Systematics in the size structure of prairie pothole lakes through drought and deluge



[1] This study examines the response of a complex lake-wetland system to variations in climate. The focus is on the lakes and wetlands of the Prairie Coteau, which is part of the larger Prairie Pothole Region of the Central Plains of North America. Information on lake size was enumerated from satellite images and aerial photos and yielded power law relationships for different hydrological conditions. Of particular interest is a recent drought and deluge sequence, 1988–1992 and 1993–1998. Results showed that lake sizes followed well-defined power laws that changed intra-annually and interannually as a function of climate. The power laws for spring seasons in 1987, 1990, 1992, 1997, and 2002 have a relatively constant slope. However, slopes changed with time within each year. These lines produced from Landsat images and aerial photos describe a systematic variation in sizes for lakes ranging in area from 100 m2 to more than 30,000 m2. This tendency for lake size to follow a power law coupled with area measurement from aerial photos taken on 29 July 1939 provides a basis for reconstructing the distribution of pothole lakes in summer 1939, near the end of the Dust Bowl drought. The study shows that the areas of smaller lakes are profoundly affected seasonally by the spring snowmelt and evaporation. The areas of larger lakes are influenced more slowly by longer- term periods of drought and deluge.

1. Introduction

[2] Studies have shown that the water areas of lakes and wetlands follow a power law [Rapley et al., 1987; Wetzel, 1990; Birkett and Mason, 1995; Meybeck, 1995; Lehner and Doll, 2004]. This systematic size structure was first pointed out by Korcak [1940]. Korcak's power law showed that areas of geographical objects follow a hyperbolic distribution with a power law function. Plotting the numbers of lakes of different sizes versus lake area classes in logarithmic coordinates produces a straight line. This idea was tested by Kent and Wong [1982] in 1982 for about 2500 lakes in Canada. Now, power law relationships between the frequency of lakes of different sizes and water areas have been elucidated and have been applied in global and regional assessments of surface water resources [Lehner and Doll, 2004; Downing et al., 2006; Sagar, 2007]. There are obvious systematics in the change in the frequency of lakes of different sizes, and remarkable coincidences in the slopes of the regression lines for different areas [Downing et al., 2006].

[3] With power law relationships, complex heterogeneity among lakes can be rationalized by just a few parameters. For example, Downing et al. [2006] show how such relationships provide a simple way of comparing differences in lake numbers among regions. Our study focuses in particular on lake and wetland systems in the Prairie Pothole Region (PPR) of the Central Plains of the United States and Canada. The PPR contains literally millions of lakes and wetlands, which are sensitive to climate variability and likely to be impacted by global climate change [Larson, 1995; Sorenson et al., 1998]. While much is known about the fundamental hydrology of lakes and wetlands of the PPR [e.g., Winter and Rosenberry, 1998; Euliss and Mushet, 1996; Johnson et al., 2004], the size and complexity of the PPR, coupled with the large numbers of lakes make broad regional/temporal assessments difficult. If consistent multitemporal power law relationships can be shown to hold, it should be possible to use historical data (e.g., climate data and aerial photography) to understand the regional response of lakes and wetlands to important periods of drought and deluge in the past.

[4] Our study examines pothole lakes and wetlands located along the northern tip of the Prairie Coteau in South Dakota, where pothole lakes and wetlands are numerous. Of particular interest is the behavior of this lake-wetland complex through an interesting transition from a period of drought (1988–1992) to one of deluge (1993–1998), and at the end of the Dust Bowl drought of the 1930s. One challenge with understanding hydrologic conditions during Dust Bowl times is the lack of data on small lakes and wetlands.

2. Methodology

2.1. Description of the Study Area

[5] The PPR of central North America is unique because of the extremely large numbers of small lakes, ponds, and wetlands and their ecological significance. This approximately 750,000 km2 region extends from southern Canada (Alberta, Saskatchewan, and Manitoba) to the northern United States (Montana, North Dakota, South Dakota, Nebraska, Minnesota, and Iowa). The exact numbers of ponds, lakes, and wetlands are unknown but could easily be several million [Larson, 1995]. Although the PPR represents about 10% of the total breeding and nesting area for waterfowl in North America [Guntenspergen et al., 2002], it produces more than 50% of all breeding ducks and more than 60% of continental mallards [Batt et al., 1989].

