## 1. Introduction

An accurate determination of the water and energy flux densities (customarily simplified by fluxes) at the earth’s surface, soil temperature and moisture states, and the corresponding heat and water fluxes in the uppermost soil layer of up to 10 m or so is essential for agrometeorological applications, numerical weather prediction (NWP), and ecosystem and climate system modeling (e.g., Robock et al. 1997). These quantities are typically predicted by soil models embedded in numerical models used in these kinds of applications (e.g., Dickinson et al. 1993; Robock et al. 1995; Huang et al. 1996). Usually, such soil models are based on the governing balance equations for soil heat and water fluxes that include water extraction by roots, freezing of soil water, and melting of soil ice. From a mathematical point of view these equations are partial differential equations (PDEs) that have to be solved by numerical schemes (e.g., Jacobson 1999; Pielke 2002). Here, finite-difference (FD) schemes like the semi-implicit Crank–Nicholson scheme (CNFD) are frequently applied (e.g., McCumber and Pielke 1981; Savijari 1992; Kramm et al. 1996; Dai et al. 2003; Kumar and Kaleita 2003).

From a theoretical point of view Crank–Nicholson-type FD schemes can be accompanied by numerical difficulties when discontinuities appear (e.g., Donea and Huerta 2003). Such discontinuities may occur with respect to the coexistence of water and ice and water fluxes at the internal interface between frozen and unfrozen ground (freezing line) in soils of high latitudes, for instance. At this interface, moreover, phase transitions cause the release of latent heat or the consumption of heat, changing the temperature. Capturing the depth of the freezing line, however, is essential in determining soil temperatures and moisture (e.g., Montaldo and Albertson 2001) and the energy and water fluxes at the earth’s surface (e.g., Cherkauer and Lettenmaier 1999) because temperature variations diminish throughout a great extent of the soil below (e.g., Luo et al. 2003). Once frozen ground is thawed, it acts as a porous medium and permits water to percolate. Advection of water through soil layers enhances soil heat capacity, and soil processes may become advection dominated. Hence, modeling frozen ground can be seen as a twofold problem: namely to correctly predict the depth of the freezing line and to accurately predict soil heat and water fluxes when the ground turns into a porous medium.

Recently, Mölders et al. (2003a) included frozen ground processes in the soil model of the Hydro–Thermodynamics Soil–Vegetation Scheme (HTSVS; Kramm et al. 1996), which was based on the CNFD and originally developed for dry deposition studies. A first long-term evaluation with routine observations at a midlatitude lysimeter station, which comprises 2050 consecutive days where occasionally the ground was frozen in winter, showed that including frost effects improved soil temperature predictions (Mölders et al. 2003b). Predicted soil temperatures are within 1–2-K accuracy (e.g., Kramm 1995; Mölders et al. 2003b; Narapusetty and Mölders 2005). Occasionally, compared to observations, a slight shift in phase of the freezing line was analyzed (e.g., Fig. 9 in Mölders et al. 2003b). However, a thorough examination whether using the CNFD is appropriate after including phase transitions between soil water and ice is still pending for extended freeze–thaw cycles, as they occur daily in the active layer at high altitudes and in subarctic or arctic regions in summer.

Usually, observations of a routine network serve for soil model evaluations. Routine soil temperature measurements, however, are of limited accuracy (±0.5 K) and low spatial and temporal resolution compared to those gained in sophisticated field campaigns. Therefore, an evaluation using routine data will never yield results as good as those using data from special field campaigns (e.g., Slater et al. 1998; Mölders et al. 2003b). Moreover, in any soil temperature observations, installation of temperature sensors can disturb the soil horizons, which are assumed undisturbed in the model. The temperature field is possibly influenced by the sensors with decreasing impact with increasing depth. Consequently, it is indispensable to assess the timing of predicted soil temperature peaks and position of the freezing line, not only by observations (necessary condition) but also by independent theoretical–numerical results (sufficient condition).

Therefore, we incorporate a theoretically advanced nonoscillatory numerical finite-element (FE) scheme—namely a Galerkin weak finite-element (GWFE) method—into HTSVS to solve the soil heat and water balance equations in a quite independent manner. Despite various comparisons of tasks in which analytical solutions have shown that GWFE schemes perform well in capturing discontinuities and have high phase fidelity (e.g., Fig. 1), their huge computational burden has made them impractical for agrometeorological studies, NWP, and ecosystem and climate modeling studies where quick turnaround times are essential. Theoretical strengths, however, make predictions using the GWFE scheme attractive as a sufficient condition in assessing the performance of simulations performed with the CNFD scheme.

