Power Scaling of Ice Floe Areas in the Weddell Sea, Southern Ocean With a Summary of Previous Ice Floe Scaling Studies

doi:10.21203/rs.3.rs-3784463/v1

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Power Scaling of Ice Floe Areas in the Weddell Sea, Southern Ocean With a Summary of Previous Ice Floe Scaling Studies

https://doi.org/10.21203/rs.3.rs-3784463/v1

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The distribution of cumulative number as a function of floe area of seasonal ice floes from four satellite images covering the summer season (November - February) in the Weddell Sea, off Antarctica, during the summer ice breakup were well fit by two scale-invariant power functions. For a power function of the form N = C x^-β, the scaling exponents -β for the larger floe areas range from − 1.5 to -1.8. Scaling exponents -β for the smaller floe areas range from − 0.8 to -1.0. The inflection point between the two scaling regimes ranges from 58 x 10⁶ to 155 x 10⁶ m² and generally moves from larger to smaller floe areas through the summer season. We propose that the two power scaling regimes and the inflection between them are established during the initial breakup of sea ice solely by the process of fracturing. Floe areas range from 3 x 10⁶ to 550 x 10⁶ m². The distributions of floe size regimes retain approximately the same scaling exponents as the floe pack evolves from larger to smaller floe areas from the initial breakup through the summer season, due to scale-independent processes of fracturing, grinding and melting. The scaling exponents for floe area distribution are in the same range as those reported in previous studies of Antarctic and Arctic floes. A probabilistic model of fragmentation is presented that generates a single power scaling distribution of fragment size.

Ice floes

power distribution

scaling

Antarctic

Southern Ocean

In winter season (March-October), sea ice forms and floats on the surface of the Southern Ocean (Fig. 1) and ranges up to 3 m thick (Shi et al., 2021). In November, the ice begins to break up into free floating angular fragments of different shapes and sizes. A floe is defined as a fragment of sea ice whose upper surface area is larger than 1 m² (Gherardi and Lagomarsino, 2015). Newly formed floes exhibit rough angular edges, wavy edges, and sharp corners. The surface area of the floes observed in the Weddell Sea in November 2016 range in area from 4 x 10⁶ to 550 x 10⁶ m². Once formed, the floes sizes are reduced by additional fracturing caused by impact, crushing, and grinding due to wave motion, wind, and ocean currents and by melting and smoothing along their edges and faces. By the end of the summer season, as observed in the Weddell Sea in February 2015, the surface area of the floes range from 3 x 10⁶ to 390 x 10⁶ m². Larger floes are surrounded by smaller floes which cushion the larger floes protecting them from further fracturing (e.g., Fig. 4a). The cushioning extends over a wide range of size scales. In the process of fracturing, only particles of approximately the same size or larger can fracture any given particle (Sammis & King, 2007). This observation is the basis for the probabilistic model put forward in Geise et al. (2017) to explain the power distribution of floe sizes in the Arctic Ocean. The size distribution of floes and its evolution over the Antarctic summer is the subject of this paper. This topic is of relevance to marine vessels that encounter floes, to the calculation of sea ice albedo, to the determination of Antarctic heat exchange which is strongly influenced by ice concentrations and the amount of open water between floes (Worby & Allison, 1991), and to photosynthetic marine organisms which are dependent upon sunlight penetrating the spaces between floes (Grose & McMinn, 2003).

Previous studies of the distribution of floe surface areas plot the cumulative or the non-cumulative number as a function of either the floe diameter or the upper surface area of the floes, both of which are fit by a power function. The power function fit to a cumulative distribution is:

$$N=C {x}^{-\beta }$$

where N is the number of floes greater than size x, -β is the scaling exponent, and C is a constant of proportionality. C is also referred to as the activity level and is equal to the value of N when x = 1.

In many studies, the upper floe surface area was approximated by the caliper-diameter method where one approximates the diameter d of a circle of equivalent area A using the relation A = 0.7854 d ². Note, the diameter is a one-dimensional measure of a two-dimensional object (upper floe surface area). Since floe upper surfaces are rarely circular, the equivalent diameter of a floe may be a poor approximation of floe area and may lead to a different scaling exponent for the distribution of floe areas. The scaling exponents of the cumulative distribution of floe area can be converted to obtain the equivalent scaling exponent of the caliper-diameter method by multiplying the scaling exponent of the area method by two (Stern et al., 2018). A more precise method, the group pixel method, groups image pixels of ice and water and uses these to determine areas. A few previous studies have used the grouped pixel method to approximate the area (a two-dimensional measure) of each upper floe surface.