[6] The pothole lakes, ponds and wetlands are mostly isolated from stream networks. Major inflows of water come from snowmelt and summer precipitation, whereas the greatest loss of water is due to evaporation [Winter et al., 2001]. With the high rates of evaporation in summer and variable precipitation from year to year, the lakes and wetlands exhibit significant variability in size.

[7] During the last century, parts of the PPR experienced two extreme droughts, including the Dust Bowl drought of the 1930s, and a shorter and more recent one from 1988 to 1992. This latter drought was followed by the most significant wet period of the century beginning in 1993 [Winter and Rosenberry, 1998]. Currently, the region is arid again and people are concerned with the possible reoccurrence of Dust Bowl–type conditions [Andreadis et al., 2005; Fye et al., 2003; Stahle et al., 2007].

[8] Our 5750 km2 study area is located in the PPR at the northern tip of the Prairie Coteau of northeastern South Dakota (Figure 1). During the Wisconsin deglaciation, stagnation of ice on the Prairie Coteau upland produced hummocky topography and ultimately thousands of closed-basin pothole lakes and wetlands. Bedrock in this area is Cretaceous shale, which is mantled by more than 120 m of glacial till. The region is important for agriculture. Flat areas are typically used for dry-land farming, while more hummocky land is used as rangeland. In recent years, there have been aggressive federal programs to remove marginal land from agricultural uses to enhance waterfowl production.

Figure 1.

Prairie Coteau study area in South Dakota.

[9] Figure 1 also shows several large recreational lakes. The most important are lakes of the Waubay Lakes chain, located in the center of the triangular study area. Available stage hydrographs for Waubay Lake and other nearby lakes (e.g., Enemy Swim Lake, Bitter Lake, Spring Lake) show the dramatic change in water levels, as the drought ending in 1992 gave way to wetter conditions over the next 6 years. For example, the stage of Waubay Lake increased 1.1 m from May 1991 to September 1993, and an overall 5.4 m from May 1991 to May 1998. No similar stage data exist for the thousands of smaller pothole lakes in this region. With the exception of interlake transfers, the Waubay Lakes system is closed hydrologically.

[10] This area of South Dakota has a subhumid to subarid continental climate with short hot summers, long cold winters, low precipitation, and high evaporation. Average annual precipitation ranges from 51.8 cm (water years 1961–1990) to 59.7 cm (water years 1991–1998). Average temperatures range from −17.2°C in January to 29.4°C in July. Figure 2 shows precipitation data with rainfall and snow separated from 1986 to 2002 at the Waubay National Wildlife station, Day County, South Dakota. As is evident, a relatively small proportion of the annual precipitation (approximately 14%) occurs as snow (Figure 2). However, snowfall amounts can be quite variable. For example, snowfall amounts in 1993–94 and 1996–1997 were significantly greater than those in 1989–1090 and 1991–1992. Summer rainfall is also variable from month to month in different years.

Figure 2.

Monthly precipitation data from the Waubay National Wildlife meteorological station in Day County, South Dakota, from 1986 to 2002.

2.2. GIS and Remote Sensing Approaches to Evaluate Pothole Lakes

[11] Most previous studies of lakes and wetlands in the PPR commonly have relied on careful observations of a few clusters of pothole lakes and wetlands. This study examines the systematics of thousands of pothole lakes within the study area. For example, during droughts there are about 2000 lakes greater than 0.5 ha. During periods of deluge, this number can increase to about 10,000. Given the large number of lakes involved, we relied on remote sensing techniques for lake size estimates, which are compatible with the scope of the observational problem.

[12] Since the 1970s, aerial photos and satellite images have been used for mapping lake areas [Lunetta and Balogh, 1999; Ozesmi and Bauer, 2002; Pietroniro and Prowse, 2002; Sawaya et al., 2003; Schmugge et al., 2002; Stewart et al., 1980]. Such data archives, when coupled with the processing capabilities of a geographical information system (GIS), provide a powerful approach for examining how lake surface areas change with time.

[13] Landsat imagery was used to provide historical data on the occurrence of pothole lakes for various times from 1987 to 2002. Using ArcGIS, lake areas were determined automatically by the advanced image classification and image processing methods described in detail by B. Zhang (A CART based sub-pixel method to map spatial and temporal patterns of lakes and wetlands within the Prairie Pothole Region, paper presented at University Consortium for Geographic Information Science Summer Assembly, 2007, available at Because of the moderate spatial resolution of about 30 m, Landsat images were most useful in studying pothole lakes larger than 0.7 ha.