In this paper, the PDEs are rearranged for a numerical solution within the framework of a GWFE scheme. Simulations performed with the CNFD scheme are evaluated by both 1) observations available around Council, Alaska, and 2) the results of the simulations with the theoretically advanced and quite independent GWFE scheme. Herein, results of simulations with the GWFE scheme serve to analyze the source for any discrepancies. Additionally, cell Peclet numbers (ratio of advection to diffusion) are used to determine under which circumstances the CNFD scheme performs well or not. Advantages and disadvantages of both schemes are discussed.

## 2. Method

### a. Brief description of the HTSVS soil model

*η*,

*T*, and

_{S}*χ*are volumetric water content, soil temperature, and water extraction by roots, and

*D*

_{η,υ},

*D*

_{η,w}, and

*D*

_{T,υ}are the transfer coefficients with respect to water vapor, water, and heat, respectively. Further,

*K*=

_{W}*K*is the hydraulic conductivity, where

_{WS}W^{c}*K*is the saturated hydraulic conductivity,

_{WS}*c*= 2

*b*+ 3 is the pore disconnect index, and

*W*=

*η*/

*η*is the relative volumetric water content with

_{S}*η*being the porosity of nonfrozen soil. Furthermore,

_{S}*η*

_{ice},

*t*, and

*z*are the volumetric ice content, time, and depth, respectively. Thermal conductivity

*λ*is a function of water potential, parameterized in accordance with McCumber and Pielke (1981). For

*T*≤ 273.15 K, a mass-weighted thermal conductivity is calculated depending on the volumetric water and ice content using the value obtained by the aforementioned equation for the water, and 2.31 J m

_{S}^{−1}s

^{−1}K

^{−1}for the ice fraction. The latent heat of fusion and vaporization are denoted

*L*and

_{f}*L*. Furthermore (Mölders et al. 2003a),

_{υ}*ρ*,

_{S}*ρ*,

_{W}*ρ*

_{ice},

*ρ*,

_{a}*c*,

_{S}*c*,

_{W}*c*

_{ice}, and

*c*being the density and specific heat of the dry soil material, water, ice, and air, respectively.

_{p}*T*≤ 273.15 K is given by

_{S}### b. Original numerical scheme

The coupled PDEs (1) and (2) are solved numerically using the implicit Crank–Nicholson finite-difference scheme (see appendix A) in conjunction with a Gauß–Seidel iteration technique (e.g., Kramm 1995; Kramm et al. 1996). In the latter, the convergence criteria are 0.0001 m^{3} m^{−3} and 0.01 K for volumetric water content and soil temperature (Kramm 1995). A logarithmic coordinate transformation is required for equal spacing of the grid to use centered finite differences. This means soil temperature and volumetric water content are predicted on a logarithmic grid with increasing spacing with depth.

### c. Theoretically advanced numerical scheme

*ϑ*stands for the viscosity coefficients. Applied to Eqs. (1) and (2), the advection velocity is

*ϑ*

_{11}= (D

_{η,v}+ D

_{η,w}),

*ϑ*

_{12}= D

_{T,v},

*ϑ*

_{21}= (

*λ*+

*L*

_{υ}ρ_{w}D_{T,υ})/

*C*, and

*ϑ*

_{22}=

*L*

_{υ}ρ_{w}D_{η,υ}/

*C*; and the source terms are

*S*

_{3}= −(

*χ*/

*ρ*) − (

_{w}*ρ*

_{ice}/

*ρ*)(∂

_{w}*η*

_{ice}/∂

*t*) −

*W*(∂

^{c}*K*/∂

_{ws}*z*) −

*K*ln

_{w}*W*(∂

*c*/∂

*z*) +

*K*(

_{w}*c*/

*η*)(∂

_{S}*η*/∂

_{S}*z*) and

*S*

_{4}= [(

*L*

_{f}ρ_{ice})/

*C*]∂

*η*

_{ice}/∂

*t*, respectively. Thus, one obtains

*C*/∂

*z*is assumed as constant when determining the transfer coefficients and source terms. The associated error remains negligibly small even along soil type discontinuities because the volumetric heat capacities of the various mineral and wet organic soils only differ slightly in permafrost regions and permafrost soils are typically at or close to saturation (Hinkel et al. 2003).

Equations (6) and (7) are now discretized by a GWFE method (see appendix B) and solved simultaneously by a Gauß–Seidel iteration technique using the same convergence criteria as for the CNFD scheme.

