The soil or snow surface temperature (TS0, K) was treated as upper boundary of the soil column. To derive TS0, a physically-based surface energy balance scheme for different land surface cover types was coupled to FPM, and was formulated as: 11-αQsi+Qli+Qle+Qh+Qe=Qc where Qsi is the incoming shortwave radiation, Qli is the incoming longwave radiation, Qle is the emitted longwave radiation, Qh is the turbulent exchange of sensible heat, Qe is the turbulent exchange of latent heat, and Qc is the energy transport due to conduction. All the above energy terms have units of Wm-2. The α (–) is the surface albedo, obtained as a fraction-weighted average of albedo from snow-free (αg) and snow-covered (αsn) areas. 2α=(1-SCF)⋅αg+SCF⋅αsn where SCF (–) is the snow cover fraction (see Sect. ), and αg is from MODIS products whenever snow is not present (Table ).

The Qsi and Qli are either from observations or from a reanalysis dataset (Table ). The Qle is computed under the assumption that ground (snow) emits as a gray body: 3Qle=-εsσTs04 where εs (–) is the surface emissivity, and σ is the Stefan–Boltzmann constant of 5.67×10-8 Wm-2K-4. The εs was set to 0.92 for soil surface, and 0.98 for snow surface .

The turbulent exchange of sensible heat Qh is given by : 4Qh=ρacpDh(Ta-Ts0)5ρa≈PRdTa where ρa is the air density (kgm-3), cp=1004 Jkg-1K-1 is the specific heat of air, P (Pa) is atmospheric pressure, Ta (K) is near-surface air temperature, and Rd=0.287 kJkg-1K-1 is specific gas constant. The exchange coefficients for heat Dh (–) can be estimated as: 6Dh=κ2uzln⁡z/z02 where κ=0.4 is the Von Karman's constant, uz (m) is the wind speed at the instrument height z (m), and z0 (m) is the roughness length as 0.015 m for soil surface and 0.001 m for snow surface following .

The latent heat flux was calculated via the Priestley–Taylor method . 7Qe=S⋅αptΔ(Qn-Qc)Δ+γ where S (–) is the evaporation stress factor, αpt (–) is the Priestley–Taylor coefficient, Qn (Wm-2) is the net radiation, Δ (PaK-1) is the slope of the saturation vapor pressure-temperature curve, and γ (PaK-1) is psychrometric constant. Δ is determined following : 8Δ=4098es(Ta-Tfrz+237.3)2 where Tfrz=273.15 K is the freezing point temperature, and es (Pa) is saturation vapor pressure derived following : 9es=611exp⁡17.3Ta-TfrzTa-Tfrz+237.3 the γ is calculated using the formula proposed by : 10γ=cpPεLv where Lv=2.471×106 Jkg-1 is the latent heat of vaporization, and ε is a constant of 0.622 . For each grid, the latent heat flux is treated separately for the bare ground surface, vegetation, and snow. The detailed parameterizations are in Appendix . In current FPM, Monin–Obukhov similarity theory (for Qh) and Priestley–Taylor method (for Qe) were combined to improve simulation efficiency as some previous studies . We acknowledge that using two different theories may introduce additional uncertainties.

Heat conduction through the snow layer or/and ground surface (Qc) was given as : 11Qc=(Ts0-Ts)zsks-1,snow-free(Ts0-Ts)zsnksn-1,snd≥0.1(Ts0-Ts)zsnksn+zsks-1,0<snd<0.1zs+zsn=0.1 where Ts (K) is the temperature at 0.1 m depth, and is either from snow (if snd≥0.1 m) or from soil. The ks and ksn (Wm-1K-1) are the thermal conductivity of the soil and snow at the depth of zs and zsn, respectively.

The surface energy balance and heat conduction constitute a coupled non-linear equation system, and was solved iteratively for the surface temperature Ts0, using the Newton–Raphson method.