[14] Digital orthophoto quarter quadrangles (DOQQ) from 2003 with about 1 m resolution were available for parts of the study area. We used this higher-resolution imagery to check the lake area estimates from Landsat, and to examine the validity of one power law derived from a combination of the Landsat and DOQQ imagery. For this analysis, we selected a small subarea (sampling area 2003, Figure 1) having a density of pothole lakes comparable to the larger study area. All of the pothole lakes in the sampling area were delineated manually on the corresponding DOQQ with water areas determined using GIS tools.

[15] Figure 3 provides examples of the two different types of imagery. Figure 3a is the classified water area image from a Landsat scene. The area covered by water is estimated by summing all water pixels for each lake. Figure 3b shows examples of easily identifiable pothole lakes on the DOQQ for sampling area 2003. Imagery of different resolution, thus, provides water areas for pothole lakes of differing sizes. Estimates from Landsat images encompass lakes ranging in size from Waubay Lake (the largest) to lakes as small as several Landsat pixels. Estimates from DOQQ provide area measurements for lakes smaller than one Landsat pixel (about 900 m2) but larger than 100 m2. Generally, estimates of lake area from the Landsat imagery were shown to be quite comparable to the estimates from DOQQ. With lakes of just a few Landsat pixels (i.e., 0.7 ha or less), areas could be underestimated (Zhang, presented paper, 2007). This problem did not affect the development of power law lines because only a few size categories were biased.

Figure 3.

Pothole lakes (a) on 29 July 2002, Landsat image, and (b) on 4 July 2003, DOQQ image. Sampling area 2003 is outlined in red.

[16] The large database of lake areas developed from the Landsat and DOQQ imagery provided the basis for our assessments. In general, one power law line was produced from one image. Because DOQQ imagery provided only one-time coverage in 2003, there is only one Landsat and one DOQQ image that can be used together to provide a composite power law.

2.3. Power Laws

[17] Once lake area information is extracted, a power law can be developed, relating the numbers of lakes of a given size, to size classes (bins) on log-log scales. Parameters of fitted regression lines (e.g., slopes and intercepts) provide a basis for comparison. The most common technique for fitting a line through points is linear regression, which provides least squares estimates of parameters. This statistical approach is simple to use under Gauss-Markov assumptions of normality with a zero mean and a constant variance for random errors. However, this assumption was not satisfied by the data in this study. An alternative Bayesian hierarchical linear regression model was used. Unlike least squares models, which produce best estimators for unknown fixed model parameters, Bayesian models generate posterior distribution samples of model parameters directly on the basis of known data. Thus, they produce a more intuitive interpretation of confidence intervals of the model parameters, which are of particular interest [Rachev et al., 2008].

3. Results

3.1. Power Laws for Lakes as a Function of Time

[18] A series of Landsat images is analyzed to provide power law lines at different times through the period of interest from 1990 to 2002. Four power laws are developed for Landsat images from 1990, 1992, 1997, and 2002 (Figure 4a). The linear trends in numbers of lakes as a function of the various size classes are remarkably consistent and validate the power law relationship. Figure 4a also provides visual evidence of the impact of the marked changes on the size structure of the pothole-lake system in shifting from drought to deluge. For instance, in the 0.9 ha bin (containing all potholes lake with areas larger than 0.9 ha but smaller or equal to 0.99 ha), the numbers of lakes increased from about 370 on 6 May 1992 to 780 on 4 May 1997.

Figure 4.

Power law lines of pothole lakes from 1990 to 2002: (a) interannual patterns and (b) intra-annual patterns. The bin size is 0.09 ha.

[19] The Landsat images used to produce the lines in Figure 4a were from late April to early May in the various years. The structure of the pothole-lake system is examined at about the same time each year to make results seasonally comparable. Prairie pothole lakes exhibit distinct variability in size within the same year (Zhang, presented paper, 2007). In Figure 4a, the lines for 1997 and 1992 are the upper and lower bounds, respectively, of these power law lines during the late spring. This range reflects the variability due to the drought of 1988–1992 and a deluge reaching a maximum in 1997.

[20] Another important feature of these lines is that all are nearly parallel. This result implies that the size structure of the pothole-lake system remained constant even though climatic conditions were significantly different. In other words, the numbers of lakes and wetlands of different sizes exhibited a consistent pattern of size abundances through the different years.