## 3. Experimental design

### a. Observational data

Soil temperature data at different depths are available at various sites around Council, Alaska, from spring to fall 1999, 2000, 2001, and 2002 (Table 1). Thus, they permit evaluating various thaw–freeze cycles during thaw-up and freeze-up and several seasonal thaw–freeze cycles. The sites at 64°53.47′N, 163°38.61′W, 140 m MSL (HyyC1); 64°50.60′N, 163°42.32′W, 50 m MSL (HyyC2); 64°44.76′N, 163°53.61′W, 110 m MSL (HyyC3); and 64°50.499′N, 163°41.591′W, 56 m MSL (RyyT) are typical for warm permafrost in tundra regions. A nonpermafrost site located at 64°54.456′N, 163°40.469′W, 96 m MSL (RyyF), also run by the permafrost observatory, is covered by forest.

In our nomenclature, the first letter refers to the data origin (Hinzman 2005; V. E. Romanovsky 2005, personal communication), the letters yy stand for the last two digits of the year, and the last letter denotes the name of the site—specifically, C1, C2, and C3 for the Council sites of the Arctic Transitions in the Land–Atmosphere System (ATLAS) data (Hinzman 2005), and T and F for the tundra and forest site data from the permafrost observatory (V. E. Romanovsky 2005, personal communication). RyyTN denotes measurements taken a few meters away in the same region as RyyT.

At the tundra sites, the soil vertical profile is moss from the surface to 0.12 m, dead moss from 0.12 to 0.22 m, peat from 0.22 to 0.30 m, and silt from 0.30 to 15 m. The same soil vertical profile exists at the forest site, but soil layer thicknesses range from the surface to 0.08 m, 0.08 to 0.25 m, 0.25 to 0.65 m, and 0.65 to 5 m, respectively. Table 2 summarizes the soil physical parameters determined for Council.

### b. Simulations

*η*

_{total}is the total soil volumetric (liquid and solid) water content.

The above procedures guarantee that simulations with the CNFD and GWFE scheme have the same initial and boundary conditions independent of the numerical scheme used. Consequently, all differences between these simulations result only from the numerical schemes applied. The results of the simulations are addressed by the names of the datasets discussed in section 3a. Soil layers are counted from the surface to the bottom of the model.

### c. Analysis

In the CNFD, before integrating, a logarithmic coordinate transformation is introduced to apply equal spacing and central differences for appropriate finite-difference solutions (e.g., Kramm et al. 1996). On the contrary, the GWFE scheme works well with any grid spacing. Therefore, it is formulated on *z* coordinates that can be freely chosen. However, for easy comparison of CNDF and GWFE results, simulations with GWFE use the same grid as the CNFD simulation. In principle, logarithmic grids are favorable (e.g., Pielke 2002), but a free choice of the model layer depths may be desirable for easy comparison to observations for evaluation studies. For comparison of simulated and observed soil temperatures, simulated quantities are interpolated to the observational depths. To assess the uncertainty introduced by interpolation another set of simulations with GWFE is performed with a grid that coincides with the observation levels.

Soil temperatures simulated by the CNFD scheme are evaluated by the observations available (Table 1) and results obtained with the theoretically advanced GWFE scheme. The superiority of the GWFE scheme is demonstrated by means of comparison to problems that have known analytical solutions.

Discrepancies between the simulated and observed soil temperatures are evaluated by root-mean-square errors (RMSEs). Improvement indices are calculated to judge the potential of the advanced numerical scheme. CPU time differences are documented for comparison of the computational burden of the numerical schemes (Table 3). The different advection and diffusion regimes within each element are recognized by means of nondimensional cell Peclet numbers.

## 4. Results and discussions

### a. Reproduction of analytical solutions

To demonstrate that the GWFE method is a theoretically advanced numerical scheme various tests are carried out with conditions for which analytical solutions exist. Generally, the GWFE scheme reproduces well the analytical solutions with exact phase and peak magnitude. On the contrary, the CNFD scheme often shows diffusive solutions with shifts in phase, smoothed peaks, and oscillations (e.g., Fig. 1). Based on these studies we conclude that 1) GWFE is a nonoscillatory, advanced scheme that reproduces well the phases and peaks and that 2) GWFE can be used to further evaluate the soil temperatures simulated by the HTSVS soil module with CNFD where observational data alone do not indicate the reasons for discrepancies.

### b. General remarks

Results from simulations with the GWFE scheme interpolated to observational levels and those obtained from simulations with the GWFE scheme performed on the same grid as the observations differ marginally (up to 0.2 K; therefore, not shown). The RMSEs between these simulations are less than 0.1 K. Thus, the uncertainty introduced by the interpolations is negligible.

