Introduction
Information regarding the presence and maximum penetration of soil freezing is necessary for a variety of climate-sensitive engineering applications. The depth of soil freezing varies greatly from season to season and region to region. As a result, building codes must consider the extreme freezing penetration events to assure that footings and utilities are buried at an adequate depth. In addition, information on soil frost penetration is relevant in agricultural (e.g., van Es et al. 1998) and flood forecasting (Molnau and Bissell 1983) applications. Given this relevance, it is not surprising that many methods have been developed for estimating soil freezing. The methods described by Berggren (1943), Harlan (1973), Cary et al. (1978), Benoit and Mostaghimi (1985), Gusev (1985), Flerchinger and Saxton (1989), Jansson (1991), and Gusev and Nasonova (1997) provide an idea of the history and range in complexity of soil freezing models. Kennedy and Sharratt (1998) compared the performance of four of these soil freezing models and found that the finite difference models of Flerchinger and Saxton (1989) and Jansson (1991) simulated maximum frost depth with reasonable accuracy, while the heat flux balance methods of Benoit and Mostaghimi (1985) and Gusev (1985) tended to overpredict frost depth.
Despite the relevance of frost depth information and the lack of in situ frost depth observations, few have attempted to create a model capable of simulating maximum frost penetration at a nationwide network of stations. National scale soil freezing maps appear in the 1941 Yearbook of Agriculture (USDA 1941) and Sowers (1979). Unfortunately, the data used in these maps are unofficial, unreferenced, and/or antiquated (1899–1938). A related national “climatology” of 100-yr return period air freezing indices was developed by Steurer and Crandell (1995). Using this value, maximum annual soil freezing depths can be inferred based on the Berggren Equation (Berggren 1943).
For many applications, the Berggren Equation estimates the winter maximum frost depth with sufficient accuracy (Gel’fan 1989). However, the model requires that snow depth be assumed constant and neglects seasonal changes in soil water content. These assumptions are unrealistic in many parts of the United States. Furthermore, model output is limited to the maximum depth of soil freezing, precluding its use in applications where the daily progression of frost depth is required.
These limitations are addressed by physically based soil freezing models, such as the Simultaneous Heat and Water (SHAW) model (Flerchinger and Saxton 1989; Flerchinger 1991; Flerchinger et al. 1994, 1996). The state-of-the-art SHAW model was designed for hydrological applications and thus considers such factors as evaporation, snow depth, runoff, and soil water profiles in addition to soil freezing depth. The model assumes a one-dimensional vertical soil profile, which extends upward through multiple layers representing undisturbed soil, tilled soil (a maximum of 20 soil layers can be considered), vegetative residue, snow, and the plant canopy. Required meteorological and site characteristic inputs to the model are extensive (Table 1). Based on hourly (or daily) meteorological observations, heat and moisture fluxes can be obtained for the upper model boundary, which in turn are used to compute the fluxes between layers. Equations in the model are solved implicitly with the Newton–Raphson method (Flerchinger and Saxton 1989).
As opposed to the empirical Berggren approach, the amount of water in the soil is an integral part of the SHAW model’s simulation of the daily progression of soil freezing. In addition to the effect of soil moisture on soil thermal conductivity and latent heat release during freezing, water movement also plays a role in the process of soil freezing, particularly when the soil is near saturation. Water has a tendency to be attracted to the boundary between frozen and unfrozen soil. The movement of water to this freezing front results from thermal gradients, which induce water potential gradients within the soil and thus further water movements.
Although the SHAW model addresses each of the relevant physical processes that govern soil freezing, its extensive data requirements limit its use to a very few heavily instrumented locations. An intermediate class of models is referred to as heat flux balance methods by Kennedy and Sharratt (1998). Examples of this type of approach are given by Benoit and Mostaghimi (1985) and Gusev (1985). The data requirements of these approaches can be fulfilled by the data available from the Cooperative Observer network, making them attractive for estimating soil freezing on a national scale. However, in a comparison of the ability of these models to estimate maximum frost depth, Kennedy and Sharratt (1998) found that they tended to overpredict the depth of soil freezing due to their neglect of volumetric heat content. Furthermore, the Gusev (1985) model does not allow for thawing at the soil surface. Although presumably this has only minor consequences for maximum frost penetration, it limits the model’s use in other applications.
