Thawing and Freezing Indices in the Active Layer

Thawing and freezing indices in the active layer

D.W. Riseborough Geological Survey of Canada

ABSTRACT: Several schemes for relating climatic conditions to the ground thermal regime in permafrost use a transfer function (“n-factor”) relating the thawing and freezing indices (degree-day totals above and below 0°C, respectively) in the air to the values at the ground surface. However, temperatures are often measured a few centimeters in the ground, due to the problems inherent in capturing the true temperature at the interface. The effect of measuring below the ground surface is analysed here using the Stefan analytical model for thawing soil. The derived relationship between the subsurface thawing index and depth below the surface is shown to agree with some field data. An empirical numerical study shows that a similar relationship holds for the freezing index. Comparisons with field data show that this relation does not usually hold in field conditions, likely due to inho- mogeneous soil properties. Implications for the interpretation of near-surface temperature measurements are discussed.

1 INTRODUCTION n factors to determine the characteristic behaviour of different land-covers (Klene et al. 2001, Jorgenson Ground surface temperature is a key parameter in the and Kreig 1988, Taylor 1995, 2000). N factors relate relationship between climate and the thermal regime seasonal air temperature to ground surface tempera- of the active layer and permafrost (Lachenbruch et al. tures (Lunardini 1978). For example, the surface 1988). Unfortunately, ground surface temperature is thawing n factor is defined: difficult to measure due to the radiative and convective energy flows at the surface and difficulty in maintain- ITS nTS (2) ing a sensor in position. Because of these difficulties, ITA nominal surface temperature measurements are often taken a few centimeters below the surface. As auto- where nTS surface thawing n-factor; and ITA air matic data acquisition systems have become more thawing index. affordable and compact, efforts to measure such near N factor values will be influenced by burial depth surface ground temperatures are increasing (Klene and by soil thermal properties: As the database of et al. 2001, Smith et al. 2001, Hoelzle 1999, Taylor ground temperature data grows, the effect of these 1995, 2000). influences on n factor values needs to be evaluated. In Many simple models of climate – permafrost inter- this paper, the temperature index concept is extended action divide the year into freezing and thawing sea- from the surface into the ground, to determine the fac- sons, using the freezing index (IF) and thawing index tors that influence the relationship of near-surface (IT) to summarize the annual regime. A seasonal index indices to surface indices. The analysis may also aid is defined as the seasonally integrated temperature, other interpretations of field data. which is approximately equal to the sum of daily Assuming that the dominant process in the active mean temperatures for the duration of the season. For layer is phase change associated with freezing and example, the thawing index (following Klene et al. thawing, the Stefan model is used to develop a solu- 2001) is: tion for thawing index versus depth. After evaluation of the analytical model using numerical simulation, us us Ϸ the results are compared with field data. ITTdtTTS∫ S F∑ S (1) 0 0 where TF temperature of the freezing point (0°C); 2 MODELLING TEMPERATURE INDICES TS surface temperature °C, S duration of the IN THE GROUND thawing season, days; and TS daily mean surface temperature. An equivalent equation applies to the One of the simplest models of ground freezing freezing index. and thawing is the Stefan model (Romanovsky and An important application of temperature index data Osterkamp 1997). This provides a simple equation on near surface temperatures is to compile values for that ignores sensible heat, assuming all heat flow is

953 used to supply heat to the freezing or thawing front. ITS can therefore be estimated by linear extrapola-

The Stefan equation applied to the accumulated thaw- tion of ITD if IT is known at two depths, or if IT for ing degree-day total (the thawing index IT) is: one depth is known together with the active layer depth (where ITD 0). Substituting equation (3) for X in equation (10), the index I at depth D below the 2kITTS TD X (3) surface can be calculated from L 2 IIDLkTD() TS/(2 T ) (11) where X thaw depth; kT unfrozen thermal con- ductivity of the soil; L soil volumetric latent heat of The slope of the straight line between I (at the fusion; and I surface thawing index (at depth 0). TS TS surface) and I 0 at the base of the active layer is: The thawing index at a depth D below the surface T but within the active layer (i.e. D X), must satisfy I I 0 equation for thaw below that depth, so that TD TS Z 2kI 0 TTS (12) 2kI L XD TTD (4) L Lk/(2 T ) where D Depth below the surface (within the active layer); and ITD Thawing index at depth D. 3 NUMERICAL MODELLING From equation (3):

