Permafrost and its spatiotemporal variation considerably influence the surface and sub-surface hydrological processes, biogeochemical cycles, fauna and flora growth and cold region engineering projects in the Three-River Source Region (TRSR), Qinghai-Tibet Plateau. However, the dynamics of permafrost over a relatively long term duration (e.g. >100 years) in the TRSR is not well quantified. Thus, the spatial and temporal variations of the temperature at the top of the perennially frozen/unfrozen ground (TTOP), active layer thickness (ALT) in permafrost regions and the maximum depth of frost penetration (MDFP) in the seasonally frozen ground of the TRSR during 1901-2020 were simulated using the TTOP model and Stefan equation driven by the widely used reanalysis Climatic Research Unit 4.05 dataset. Results revealed that the permafrost in the TRSR over the past 120 years did not degrade monotonically but experienced considerable fluctuations in area with the decadal oscillations of climate warming and cooling: shrinking from 263.9 x 103 km2 in the 1900s to 233.3 x 103 km2 in the 1930s, expanding from 232.3 x 103 km2 in the 1940s to 260.9 x 103 km2 in the 1970s and shrinking again from 254.1 x 103 km2 in the 1980s to 228.9 x 103 km2 in the 2010s. The regional average TTOP increased from -1.34 & PLUSMN; 2.74 & DEG;C in the 1910s to -0.48 & PLUSMN; 2.69 & DEG;C in the 2010s, demonstrating the most noticeable change for the extremely stable permafrost (TTOP 3.0 m by 12% from 1901 to 2020. Notably, minor changes were observed for the regional average MDFP, probably due to the increase in the area proportion of MDFP 3.5 m (owing to the transformation of permafrost to seasonally frozen ground) by 7.39% and 4.77%, respectively. These findings can facilitate an in-depth understanding of permafrost dynamics and thus provide a scientific reference for eco-environment protection and sustainable development under climate change in the TRSR

Active layer thickness (ALT) of permafrost changes significantly under the combined influence of human activities and climate warming, which has a significant impact on the ecological environment, hydrology, and engineering construction in cold regions. The spatial differentiation of Active layer thickness and its influencing factors have become one of the hot topics in the field of cryopedology in recent years, but there are few studies in the Da Hinggan Ling Prefecture (DHLP). In this study, the Stefan equation was used to simulate the Active layer thickness in the Da Hinggan Ling Prefecture, and the factor detection and interaction detection functions of geodetector were used to analyze the factors affecting the spatial differentiation of Active layer thickness from both natural and humanity aspects. The results showed that Active layer thickness in the Da Hinggan Ling Prefecture ranges from 58.82 cm to 212.55 cm, the determinant coefficient R-2, MAE, RMSE between simulation results and the sampling points data were 0.86, 11.25 (cm) and 13.25 (cm), respectively. Lower Active layer thickness values are mainly distributed higher elevations in the west, which are dominated by forest (average ALT: 136.94 cm) and wetlands (average ALT: 71.88 cm), while the higher values are distributed on cultivated land (average ALT: 170.35 cm) and construction land (average ALT: 176.49 cm) in the southeast. Among the influencing factors, elevation is significantly negatively correlated with ALT. followed by summer mean LST, SLHF and snow depth. NDVI and SM has the strong explanation power for the spatial differentiation of ALT in factor detection. Regarding interactions, the explanatory power of slope boolean AND snow depth is the highest of 0.83, followed by the elevation boolean AND distance to settlements. The results can provide reference for the formulation of ecological environmental protection and engineering construction policies in cold regions.

The modeling of seasonal ground ice (SGI) freeze/thaw a common feature in boreal peatlands, has often been completed using a unidirectional approach, where melting is driven by energy inputs from the surface. However, bi-directional melt is known to occur, and can potentially increase the spring melt rate. Accurate modelling of the timing of ice-free conditions in peatlands is important because SGI can impede spring infiltration and lead to substantial spring snowmelt runoff from peatlands. However, when modelling melt only from above, erroneous results in the model estimation of ice-free conditions can occur, which can lead to knock on-effects for modelling peatland hydrological function. Furthermore, as the climate warms, it is unclear how this role of SGI may change in the future. This study used the Stefan Equation to model unidirectional and bi-directional melt to assess which performed better in modelling the timing of ice-free conditions compared to observed values (BI: 3.9 +/- 2.1 days, UNI: 9.0 +/- 4.7). Including bi-directional melt improved model performance by reducing this difference by approximately 5 days. Model performance for SGI freeze/thaw cycles were similar, with BI being slightly more accurate in freezing (RMSE:2.7 cm versus 3.3 cm) and melting (RMSE: 2.6 cm vs 3.7 cm) compared to the unidirectional approach. While the model improvement in the timing of ice-free conditions was substantial, careful consideration is needed in determining when a peatland is functionally ice free in future modelling studies. The Stefan Equation was found to be most sensitive to changes in soil moisture, compared to ground surface temperature and peat porosity, likely due to the relationship between thermal conductivity and frozen and liquid water content. Comparisons with future climate change projections suggest that the timing of ice-free conditions 'could shift by as much as 2 weeks earlier in the 2050's and by almost a month earlier in the 2080's. However, the timing of snowfall, and rain on snow events continues to be a source of model uncertainty. Future studies should work to investigate the potential positive feedbacks this could create. In conclusion, the Stefan Equation presents a relatively easy path for incorporating bi-directional melt into peatland models. This process should be included in peatland ecohydrological models in order to properly model the timing of melt and ice-free conditions.

