The UVic ESCM is a climate model of intermediate complexity with a full three-dimensional ocean general circulation model coupled to a simplified moisture–energy balance atmosphere and thermodynamic–dynamic sea-ice model . The model contains a full realization of the global carbon cycle. The terrestrial carbon cycle is simulated using the Top-down Representation of Interactive Foliage and Flora Including Dynamics (Triffid) dynamic vegetation model. Triffid is composed of five plant function types: broadleaf trees, needleleaf trees, shrubs, C3 grasses, and C4 grasses. These plant function types compete with one another for space in each grid cell based on the Lotka–Volterra equations . The simulated plants take up carbon through photosynthesis and distributed acquired carbon to plant growth and autotrophic respiration. Dead carbon is transferred to the soil carbon pool as litter fall and is distributed in the soil as an exponentially decreasing function of depth. Production of plant litter (and therefore new soil carbon) in Triffid is a function of temperature, plant function type, soil water availability, and atmospheric CO2 concentration .

The ocean inorganic carbon cycle is simulated following the protocols of the ocean carbon cycle model intercomparison project . Dissolved inorganic carbon is treated as a passive tracer by the model and carried throughout the ocean following ocean circulation . Ocean biology is simulated using a nutrient–phytoplankton–zooplankton–detritus ocean biology scheme . The slow feedback between ocean alkalinity and calcite dissolution is simulated using an oxygen only representation of respiration of organic matter in sediments . The simplified atmospheric scheme makes it possible to alter the equilibrium climate sensitivity of the model . This is accomplished by altering the outgoing long-wave radiation to space as a function of global average near-surface air temperature anomaly .

The version of the UVic ESCM used here is based on the frozen ground version documented in and . This version of the model has a deep subsurface extending down to 250 m depth and is composed of 14 vertical layers. These layers are of unequal thickness and become exponentially thicker with depth. The top eight layers (10 m) are hydraulically active and top six layers (3.35 m) are active in the carbon cycle. In the hydraulically active layers the subsurface porosity and permeability is prescribed based on the sand, silt, clay, and organic matter content of the grid cell. These gridded data are interpolated from the International Satellite Land Surface Climate Project Initiative II . The model accounts for the effect of soil valence forces on freezing point and the fraction of frozen and unfrozen water in soil is computed based on equations that minimize Gibbs free energy . The thermal conductivity of each soil layer is determined by the sand, silt, clay, water, ice, and organic carbon fraction of the layer .

Figure  displays maps of the estimated soil carbon density in the top 3 m of soil in the northern hemispheric permafrost region as presented in , compared to simulated soil carbon density in the top 3.35 m of the permafrost region as simulated by the UVic ESCM (using standard model parameter values). The maps show that the UVic ESCM generally simulates reasonable values for the density of carbon in the permafrost region but with substantial spatial biases. The model has too much carbon in northern fraction of the Fennoscandia peninsula, southern Alaska, and near the Lena River basin. The model does not capture the large permafrost carbon density in the Hudson Bay lowlands and permafrost (and therefore permafrost carbon) is absent from the Labrador peninsula, a bias common to many Earth system models . However, the model is able to capture some of the geographic features of the permafrost carbon pool including the high carbon density in northwestern Russia and the low carbon density in the eastern Canadian Arctic and Arctic archipelago.

The saturation factor from Eq. () was used to tune the total amount of carbon in the permafrost region such that in the default version of the model it matches the total from very closely. Therefore, in year 1995 the simulated permafrost region has 1035 Pg C in the top 3.35 m, equal to the best estimate for the carbon in the top 3 m of permafrost soil provided by . Carbon held in perennially frozen soil layers makes up 49 % of the carbon in the permafrost region in the UVic ESCM. This metric, which was not tuned, is very close to the estimate of ∼ 50 % provided by .

We have chosen to perturb four parameters that describe the permafrost carbon pool: (1) the quantity of soil carbon in the top 3 m of soil in the permafrost region, taken from ; (2) the permafrost decay rate constant κp, computed from mean residence time of the slow permafrost soil carbon pool from ; (3) the available fraction of permafrost carbon computed from the combined size of the fast and slow soil carbon pools in measured permafrost soils samples from ; and (4) the passive pool transformation rate κtf, estimated from . We also perturb two physical climate parameters: the climate sensitivity and the arctic amplification factor.

Besides the parameters we have chosen to perturb, many other parameters in the UVic ESCM could affect the magnitude of the release of carbon from permafrost soils. In particular parameters from the Triffid dynamic vegetation model that control net primary production determine the input of carbon into the soil, and therefore the net change is soil carbon in response to warming. However, for this study we have chosen to focus on uncertainty inherent to the permafrost carbon system instead of taking a global focus implied in perturbing the whole terrestrial carbon cycle e.g..