Typically Peclet numbers are low in frozen ground and comparatively higher in the unfrozen part of the active layer. The Peclet numbers identify H99C1, H99C2, H02C3, R99T, R00T, R01F, and R02F as moderate advection and H00C1, H00C2, H00C3, H01C1, H01C2, H01C3, R01TN, and R01T as diffusion-dominated episodes (Fig. 2).

Generally, computational time will increase from 1.6 to 2.8 times if the GWFE scheme is used instead of the CNFD scheme (Table 3). In the GWFE method, additional memory requirements and matrix operations such as multiplication, successive elimination of variables to solve the linear system using decomposition of the system matrix into lower and upper triangular matrices (known as LU decomposition), and the following back substitution procedures involving the global matrices 𝗠, 𝗖, and 𝗗 (discussed in appendix B) increase computational time. The magnitude of increase depends on the number of soil layers (size of the matrices) and the length of the episode (cf. Tables 1 and 3). Note that in our simulations, there are 11 soil layers at maximum; that is, we deal with 11 × 11 global matrices at each time step. However, for short-term simulations, the shorter spinup time and the high phase fidelity and additional accuracy in determining the soil state variables and fluxes with encouraging convergence properties make the GWFE method attractive (cf. Table 3).

### c. CNFD versus observations

Initialization does not mean that the soil conditions are initially in local thermodynamic equilibrium because of measurement errors, interpolation from the observational levels to the logarithmic grid of the soil model, and the assumptions on soil initial moisture. Thus, any soil model needs to spin up. Simulations with the CNFD scheme require some days for spinup in the deeper soil layers (e.g., Fig. 3). Spinup time will be especially long if the soil is partly frozen. Obviously, along the freeze–thaw line phase transitions occurring to achieve local thermodynamic equilibrium and the diffusive nature of the CNFD scheme trigger slight oscillations that slowly smooth out when locally, temporally (at a level and time step) thermodynamic equilibrium is achieved. In the following, results after spinup are discussed.

The CNFD scheme captures soil temperatures changes relatively better for episodes starting in spring (H00C1, H00C2, H00C3, H01C1, H01C2, H01C3, R00T, R02TN, and R00F) than for those starting in summer when soils are warmer (H99C1, H99C2, H99C3, R99T, H02C1, H02C2, H02C3, R01T, R01TN, R01F, and R02F) (Fig. 4). Note that, in spring, soil temperatures are still below the freezing point. If frozen ground thaws, RMSEs averaged over all layers will reach up to 2.2 K, and individual absolute differences up to 3.4 K. RMSEs (averaged over all layers) are as large as 2 K where/when no thawing of the ground occurs. Looking at the individual layers, the RMSEs are the greatest (up to 2.9 K) in the layer that contains the freezing line.

In nature, sudden soil temperature increases often occur in the upper 0.15–0.2 m beneath the surface after the insulating snowpack melts. After such a sudden warming, the diffusive nature of the CNFD scheme shifts the soil state toward (up to 1.5 K) warmer conditions (e.g., Fig. 5a; after day 52) yielding an overestimation of thaw rates. The cooling related to phase transition is overcompensated by the diffusive nature and triggers slight oscillations. Then, if in nature an episode of sudden cooling occurs, the CNFD scheme will shift into a mode of underestimating soil temperatures by up to 1 K (e.g., Fig. 5a; after day 65). Thus, we conclude that the CNFD scheme will have difficulties reproducing sudden changes, especially if appreciable phase transitions occur. The results converge to the observations after some time (e.g., Fig. 5).

If warming occurs constantly, but slowly over a relative long time and leads to a thawing event, simulations with the CNFD scheme will overestimate soil temperatures about 2 K (e.g., Fig. 3; after day 57). Then, if a slight slow cooling occurs, the overestimations will remain with a slight phase shift and decrease slightly over time. After the thawing event, the simulations with the CNFD scheme converge to the observations (e.g., Fig. 3). The time needed for this “catchup” after slow warming–cooling events is shorter than for sudden temperature change events (cf. Figs. 3 and 5).

For diffusion-dominated regimes, that is, low Peclet numbers, simulations with the CNFD scheme produce acceptable results (e.g., Fig. 6a). As the active layer thaws, cell Peclet numbers increase. Consequently, since the CNFD scheme is diffusive, soil temperature simulations lose quality at temperatures around the freezing line (e.g., Fig. 7a). The results show a phase shift in the peak temperature compared to the observations (e.g., Fig. 3). The maximum phase shift found for our cases amounts to 7 days.

