A thin layer of soil used in many coupled global climate models does not resolve the heat reservoir represented by underlying ground material. Under representation of this feature leads to unrealistic simulation of temperature dynamics in the active layer and permafrost. Using the Community Land Model (CLM3) and its modifications we estimate a required thickness of soil layers to calculate temperature dynamics within certain errors. Our results show that to compute the annual cycle of temperature dynamics for cold permafrost, the soil thickness should be at least 30 meters. Decadal-to-century time scales require significantly deeper soil layers, e.g. hundreds of meters. We also tested a new geometrical configuration of the soil layer geometry which is called slab permafrost. This configuration is represented by a thick soil layer underneath the traditional resolved soil layer. The model configuration with 30 m deep resolved soil layer and a 30 to 100 m thick slab shows results that favorably compare with our benchmark model which has a fully resolved 300 m-deep soil layer.
 Permafrost regions occupy approximately 25% of land in the Northern Hemisphere [Brown et al., 1997]. This is a climatologically important feature of the Northern high latitudes. Changes in permafrost are likely to have an impact on ecosystems, hydrology and infrastructure. The local communities of the North are very vulnerable to the thawing of permafrost and to associated changes in the northern environment.
 In cold permafrost, a detectable annual cycle can be measured at depths of 15–20 m [Yershov, 1998]. This indicates that for proper simulation of the annual cycle in a soil model (with zero heat flux at the lower boundary conditions) a soil layer which is at least 20–30 m deep is required. Longer time scales will obviously require deeper soil layer thickness. The position of the bottom boundary in a soil model in a climate model has an impact on the soil heating storage capacity [Stevens et al., 2007]. In the present article we estimate a minimum total soil layer thickness that is required to reasonably describe soil temperature dynamics across a range of different time scales. Using a linear heat diffusion equation we analytically estimate the error of simulations in a shallow soil layer model compared with the exact formulation in a semi-infinite domain. This study can be considered as a test of what kind of biases different configurations of model soil layers can introduce in the simulation of permafrost. A new configuration for climate models is suggested and tested with a thick slab soil layer underneath the traditional GCM upper soil model. This configuration can potentially save time without loss of quality of simulations. All of our runs are driven by the same forcing, namely, NCEP/NCAR reanalysis persistent year 1998 forcing, for a location roughly corresponding to typical observed Alaskan North Slope conditions.
2. “Back of the Envelope” Calculations
 We intentionally decided to simplify processes during soil freezing by omitting any water phase change effects, although we acknowledge that these effects are very important. This simplification is valid for model temperature dynamics below the active layer depth where the unfrozen liquid water content is usually negligible. To derive an analytical solution to the heat equation, we assume that the thermal properties for the thawed and frozen ground material are the same.
2.1. Analytical Solution for Diffusion Equation in Limited and Unlimited Domains
 In this section, we consider a one dimensional heat equation
in a bounded interval z∈[0,H]. Here D is a diffusion coefficient. Given the boundary condition at the surface z = 0 and some depth z = H:
we look for a periodic solution of (1) in time t. Here, the constant A is the amplitude of temperature oscillations on the soil surface, ω = 2π/τ is the frequency of the forcing at the boundary and τ is the period of the forcing. We also consider a bounded solution T∞(z, t) of the heat equation (1) in a semi-infinite domain [0, ∞] with the same boundary condition on the surface.
where h = is a characteristic vertical damping length. The solution of (1) for the limited domain [0, H] is:
We define the solution error due to using limited domain in the following way:
This error as a function of the total soil layer depth H and the period τ of the surface temperature oscillations is shown in Figure 1 (top). The error is small when the soil layer depth is very shallow or when it is very deep. When the model soil layer is very shallow (and therefore heat storage of the soil layer is small) the solution is completely controlled by the surface forcing for large enough τ (which in this case serves as a measure of persistence of the forcing). The warming/cooling periods in this case are slow enough and also they last long enough time so that because of the too low heat capacity soil temperature simply follows the changes in the atmospheric temperature. The error becomes small again when the model total soil layer depth is much thicker than the corresponding characteristic damping length. In this case the bottom boundary does not “feel” the surface, nor does the model domain “feel” the effect of the bottom boundary. There is a maximum in the error when the total soil layer depth is comparable with the vertical damping length. In this case the bottom boundary has a maximum effect on the model solution. The heat capacity of the total model soil layer is still too low because of the no-flux lower boundary condition. Therefore the heat, which normally goes to the deeper soil, keeps accumulating in the upper soil layers. This results in a too high sensitivity of soil to changes in the surface forcing. This would be the main point of the article: depending on the timescale of interest the lower boundary should be placed far enough from the surface so that the total soil depth is much greater than the corresponding damping length h. The location of the error maximum is always the same - near the soil layer bottom (Figure 1, bottom). The timescale corresponding to this maximum gets longer as the total model soil layer depth deepens. For the 30 m deep total soil layer shown in Figure 1 (bottom) the timescale of maximum error is at approximately 200 years. For a 3–4 m deep soil layer the picture looks very similar (not shown) with a maximum error at about 2 years. After reaching the maximum, the error starts decreasing because for very long timescales the temperature dynamics of the soil layer becomes fully controlled by the surface forcing.