The quantity of carbon in permafrost soils is controlled by changing the saturation factor S presented in Eq. (). Calibration simulations were conducted with the UVic ESCM to derive a functional relationship between S and the quantity of carbon in permafrost soils. The probability distribution function (PDF) for the permafrost carbon quantity (in the top 3 m of soil) was taken as a normal distribution with a mean of 1035 Pg C and a standard deviation of 75 Pg C, taken from . The permafrost carbon decay rate is derived from the mean residence time of the slow carbon pool in permafrost soils. The permafrost decay rate is taken to be normally distributed with a mean of 7.45 years and a standard deviation of 2.67 years, with values taken from . reports the size of the fast, slow, and passive pool of soil organic carbon separately for organic, shallow mineral (< 1 m), and deep mineral (> 1 m) soils. Here these three categories of permafrost carbon have been combined to produce a single value for the available fraction. The sum of three weighted gamma distributions with each distribution respectively describing the PDF of the organic, shallow mineral, and deep mineral soils are used to describe the available fraction. The weights for the PDFs were derived from the relative fraction of permafrost soil carbon in organic, shallow mineral, and deep mineral soils from . The parameter values for the PDFs were derived by fitting gamma functions to the data in Fig. 3 of . The passive pool transformation rate is very poorly constrained as the incubation experiments analyzed by were unable to constrain the parameter's value (the contribution from the passive carbon pool was too small to be detected). The value of the parameter was estimated from the 14C age of the passive carbon pool from midlatitude soils . The mean residence time at 5 ∘C was estimated at 300 to 5000 years with a best guess of 1250 years yielding a passive pool transformation rate of 0.25 × 10-10 to 4 × 10-10 s-1, with a best guess of 1 × 10-10 s-1. The PDF was taken as uniform in base-two log space.

Arctic amplification can be changed in the UVic ESCM by changing the meridional diffusivity of the simplified atmospheric model . Here the Arctic amplification factor was taken to be normally distribution with a mean of 1.9 and standard deviation of 0.2 . Many studies have attempted to derive a PDF of equilibrium climate sensitivity for recent summary from model-based, observational, and paleoclimate evidence. Here we chose to use a PDF that captures the general features of these distributions with a mean of 3.25 ∘C for doubling of CO2 and the 5th and 95th percentile 1.7 and 5.2  ∘C respectively . The PDFs for all six perturbed parameters are shown in Fig. .

The Latin hypercube sampling, described in the introduction, was used to create the parameter sets. Each PDF was sampled from 25 equal-probability intervals and the value selected from each interval was randomly matched to one of the values selected from each of the other PDFs to create a “cube” containing 25 parameter sets. This sampling was repeated 10 times to create 10 cubes for a total of 250 model variants. Each of these variants was spun up for 5000 years under estimated year 1850 forcing to generate the permafrost carbon pool. Each model variant was forced with historical forcing followed by each of the four representative concentration pathways (RCPs) used in the fifth assessment report of the Intergovernmental Panel on Climate Change (IPCC AR5). The simulations were carried out with prescribed atmospheric CO2 concentrations and compatible anthropogenic CO2 emissions were diagnosed as a residual of the carbon cycle.

The old permafrost carbon capable version of the UVic ESCM was able to quantify the previously unaccounted for temperature effect of the permafrost carbon feedback by comparing model simulations with and without permafrost carbon . This has become much more difficult with the introduction of the permafrost carbon pool diffusion module. The soil carbon diffusion scheme causes the active layer to accumulate more soil carbon than in the model version without a prognostically generated permafrost soil carbon pool. Consequently we can no longer easily “turn off” the permafrost carbon. Therefore we have chosen to conduct experiments which quantify the permafrost carbon feedback in terms of carbon released from permafrost affected soils. As carbon released from permafrost soil displaces fossil fuel carbon in the carbon budget , we feel this is the most policy-relevant metric.

Twenty-five model variants (one cube) were projected 8000 years into the future under continued RCP 4.5 and 8.5 forcing. For this experiment only the four permafrost carbon parameters were perturbed, and climate sensitivity and arctic amplification were held at their model default values. For each scenario the models were forced with prescribed atmospheric CO2 concentration until peak CO2 concentration was reached (year 2150 for RCP 4.5 and year 2250 for RCP 8.5). Thereafter the simulated CO2 emissions were set to 0 and atmospheric CO2 was allowed to freely evolve. All other RCP forcings follow their prescribed trajectory until year 2300 and subsequently are held constant. The simulations were continued until the year 10 000 of the common era, 8000 years into the future. Expecting non-CO2 forcings to be constant for thousands of years following year 2300 is highly idealized, but this method was seen as the simplest approach for evaluating the long-term response of the permafrost carbon pool to anthropogenic forcing.