As the thawing of the active layer progresses, oscillations may occur along the freezing line (e.g., Fig. 5). Obviously, the diffusive nature of the CNFD scheme will trigger alternating thawing and cooling followed by freezing and warming if freeze–thaw cycles occur within the same model layer over several weeks. These oscillations are also visible in the simulated volumetric water and ice content fields (e.g., Figs. 5c and 5d) and the relatively long spinup for partly frozen ground.

### d. GWFE versus observations

The GWFE scheme captures soil temperatures relatively well for H99C1, H99C2, H99C3, H02C1, H02C2, H02C3, R01F, and R02F. Particularly during the R02F and H02C3 episode, when cell Peclet numbers are relatively high (Fig. 2), the scheme leads to moderate RMSEs (up to 1.3 K, Fig. 4) and captures well the temporal evolution of soil temperatures, that is, the phase of the temperature peaks (e.g., Figs. 6b and 7b). The H02C3, R01F, and R02F episodes are characterized by consistently high Peclet numbers and low RMSEs (e.g., Figs. 2 and 4), and the freezing line is well captured (e.g., Fig. 7b).

The GWFE scheme will show a stable nature with hardly any diffusion if episodes start in spring when soil temperatures are below freezing. However, if the episode starts in spring at relatively cold temperatures and soil temperature rises rapidly within a short time, particularly in the upper layers beneath the surface, the scheme will overestimate the soil temperature as compared to the observations (e.g., Fig. 3). Nevertheless, it still reproduces the pattern of peaks in the temperature maxima and minima correctly and nearly in phase (e.g., Fig. 3). Quick temperature rises are captured well in the upper layers, but in the lower levels the rise is considerably underestimated and a slight diffusion is noticed (Fig. 5b).

The main advantages of the GWFE scheme are that it 1) captures the right phase for the temperature peaks (e.g., Fig. 3), 2) generates nondiffusive solutions (e.g., Fig. 5), 3) is able to handle high cell Peclet number regimes relatively well (e.g., Figs. 1, 3 and 5), and 4) well captures the position and variability of the freezing line (e.g., Fig. 7b). A major disadvantage of the GWFE scheme is its high computational burden (Table 3). Generally, the GWFE scheme seems to marginally overestimate the peaks in diffusion-dominated regimes during the simulation periods.

### e. Performance evaluation of CNFD by GWFE

As pointed out above, both the CNFD and GWFE schemes simulate the soil temperatures acceptably at all depths for all episodes tested (cf. Table 3; Fig. 4). On average, the RMSEs are comparably lower for the simulations using the GWFE than those applying the CNFD scheme, particularly for high Peclet number regimes (Table 3; Figs. 2 and 4). On average, simulations with the GWFE scheme produce up to 1 K lower overall RMSEs than those with the CNFD scheme. For the simulations with the CNFD scheme the overall RMSE (averaged over all episodes and depths) is about 1.6 K, while that with the GWFE scheme amounts to 1.1 K.

Obviously, the quality of simulations with the CNFD scheme depends on the soil regime (e.g., Figs. 6 and 7). Under diffusion-dominated regimes, simulations with the CNFD scheme perform well and provide results of the same quality as those using the GWFE scheme (e.g., Fig. 6). For moderate advection regimes (e.g., Fig. 7) and for soil temperatures ranging between −2° and 2°C, peaks in soil temperature simulated with the CNFD scheme often show a phase shift (up to 7 days) with deviations of ±2.5 K from the observations and the results of simulations using the GWFE scheme (e.g., Fig. 3).

Comparison of the results obtained with the CNFD and GWFE schemes documents that the long spinup can be attributed to the CNFD scheme (diffusive nature, difficulties with discontinuities, tendency to oscillations) rather than to the uncertainty involved in the initialization (e.g., Fig. 3). It also shows that the phase shift reported for midlatitudes (Mölders et al. 2003b) and also found in this study when using the CNFD scheme (e.g., Fig 3) results from the diffusive nature of the CNFD scheme rather than from the parameterized/considered physical processes.

After spinup, the distributions of volumetric water and ice content slightly differ for the simulations with the two different numerical schemes. The differences will be the greatest (up to 0.175 m^{3} m^{−3}) where diffusion is the greatest and close to the freezing line if the latter remains in nearly the same position for a relatively long period of time (e.g., Fig. 5). In this case, the simulations with the CNFD scheme predict lower ice content than those with the GWFE scheme (e.g., Figs. 5c and 5d). Consequently, the volumetric heat capacity is notably higher for the simulations with the GWFE than for the CNFD scheme [cf. Eq. (3)] with impacts on soil temperature.