Based on the frequency of requests for soil freezing information received by the Northeast Regional Climate Center (NRCC), DeGaetano et al. (1996) developed a physically based soil freezing model with application to the northeastern United States (hereinafter the NRCC model). The design of the model was guided by the availability of meteorological data at the national network of stations with the greatest spatial density. Thus, meteorological input was limited to daily maximum and minimum temperature and snow depth. In addition, daily observations of liquid-equivalent precipitation and snowfall were used to estimate snow density empirically from snow depth. The NRCC model can best be classified as a hybrid of the more complex finite-differencing approaches (e.g., SHAW) and the simpler heat flux balance methods. As such, it blends the desirable characteristics of each model group. Meteorological input requirements are limited. However, instead of solving a set of equations representing discrete modes of heat transfer, as is done by Benoit and Mostaghimi (1985), the NRCC model employs a coarse finite-differencing scheme using soil layers of variable depths.
Like both classes of models, the NRCC model assumes one-dimensional heat flow. This is shown schematically in Fig. 1. In this figure, depths (m) below or above (in the case of snow and/or air) the surface are indicated by Z, and temperatures (°C) are indicated by T. Subscripts indicate snow (s), frozen soil (f), the soil surface (O), and the lower boundary (D). The subscript “y” refers to the value observed or estimated for the previous day. The model assumes that the flux of heat through the lower boundary is negligible. At the lower boundary, ZD, which is set at a depth of 2 m, a daily “deep” temperature TD is specified as a function of the average air temperature over a period from the previous April through the March following the winter in question, the 25th percentile January through March snow depth for the current winter, and the combined thermal diffusivity of the snow and soil. Specifying TD over the period from April through March assured proper initialization of this boundary temperature prior to the start of the freezing season while also accounting for the effects of the previous winter on the rate of summer warming. Because weather data for the entire season are used to specify the lower boundary conditions, the model is diagnostic rather than prognostic.
The upper boundary condition is given by the observed average daily air temperature. Here, the assumption is made that the average daily air temperature is representative of the temperature of the snow surface. The snow depth (Zs) gives the thickness of the first layer in the snow/soil system (Fig. 1). In the absence of snow cover, the air temperature is assumed to equal the temperature at the upper surface of a 1.0 × 10−3-m laminar layer, the thermal properties of which are characteristic of still air. To avoid the assumption of equality between the air and soil surface temperature, the model treats this laminar layer as part of the air–soil system and computes a soil surface temperature to achieve balance between the relevant heat fluxes. The assumption of isothermal conditions in the air layer between the shelter and the top of the laminar layer likely introduces some small error in the daily frost penetration depths. Nonetheless, it avoids the use of an empirical n-factor as in Eq. (1). Progressing downward, soil layers of variable depth are defined by frozen and unfrozen zones, the boundaries of which are at 0°C. A maximum of three soil layers (one frozen and two unfrozen) is allowed by the model.
The variables used in Eqs. (2)–(5) have been defined previously, with the exception of the latent heat of freezing Lf (J m−3); soil porosity (ϕ); thermal conductivities (W m−1 °C−1) of snow Ksnow, frozen soil Kfroz, and unfrozen soil Kdeep; and the change in heat storage terms ΔQ (W m−2). Equations (2)–(5) are solved numerically for the prognostic variables T0 and Zf. Nearly saturated soil moisture conditions are assumed at all times. This assumption is quite reasonable during the soil freezing season in the northeastern United States. In Fig. 1, ΔQU is represented by the hatched and cross-hatched areas between the two consecutive daily average temperature profiles. Similarly, ΔQL is shown by the speckled and dotted regions.
Only one of three possible soil freezing states is illustrated in Fig. 1. In this state, a layer of frozen soil extends from the surface to some depth Zf. The other possible states are that the soil may remain unfrozen from the surface to the lower boundary ZD, or a layer of frozen soil may exist between two layers of unfrozen soil. In addition, five transition modes are possible, corresponding to the transitions between the three basic states, with the exception of the transition from an unfrozen to a buried frozen layer, which is not physically realizable.