2 Simulations were performed using a numerical model XL (TONE, developed by L. Goodrich) to evaluate the ITS (5) 2kT validity of equations (9) to (12). The annual thermal regime was simulated for several points along a climatic From equation (4): gradient. Two different soil conditions were examined, with thermal properties calculated using Johansen’s 2 equations (Farouki 1980): ()XDL ITD (6) 2kT • Case 1: A uniform coarse-grained material with 49% 1 1 volumetric water content (kF 2.886 Wm K ; 1 1 In analogy to the surface n factor, a subsurface n kT 1.488 Wm K ) factor nTD can be defined as: • Case 2: As above, overlain with a 0.2 m thick organic layer. The organic layer comprised a 0.14 m thick layer 1 1 I saturated with 89% water (kF 1.803 Wm K ; TD nTD (7) 1 1 I kT 0.535 Wm K ), overlain by a 0.06 m thick TS layer that was assumed to dry during summer and 1 1 saturate during winter (kF 1.803 Wm K ; kT From equations (5) and (6): 0.094 Wm 1 K 1). To approximate the behaviour of this material in the model, the thermal conduc-  ()XDL 2    tivity and volumetric heat capacity of the unfrozen  2k    2 T D organic layer was assumed to be that for organic nTD 1  (8)  XL2   X  soil with 30% water by volume, while kF and L were   assumed to be for saturated soil (89% water).  2kT  Case 1 allows a straightforward comparison of simulation results with equations (11) and (12), while case Note that nTD 1 when D 0 and nTD 0 when D X. From equation (8): 2 demonstrates the effect of layers with different thermal properties. For the simulation runs: D nTD 1 (9) • Air temperature was a sine wave with a period of X 365 days and an amplitude of 22°C. Mean annual temperatures were varied from 8°C to 12°C, From equations (7) and (8): at 2°C intervals, generating 11 cases along each climatic gradient.  D  2 • The thawing n factor nT was assumed 1.0, with ITD1  I TS (10) X the freezing n factor nF 0.5.

954 • Simulation time steps were 40 minutes; element size was 0.02 m to 1 m depth, increasing to 0.6 m at 16 m (the base of the grid). The model was run until the temperature profile differed between annual cycles by less than 0.001°C, ensuring that the initial temperature condition did not influence results.

3.1 Case 1: uniform soil

For brevity, results are presented for thawing cases only; equivalent results were obtained for freezing simulations, as discussed below. Simulated depth profiles of ITD are shown in Figure 1. For the permafrost cases, profiles are nearly linear, with slight departures from linear close to the base of the active layer (where all indices approach 0). These departures may be due to the effect of accumulating numerical errors, and the effect of departures from the Sefan Figure 1.ITD for thermal regimes simulated in case 1. Each model due to upward freezing at the end of the thaw line represents result for a given mean annual temperature. season. For the non-permafrost cases, the profiles of IT are far from linear, and do not approach 0 at the base of the active layer. The assumption of a linear temperature gradient is also reasonable during the thaw period in non-permafrost: The profile of ITD is not linear in the non-permafrost cases because the profile includes thawing degree-days that are not involved in phase change. The mean annual temperature at the base of the seasonal freezing layer is equal to TTOP. In non permafrost:

II T TX FX andI 0 TOP P FX (13) ∴ ITPTX TOP

Figure 2.ITP versus depth for non-permafrost where I thawing index at the base of the seasonal TD TOP TX regimes from case 1. freezing layer; IFX freezing index at the base of the seasonal freezing layer; and P the annual period (365 days). IT/365, since in non-permafrost IF 0 at the base of the active layer and). Figure 2 shows that the profile of ITD can be transformed into an approximately linear function of ITPTD TOP . Similar arguments hold for the mirrored conditions: in the non-permafrost case, a simple relationship holds forI FD while in permafrost ITPFD TOP yields a straight-line relationship with depth within the active layer. From equation (12), the profiles in Figures 1 and 2 should be parallel, with a slope of 24.13 K0.5 day0.5 m1. Figure 3 shows the relationship between the slope of the profile for each simulated regime (determined by least squares regression) and TTOP. Results for IFD are shown on the same figure (for which equation (12) gives a slope of 17.33 K0.5 day0.5 m1). Lines representing the profile slopes from equation (12) are shown for comparison. For both IFD and ITD, Figure 3. Slopes of ͙I for permafrost regimes of figure 1 the discrepancy between the numerical results and and non-permafrost regimes shown in figure 2.