Compared to the results obtained with the GWFE scheme, using the CNFD scheme, on average, yields slightly (up to 0.1 m^{3} m^{−3}) lower volumetric water content in the thawed active layer (e.g., Fig. 8).

In frozen ground at temperatures close to the freezing line, the GWFE scheme well captures observed sudden rapid increases in soil temperature (e.g., after a snowpack vanishes), while the CNFD scheme produces oscillations (e.g., Fig. 5). The discrepancies found may only be partly attributed to the CNFD scheme. The fact that the GWFE scheme well captures the observed maximum temperature in the third layer, for instance, but underestimates it in deeper layers (Figs. 5a and 5b) provides evidence that the physical parameters for the dry soil material may not be as representative of the fourth layer as they should be. Natural heterogeneity within the soil may be a reason. Thus, we conclude that incorrect soil physical parameters contribute to the discrepancies between the CNFD (GWFE) simulations and the observations found for the fourth layer (Fig. 5b); that is, here the discrepancies are not solely due to the numerical scheme.

Obviously, the GWFE scheme is less sensitive to melting/freezing than the CNFD scheme. The fact that, in layer 3, the GWFE scheme well reproduces the sudden increase and then decrease (Fig. 5) is evidence that the parameterization of the freezing–melting term (which is part of the source term) is not the cause for the discrepancies found in the simulations with the CNFD scheme. Both simulations use the same parameterization and differ only by the numerical scheme applied. Thus, the CNFD scheme produces the discrepancies with the observations under these soil conditions.

In thawing seasons with slow-warming/slow-cooling events, the CNFD scheme tends to overestimate soil temperatures and yields a slight phase shift, but the results converge to the observations and those obtained with the GWFE scheme. As pointed out above, the time required by the CNFD scheme for this “catchup” is shorter for slow-warming/slow-cooling events than for sudden temperature changes (cf. Figs. 3 and 5).

In most cases, volumetric water content predicted with the CNFD and GWFE schemes differ by about 0.1 m^{3} m^{−3} (Fig. 8). The same is true for soil ice content. However, along the freezing line volumetric water content may differ considerably (up to 0.3 m^{3} m^{−3}) when oscillations occur in the simulations with the CNFD scheme (e.g., Fig. 5). This finding is consistent with the temperature differences found between simulations with the CNFD and GWFE scheme. As for soil temperature, there also exist phase shifts in the onset time of thawing of up to 7 days. Moreover, the time needed to complete the thawing of a model layer differs slightly depending on the numerical scheme (e.g., Fig. 5). Since the GWFE scheme has shown itself to have high phase fidelity and represent well the magnitude of the peaks for the test problems with analytical solutions and soil temperature observations, we have to assume that it also captures well those in volumetric water content. Under this assumption we must expect that extremes in volumetric water content will be underestimated on the high end by HTSVS with the CNFD scheme if sudden changes occur.

## 5. Conclusions

This study examines whether the Crank–Nicholson finite-difference scheme will still be suitable in the soil model of HTSVS if discontinuities in soil ice and water exist. Such discontinuities are associated with phase transition processes occurring during freezing and thawing in the active layer of permafrost soils, or in midlatitudes during frost episodes in winter. Since routine observation have limited accuracy and coarse spatial and temporal resolution, it is indispensable to assess the timing of predicted soil temperature peaks and position of the freezing line, not only in observations (necessary condition), but also by independent theoretical–numerical results (sufficient condition). Therefore, simulations performed with the soil model of HTSVS using the CNFD scheme are evaluated by soil temperature observations reported around Council, Alaska, and by means of simulations with a Galerkin weak finite-element method. In doing so, we implement the GWFE scheme in HTSVS to solve the soil heat and water balance equations in an independent manner.

Results obtained with the GWFE scheme show high phase fidelity, peak accuracy, and capture discontinuities for all soil regimes. Under diffusion-dominated regimes, the CNFD scheme provides acceptable results of the same quality as the GWFE scheme. The CNFD scheme has some slight difficulties in simulating correctly the 1) moderate advection-dominated regimes, (2) soil temperature regimes between −2° and 2°C, and 3) position and variability of the freezing line. Taking into account that 1) routine measurements of soil temperatures have errors of ±0.5 K, on average, and that 2) the differences between observations and simulations are relative small (less than 3.4 K absolute, and 1.6 K on average), results obtained by HTSVS with the CNFD scheme have to be considered as acceptable. Nevertheless, the performance skills and improvement indices indicate that uncertainty in modeling frozen ground can be reduced by using GWFE. This fact and the much shorter spinup time required for the GWFE (1 day) than CNFD (several days) scheme makes the former scheme attractive for short-term applications where CPU time is of minor importance. The small difference in RMSE and the greater computational efficiency of the latter suggest that, in climate modeling, the CNFD scheme will provide acceptable soil temperatures, if seasonal averages are considered.