 We can define the relative amplitude error as the absolute value of the difference between the amplitudes of the exact solution for the semi-infinite domain and that of the finite domain divided by the amplitude of the exact solution. It can be shown that the behavior of the amplitude error non-monotonically depends on the total soil layer depth if we look at a fixed depth. We will see this effect later on in our numerical experiments.
2.2. Characteristic Vertical Damping Lengths for Different Time Scales
 The damping length h enters both the solution in the limited and unlimited domains. Frequency of the time signal is an important factor affecting this damping length. Therefore this simple formula can be used to assess how deep the soil layer in a model should be depending on a time scale of an arbitrary external forcing. Let us define the constants in the following way: D = d010−7m2/s, ω = 2π/T = 2π/n × 86400 × 365 s−1 ≈ (2*10−7/n) s−1. Here for typical soils d0 would vary between 1–20 and n would denote number of years in a period of the applied periodic forcing. The formula for the damping length then becomes the following h ≈ . For d0 = 10 and n = 365−1 (diurnal cycle) we get h ≈ 0.15m. For d0 = 10 and n = 1 (annual cycle) we get h ≈ 3 m for the damping height of the seasonal cycle, consistent with measurements by Osterkamp and Romanovsky  in cold permafrost showing the damping factor of about 3 at 3m depth. This is why the soil layer thickness in the standard CLM3 (3.4 meters) is too shallow to approximate the semi-infinite domain problem. The above estimates show that even for seasonal timescales the soil layer depth should not be smaller than 25–30 m. For timescales longer than a century we need to use 10 times thicker soil layer − 200–300 m. Millennial scales will require even deeper soil layer − 500 m and more.
3. Model Sensitivity Tests With the CLM3
 Our error estimates are tested using the CLM3 with different soil layer configurations driven by the same forcing. The reference solution is calculated using the model with total soil layer thickness equal to 300 m. The upper 3.4 m of soil in this model are identical to the standard CLM3 model configuration with 10 layers. The thickness of added soil layers underneath is exponentially increasing from 1.2 to 7 m. The total number of layers added is 90. The reference solution inherits all the problems typical for the given model. We discuss some of the problems of the current version of the CLM3 in our other paper [Nicolsky et al., 2007]. All the model configurations used for comparison are identical to the CLM3 soil model in the upper 3.4 meters. The CLM3 with 300 m thick soil is spun-up for 2000 years before we start collecting the statistics of the reference solution.
 We conduct runs with different soil layer thicknesses (3.4 m, 10 m, 30 m, 100 m) and different periodic forcings with periods: 1 year (annual cycle), 20 years, 60 years, 200 years, 600 years, 2000 years. For the annual cycle we use the NCEP forcing that comes with the CLM3 in the persistent 1998 regime. All other periods represent artificially added perturbations with 5°C amplitude in the surface temperature oscillating with the corresponding periods. This is done in order to mimic different timescales of the long-term variability of the atmospheric forcing.
 Slab soil layer configurations, upper level resolved soil layers with a thick layer underneath, were also tested. The tested configurations included three different slab layer depths (30 m, 100 m, 300 m) underlying 2 resolved soil layer depths (3.4 m and 30 m). We are evaluating this type of configuration as a computationally cheaper alternative to the more straightforward solution adding a large number of deep layers. The upper layers are run in their usual mode (the resolved soil) while the layer underneath is represented by just one very thick mineral soil layer (ranging from 30 to 300 m in thickness), and one more thinner layer of soil to implement the no flux lower boundary condition. The slab configurations are marked as (X + Y)m in the text below where X and Y show depths of the resolved and slab soil layers correspondingly.
 To quantify the accuracy of our model in different configurations we calculate the “error” of the solution at 1 m and 3 m for different configurations assuming the solution with 300 m soil thickness represents the reference solution. The error is defined as a root mean square deviation of the approximate solution from the reference solution divided by the root mean square of the reference solution. Another measure of the quality was the error of the solution at different timescales. For the reference solution we calculate the amplitudes of the solution at frequencies corresponding to the periods of long-term oscillations in the forcing at 1 m and 3 m depths. The relative amplitude error at a certain frequency is calculated by dividing the absolute difference between the amplitudes of the approximate and the reference solution by the amplitude of the reference solution. The amplitude error will indicate how different timescales are reproduced in models with different configurations of soil layers. Looking at the conventional solution error as defined at the beginning of this section may not be enough since it will include all timescales.
