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.- Preprint
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Status: final response (author comments only)
- CC1: 'Comment on egusphere-2025-1817', Elchin Jafarov, 17 Jul 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 - RC2: 'Comment on egusphere-2025-1817', Anonymous Referee #2, 26 Sep 2025
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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- 1
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.