The permafrost in the Qilian Mountains (QLMs), the northeastern margin of the Qinghai-Tibet Plateau, changed dramatically in the context of climate warming and increasing anthropogenic activities, which poses significant influences on the stability of the ecosystem, water resources, and greenhouse gas cycles. Yet, the characteristics of the frozen ground in the QLMs are largely unclear regarding the spatial distribution of active layer thickness (ALT), the maximum frozen soil depth (MFSD), and the temperature at the top of the permafrost or the bottom of the MFSD (TTOP). In this study, we simulated the dynamics of the ALT, TTOP, and MFSD in the QLMs in 2004-2019 in the Google Earth Engine (GEE) platform. The widely-adopted Stefan Equation and TTOP model were modified to integrate with the moderate-resolution imaging spectroradiometer (MODIS) land surface temperature (LST) in GEE. The N-factors, the ratio of near-surface air to ground surface freezing and thawing indices, were assigned to the freezing and thawing indices derived with MODIS LST in considerations of the fractional vegetation cover derived from MODIS normalized difference vegetation index (NDVI). The results showed that the GEE platform and remote sensing imagery stored in Google cloud could be quickly and effectively applied to obtain the spatial and temporal variation of permafrost distribution. The area with TTOP < 0 degrees C is 8.4 x 10(4) km(2) (excluding glaciers and lakes) and accounts for 46.6% of the whole QLMs, the regional mean ALT is 2.43 +/- 0.44 m, while the regional mean MFSD is 2.54 +/- 0.45 m. The TTOP and ALT increase with the decrease of elevation from the sources of the sub-watersheds to middle and lower reaches. There is a strong correlation between TTOP and elevation (slope = -1.76 degrees C km(-1), p < 0.001). During 2004-2019, the area of permafrost decreased by 20% at an average rate of 0.074 x 10(4) km(2)center dot yr(-1). The regional mean MFSD decreased by 0.1 m at a rate of 0.63 cm center dot yr(-1), while the regional mean ALT showed an exception of a decreasing trend from 2.61 +/- 0.45 m during 2004-2005 to 2.49 +/- 0.4 m during 2011-2015. Permafrost loss in the QLMs in 2004-2019 was accelerated in comparison with that in the past several decades. Compared with published permafrost maps, this study shows better calculation results of frozen ground in the QLMs.

Frozen ground degradation resulting from climate warming on the Tibetan Plateau has aroused wide concern in recent years. In this study, the maximum thickness of seasonally frozen ground (MTSFG) is estimated by the Stefan equation, which is validated using long-term frozen depth observations. The permafrost distribution is estimated by the temperature at the top of permafrost (TTOP) model, which is validated using borehole observations. The two models are applied to the upper Yellow River Basin (UYRB) for analyzing the spatio-temporal changes in frozen ground. The simulated results show that the areal mean MTSFG in the UYRB decreased by 3.47 cm/10 a during 1965-2014, and that approximately 23% of the permafrost in the UYRB degraded to seasonally frozen ground during the past 50 years. Using the climate data simulated by 5 General Circulation Models (GCMs) under the Representative Concentration Pathway (RCP) 4.5, the areal mean MTSFG is projected to decrease by 1.69 to 3.07 cm/10 a during 2015-2050, and approximately 40% of the permafrost in 1991-2010 is projected to degrade into seasonally frozen ground in 2031-2050. This study provides a framework to estimate the long-term changes in frozen ground based on a combination of multi-source observations at the basin scale, and this framework can be applied to other areas of the Tibetan Plateau. The estimates of frozen ground changes could provide a scientific basis for water resource management and ecological protection under the projected future climate changes in headwater regions on the Tibetan Plateau.

Wilhelm et al. (2015) employed the widely used Stefan and Kudryavtsev equations to predict the maximum active-layer thickness (ALT) on Amsler Island, Western Antarctic Peninsula. Their predictions far exceed the observations of ALT reported from other parts of the region. Here, I demonstrate that the values of ALT are significantly overestimated by the predictive equations because the authors incorrectly assumed that little or no latent heat of phase change is absorbed during thawing. Although the area is the warmest in the Antarctic Peninsula region, with a rapid increase in air temperature and permafrost temperatures close to 0 degrees C, the active layer is likely to be substantially thinner than values predicted by Wilhelm et al. (2015). Copyright (c) 2016 John Wiley & Sons, Ltd.

An adapted version of the Stefan equation (CLIFFSE) was tested to predict lateral progression of the frost front into cohesive sediments that form coastal cliffs along the north shore of the maritime estuary and gulf of the St Lawrence River (Quebec, Canada). The equation was adapted to accommodate the influence of cliff erosion on lateral penetration of freezing and thawing into vertical cliff faces. As the cliff erodes, freezing and thawing are initiated from the newly revealed surface. Frost progression and erosion were measured with an automated thermal erosion pin system. Measured observations agreed with predictions from the adapted equation (78 to 99% of the variability explained). Erosion associated with thawing front progression during winter warm spells led to a relative reduction in the frost front depth. Subsequently, progression of the frost front into the cliff contributed to an additional 50 cm of sediment freezing and erosion by the end of the cold season, which was not predicted by the original Stefan equation. Our findings support the hypothesis that multiple warm spells influence the amount of lateral penetration of the frost front in vertical cliffs. Copyright (c) 2015 John Wiley & Sons, Ltd.