[21] Figure 4b displays power law relationships at different times of the years 1990, 2001, and 2002. Although a linear power law relationship still holds, the lines are not parallel at different times in the same year. Lines from summer or fall in a given year are lower and flatter compared to the corresponding spring season. This consistent pattern of change in the power laws through the year suggests that the areas of small wetlands and lakes are being impacted preferentially during hot summer months.

[22] In Figure 4, linear trends are only significant for lakes having areas less than 9 ha. It is likely that the variance increase for the larger lakes is due to the relatively smaller number of lakes in certain size classes as the lake sizes become larger. A regression line for the large lakes alone would be essentially flat. The departure of a few data points from the fitted lines for the very smallest lakes is caused by errors in area estimates due to the coarse resolution of Landsat images.

3.2. Parameters of the Power Law Lines

[23] The statistical fitting of lines provides a set of descriptive parameters for each of the power law lines. Figures 5 and 6 provide a statistical description of slopes and intercepts for different power law lines interannually and intra-annually by box plots of their posterior distributions. For each distribution, the box plot gives the upper and lower bounds of the 95% confidence interval shown as the top and bottom line segments. The rectangle inside this interval shows samples within 25% (lower bound of the rectangle) and 75% (upper bound of the rectangle) quintiles of the distribution, and the solid dark line segment represents the median of the distribution.

Figure 5.

Box plots of the distributions of slopes and intercepts of power law lines simulated by Bayesian linear models. These lines are approximately parallel.

Figure 6.

Box plots of the distributions of slopes and intercepts of seasonal power law lines simulated by Bayesian linear models. These lines are not parallel.

[24] Figure 5 shows how the intercepts and the slopes of power law lines in late spring changed under different climatological conditions for different years. The median values of the intercepts ranged from 3.32 to 3.65, and the median values of the slopes ranged from −1.59 to −1.80. The confidence intervals provide a measure of the significance of differences in these parameters. If any two 95% confidence intervals overlap each other, it means that these two parameters cannot be differentiated at the 5% significance level. With lines from the two extreme years, 6 May 1992 (drought), and 4 May 1997 (deluge), the intercepts of the power law lines were different at the 5% significance level but their slopes were almost identical. Essentially, these two lines are close to parallel with a difference of about 0.3. Thus, the number of lakes on 4 May 1997, within each bin was consistently about two (100.3) times the number observed on 6 May 1992. Examination of Figure 2 indicates that winter 1992 was an extremely poor snow year, while 1997 had large snow accumulations.

[25] In Figure 5, the power law line for 15 April 1990, has the largest negative slope as compared to the smallest on 18 May 2002. Only these two values are different from each other at the 5% significance level, because the 95% confidence intervals do not overlap. We think that the differences in these slopes are caused by slight differences in the time in spring when the images were acquired. Even though these Landsat images were selected to be reasonably synchronous among the various years, the small time gap from 15 April to 18 May for these two lines could be showing the influence of intra-annual variability rather than the intended depiction of interannual variability. Notwithstanding the slight differences in time, and the variability in climate through this period, these power law slopes for late spring are surprisingly similar with a slope value about −1.7. It is likely that the tendency for snowmelt to maximize the number and surface areas of smaller lakes during spring pushes the slopes toward the higher end of their natural range, providing relatively consistent values. The numbers of power lines developed are obviously too few to provide a more definitive explanation.

[26] The sensitivity of lake surface areas to variability in climate implied by Figure 5 is shown convincingly in Figure 6 by box plots showing the distributions of slopes and intercepts of the power law lines for different seasons. For each of the 3 years, the lines for spring had higher intercepts and higher slopes than lines developed at later times in each year. Furthermore for most early season distributions, these parameters are different at the 5% significance level when compared with later-season box plots shown in Figure 6. The highest intercept value of 3.57 and the highest slope value of −1.80 came from 15 April 1990, compared with the lowest intercept of 2.34 and the lowest slope value of −1.24 on 5 August 1990. As indicated by Figure 4b, the change in slope from April to August 1990 is due to preferential reductions in the areas of the smaller lakes during a period of drought. Model studies are presently underway to explain why the April to August change in these power line laws are so marked relative to other intra-annual changes.

[27] This pattern of variation shows a seasonal influence on power laws generated from spring through summer. The structure of the pothole lake system changes both in terms of the numbers of lakes of a given size and the size relationships within the entire family of lakes. Generally, with the temperature and evaporation rate rising dramatically from spring to summer, the area of smaller lakes declined to a much greater extent than larger lakes. These changes cause the power law line to move downward and become flatter. Given these seasonal effects, care should be exercised in using a single power law to estimate how smaller lakes are responding.

