additionally, towards the end of the introduction, the authors refer to the abrupt thaw model introduced by Nitzbon et al. (2020). If the goal of this study is to model abrupt thaw processes, it would be helpful to clarify what new insights this model offers beyond those provided by Nitzbon et al. (2020). What additional understanding or capabilities does this approach contribute to the study of abrupt thaw? Furthermore, elaborating on how this model connects to the broader issue of permafrost carbon feedback, particularly in terms of coupling with lake carbon emissions, would strengthen the study’s relevance to permafrost climate feedback.
Citation: https://doi.org/10.5194/egusphere-2025-1817-CC1 -
AC1: 'Reply on CC1', Constanze Reinken, 19 Dec 2025
Dear Dr. Jafarov,
we sincerely thank you for your comments and will make sure to incorporate your suggestions:
- We will add one or two paragraphs outlining technical details of how the model could be incorporated or used for ESMs, which would mainly work through providing the ESMs with water area fractions for grid cells in thermokarst-affected regions. We will mainly use the example of the MPI-ESMs land surface component JSBACH / ICON-Land, which some co-authors are experts on.
- The main differences are that Nitzbon et al. 2020 is not stochastic and consists of tiles that each contain a one-dimensional vertical representation of the subsurface that needs to be initialized and parameterized. With our approach, on the other hand, we try to bypass the necessity of such sub-surface initializations and parameterizations by using stochastic processes and only looking at the changes of the land surface. We realize now that the placement of the mention of Nitzbon et al. 2020 within the text is confusing and might suggest that this model is also stochastic. We will rearrange this part of the text and try to make the differences between our approach and Nitzbon et al. 2020 more clear.
- In the “Introduction” and “Conclusions” we will add more insight into how the model can potentially contribute to the estimation of permafrost carbon emissions by drawing connections to previous work and knowledge on carbon emissions from high-latitude lakes. For instance, we will add details on how the model’s projections of changes in water area fraction and lake size distribution can help estimate methane vs. carbon dioxide emissions by giving an indicator for surface water area and lake coastline lengths and depths, which influence carbon emission pathways and processes of microbial decomposition (e.g. Rehder et al. 2023, Turetsky et al. 2020).
Citation: https://doi.org/10.5194/egusphere-2025-1817-AC1
-
AC1: 'Reply on CC1', Constanze Reinken, 19 Dec 2025
-
RC1: 'Comment on egusphere-2025-1817', Anonymous Referee #1, 16 Sep 2025
This is a well-structured paper that addresses a critical gap in Earth System Modeling (ESM) by proposing a novel stochastic approach to represent thermokarst lake dynamics at a regional scale. The model is conceptually sound, clearly explained, and the exploration of different dynamic regimes is a particular strength. The main limitations are openly acknowledged and are largely tied to the current state of observational data, which is a field-wide challenge, not a flaw specific to this study. Regarding the specific content of this manuscript, I have the following comments and suggestions.
1. The application of Geometric Brownian Motion (GBM) and Poisson processes to model lake area change and formation/drainage events is innovative and well-justified within the context of landscape-scale heterogeneity and stochasticity. This is a significant step beyond previous deterministic or analytical models.
2. The three idealized regimes (Complete Drainage, Oscillation, Stabilization) effectively demonstrate the model's behavioral range and provide a useful conceptual framework for understanding possible long-term trajectories of thermokarst landscapes. The links to real-world examples for each regime are well-made.
3. The current merging algorithm is identified correctly as a significant weakness. The assumption that merged lake area is the sum of the two original areas and that the new lake is perfectly circular is physically unrealistic and computationally expensive. It likely leads to a drastic overestimation of lake sizes and an underestimation of lake numbers.
4. The high percentage of unusable data points (30-33%) in the remote sensing dataset severely impacts the robustness of the parameter estimation. This uncertainty propagates through the observation-based simulations and limits the confidence in the derived parameters (λf, λd, μ, σ).
5. A crucial result is that the stochastic component (σ) dominates the deterministic drift (μ) in the observed period. This implies that, for the 2000-2020 period in this region, random environmental variability was a stronger driver of annual lake area changes than any clear climate-driven trend, explaining the lack of strong correlations.
6. The inability to find significant correlations between model parameters and climate variables (TDD, P) is a notable negative result. While honestly reported, it highlights the current impossibility of confidently projecting lake dynamics under climate change scenarios with this model, as intended in the abstract. This is a major constraint on its immediate application in ESMs.
7. The observation-based simulations project water area fractions increasing to over 50%, which is acknowledged as rare. This, combined with the high volatility, suggests the model parameters derived from 20 years of data may not be stable or representative of centennial-scale dynamics, potentially overestimating expansion.
8. The suggestion that the idealized simulations could be interpreted as spanning 10 ka with a 10-year time step is helpful for context, but the parameters would then be "per decade." This should be stated explicitly in the text to avoid confusion (e.g., in Table 1, add a note: "Parameters are per year; for a 10-year time step interpretation, values would be per decade").
9. The comparison with the van Huissteden et al. (2011) model is good. The explanation for the differing results (their reliance on a prescribed river network vs. your data-driven drainage rates) is plausible and highlights the advantage of your approach, but also its current data dependency.