955 Figure 4. Square root of IT versus depth for thermal regimes simulated in case 2 (layered soil). Figure 5. Square root of nTD versus relative depth (D/X within the active layer for 11 regimes simulated in case 1. equation (12) is smallest near TTOP 0, where the slope is underestimated by approximately 5%, Despite these limitations, case 2 shows that this trans- increasing to 20–25% in warmer or colder soils. formation is useful in the interpretation of field data, The discrepancy is due to the neglect of sensible heat since soil profiles typically involve changes in bulk- in the Stefan model (Romanovsky and Osterkamp density and water content. 1997). At TTOP 0°C, the neglected sensible heat is predominantly that within the seasonal freezing (active) layer; above or below 0°C, additional heat is neglected 4 COMPARISON WITH FIELD DATA as sensible heat is extracted from or supplied to the soil below. Ground temperature data for a number of sites were While I and ITP are linear with depth, TD FD TOP examined to evaluate whether profiles of I or I measurements below the surface cannot be extrapolated T F can be linearized using equation (10). This would then to the surface using equation (11), due to the limita- be a useful method to extrapolate the indices to the tions of the Stefan model. None the less, transforming surface. Data sources include the Norman Wells the results in Figures 1 and 2 from actual depth to rel- Pipeline thermal monitoring program (MacInnes et al. ative depth within the active layer (D/X), and from 1990) and the archives for the Circumpolar Active I to n , the similarity of results for all simula- TD TD Layer Monitoring program (CALM, currently at tions (shown in Figure 4) suggest that equations (9) http://www.geography.uc.edu/ϳkenhinke/CALM/). and (10) have utility. Data for Deadhorse (Figure 6) are for a site that remains saturated throughout the year (Romanovsky 3.2 Case 2: layered soil and Osterkamp, 1997). Data for Norman Wells Pipeline site KP182 (Figure 7) are for a forest-fire

Figure 5 shows depth profiles of ITD for the thermal affected site with ice rich silty clay till. At this site, regimes simulated using the conditions defined for active layer is thinning following vegetation recovery: case 2. From the surface downward, the slope of the it was assumed that TTOP Ϸ 0°C. Data for the Parsons profile declines in each layer, since kT changes more Lake site (Figure 8) are for a site with earth hum- quickly than L with increasing water content. As with mocks over well-drained morainal loam. case 1, the profiles for the permafrost cases are approx- Data manipulation for ITPFD TOP was not straight- imately linear through each layer, although slopes differ forward, since the sensor with the minimum freezing slightly. The neglect of sensible heat in equation (11) index was often within the active layer (due to inter- precludes its use in resolving the profile through indi- annual variability or upwards freezing at the base of vidual layers. Further, calculation of the ratio of slopes the active layer): in these cases, (IF TTOPP) values in the different layers (dry/wet organic; wet organic/silt) were estimated using the minimum IF value. Years that showed that the ratio differs from that suggested by appeared to have corrupted data were not included in equation (12), and varies with the thermal regime. the analysis.

956 Figure 8. Profiles of ͙ITD from CALM data for Parsons Lake, Northwest Territories, Canada.

value for the surface sensor: This may represent a prob- lem with sensor exposure, or a sharp change in thermal properties. Profiles at KP182 (figure 7) include two features not found at the other sites: This site shows a lower gradient in IF near the surface than deeper in the profile (albeit based on data for a single sensor), while the profile of IT has a marked different slope in each year. Profiles of I for Parsons Lake, Northwest Territories Figure 6. Profiles of square root functions for IT (A) and T IF (B) from CALM data for Deadhorse Alaska. exhibit a significant change in slope with depth, with no linear sections, although the profiles are almost identical in the two years of the record. Profile slopes are similar year to year at individual sites (with the exception of IT at KP182), suggesting that the thermal properties and processes in the active layer are stable over time. Most profiles (except IT as Deadhorse) show an increase in slope near the base of the active layer likely as a result of increasing water content and soil bulk density with depth.