The significant influences of snow cover on soil thermal regime have been well documented . Over the TP, snow cover is minor, with a mean annual snow depth of about 0.002 m (and about 0.01 m in winter) according to the ground observations from the China Meteorological Administration network . Consequently, the snow effects are relatively minor in this region. The required degree of model complexity depending on the intended applications. For this reason, a simple snow scheme was incorporated into the initial version of FPM to represent the influences of seasonal snow on soil thermal regime. We acknowledge that the application of the FPM is not recommended in regions with prevalent snow due to its limited capability in simulating snow processes. The snow layer was discretized into multiple layers with a vertical resolution of 0.01 m snow depth (snd) for heat transfer simulations. The snow densification algorithm from was introduced here. 12ρsnt+Δt=(ρsnt-ρsnmax)⋅exp⁡-0.24Δt+ρsnmax where ρsnmax=300 kgm-3 is the maximum snow density, and Δt is the simulation time step in day. The fresh snow density was set as 100 kgm-3.

The snow albedo is treated separately for non-melting and melting conditions following . For non-melting conditions, a linear decrease is assumed for αsn, while an exponential decrease is assumed for melting snow due to the presence of liquid water. 13αsnt+Δt=αsnt-0.008Δt,non-meltingαsnmin+αsnt-αsnmine-0.24Δt,melting αsnmax,Δsnd≥0.01m where αsnmax=0.85 denotes the maximum snow albedo or the fresh snow albedo, while αsnmin=0.50 is the minimum snow albedo for old snow, and Δsnd (m) refers to the change in snow depth per time step. If there is a significant snowfall, i.e. Δsnd≥0.01 m, snow albedo was reset to the maximum by assuming the snow surface is completely overlaid by the fresh snow.

The snow cover fraction was given as 14SCF=min⁡sndsndcr,1 where sndcr=0.1 m is the minimum snow depth that ensures complete coverage of the grid cell.

The snow volumetric heat capacity (CVsn, Jm-3K-1) and thermal conductivity are treated as functions of snow density (ρsn, kgm-3) following : 15CVsn=CViρsnρi16ksn=kiρsnρw1.88 where CVi=1.93×106 Jm-3K-1 is volumetric heat capacity for ice, ki=2.22 Wm-1K-1 is the thermal conductivity of ice, ρi=920 kgm-3 is ice density, and ρw=1000 kgm-3 is water density. Snow ablation was heavily simplified by directly using the snow water equivalent from reanalysis. Snowfall temperature equals the near-surface air temperature, capped at 273.15 K. During snowmelt, any simulated snow temperature exceeding 273.15 K is reset to this threshold.

The ground temperature T (K) changes over time (t) and depth (z, m), and is numerically solved by the heat conduction for energy transfer and phase change determined by Fourier's law: 17C∂T∂t=∂∂zk∂T∂z The latent heat during phase change is taken into account through an apparent volumetric heat capacity: 18C=CVs+Lfρw∂θu∂T where C and CVs are the apparent volumetric heat capacity and volumetric heat capacity of soil (Jm-3K-1), respectively, Lf=0.334×106 Jkg-1 is mass specific latent heat of water, θu (m3m-3) is the volumetric unfrozen water content or super-cooled water, and T is from the previous step. The unfrozen water was parameterized following : 19θu=θsat103LfT-TfrzgTψsat-1b where g=9.80665 ms-2 is the acceleration due to gravity. The θsat (m3m-3), ψsat (mm), and b is the saturated soil moisture, saturated matric potential, and Clapp–Hornberger parameter, respectively. They are all derived based on the soil material properties (Appendix ). FPM implements the nonlinear heat-transfer equations in Cartesian coordinates.

Thermal properties of the soil are assumed to be a weighted combination of different components of the soil column. The CVs is calculated from the volumetric fractions of the constituents as follows : 20CVs=∑nfnCVn where subscripts n=m, o, w, i, a, and g refer to soil constituents of mineral, organic, water, ice, air, and gravel. In this context, fn (m3m-3) and Cn (Jm-3K-1) represent the volumetric content and volumetric heat capacity for each component (Table , Appendix ). Similarly, the thermal conductivity of the soil ks (Wm-1K-1) is calculated following : 21ks=∑nfnkn2 where kn is the thermal conductivity (Wm-1K-1) for each soil component.