The model is initiated in the unfrozen state and continues in this manner until T0 falls below 0°C. At this point, the transition to frozen soil mode is activated. Provided the temperature remains below 0°C on subsequent days, the model operates in the frozen soil mode. In this state, both soil freezing and thawing occur at the bottom of the frozen layer. When T0 exceeds 0°C, the model makes a transition to either the unfrozen or surface thaw state. In the surface thaw state, the layer of frozen soil is allowed to thaw both from its top and bottom. The temperature throughout the buried frozen layer that results is assumed to be constant at 0°C. For subsequent occurrences of T0 < 0°C, freezing occurs at the both the top and bottom of the buried frozen layer.
Despite favorable correspondence between measured and observed frost depths in the Northeast (DeGaetano et al. 1996, 1997), the original NRCC model is not applicable across the United States. In particular, winter soil moisture conditions in the more arid northcentral and northwestern United States can be substantially drier than is assumed by the model. This influences the dynamics of soil freezing very substantially. For example, Fig. 2 shows maximum soil freezing depths for the winters of 1984/85 through 1996/97 at Ithaca, New York, as simulated by the SHAW model for different fixed moisture contents. The maximum depth of freezing in each year increases as the water content decreases from 15% (dry) to 5% (extremely dry). A general increase in maximum annual frost depth is noted as water content decreases from 35% (near saturation) to 15%.
NRCC model refinements
For the NRCC model to be applicable in drier climates, it is necessary both to specify the degree of dryness in a given winter, consistent with data limitations, and to capture the effects of lower soil water content on thermal conductivity and reduced latent heating.
Water budget
The methods of Palmer (1965) and Thornthwaite (1948) (also see Alley 1984) were used to compute volumetric soil water content on the spatial scale of climate divisions (Guttman and Quayle 1996). Each state is divided into divisions ranging in number from 1 to, at most, 10 climate divisions. Divisions generally represent drainage basins or crop-reporting districts. For each division, an average soil profile is defined by two layers. The top layer (SS) contains up to 2.5 cm of soil water, with the remaining soil water contained in the lower layer (SU). Water first enters the top layer, which must fill to capacity before any water infiltrates into SU. Water leaving the system evaporationally is withdrawn from the top layer first before being drawn up from below. Precipitation, evapotranspiration, and, to some extent, runoff dominate this simple water budget.
Monthly climate division precipitation totals are the source of water input to the budget. These totals represent the average precipitation received at all reporting sites within a division. Climate division precipitation totals are updated operationally on a monthly basis and have been archived from 1895 onward.
Monthly climate division evapotranspiration totals were obtained using the method of Thornthwaite (1948). This empirical procedure assumes a direct relationship between monthly average temperature
The value of ETP given by Thornthwaite’s method is based on 12 h of daylight and maximum soil moisture availability. Therefore corrections for day length and soil moisture deficit are applied to obtain the actual monthly evapotranspiration total (ET). The daylength correction is the average monthly hours of daylight divided by 12, and the soil moisture correction is the ratio of actual to saturated volumetric water content (Palmer and Havens 1958). Based on these values of ET and precipitation (P), changes in monthly soil water content are tracked using a bookkeeping approach. Soil moisture recharge occurs when P exceeds ET, with the new soil moisture expressed as the sum of the previous month’s moisture content and the difference (P − ET). When recharge exceeds the soil’s capacity for holding water, the excess water is lost as runoff.
Use of the Palmer soil water budget requires information concerning the previous month’s soil moisture storage. This is problematic when initializing the budget. To address this issue, an iterative scheme was developed to search through the P and ET data for the earliest month when P exceeded the sum of available water capacity and monthly ET. The budget could then be initialized at AWC during this month. When a single month meeting this criterion could not be identified, the search and initialization procedure was applied using 2- or 3-month P and ET totals. In all climate divisions, the water budgets were initialized prior to 1930.
Thermal conductivity adjustment
The E term is typically around 4.0, as validated through experimentation, and Δz is the depth of the soil layer (2.0 m) (Campbell 1985).
Because water content can vary through the winter, thermal conductivity is recalculated during each month. The median thermal conductivity (April–March) is used to calculate the deep temperature wave.
Refined model validation
Frost depth measurements at several central and western U.S. sites were available for comparison with the revised NRCC model simulations. Temperature and precipitation (both liquid and snow) data for these sites were available from collocated National Weather Service Cooperative Observer Network stations. Soil characteristics were determined using the U.S. Department of Agriculture soil survey data.