5 DISCUSSION AND CONCLUSION

Model results suggest that using near-surface measurements to estimate surface indices will underesti- mate their value by an amount that increases with the square of measurement depth as a proportion of active layer thickness. From equation (11), the effect will

be greater for soils with a larger ratio of Lk/2 T . Estimates using Johansen’s equations (Farouki 1980)

Figure 7. Profiles of IT and IF functions for Norman Wells for kT suggest that Lk/2 T varies by a factor of about Pipeline monitoring site KP 182, within an area affected by 2 for a given mineral soil, while its value is an order of forest fire. magnitude higher for organic soils. In a typical thawing mineral soil with IST of 1000 degree days, ITD would At Deadhorse (figure 6), profiles include a surface be reduced by 20 to 30 degree days at 2 cm depth, or temperature measurement (0.0 m depth). With the 100 to 155 degree days at 10 cm depth. In organic soil exception of IFD for 1990, the profiles for the buried however, ITD would be reduced by 70 to 90 degree days sensors are nearly linear, but do not extrapolate to the at 2 cm depth, or by 300 to 400 degree days at 10 cm

957 depth. This is a significant fraction of the seasonal REFERENCES surface index, with an effect that will be relatively greater as the surface index declines. Unfortunately, Hinkel, K.M., Paetzold, F., Nelson, F.E. and Bockheim, J.G. installation within 1 or 2 cm of the surface in low- 2001. Patterns of soil temperature and moisture in the density organic surface materials would likely expose active layer and upper permafrost at Barrow, Alaska: sensors to radiation. 1993–1999 Global and Planetary Change 29: 293–309. Hoelzle, M., Wegmann, M., and Krummenacher, B. 1999. Comparisons demonstrate that the model does not Miniature temperature dataloggers for mapping and translate perfectly to field data. These limitations are monitoring of permafrost in high mountain areas: first likely due primarily to the heterogeneity of thermal experience from the Swiss Alps. Permafrost and properties in the ground. Almost all cases examined Periglacial Processes 10(2): 113–134. show profiles that exhibit a curvature indicating that Farouki, 0. 1980. Thermal Properties of Soils. CRREL Mono- L/k increases with depth, as would usually happen graph 81-1. U.S. Army Cold Regions Research and Engi- with increasing water content and bulk density. Thermal neering Laboratories, Hanover, New Hampshire. 135 p. properties are unlikely to be uniform with depth, and Jorgenson, M.T. and Kreig, R.A. 1988. A model for in many situations will vary through with time, due mapping permafrost distribution based on landscape to moisture content variation (Hinkel et al. 2001) component maps and climatic variables. Fifth Inter- national Conference on Permafrost, Trondheim, Norway: and the effect of unfrozen water (Romanovsky and 176–182. Osterkamp 2000). The effect on index profiles should Klene A.E., Nelson, F.E., Shiklomanov, N.I., and Hinkel, be much more pronounced in the transition between K.M. 2001. 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Thawing of Even with these limitations, transforming the pro- the active layer on the coastal plain of the Alaskan file of IT and IF may produce useful diagnostic infor- Arctic. Permafrost and Periglacial Processes 8: 1–22. mation about the processes occurring within the active Romanovsky, V.E. and Osterkamp, T. E. 2000. Effects of layer. Changes in slope with depth, or from year to unfrozen water on heat and mass transport processes year will likely correlate with changes in thermal prop- in the active layer and permafrost. Permafrost and erties. Further, translating nominal surface indices Periglacial Processes 11 (3): 219–239. measured at different depths (or in different substrates) Smith, S.L, Burgess, M.M., and Nixon, F.M. 2001. to a common depth using equation (11) may improve Response of active-layer and permafrost temperatures correlations between n factor values obtained in other- to warming during 1998 in the Mackenzie Delta, wise similar environments where burial depth or sub- Northwest Territories and at Canadian Forces Station Alert and Baker Lake, Nunavut. Geological Survey of strate properties differ. Canada, Current Research , 2001-E5: 20 p. Taylor, A.E., 1995. Field measurements of n-factors for natural forest areas, Mackenzie Valley, Northwest Territories, ACKNOWLEGMENTS Geological Survey of Canada Current Research. Geological Survey of Canada, 1995-B: 89–98. Taylor, A.E., 2000. Relationship of ground temperatures to I am grateful to Margo Burgess, Charles Tarnocai, and air temperatures in forests. The physical environment Tom Osterkamp for their permission to use the field of the Mackenzie Valley, Northwest Territories: a base data presented in this paper, to the anonymous review- line for the assessment of environmental change; ers for their helpful suggestions, and to Mike Smith LD Dyke and CR Brooks eds. Geological Survey of and Sharon Smith for pre-submission reviews. Canada Bulletin 547: 111–117.

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