10. The definition of "abrupt drainage" as a complete (>90% loss) and rapid event is clear. However, the discussion of results from other studies (Jones et al., 2011, 2020) that use different thresholds (e.g., >25% loss) is slightly confusing. A small table summarizing different study's definitions and converting their rates to a common framework would be helpful.
11. The axis titles of many figures are obscured, or the figures are not very clear, for example, in Figures 5, 6 and B1.
12. There are some spelling errors in the manuscript. For instance, the first reference should read "thermokarst lake" instead. The authors should carefully proofread the manuscript to avoid such errors.
Overall, this work is a valuable and necessary contribution to the field. It moves the conversation forward from deterministic single-lake models and analytical equilibrium solutions to a dynamic, stochastic, landscape-scale framework. While current data limitations restrict its predictive power, the model provides a powerful tool for hypothesis testing and a clear roadmap for what data is needed to eventually integrate these processes into climate projections.Citation: https://doi.org/10.5194/egusphere-2025-1817-RC1 -
AC2: 'Reply on RC1', Constanze Reinken, 19 Dec 2025
Thank you for your comments. We sincerely appreciate your positive feedback. We are happy to address and implement the suggestions for improvement, mainly mentioned in comments 10. to 12.
For detailed point-by-point replies please refer to the supplementary document.
-
AC2: 'Reply on RC1', Constanze Reinken, 19 Dec 2025
-
RC2: 'Comment on egusphere-2025-1817', Anonymous Referee #2, 26 Sep 2025
-
AC3: 'Reply on RC2', Constanze Reinken, 19 Dec 2025
We sincerely thank you for your constructive feedback. We strongly appreciate your extensive recommendations and suggestions for the improvement of both the model as well as analysis. We are happy to incorporate these suggestions as far as we can, and, as recommended, will prioritize adding basic analytical results with Fokker-Planck or partial moment equations (iii) as well as replacing the merging algorithm with a stochastic and geometrically-consistent alternative (iv). The GBM noise scaling (i) does not need to be corrected in the model as this was only an error in our description. We will correct the text accordingly. We do not yet see a way to formalise the event rates differently (ii) without adding additional degrees of freedom to the model, but we are happy to put further thought into this.
Implementing the suggestions from comments 2. to 5. (incl. recommendations (ii) and (iv)) requires substantial changes to our model. While we are in principal open to work on each suggestion/recommendation, it is difficult to judge in advance how successful and feasible each of these changes will be, especially considering our capacities and time. We argue that, even without these changes to the model, a publication of the paper would still be beneficial for the permafrost modelling community and invite further model development led by other researchers with more expertise in fields relevant to the respective approaches.
For detailed point-by-point replies to the comments please refer to the supplementary document.
-
AC3: 'Reply on RC2', Constanze Reinken, 19 Dec 2025
Peer review completion
Journal article(s) based on this preprint
Data sets
Surface water data: Supplementary Dataset used in: Reinken et al.: Stochastic Modelling of Thermokarst Lakes: Size Distributions and Dynamic Regimes Ingmar Nitze, Todd Nicholson https://doi.org/10.5281/zenodo.15011121
Model code and software
Thermokarst Lake Model (TLM) Constanze Reinken https://github.com/cowniflow/TLM
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Constanze Reinken
Victor Brovkin
Philipp de Vrese
Ingmar Nitze
Helena Bergstedt
Guido Grosse
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(6335 KB) - Metadata XML
The study presents a new stochastic model to simulate the formation, expansion, and drainage of thermokarst lakes in Arctic permafrost regions. These lake processes are significant for carbon and energy fluxes, and are often underrepresented in Earth System Models (ESMs). The authors employ a probabilistic framework, where lake formation and abrupt drainage are modeled as Poisson processes, while size variations follow Geometric Brownian Motion. The model is calibrated using high-resolution satellite data and tested through both idealized simulations and observation-based scenarios from Siberia's Yana-Indigirka Lowland. Three dynamic regimes are demonstrated: complete drainage, oscillation, and stabilization of lake areas. The model’s simplicity and flexibility make it potentially integrable into ESMs, which could lead to more accurate projections of permafrost landscape changes and associated climate feedbacks. The study underscores the importance of remote sensing for parameterization and future model refinement. Overall, the study is well-conceived and clearly written. I have a few comments that could help clarify the model’s potential applications and provide more detail on how the authors envision its integration into Earth System Models (ESMs).
The authors briefly mention the challenges associated with incorporating phase change processes into their model. Expanding on this point would improve the manuscript, particularly by discussing how this stochastic model could be integrated into ESMs. For example, assuming a grid cell resolution of 0.5° or coarser, would the model estimate the ratio of land to water and apply distinct terrestrial and aquatic parameterizations based on that ratio? Providing more technical insight into how the model could be coupled with ESMs, specifically regarding the integration of thermal and hydrological processes, would be valuable.
Additionally, towards the end of the introduction, the authors refer to the abrupt thaw model introduced by Nitzbon et al. (2020). If the goal of this study is to model abrupt thaw processes, it would be helpful to clarify what new insights this model offers beyond those provided by Nitzbon et al. (2020). What additional understanding or capabilities does this approach contribute to the study of abrupt thaw? Furthermore, elaborating on how this model connects to the broader issue of permafrost carbon feedback, particularly in terms of coupling with lake carbon emissions, would strengthen the study’s relevance to permafrost climate feedback.