The release of carbon from permafrost soils for each RCP and for each of the 250 model variants is shown in Fig. . Average values and ranges for this quantity are given for all RCPs in Table . Model results in this section are quoted as the mean value of all model variants with the 5th and 95th percentile range in brackets. This is equivalent to the “very likely” range from IPCC AR5, although the numbers here are of course conditional on the model structure and parameter PDFs chosen. By year 2100 the model estimates that 56 (13 to 118) Pg C will be released under RCP 2.6 and 102 (27 to 199) Pg C released under RCP 8.5. By year 2300 the model estimates that 91 (32 to 175) Pg C will be released under RCP 2.6 and 376 (159 to 587) Pg C released under RCP 8.5. These results are generally consistent with the inter-model range of 37 to 174 Pg C, mean of 92 Pg C, by year 2100 under RCP 8.5 from .

The emission rate of CO2 from permafrost soils is shown in Fig.  and peak emissions for each RCP given in Table . Peak emissions under RCP 2.6 is 0.56 (0.13 to 1.29) and under RCP 8.5 is 1.05 (0.28 to 2.36) Pg C a-1. The timing of peak emissions of CO2 from permafrost soils varies by model variant and scenario followed (Fig. ) but generally occurs in the mid- to late 21st century or early 22nd century in the case of RCP 6.0. The emission rate from permafrost soils is a function of both the rate of permafrost thaw and the depletion of the available fraction of permafrost carbon in thawed soils. The similar trajectories of emissions in the early to mid-21st century for the different RCP scenarios is consistent with the lag between forcing and response of the permafrost system. These simulated peak emission rates are of similar magnitude to modern land use change emissions, 0.9 ± 0.8 Pg C a-1 averaged over the year 2000 to 2011 period . Even in the most extreme bound emissions from permafrost carbon are projected to be far lower than modern CO2 emissions from fossil fuel burning and cement production (9.5 ± 0.8 Pg C a-1 in 2011; ).

The permafrost carbon feedback's effect on climate change will ultimately be determined by how large the release of carbon from permafrost soils is relative to the cumulative fossil fuel emissions . This notion follows from the near-linear relationship between cumulative emissions of CO2 and change in global temperature , a relationship that emerges from the interaction of atmospheric and oceanic processes with the land surface source or sink effectively acting in the same manner as fossil fuel emissions . The release of carbon from permafrost soils relative to the diagnosed cumulative emissions for each model variant and RCP scenarios is shown in Fig. . The relative emissions are highest under RCP 2.6 where emissions from permafrost soil are 13 (2 to 39) % of fossil fuel emissions in 2100 and 21 (5 to 54) % of fossil fuel emissions by 2300. Under RCP 8.5 carbon released from permafrost soils is only 2 (0.5 to 5) % of fossil fuel emissions in 2100 and 8 (3 to 14) % of fossil fuel emission by 2300. RCPs 4.5 and 6.0 fall between these bounds with 7 (1 to 16) and 4 (1 to 10) % respectively by 2100 and 14 (3 to 29) and 12 (3 to 24) % respectively by 2300. These results suggest the permafrost carbon feedback to climate change will be more important in a relative sense to the magnitude of climate change in scenarios with substantial mitigation, consistent with previous studies e.g..

In year 1850 the UVic ESCM has a northern hemispheric permafrost area (including the Tibetan plateau) of 14.87 million km2, comparing well to the total of continuous and discontinuous permafrost area in the natural world e.g.. By year 2100 the northern hemispheric permafrost area has been reduced by 5.91 (2.25 to 8.43) million km2 under RCP 2.6 and 9.30 (7.49 to 9.90) million km2 under RCP 8.5. By 2300 a small recovery of permafrost area occurs under RCP 2.6 with a net reduction from year 1850 of 4.78 (1.71 to 8.13) million km2, while the loss of permafrost area continues until at least year 2300 under the other RCPs (Table ).

The relative importance of uncertainty from each perturbed model parameter to the overall uncertainty can be evaluated by computing the correlation coefficient between the parameter value and the value of some model output e.g. In Fig.  the correlation between each of the six perturbed model parameters and release of carbon from permafrost soils under RCP 8.5 by 2100 is shown. This particular metric was chosen as it has become the benchmark to compare simulations of the permafrost carbon feedback e.g.. The two highest correlations are for the initial available fraction with an R value of 0.78 and climate sensitivity with an R value of 0.51. Correlations for the other perturbed parameters are less that 0.13. These correlations suggest that reducing the uncertainty in the release of carbon from permafrost soils by 2100 requires better quantification of the size of the fast and slow carbon pools in permafrost soils. Also important is reducing the uncertainty in climate sensitivity, already a paramount, if intractable, problem in climate science e.g.. The four other perturbed parameters are relatively unimportant for reducing uncertainty to year 2100.