For temporal discretization, the model employs a first-order backward Euler scheme with a daily time step. We selected this unconditionally stable implicit method to overcome the strict step-size limitations typically associated with explicit schemes. By preventing numerical divergence even with a relatively coarse temporal resolution, this approach allows for a computationally economical implementation that maintains sufficient fidelity for capturing long-term thermal dynamics. The spatial derivatives are discretized using the control volume method, yielding a tridiagonal system of algebraic equations at each time step that is efficiently solved via the Thomas algorithm.

To maintain numerical stability and reduce computational cost, the soil vertical grid size increased from 0.01 m for subsurface to 5.0 m for deep soil (Table ), similar to the other models . The soil column with a total depth of 150 m is discretized to 172 layers. We use the soil texture of the Global Soil Dataset for Earth System Models (GSDE) as it additionally provides the soil gravel content, which is prevalent over the TP . GSDE has 8 soil layers with a total depth of 3.8 m. The information of the last layer was extended to deeper soil, but was refined based on the bedrock depth from . Note that, we assume soil organic matter is absent for soil below 3 m as is normally done in Earth system models (e.g. ).

Soil moisture can significantly affect the dynamics of the soil thermal regime through evapotranspiration and by altering soil thermal properties . However, in the permafrost regions of the TP, soil moisture exhibits marked heterogeneity and is difficult to accurately represent in models. This challenge stems from uncertainties in soil datasets and climate forcing, as well as the inherent complexities of the rugged terrain. For the current version, the static soil moisture is used. To specify the vertical water distribution within the soil column, we used sub-grid parameterizations from SURFEX and CryoGridLite . Here, we distinguished four hydrological layers in the subsurface, from the uppermost surface to deep layer, including the: (1) root zone; (2) vadose layer, extending from the root layer downward to the lower boundary or saturated table/bedrock (if present); (3) saturated layer, which lies between the depth of groundwater table and bedrock; and (4) bedrock layer, extends down to the lower boundary. Note that the presence and depth of the saturated layer is estimated based on the groundwater table information from . The root depth is set between 0.05 m for bare soil and 0.5 m for high vegetation, and the typical depth for different vegetation cover is from GEOtop .

In the root layer, the water content θR (m3m-3) is estimated as the ensemble mean of five remote sensing-based products (Table , details see Sect. ). The water content for the vadose layer θv (m3m-3) is determined based on field capacity θfc (m3m-3) and soil porosity ϕ (m3m-3), and an ensemble range is used (see Sect.). Please see Appendix  for the parameterizations of soil properties. In the saturated layer, the water content (θsat) is equal to ϕ. The water content of 0.05 m3m-3 was used for the bedrock .

Although the use of static soil moisture models is common practice for investigating long-term permafrost changes among permafrost researchers (e.g. ), and four vertical water distribution schemes were implemented in FPM, this estimate is subject to large uncertainty. To allow the propagation of this uncertainty into model results, we introduced both wet and dry variants of the default parameters (see Sect. ). The ensemble simulations – allowing degrees of parameter uncertainties – are widely used for Earth system studies as they generally outperform individual simulations . In this study, the ensemble simulation is produced using reasonable ranges of parameters (Table ). Both the ensemble range of soil moisture in root layer and vadose zone are assumed here to allow possible uncertainties. The θR is derived as an ensemble mean of five state-of-the-art products (Table ), and the ensemble was produced by +/- (wet/dry soil) standard deviation of all these products. The baseline of θv was determined as the mean of θfc and θsat in previous studies (e.g. ). We followed this algorithm, but introduced a dry and wet variants of the θv parameter used (Table ). We emphasize that the ensemble was chosen after considerable tests and comparisons but ultimately remains a subjective choice at this time. The range of the above selected parameters are used for the 45-member ensemble simulation to represent a wide range of permafrost conditions. This approach prevents the simulation from capturing seasonal and long term evolution linked to changes in the soil water content which can be an important driver of permafrost dynamics.

The simulation was conducted at a time step of 1 d. A lower boundary condition of geothermal heat flux from is used (Table ). To ensure the convergence of soil temperature profile, the model was initialized through a 1000 year spin-up process. This was achieved by cyclically applying the climate forcing data from the first decade (July 1950 to June 1960) one hundred times. Simulation performance was measured by the mean bias (BIAS) as well as root mean square error (RMSE), and more details can be found in Appendix .