[28] In summary, the statistical analyses based on power law relationships show how the size structure of the pothole-lake complex varies on an intra-annual and interannual basis. Indications are that the relative abundance of the lakes was similar in springtime but changed through the year. Therefore, it is important to account for seasonal effects in developing power laws on lake size distributions.

3.3. Range of Validity of Power Laws

[29] Although the power law relationships developed from the Landsat images are suggestive of scaling behavior in terms of water area, the actual range in the size of lakes is not particularly large. The DOQQ imagery provides a basis for extending the power law relationship to include data for a number of smaller lakes, albeit within a much smaller study area. Unlike the previous processing, lake areas were measured by manually outlining water areas and by calculating areas with GIS tools.

[30] Lakes within a small 59 km2 test area (sampling area 2003, Figure 1) were analyzed using the DOQQ from 4 July 2003, and GIS tools. The test area is located within the larger study area and includes lakes with water areas ranging from 100 m2 to 10,000 m2. This smaller study area comprises about 1.4% of the larger study area. The power law relationship developed from lakes in this smaller study area is plotted together with that estimated previously from the Landsat image of 29 July 2002 (Figure 7a).

Figure 7.

(a) Power law lines from Landsat and DOQQ combined to form a single power law. (b) Comparisons of power law lines among the years 1939, 1990, 1992, and 1997. The 1939 power law is based on an extrapolation of lake areas measured from aerial photography. The bin size is 100 m2.

[31] The size of the test region analyzed by DOQQ is much smaller than the region studied with Landsat. With many fewer total lakes in the test area, the two power laws plot at different places on Figure 7a. The DOQQ power law is defined by solid black circles (bottom left, Figure 7a) versus that from Landsat defined by solid green triangles. It is, however, possible to normalize the area of coverage of the DOQQ to the same area (4365 km2) as the Landsat image. This normalization effectively moves the DOQQ-derived line vertically (red circles, Figure 7a). Remarkably with the normalization, the two line segments essentially coalesce to describe a single power law.

[32] This result suggests that areas of prairie pothole lakes in the Prairie Coteau region obey the power law relationships observed for lakes in other places. This relationship describes consistent size behavior in lakes ranging in area from 100 m2 to more than 30,000 m2. For small patches of surface water less than 100 m2 in area, we lose the ability to measure lake areas from aerial photos.

3.4. Power Laws in the Reconstruction of Conditions During the Dust Bowl Drought

[33] The consistent power law behavior of pothole lakes in the Prairie Coteau region provides a logical basis for interpreting the structure of the lake complex in 1939 from a small collection of aerial photographs of that period. The 10-year Dust Bowl drought of the 1930s is the most famous North American drought of the 20th century. It devastated the agricultural economy of the Great Plains for about a decade with health and social impacts lingering for years afterward. Researchers have suggested that abnormal deficits in precipitation and high temperatures were related to extreme anomalies of Pacific Ocean sea surface temperatures (SSTs) [Schubert et al., 2004; McCabe et al., 2004]. Laird et al. [1996] considered the drought of the 1930s as unremarkable in relation to others of greater intensity and frequency before AD 1200 in the Great Plains.

[34] While some hydrological and meteorological records are available for the early 1900s in South Dakota, the fate of the pothole lakes during the Dust Bowl drought is not well known. We examine the impacts of drought on lakes located in a second 90 km2 test area (sampling area 1939, Figure 1) using the aerial photos of from 29 July 1939. This sampling area overlaps the area used previously with recent DOQQ imagery (Figure 1). Using the 1939 photography, we measure the water area of lakes and wetlands greater than 100 m2 in order to define the size structure of the pothole-lake system.

[35] Figure 8 provides illustrative examples of comparable segments of the digitized 1939 imagery with the DOQQ. Thus, it is possible to compare the water distribution in potholes near the end of the Dust Bowl drought with more recent flooded conditions reflected in the DOQQ of 4 July 2003. The contrast between the areal extent of pothole lakes in 2003 and that in 1939 is striking. Some large pothole lakes were completely dry in 1939.

Figure 8.

Comparison of lakes from aerial photos taken in (a) 1939 and (b) 2003. The lakes in Figure 8a are commonly surrounded by a ring of evaporates. The area represented by the photographs is the sampling area 1939 (Figure 1).