## Acknowledgments

We thank U. S. Bhatt, G. Kramm, J. E. Walsh, and the anonymous reviewers for helpful discussions; V. E. Romanovsky and L. D. Hinzman for data access; and the NSF for support under Contract OPP-0327664. The University of Alaska Fairbanks Graduate School supported the first author with a scholarship.

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## APPENDIX A

### Crank–Nicholson Finite-Difference Scheme

*ϑ*

_{11},

*ϑ*

_{12},

*ϑ*

_{21}, and

*ϑ*

_{22}are the viscosity coefficients given in section 2c;

*S*

_{1}= −(

*χ*/

*ρ*) − (

_{W}*ρ*

_{ice}/

*ρ*) [(

_{W}*η*

^{j+1}

_{ice}−

*η*

^{j}_{ice})/Δ

*t*];

*S*

_{2}= (

*L*

_{f}ρ_{ice}/

*C*)[(

*η*

^{j+1}

_{ice}−

*η*

^{j}_{ice})/Δ

*t*]; and Δ

*z*and Δ

*t*are the thickness of a model layer and time step. The superscript

*j*and subscript

*i*indicate the time step and soil model layer. The weights, Θ and (1 − Θ), for the present and past time step spatial approximations amount to 0.5 in the CNFD scheme (Kramm 1995; Jacobson 1999).

## APPENDIX B

### Galerkin Weak Finite-Difference Scheme

*η*and

*T*can be approximated by

_{S}*η*

_{1},

*η*

_{2},

*T*

_{S}_{1}, and

*T*

_{S}_{2}are the approximated values at the nodes of the linear element through so-called shape functions

*N*

_{1}and

*N*

_{2}, and

**N**is the shape function vector. The source terms

*S*

^{a}_{3}and

*S*

^{a}_{4}are approximated analogously. Thus, one obtains

*η*and

^{a}*T*are approximated by multiplying these equations with a test function

^{a}_{S}*N*and integrating over the entire vertical soil column (e.g., Gallagher et al. 1975; Hinton and Owen 1977; Smith and Griffiths 1988)

^{b}*n*stands for the total number of grid elements. A Galerkin method is used to postulate the test function

*η*(e.g., Finlayson 1972; Smith and Griffiths 1988) to solve Eq. (B3). The requirement for the order-of-shape function is reduced by applying Green’s theorem that yields the so-called weak formulation. Thus, Eq. (B3) formulated in the GWFE method reads

^{a}**N**

^{T}is the transpose of the shape functions vector for the vector multiplication over the linear element. By applying the concept of isoparametric elements associated with a five-point Gauß–Legendre quadrature integration and by matrix operations (e.g., Hinton and Owen 1977), one obtains

**=**

*η***N**{

*} and*

**η**^{a}**T**

_{S}=

**N**{

**T**

*} are consistent with the generic FE approximation.*

^{a}_{S}*dη*/

*dt*and

*dT*/

_{S}*dt*are discretized in the framework of the so-called Θ-family methods using a second-order-implicit Padé approximation (e.g., Donea and Huerta 2003), as

Averaged cell Peclet values for (a) the ATLAS sites and (b) the permafrost observatory sites.

Citation: Monthly Weather Review 134, 10; 10.1175/MWR3219.1

Averaged cell Peclet values for (a) the ATLAS sites and (b) the permafrost observatory sites.

Citation: Monthly Weather Review 134, 10; 10.1175/MWR3219.1

Averaged cell Peclet values for (a) the ATLAS sites and (b) the permafrost observatory sites.

Citation: Monthly Weather Review 134, 10; 10.1175/MWR3219.1

Soil temperatures as obtained for the fifth layer for R00T using the CNFD and GWFE schemes.

Citation: Monthly Weather Review 134, 10; 10.1175/MWR3219.1

Soil temperatures as obtained for the fifth layer for R00T using the CNFD and GWFE schemes.

Citation: Monthly Weather Review 134, 10; 10.1175/MWR3219.1

Soil temperatures as obtained for the fifth layer for R00T using the CNFD and GWFE schemes.