Reynolds Creek Watershed, Idaho
Extensive soil freezing studies conducted in the 1970s and 1980s at several observation sites in the Reynolds Creek Watershed of southwestern Idaho provided the most thorough set of available verification data. Data were used from three sites, Reynolds Creek, Reynolds Mountain, and Lower Sheep Creek. Because a cooperative station is collocated with the Reynolds Creek observation site, daily snowpack and snowfall data were available there. Daily snowpack and snowfall data were not available for Reynolds Mountain or Lower Sheep Creek but were estimated from the approximately biweekly snow observations reported by Hanson et al. (1988). At each site, soil characteristics were described by ϕ = 0.40, FC = 0.30, and φm = 0.2.
Simulations based on the Reynolds Creek weather data verified the revised NRCC model’s ability to model maximum frost depths, timing of maximum depths, and progression of frost depths through the course of winter. Representative trials for the 1977/78 and 1978/79 winter seasons are shown in Figs. 3 and 4. Minimal soil freezing was observed during the winter of 1977/78 (Fig. 3). During this winter, maximum modeled and observed soil freezing occurred in late November, with the observed maximum depth reaching 14 cm and the modeled depth at about 17 cm. Through most of December, both the model and observations indicate frost-free conditions. A short period of frozen soil is simulated by the model in early January (14-cm depth) and confirmed by the observations (9-cm depth). The week-long period of shallow (<10 cm) frozen soil given by the model in late December is not evident from the observations. During each of these three periods, the frost depths simulated by the original NRCC model are similar to the revised values, despite dry (10% water content in December) soil moisture conditions.
Figure 4 shows a winter experiencing prolonged and relatively deep soil freezing. During this winter, the simulation shows remarkable correspondence to the observed frost depths. In both cases, soil freezing was initiated in early November, with frozen conditions remaining almost uninterrupted through late February. With regard to the maximum frost depth, the model-derived depth of 74 cm compares favorably with the 80-cm depth observation. Both the observed and modeled value occur on February 4. The original NRCC model gives a much deeper (104 cm) frost depth during this dry (θ = 0.10) winter.
The sensitivity of several of the parameterizations incorporated into the revised model was evaluated using data from the two seasons presented in Figs. 3 and 4. Table 2 shows the results of this analysis for the five empirical coefficients used in the Campbell (1985) conductivity equation as well as θ, φm, and ϕ. In each trial, these eight parameters were increased (or decreased) from their original value by 20% and 50%. Overall, the model is most sensitive to ϕ, because a 20% change in this value results in as much as an 18% difference in maximum frost depth. Although a 20% change in the coefficient C produces an 11% change in maximum frost depth during 1977/78 (Fig. 3), changing this parameter had little effect on the model output during 1978/79, when the freezing was extensive. A similar disparity was found for φm. This is expected given the reliance of C on clay content. A 20% modification of the A coefficient or θ resulted in a change of maximum soil freezing depth of about 5%, while a larger 50% perturbation of these parameters was associated with frost depth differences in the range of 6%–10%.
Collectively, it appears that the model is most sensitive to the proportion of soil volume composed of air and water. Incorrect specification of φm and Campbell’s A coefficient accounts for relatively small (about 10%) differences in the estimated frost depth. This is in agreement with a more extensive analysis of the sensitivity of the original NRCC model to differences in φm and ϕ given by DeGaetano et al. (1997), which found at most a 5% change in the maximum depth of soil freezing for clay contents in range of 2%–50%. Given these results and the lack of site-specific soil information at most weather stations, it would be prudent to compare frost depths based on a range of porosities to characterize the uncertainty due to differences in the assumed ϕ value.
Trials conducted at Reynolds Mountain and Lower Sheep Creek, Idaho also matched the observations relatively well. Unfortunately, the lack of daily snow data required that snow input be extrapolated, which clearly introduces errors into the comparisons. Only maximum frost depths were analyzed for this reason. Nonetheless, modeled maximum frost depths consistently occurred within three days of the observed maximum frost depths for the three freezing seasons simulated. During these seasons, which correspond to the data given by Hanson et al. (1988), the differences between observed and modeled maximum frost depth averaged 5.0 and 6.7 cm at Lower Sheep Creek and Reynolds Mountain, respectively (Fig. 5). The consistent underestimation of maximum frost depth at Reynolds Mountain is likely due to spatial variations in snow depth between the frost depth and precipitation observation sites. Clearly, the ability to interpolate soil freezing depth estimates spatially in mountainous regions is complicated by these microclimatic differences.