[36] To provide a regional comparison, the 1939 relationship was normalized by the area ratio (log 4365/90), which effectively adjusts the numbers of lakes by 1.69 log cycles. The result of this normalization process is shown in Figure 7b.

[37] Because power law lines are robust across a broad range of lake sizes, it is appropriate to extrapolate relationships beyond the data for mainly small lakes in the photographs. Our previous analyses provide justification to extrapolate this curve to larger lakes, which yields the estimated power law for the overall region for 1939. To provide context for the 1939 result, the power law relationships are plotted for 1992 (drought) and 1997 (deluge) estimated from Landsat images. This single plot thus represents the distributions of lake areas at the end of the three most significant hydrological extremes of the last century. The estimated power law line for 1939 is comparable to the other lines in terms of slope.

[38] The response of the large lakes (>10,000 m2) through the Dust Bowl drought and observed in 1939 stands in stark contrast to their behavior through the second worst drought of the century (lines 1990 and1992), and the greatest deluge (line 1997). Figure 7b suggests the extreme variability in the abundances of lakes of a given size for these different times. For instance, in 1939 about seven lakes had an area of about 10,000 m2. During the drought represented by data from 1990 and 1992, there were 15 and 25 lakes of this size, respectively. During the subsequent deluge in 1997, there were about 50 lakes of this size.

[39] The smaller lakes (e.g., 100 to 1000 m2) show different patterns of variability. Interestingly, the drought of 1990 produced a greater impact on these smaller lakes than was evident with the 1939 results.

4. Discussion

[40] The pothole-lake system of the Prairie Coteau exhibits relationships between lake numbers and sizes that are similar to those observed for lake systems around the world [Lehner and Doll, 2004; Downing et al., 2006] or for regional collections of lakes [e.g., Downing et al., 2006; Kent and Wong, 1982]. In our study area, the power law relationships hold over about 3 orders of magnitude. In this respect then, our study confirms previous observations pointing to systematics in the size structure of lakes.

[41] Our study is distinguished from others in terms of temporal trends. The Landsat data archive shows how the structure of the lake complex changes with short-term and long-term climate effects. For any snapshot in time, a single power law relationship holds for the collection of lakes. This relationship however changes in time as the pothole-lake system responds dynamically to changes in precipitation and evaporation.

[42] The role of precipitation and evaporation as drivers of the response of the lake complex can be examined more explicitly by examining the correlation between parameters of the power laws, slopes and intercepts, and a common measure of water availability: precipitation minus evaporation [Winter and Rosenberry, 1998]. In effect, we test whether characteristics of the power laws at some time of interest (ti) are related to the total availability of water through a period of months or years preceding the time of interest.

[43] This paper already has shown large lake areas to be dependent upon water availability over several preceding years. Similarly, small-lake areas are dependent upon the water availability over the previous several months. Using monthly precipitation (P) data from the Waubay National Wildlife meteorological station and estimated monthly evaporation (E) data from nearby Lake Traverse [Vining, 2003], P minus values (PE) (m month−1) were calculated from 1980 to 2002. Given the known date associated with each power law line of interest, a total PE value is calculated for some specified number of preceding months or years (Table 1). A series of x-y plots is created to correlate the water availability estimate, PE, with three parameters derived from the equations of the power law lines, namely: (1) the numbers of small lakes having an area of 0.4 ha, (2) the numbers of large lakes having an area of 8.0 ha, and (3) the slope.

Table 1. Tabulation of Parameters for 11 Power Law Linesa
Datelog (Number of Lakes)bSlopeTotal PE (m)
0.4 ha8.0 ha3 months3 years
  • a

    Included are the log10 values of the number of lakes with a 0.4 ha area, the log10 values of the number of lakes with a 8.0 ha area, slopes, and water availability, represented as the previous 3-month total PE and the previous 3-year total PE.

  • b

    The number of lakes with areas of 0.4 and 8.0 ha is calculated from the equations of the relevant power law lines.