Citation: Monthly Weather Review 134, 10; 10.1175/MWR3219.1

Comparison of simulated and observed soil temperature RMSEs using the CNFD and GWFE schemes as obtained for various sites. The solid line shows the results obtained with the CNFD scheme and the dashed line indicates that obtained with the GWFE scheme for (a) the ATLAS sites and (b) the sites run by the permafrost observatory.

Citation: Monthly Weather Review 134, 10; 10.1175/MWR3219.1

Comparison of simulated and observed soil temperature RMSEs using the CNFD and GWFE schemes as obtained for various sites. The solid line shows the results obtained with the CNFD scheme and the dashed line indicates that obtained with the GWFE scheme for (a) the ATLAS sites and (b) the sites run by the permafrost observatory.

Citation: Monthly Weather Review 134, 10; 10.1175/MWR3219.1

Comparison of simulated and observed soil temperature RMSEs using the CNFD and GWFE schemes as obtained for various sites. The solid line shows the results obtained with the CNFD scheme and the dashed line indicates that obtained with the GWFE scheme for (a) the ATLAS sites and (b) the sites run by the permafrost observatory.

Citation: Monthly Weather Review 134, 10; 10.1175/MWR3219.1

Soil temperatures as obtained and observed for the (a) third layer and (b) fourth layer for R01TN and soil volumetric water and ice content as obtained for the (c) third layer and (d) fourth layer for R01TN. Note that no observed volumetric soil ice and water content data are available. Thawing starts on day 53 and 58 in the third and fourth layer, respectively.

Citation: Monthly Weather Review 134, 10; 10.1175/MWR3219.1

Soil temperatures as obtained and observed for the (a) third layer and (b) fourth layer for R01TN and soil volumetric water and ice content as obtained for the (c) third layer and (d) fourth layer for R01TN. Note that no observed volumetric soil ice and water content data are available. Thawing starts on day 53 and 58 in the third and fourth layer, respectively.

Citation: Monthly Weather Review 134, 10; 10.1175/MWR3219.1

Soil temperatures as obtained and observed for the (a) third layer and (b) fourth layer for R01TN and soil volumetric water and ice content as obtained for the (c) third layer and (d) fourth layer for R01TN. Note that no observed volumetric soil ice and water content data are available. Thawing starts on day 53 and 58 in the third and fourth layer, respectively.

Citation: Monthly Weather Review 134, 10; 10.1175/MWR3219.1

Comparison of simulated and observed soil temperatures for H99C3 as obtained for the simulations with the (a) CNFD scheme, (b) GWFE scheme, and (c) differences between CNFD and GWFE.

Citation: Monthly Weather Review 134, 10; 10.1175/MWR3219.1

Comparison of simulated and observed soil temperatures for H99C3 as obtained for the simulations with the (a) CNFD scheme, (b) GWFE scheme, and (c) differences between CNFD and GWFE.

Citation: Monthly Weather Review 134, 10; 10.1175/MWR3219.1

Comparison of simulated and observed soil temperatures for H99C3 as obtained for the simulations with the (a) CNFD scheme, (b) GWFE scheme, and (c) differences between CNFD and GWFE.

Citation: Monthly Weather Review 134, 10; 10.1175/MWR3219.1

Soil water content (m^{3} m^{−3}) as obtained by CNFD and GWFE with H99C2.

Citation: Monthly Weather Review 134, 10; 10.1175/MWR3219.1

Soil water content (m^{3} m^{−3}) as obtained by CNFD and GWFE with H99C2.

Citation: Monthly Weather Review 134, 10; 10.1175/MWR3219.1

Soil water content (m^{3} m^{−3}) as obtained by CNFD and GWFE with H99C2.

Citation: Monthly Weather Review 134, 10; 10.1175/MWR3219.1

Test episodes and their data availability. Numbers in parentheses indicate length of episode in days.

Soil physical parameters used in this study. Here, *η _{s}*,

*b*,

*K*,

_{ws}*ψ*, and

_{s}*c*are the volumetric water content at saturation (porosity), pore size distribution index, saturated hydraulic conductivity, water potential at saturation, and volumetric heat capacity of dry soil material. Database courtesy of V. E. Romanovsky (2004, personal communication). Freeze/thaw characteristics were determined in accord with Romanovsky and Osterkamp (2000), applying variations in soil temperature within the active layer and near-surface permafrost in a 5-cm vertical-resolution permafrost model that is independent of HTSVS.

_{S}ρ_{S}CPU time, relative increase of CPU time, overall RMSEs, and improvement index *I* = _{GWFE}/_{CNFD} as obtained for the various simulations.