SCAN sites
Soil Climate Analysis Network (SCAN) sites at or near cooperative weather stations provided a source of verification data based on observed soil temperatures at depths of 5, 10, 20, 50, and 100 cm. From these hourly or (6-h) data, the depths of the 1°, 0°, and −1°C isotherms were linearly interpolated. This ±1°C isotherm band about the freezing point delineated reasonable bounds on the position of the 0°C isotherm.
Lind, Washington
Modeled and soil temperature–inferred frost depths at Lind, Washington, show close agreement throughout the 1994/95 winter (Fig. 6). This winter was characterized by several distinct penetrations of the 0°C isotherm, which were captured by the revised NRCC model. Based on the model, the maximum frost depth of 25.4 cm occurred in early January. Despite capturing the timing of maximum soil freezing reasonably well, this is somewhat shallower than the 35-cm maximum depth of the interpolated 0°C isotherm. Nonetheless, the modeled maximum frost depth remains within the ±1°C envelope about the freezing isotherm.
Mandan, North Dakota
Verification results using Mandan, North Dakota, data for 1996/97 provided another example of the revised NRCC model’s ability to simulate frost penetration and maximum soil freezing accurately in semiarid climates (Fig. 7). During this snowy winter, the modeled soil freezing level remains relatively constant, at about 25 cm, from late November through the end of March. The interpolated 0°C isotherm shows a similar pattern, particularly during the latter three months. Although similar in magnitude, the modeled maximum frost depth level of 32.2 cm occurs much earlier in the season than the 37-cm maximum depth of the inferred 0°C isotherm.
Midwest frost gauge observations
Frost gauge data (Ricard et al. 1976) collected at DeKalb, Illinois, during the 1998/99 winter provide a final verification of the revised NRCC model. During this winter, with nearly saturated soil moisture conditions, the modeled frost depths once again track the observations quite closely (Fig. 8). The maximum simulated frost depth of 30.4 cm agrees with the measured 23-cm maximum but occurs about 10 days earlier. However, the timings of the initiation of soil freezing in December and the late January thaw are both captured by the model. The consistent overestimation of soil freezing through the season relates to the rapid onset and penetration of soil freezing. In this case, it is likely that the temperature of the soil surface was initialized too cold in the model, producing deeper-than-observed soil freezing.
Summary
To be applicable on a national scale, the Northeast Regional Climate Center frost depth model was modified through the addition of a soil moisture budget and refinement of the model’s thermal conductivity equation. Based on model performance in near-saturated conditions and SHAW model simulations for dry soil, freeze-induced migration of water to the frozen–unfrozen boundary continued to be ignored in the revised model. This allowed the meteorological data requirements to be limited to daily temperature, precipitation (both snow and liquid equivalent), and snow cover. These data are available nationally at a relatively dense spatial resolution.
The incorporation of a water budget into the model was also predicated by these data limitations. The methods of Palmer (1965) and Thornthwaite (1948) were used to estimate soil moisture content at monthly temporal resolution. These methods use a bookkeeping approach to account for monthly variations in soil water content. Soil recharge is limited to observed precipitation, and evaporation is the sole source of water loss. Monthly evapotranspiration can be inferred using only temperature and station latitude based on the Thornthwaite method.
The ability of the revised NRCC model to simulate seasonal maximum frost depth is exceptional, based on comparisons of modeled and observed soil freezing levels at various western and central U.S. sites. The verification results given in Fig. 9 clearly support this assertion. Here, the results of 32 verification trials at western and central U.S. stations are summarized. The values at all sites fall along the 1:1 line, with mean difference (i.e., bias) of only 1.4 cm and a mean absolute difference of 5.4 cm, based on the observed frost depth sites. Both relatively shallow and deeper frost depths are estimated with similar absolute accuracy, yielding an average percent difference of 11% but better relative performance for the more significant events. These results indicate that the revised NRCC model can be used to develop a national soil freezing climatology.