19 May 19853.0800.960−1.629−0.038−0.456
23 Apr 19873.2811.134−1.6500.033−0.116
15 Apr 19903.2040.907−1.7650.042−0.861
6 May 19923.0370.905−1.639−0.012−0.417
17 Sep 19943.0611.030−1.561−0.098−0.156
4 May 19973.3641.152−1.7000.0710.067
16 Jun 20013.0770.994−1.6010.043−0.263
27 Aug 20012.9360.979−1.504−0.163−0.264
18 May 20023.1261.001−1.6330.044−0.353
29 Jul 20022.7630.925−1.413−0.101−0.465
15 Sep 20022.7430.935−1.389−0.168−0.491

[44] The number of preceding years or months that are included in the PE calculation associated with the date for a given power law line is not well known. We tested a number of time periods (1–6 years and 2–4 months) for the large and small lakes, respectively, and used those periods providing the best correlations with water availability. A period of 3 years works best for the large (8.0 ha) lakes, while 3 months is best for the 0.4 ha lakes. The total PE data calculated on this basis are shown in Table 1 along with the parameters determined from the power law.

[45] Figure 9 shows that some features of the power law are strongly dependent on water availability. The number of 0.4 ha lakes correlates well (R = 0.84) with the total PE for the preceding 3 months (Figure 9a). Similarly, the number of 8.0 ha lakes correlates well (R = 0.85) with the total PE for the preceding 3 years (Figure 9b). With a comparable scaling of data on both the x and y axes (Figures 9a and 9b), it is evident from the differences in slopes of the fitted lines that the numbers of small lakes vary much more markedly for a given change in total PE than the larger lakes. The reason for the apparent sensitivity of smaller lakes to climatic forcings will be discussed below.

Figure 9.

Regression analyses showing the relationships between characteristics of the power law lines and the water availability (i.e., precipitation minus evaporation, PE). (a) Number of 0.4 ha lakes versus the three-month total PE. (b) Number of 8.0 ha lakes versus the 3-year total PE. (c) Slope of the power law lines versus the 3-month total PE. (d) Slope of the power law lines versus the 3-year total PE.

[46] The slopes of the power law lines (Table 1) are strongly related (R = −0.85) to the total PE for the preceding 3 months (Figure 9c) but not the preceding 3 years (R = −0.001) (Figure 9d). In effect, the tendency for the numbers of small lakes to fluctuate seasonally as a consequence of short-term climatic variability is the main control on the power law slopes.

[47] This analysis forms the basis of a conceptual model of how the power law relationships for pothole-lake complex vary as a function of changing hydrologic conditions (Figure 10). For illustrative purposes a single line (hypothetical June) is shown. The relationships between the power law parameters (slopes, intercepts) and the climate drivers show how the areas of the small lakes adjust rapidly to short-term (3-month) seasonal climatic cycles. The areas of the large lakes change much more slowly in response to 3-year trends in drought and deluge. Effectively then, the relative number or numbers of small lakes changes quite rapidly, following intra-annual cycles of climate variability, producing a rapid adjustments in the slopes of the power law lines (left part of Figure 10). This behavior contrasts with the much slower adjustments in the frequency distribution of large lakes, which are responding to interannual patterns of variability. Thus, the particular slope of a power law line is determined in a complex way by both long-term and short-term variability in climate.

Figure 10.

Conceptual model of the power law relationships for a pothole-lake complex under different hydrologic conditions.

[48] In a given year, there is typically an excess of water from snowmelt in spring. Evaporation through hot, dry summers and sporadic rains produce marked reductions in the areas of the small lakes in just a few months. However, the areas of the larger lakes are much less reduced in percentage terms by this annual cycle. Differences in the relative magnitude of changes in the areas of small versus large lakes are in part a function of the geomorphology as reflected in the geometry of the lake basin [Imboden and Joller, 1984]. The following equation describes lake/wetland areas as a function of the water depth and the shape of the lake basin [Brooks and Hayashi, 2002]

equation image

where A (L2) is the area of the lake/wetland for water depth d, Amax (L2) is the maximum area of the lake at water depth dmax, d (L) is some specified lake depth, dmax (L) is the maximum water depth, and p is the basin shape parameter (dimensionless). Values of p are about one for a V-shaped basin, two for a parabolic-shaped basin, and very large (p → ∞) for an idealized cylindrical basin [Hayashi and van der Kamp, 2000]. In prairie wetlands, these researchers found p values of about two for small lakes and wetlands (∼1000–10,000 m2) and about 3 to 6 for larger wetlands (∼100,000 m2).

[49] Analyses with equation (1) show that both d/dmax and p contribute to the tendency for small lakes to exhibit greater sensitivity in area change than large lakes. For example, equal stage reductions in a shallow lake (small area) versus a deep lake (large area) for the same p value, will yield (d/dmax shallow) < (d/dmax deep). The smaller d/dmax ratio for the shallow lake results in a greater percentage change in area, as compared to the deeper lake. When equal stage reductions occur in lakes with the same maximum depths (i.e., equal d/dmax), the lake with the smaller p value exhibits a much larger percentage reduction in area than the lake with the larger p value. Thus, by virtue of their shallower depths and generally smaller p values, the areas of small lakes change much more markedly than larger lakes.