Correlations were also conducted between each model perturbed parameter value and release of carbon from permafrost soils by 2300. By 2300 the importance of the initial available fraction has decreased and has an R value of 0.36, the correlation with permafrost carbon transformation rate has increased to an R value of 0.43, and the correlation with climate sensitivity has increased to 0.64. The correlations with initial quantity of carbon in the permafrost region, permafrost carbon decay rate, and arctic amplification remain weak by year 2300, at 0.13, 0.02, and 0.11 respectively. These results demonstrate that the relative importance of uncertainty in parameters changes depending on the time frame of interest.

The low sensitivity of the release of carbon from permafrost soils to the value of Arctic amplification appears counterintuitive. However, most of the carbon held in the permafrost region is held in the region's southern extent (Fig. b), while Arctic amplification has the greatest effect over the Arctic ocean, Greenland ice sheet, and Canadian Arctic Archipelago where there is little simulated permafrost carbon.

Overall these results are encouraging as the most important factor for determining release of carbon from permafrost soils in the next century, the size of the permafrost carbon fast and slow pools, can be measured with incubation experiments e.g.. A dedicated field campaign and set of laboratory experiments to collect samples of permafrost carbon in optimal locations and conduct incubation experiments at the optimal temperatures could therefore significantly reduce uncertainty in the strength of the permafrost carbon feedback to climate change.

Climate change mitigation targets are often framed in terms of some global temperature change threshold not to be breached e.g.. Therefore examining the relationship between global temperate change and the release of carbon from permafrost soils is of interest. Figure  shows the correlation between change in global temperature and the release of carbon from the permafrost soils for all model variants and RCPs at years 2100, 2200, and 2300. The figure shows that there is a clear correlation between the two quantities at all three time horizons. However, the slope of the correlation evolves in time from 24 in 2100 to 39 in 2200 and 47 Pg C K-1 in 2300. These correlations demonstrate a key feature of the permafrost carbon system: the long time lag between forcing and response. That is, if fossil fuel emissions are eliminated and global temperature stabilizes, permafrost soils are expected to continue to release carbon for a long time.

The evolution of global mean temperature and atmospheric CO2 concentration for the multi-millennial experiments are shown in Fig. . Under continued non-CO2 RCP 4.5 forcing and with zero CO2 emissions, CO2 concentration falls until about the 28th century under all model variants and thereafter drifts slowly up or down depending on model variant. Under this scenario temperature continues to increase following cessation of emissions, becomes relatively stable for several centuries, experiences a period of renewed rapid warming in the late third millennium, and then becomes stable thereafter. Under continued RCP 8.5 forcing atmospheric CO2 declines monotonically after cessation of emissions, reaching a concentration below 1600 ppm by year 10 000 CE. Temperature continues to slowly increase following cessation of emissions, indicating that radiative forcing from atmospheric CO2 is declining to slowly to compensate for the unrealized warming of the system e.g.. Temperature change reaches a peak in the fifth millennium CE in these simulations. Thereafter, temperature begins a slow decline. The long-term evolution of temperature and atmospheric CO2 shown in these experiments is somewhat different from that in , who showed larger declines in temperature and atmospheric CO2 for comparable cumulative emissions and time frame. The continued existence of non-CO2 forcing in these scenarios and the inclusion of the permafrost carbon module are probable causes of the differences between that study and the present study, as both studies use similar versions of the UVic ESCM.

The response of the permafrost carbon pool to millennia of anthropogenically enhanced temperatures varies by scenario followed. Under RCP 8.5 the pool monotonically declines with time, with the rate of decline varying by parameter set (Fig. ). By the year 10 000 CE most of the model variants asymptote toward a carbon pool of about 10 Pg C, held around the fringes of Antarctica. Under RCP 4.5 the permafrost carbon pool begins a recovery before the year 3000 CE (Fig. ), with permafrost soil carbon reaching a nadir in the year 2411 (2254 to 2605) CE. Some of the parameter sets show renewed reduction in permafrost carbon about 2000 years after the recovery begins. The origin of this recovery, despite continued elevated global temperatures, is the creation of a large permafrost carbon pool in the Canadian Arctic archipelago and the high Russian Arctic as shown in Fig. . This region is thought to contain very little soil carbon in the modern climate (see Fig. ), a feature of the system that is captured by the model (Fig. b). Under RCP 4.5 these regions accumulate large permafrost carbon pools as they remain permafrost bound but with much higher net primary productivity from overlying vegetation. The simulations suggest that the ultimate fate of the permafrost carbon pool is highly contingent on scenario followed.