Acknowledgments
This work was sponsored by the National Association of Homebuilders (NAHB) and the U.S. Department of Housing and Urban Development through a cooperative agreement with the NAHB Research Center, Inc. Partial support was also provided through NOAA Cooperative Agreement NA67RJ-0146. We are indebted to Gerald Flerchinger for providing us with the SHAW model and with extensive help in parameterizing the model for compatibility with the NRCC model runs. Thanks also go to David Changnon and Nolan Doesken for diligently installing frost tubes and collecting soil freezing data at their weather stations.
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Schematic diagram showing the NRCC model’s frozen soil state. Heat fluxes through the centers of each layer are indicated by the bold arrows. The stippled area represents the change in energy storage ΔQL, and the hatched areas represent ΔQU
Citation: Journal of Applied Meteorology 40, 3; 10.1175/1520-0450(2001)040<0546:PSOMSS>2.0.CO;2
Maximum soil freezing depths at Ithaca, New York, based on SHAW model simulations with fixed 35% (black), 15% (shaded), and 5% (hatched) water contents
Citation: Journal of Applied Meteorology 40, 3; 10.1175/1520-0450(2001)040<0546:PSOMSS>2.0.CO;2
Observed and simulated soil freezing depths at Reynolds Creek, Idaho, during the winter of 1977/78. The solid black curve shows the modeled frost depth using the revised NRCC model. Surface thaw depths (i.e., the depth of unfrozen soil above the buried frozen layer) are indicated by the dotted black line. Frost depths obtained using the original NRCC model are shown by the dashed line. Observed frost depths are shown by the ×s
Citation: Journal of Applied Meteorology 40, 3; 10.1175/1520-0450(2001)040<0546:PSOMSS>2.0.CO;2
As in Fig. 3 but for the winter of 1978/79. Snow depth is given by the solid gray line
Citation: Journal of Applied Meteorology 40, 3; 10.1175/1520-0450(2001)040<0546:PSOMSS>2.0.CO;2
Observed (light) and modeled (dark) maximum seasonal frost depths at Reynolds Mountain and Lower Sheep Creek, Idaho. No soil freezing was observed or estimated during 1983–84 at Reynolds Mountain
Citation: Journal of Applied Meteorology 40, 3; 10.1175/1520-0450(2001)040<0546:PSOMSS>2.0.CO;2
Observed versus soil temperature–inferred soil freezing depths at Lind, Washington, during 1994/95. Model simulated soil freezing levels are indicated by the solid black line and surface thaw depths by the dashed line. The shaded area represents the ±1°C position of the 0°C isotherm (dotted line). Snow depth is given by the solid gray line
Citation: Journal of Applied Meteorology 40, 3; 10.1175/1520-0450(2001)040<0546:PSOMSS>2.0.CO;2
As in Fig. 6 but for Mandan, North Dakota, during 1996/97
Citation: Journal of Applied Meteorology 40, 3; 10.1175/1520-0450(2001)040<0546:PSOMSS>2.0.CO;2
Modeled versus measured soil freezing depths at DeKalb, Illinois, during the winter of 1998/99. Modeled frost depths are given by the solid black line, with surface thawing indicated by the dashed line. Soil freezing observations are indicated by the black dots, which represent the median of three frost tube observations. The thin line through each dot gives the range of observations. Snow depth is indicated by the gray line
Citation: Journal of Applied Meteorology 40, 3; 10.1175/1520-0450(2001)040<0546:PSOMSS>2.0.CO;2
Collective scatterplot of observed (solid symbols) and interpolated (open symbols) versus modeled maximum frost depths for 32 verification sites. Reynolds Creek watershed data are shown by circles; SCAN sites and frost tube observations are indicated by squares
Citation: Journal of Applied Meteorology 40, 3; 10.1175/1520-0450(2001)040<0546:PSOMSS>2.0.CO;2
Meteorological and site characteristic input required by the SHAW model
Change in maximum frost depth during two seasons associated with 20% and 50% changes (increases and decreases) in the empirical coefficients used by Campbell (1985) and θ, φm and ϕ, expressed as a percentage of the altered to unaltered frost depths. Unaltered frost depths of 17.7 and 73.9 cm were indicated during the 1977/78 and 1978/79 seasons, respectively