[50] The tendency for the areas of small lakes to respond differently than large lakes is also likely related to differences in various hydrologic factors relating to inflows and outflows of water. For example, surface area reductions in larger lakes in deeper basins may be attenuated by inflows of groundwater. Surface area declines in smaller lakes can be exaggerated by outflows to groundwater. Thus, the interannual variability in the power law parameters (slope, intercept value) is related to the sensitivity of smaller lakes to geomorphic factors and differences in the hydrologic processes. A more quantitative investigation of the importance of these parameters will require model studies that are now underway.

[51] For larger lakes, many years of drought are required to produce a marked reduction in their areas. The Dust Bowl drought lasted over a decade and was effective in reducing the areas of large lakes, contributing to an increase in the slope of the power law line. In the case of the shorter drought from 1988 to 1992, the impact on the large lakes was evident but less significant. The deluge from 1993 to 1998 was characterized by a decrease in the slope because the larger lakes became more numerous. Overall, there is less variability in the position of the power law line for the larger lakes. In other words, the slope of the power line curve is predominantly influenced by the interannual variation in the size of small lakes. The shaded region in Figure 10 shows a fan-shaped envelope that would encompass all the power law lines for our study area. It describes the range in behavior of the lake complex during the 20th century.

[52] These results have several important implications. Because of natural seasonal fluctuations and longer-term cycles of drought and deluge, no single power law relationship describes the pothole lake complex in this region. For example, slope is maximized by a wet spring in the midst of a long-term drought or minimized by a hot, dry summer in the midst of a multiyear deluge. Also, the concept of droughts/deluges as extreme climatic deviations from long-term averages, is probably most applicable to large lakes. This conclusion can also be inferred from paleoclimate studies of lakes of the northern Great Plains [Donovan et al., 2002], which demonstrate linkages between the hydrologic coupling of larger lakes and long-term climatic fluctuations.

[53] The size of the more sensitive small lakes varies significantly from periods of water excess to periods of water deficiency each year, which may or may not be related to the larger-scale climatic influences. Thus, wildlife, adapted to small water bodies (e.g., ducks), may be less affected by multiyear drought and deluge. The numbers of small lakes of a given size can increase rapidly as the result of a modest single seasonal snowmelt, single heavy rain or single heavy rain.

[54] Many power law relationships exhibit validity over many orders of magnitude. As suggested by Figure 7, the combination of aerial photography and Landsat observations for this study area provides observations from the largest (30,000 m2) to smallest (100 m2) lakes present. Thus, the observed range of validity of the power law is one lake of 30,000 m2 and perhaps 25,000 lakes with an area of 100 m2. If the study area was much larger, it is possible that the single, upper end lake could be much larger.

[55] Understanding the character of the natural lower limit in the power law and its temporal variability is important because the large number of smaller lakes contain the largest volume of surface water. Eventually, the power law will break down when small depressions only hold water during the wettest months. We have theoretical studies underway to examine the practical lower limit of validity of power laws in different settings.

5. Conclusions

[56] This study has shown that power law relations are useful in understanding the behavior of tens of thousands of lakes, with footprints as small as that of a family home to as large as a small city. In the study area in South Dakota, the structure in lake areas at various times of the year and through a number of years always can be described by a power law. Significant annual variability in water excesses and deficiencies, as well as longer-term cycles of drought and deluge, is reflected in variability of the power law parameters. Some combination of geomorphic and hydrologic parameters causes the area of small lakes to be more sensitive than the large lakes to short-term changes in water availability.

[57] In this lake-wetland complex, the surface areas of small lakes are much more sensitive to climate forcings than the surface areas of larger lakes. This sensitivity is due in part to the nonlinear dependence of isobath area on lake depth. Thus, knowledge of how small lakes respond to climate forcings is of critical importance in water resource assessments. However, the lower limit of the power laws remains elusive, likely to be influenced by particular conditions of water on the land surface.

[58] The next step in examining the PPR more broadly is to put together observations from climatologically and topographically diverse regions. This kind of analysis could form the basis for assessing the hydrologic response of lakes through time.


[59] This study was supported by the National Science Foundation, NSF award EAR-0440007.