Permafrost is diagnosed following its definition, i.e. the daily soil temperature of a simulation pixel remains at or below 0 °C for two or more years at any simulated depth. Instead of directly using presence/absence of permafrost for individual ensemble member, the permafrost zonation index (PZI), as a fraction (0–1) corresponds to the 45 simulation members identified the probability of permafrost presence for each grid, is introduced to represent permafrost extent here. In such, the PZI can be used to quantitatively explore permafrost changes for each grid. More details of permafrost probability estimate could be found from and . In this study, we especially focus on the thermal state of permafrost at a depth of 3 m as the near-surface permafrost treated in most land surface models , and 15 m as the permafrost mean annual ground temperature (MAGT). The permafrost extent for above 100 m is additionally used as reference. The glaciers and lakes from ERA5-Land are masked before permafrost extent analyses. The ALT is estimated from linear interpolation of daily soil temperature. In this study, the period of 2010–2023 is used as the current condition for permafrost.

FPM is driven by climate forcing from reanalysis, including: near-surface air temperature, wind speed, incoming shortwave radiation, incoming longwave radiation, atmospheric pressure, and snow water equivalent. In this study, the historical climate is taken from the ERA5-Land datasets, produced by the European Centre for Medium-Range Weather Forecasts (ECMWF, Table ). ERA5-Land is an enhanced land component of ERA5 with a spatial resolution of 0.1° and a coverage from 1950 to the present . The reanalyses were evaluated against the in situ observations (Appendix ). Note that the snowfall from ERA5-Land was reported to be over biased due to the significant drawback of precipitation represented in models . For this reason, the simulated snow influences on soil thermal regime were very likely artificially amplified.

FPM considers the influences of vegetation on permafrost via the latent heat and soil moisture etc. (Appendix ). In FPM, static vegetation is assumed and the vegetation optical depth (VOD), leaf area index (LAI), and vegetation type are required (Table ). For snow-free periods, the ground albedo is from .

The remote-sensing datasets vary in their temporal coverage, so we used the climatology to represent the long-term conditions. For the VOD and snow-free ground albedo, the daily measurements over the entire recording period were aggregated into a day-of-year climatology using the median, so as to reduce sensitivity to extreme values. The monthly LAI from was aggregated to monthly medians. Daily θR values were first aggregated into monthly averages for each dataset. These monthly values from the thawing season (June to August) were then used to compute the annual mean. For each soil moisture dataset, the average over the entire recording period was derived, and an ensemble mean across the five datasets was calculated and employed as model inputs. Note that only the measurements from the thawing season were used to derive VOD.

Our evaluation results showed the overall RMSE of daily soil temperature in the active layer was 1.8 °C with a BIAS of 0.4 °C (Fig. ). FPM showed relatively worse performance in areas with alpine swamp meadow (RMSE=2.7 °C), with warm bias in summer and cold bias in winter. This is attributed to poorly prescribed soil information, i.e. peat layer and soil moisture. At the sites with alpine desert, the overestimated soil moisture (by about 25 %) at 0.5 m depth leads to a colder simulated soil temperature in summer (Fig. a). To demonstrate the hypothesis, we conducted the additional simulations using observed atmospheric forcing and soil profile from borehole measurements. The simulated soil temperature was significantly improved by 1.9 °C, indicating the simulations could be improved with more reliable climate forcing and soil profile (Fig. ).

FPM generally has good agreement with observed ALTs with the overall RMSE of 1.0 m, and slightly underestimated ALT (BIAS=-0.1 m) due to the cold-biased summer air temperature (Figs.  and ). Following the relatively worse soil temperature, ALT was over-biased in areas with alpine swamp meadow. The additional simulation with observed forcing and soil information, again, showed more promising results (RMSE=0.7 m). The ALT bias was within ±1 m at most (64 %) evaluated cells (Fig. ).