4.1. Seasonal Cycle
 The standard CLM3 configuration with total soil layer thickness of 3.4 m does not adequately reproduce the seasonal cycle compared to the 300 m deep soil layer model. The model has strong biases – warm in the beginning of the cold season and cold in the spring (Figure 2, top). The cooling at 1 m in the model with 3.4 m deep soil layer is delayed by approximately 20 days and the model shows later recovery from the colder winter temperatures. The 3.4 m model warms up too much in the summer and cools down in the winter, compared with the 300 m deep soil model (Figure 2, bottom). This happens because in the 3.4 m model the heat cannot propagate downwards beyond the lower boundary due to the no flux condition at the bottom. Therefore only the shallow 3.4 m deep soil layer is thermodynamically active, which further helps accumulate even more heat in the model layers. Therefore the entire soil layer beneath 1 m in the 3.4 m soil model is at the freezing point in the summer. However, the amounts of unfrozen water are quite different for the 3.4 m model and the reference model with 300 m deep soil layer. For example, in the 3.4 m model within the soil layer located between 0.76 m and 1.31 m, 80% of ice turns into water during thawing, whereas in the 300 m model within the same layer about only 40% of ice melts. When we use 3.4 m deep resolved soil layer with a 30 m slab layer, the problem is opposite. The thick slab layer adds too much thermal inertia to the system, therefore the biases are just the opposite compared to the model with 3.4 m deep soil, which is also shown in Figure 2.
 The results on comparison of models with different soil layer configurations are summarized in Table 1. Analysis of Table 1 suggests that a minimum 30 m of resolved soil layer is necessary for proper simulation of the seasonal cycle. The 100 m thick soil layer model produces the same results as the model with 30 m thick soil layer, with- or without a slab layer of different depths (30, 100 or 300 m). Therefore for the reasonable reproduction of the seasonal cycle a 30 m deep resolved soil depth appears to be sufficient. Deeper soil thicknesses will be important for longer timescales as we will see later.
Table 1. Solution Error Percentage at Different Depths
3.4 + 30 m
3.4 + 100 m
3.4 + 300 m
30 + 30 m
30 + 100 m
30 + 300 m
4.2. Long Timescales
Figure 3 demonstrates the fact that a too shallow soil layer depth and therefore too low heat capacity results in too much sensitivity of the soil temperature at the lower boundary (3.4 m). We see that the model with a 3.4 m soil layer constantly overshoots and undershoots the more inert soil model with overall depth of 300 m. The difference can be as big as 4–6 K on timescales of a decade. Adding 30 m deep slab to the same 3.4 m deep soil significantly improves the solution (see Table 1). One might be inclined to choose this configuration as a compromise between the accuracy and computational efficiency, but as we know the 3.4 + 30 configuration failed the seasonal cycle test. From Table 1 we can see that the models with 30 m deep resolved soil layer and 100 m of slab underneath and the model with 100 m deep soil layer probably show best results in the long-term integrations among all the configurations, but a number of other alternative options will give similar results. For example, there is a significant improvement as we just go from 3.4 m to 30 m deep soil layer. Too thick slab layer (300 m) makes the upper soil feel the thermal inertia of the layer underneath. We can see that this configuration underestimates the variability. Table 2 gives the model amplitude error in our long-term integrations. We see that shallow soil layers (3.4 and 10) significantly overestimate variability on all timescales, especially on shorter periods. Adding a slab improves the situation, only if the slab is not too thick. Models with 100 and 300 meters thick slab layer strongly underestimate variability on shorter periods.
Table 2. Amplitude Error Percentage for Different Timescales
At 1 m Depth
At 3 m Depth
3.4 + 30 m
3.4 + 100 m
3.4 + 300 m
30 + 30 m
30 + 100 m
30 + 300 m
5. Conclusions and Discussion
 The main purpose of the article is to test sensitivity of the CLM3 to various geometries of the soil layers. The results of simulations with various total soil depths were compared with a reference solution represented by a model configuration with 300 m deep soil layer. A slab configuration represented by a resolved soil layer (3.4 or 30 m deep) with a thick layer of soil underneath is proposed and tested against the reference solution. Solution error is estimated for various time scales by forcing the model with a periodic surface forcing across a variety of frequencies that mimic annual, decadal and century timescales. The main conclusions are the following:
 (1) The standard 3.4 m soil layer used in the CLM3 significantly overestimates the temperature variability on all time scales, from seasonal to decadal and longer.
 (2) We estimate that in order to properly reproduce the seasonal cycle of the temperature a model soil layer should be at least 25–30 m deep. Decadal and longer time scales require 100 m and deeper soil layer. The best configuration is somewhat dependent on the timescale of interest.
 (3) A slab permafrost configuration with at least a 30 m deep resolved soil layer with a thick slab layer beneath could be a computationally cheaper but reasonable alternative to a deep soil model with fine resolution throughout the depth.
 (4) Deepening of the model total soil layer seems to improve the described modeling results. However, with deepening of the soil layer there may be some other constraints that may influence the quality of simulations - e.g. interaction with soil hydrology.
 The work was supported by the NSF Cooperative Agreement 0327664, and UAF CIFAR Student Grant Award. Computer simulation were carried out at Arctic Region Supercomputing Center.