FPM is found to underestimate the MAGT with an overall mean BIAS of -0.3 °C, which is aligned with the cold-biased air temperature (Fig. a). The overall MAGT RMSE was 1.0 °C (Fig. ), and about 65(93) % sites have a bias within ±1(2) °C. Although the MAGT change trend is well addressed by FPM with an RMSE of 0.22 °C per decade, it is found even greater than the observed mean MAGT trend of 0.12±0.09 °C per decade (Fig. ). This indicates that the simulated MAGT trend may not be reliable. In fact, the permafrost warming at the measured sites was relatively gentle (with a range from -0.07 to 0.3 °C per decade) compared to that in high latitudes . This is because the permafrost temperature over the TP is very “warm”, and the heat from atmosphere was consumed by phase change rather than temperature increase (also see Fig. ). The significant latent heat introduce additional challenge for reproducing MAGT trend.

Excluding glaciers and lakes, the estimated current (2010–2023) permafrost area was about 1.07±0.02×106 km2 (Fig. ) based on the ensemble simulations from FPM. This is found reasonable compared to the referenced ensemble mean of 1.15±0.12×106 km2 (Fig. ).

The ensemble simulations from FPM showed the current (2010–2023) overall mean ALT was about 2.68±0.82 m over the TP. The results are found align with the previous simulations, for example, 2.1–2.4 m (1981–2013) from GIPL2 and 2.01 m (1981–2000) from CLM . Our results indicated that about 33.0 % of permafrost regions have an ALT greater than 3 m, highlighting that the widely used land surface models and reanalyses with shallow soil column may not be sufficient for permafrost studies over the TP.

The long-term simulation showed that ALT had a inter-annual trends with a decreasing trend between 1950–1980 (-0.14 m per decade), followed by a continuous increasing trend (0.13 m per decade). Consequently, the ALT over TP increased by 0.41 m since 1980 (Fig. b). While the simulated ALT increase rate was similar to that of , it was lower than those reported by , , both of which indicated a rate of ∼0.3 m per decade. Notably, long-term observations showed that the ALT increased at a rate of 0.19 m per decade, suggesting that our results are likely underestimates .

The mean annual ground temperature at the depth of 15 m (MAGT15) is used to represent the thermal state of permafrost. The modeled overall mean MAGT15 (2010–2023) was about -1.8±2.0 °C for the permafrost regions over the TP and shows a wide range from -19.2 to 3.4 °C (Fig. a). The permafrost thermal regime is found similar to the outputs from Noah LSM, i.e. -1.6 °C . Our simulations revealed that the overall MAGT15 increased by approximately 0.22 °C since 1950, with a more dramatic warming of 0.11 °C per decade since 1980. Similar to the ALT, MAGT shows a clear cooling trend (-0.02 °C per decade) between 1950–1980 corresponding to the changes in near surface air temperature.

The permafrost warming trend was found to be MAGT-dependent, with a slower rate for warmer permafrost. For example, the permafrost warming rate was about 0.05±0.06 °C per decade for areas with MAGT15 between -1–0 °C, and 0.17±0.09 °C per decade for permafrost regions colder than -2 °C. This is consistent with observations , as heat is consumed as latent heat during the ice phase change for permafrost that is close to thawing.

The permafrost area decreased by approximately 9.5×104 km2 during 1950–2023, but increased by 4.3 % from 1950 to 1980 following cooling of the near-surface air temperature (Fig. a). Permafrost area decreased at a rate of 3.3×104 km2 per decade, or a total area of 12.4 %, since 1980. Our results showed that the model with shallow soil column would significantly underestimate permafrost area but overestimated permafrost degradation. Take the top 3 m as an example, which has been widely used in the land surface model. The estimated near-surface (top 3 m) permafrost area (7.9×104 km2) was about 26.0 % smaller compared to the ground “truth”, or 31.4 % smaller than the simulations with sufficient soil column (e.g. 100 m, Fig. b). Furthermore, by omitting permafrost below a depth of 3 m, the shallow soil model incorrectly classified areas with deep permafrost as permafrost-free. Therefore, the permafrost degradation rate was overestimated by about 42 % (3.3×104 vs. 4.7×104 km2 per decade) (Fig. ). This highlights that the current land surface models with shallow soil column can lead to significant uncertainties in permafrost simulations.

The ensemble simulation revealed that the variation in soil moisture translated into considerable influences on simulated permafrost characteristics (Fig. ), with the overall mean standard deviation was about 0.26 m in ALT and about 0.31 °C in MAGT. Notably, the input soil moisture datasets themselves exhibited substantial variability, with mean standard deviation of 0.11 m3m-3 in root zone and 0.14 m3m-3 in vadose zone (Fig. ). The propagation of input uncertainties into significant permafrost simulation bias thus highlights the essential role of obtaining more reliable soil moisture datasets for advancing our capacity to simulate permafrost changes.

We used the ERA5-Land reanalyses as model forcing. However, the ERA5-Land is found cold biased in near-surface air temperature (Appendix , Fig. ), leading to underestimated ALT as well as MAGT (Figs.  and ). In fact, permafrost simulations are hampered by reduced reanalyses quality in cold regions primarily due to inherent challenges in representing nonlinear processes involving ice, or its phase change near 0 °C . The poorly described soil column, especially the soil organic matter, put additional uncertainty for permafrost simulations.

In this study, we introduced and demonstrated the suitability of FPM for large-scale permafrost simulation. For the initial version of FPM, there are four main limitations. First, the snow scheme is simplistic, although significant influences of snow cover on soil thermal regime have been well documented . FPM currently does not consider the snow mass balance, and, therefore, additionally requires snow water equivalent as input. Further, the snow cover fraction used here was developed for high latitudes and does not consider the influences of complex terrain, which may redistribute the snow cover and lead to strong heterogeneity for the soil thermal regime. While the simple snow scheme seems adequate for the TP with little snow, considerable efforts are required to improve snow processes if FPM would applied to snow-prevalent areas, i.e. the high latitudes.

Second, FPM does not consider the water balance, and the static surface conditions, including vegetation and albedo (varies with season but remains unchanged among years), were assumed via using the remote-sensing-based climatology. This means the influences of surface and sub-surface changes are not accounted for and limited the model ability in predicting long-term permafrost changes.

Third, the fine-scale influences (i.e. debris and peat layer) are either not or simply represented here due to the simulation scale (∼10 km). This may overestimate permafrost degradation, especially for the areas near the permafrost lower limit, where the relict permafrost was found (Fig. , ).

Fourth, FPM does not consider the thermokarst processes or the so-called “abrupt thaw” raised by excess ice loss. The thermokarst was thought as local-scale tipping element that would remarkably accelerate permafrost degradation .

Our results are found comparable to previous simulations derived from the geothermal numerical models (e.g.) as well as land surface models (e.g.). FPM, as the stand-alone permafrost model, benefits from the consideration of land-atmosphere processes (i.e. the surface energy balance and vegetation effects) typically not included in geothermal numerical models, while maintaining the deep soil column and suitable numerical solver for soil phase change. In addition, the land-surface-scheme designed structure and streamlined processes (i.e. its efficient simulation of latent heat) make it suitable for large-scale ensemble simulations. Different from the other land surface scheme models with rich processes beyonds permafrost, such as SUTRA , Advanced Terrestrial Simulator , and CryoGrid 3 , which are applied to fine scales, the initial FPM aim to provide a flexible yet simple platform for large-scale permafrost simulation studies.

Previous studies reported that the permafrost processes of the Earth system model in CMIP6 is limited improved compared to the previous generation of CMIP5 (e.g. ), and current Earth system models generally have weak ability in representing permafrost . This highlights the urgent need to develop the stand-alone permafrost model. Since the required inputs are derived from global datasets, this opens the opportunity for global permafrost simulation with the FPM platform. The incorporation of a state-of-the-art snow scheme – particularly critical for high-latitude permafrost processes – will further enhance this capability. We hope with improved permafrost ground ice maps and a rigorously validated solution (addressing both physical processes and numerical solver aspects), we can implement excess ice loss processes within FPM to represent permafrost “abrupt thaw”. We hope FPM could be further improved via incorporating lateral heat transfer, as described by , making FPM a cross-scale platform for understanding diverse permafrost landscapes.