2.1 Southern Ocean

Lensu (1990) analyzed one image taken of the Weddell Sea during the summer ice pack break up from February 1990. Using ice floe surface areas, he found the distribution of cumulative number as a function of floe area was well fit by a single power function with a scaling exponent of -0.68. The floe areas ranged from 0.1 to 20 m².

Lu et al. (2008) analyzed nineteen aerial photos taken of the Prydz Bay, East Antarctic [Southern] Ocean in December 2004 through February 2005. They used the caliper-diameter method. Distributions of cumulative number as a function of floe diameter are found were well fit by a single power function with scaling exponents ranging between − 0.6 to -1.4. The floes ranged in diameter from 2 to 100 m.

Steer et al. (2008) analyzed 130 aerial images taken of the Weddell Sea in December 2004. They used the caliper-diameter method. They found the distributions of non-cumulative, log-binned histograms of caliper-diameter were well fit by two scaling exponents. A scaling exponent of -1.9 was found for floes smaller than 20 m and − 2.8 to -3.4 for floes larger than 20 m. The floes sizes ranged from 2 to 120 m.

Toyota et al. (2011) analyzed 122 helicopter images taken of the Weddell Sea on three dates from September through October 2006 and 52 images of the Southern Ocean off of Wilkes Land on three dates from September through October 2007. For both regions, they used the caliper-diameter method. In the Weddell Sea, the distribution of cumulative number as a function of floe diameter were well fit by two scaling exponents: -1.05 to -1.39 were found for floes smaller than 30–40 m and − 5.18 to -7.59 for floes larger than 30–40 m. The floe diameters ranged from 2 to 120 m. Off of Wilkes Land, the distribution of cumulative number as a function of floe diameter were well fit by two scaling exponents. Scaling exponents from − 1.03 to -1.52 were found for floes smaller than 15 to 25 m and − 3.15 to -5.51 for floes larger than 15 to 25 m. The floes sizes ranged from 2 to 100 m.

Gherardi and Lagomarsino (2015) analyzed one satellite image taken of the Weddell Sea in October 2003. They grouped white pixels for ice floes and black pixels for water in the image. They used the caliper-diameter d, of a circle of equivalent area A, to represent the upper surface area of each floe. They found the distribution of a non-cumulative log-binned histogram of caliper-diameter was well fit by a single power function with a scaling exponent of -2.0 (Gherardi and Lagomarsino, 2015, Fig. 2). The floe diameters ranged from 2 to 100 m.

Toyota et al. (2015) analyzed four helicopter and twelve satellite images taken off of Wilkes Land, East Antarctica from September-November 2012. They used the caliper-diameter method. The distribution of cumulative number as a function of floe diameter was well fit by two scaling exponents. Scaling exponents from − 2.9 to -3.1 were found for floes smaller than 100 m and from − 1.3 to -1.4 for floes larger than 100 m. The floe diameters ranged from 5 to 10,000 m.

Paget et al. (2017) analyzed six aerial images taken of East Antarctic sea ice in August 1995. They used the caliper-diameter method. They found the distributions of non-cumulative linearly-binned histograms of caliper-diameter were well fit by power functions with scaling exponents ranging from − 1.9 to -3.5 (Paget et al., 2017, Fig. 5). The floes sizes ranged from 1 to 150 m.

2.2 Arctic Ocean

Rothrock and Thorndike (1984) analyzed seven aerial and satellite images taken of the Beaufort Sea from March to October 1973–1975. They used the caliper-diameter method. They found the distributions of cumulative number as a function of floe diameter were well fit by a single power function with scaling exponents ranging from − 1.7 to -2.5. The floe diameters ranged from 1,000 to 20,000 m.

Matsushita (1985) analyzed one satellite image taken of the Sea of Okhotsk in December 1984. He used the caliper-diameter method. The distribution of cumulative number as a function of floe diameter was well fit by a single power function with a scaling exponent of -2.2. The floe diameters ranged from 5 to 30 m.

Holt and Martin (2001) analyzed fifteen satellite images taken of the Beaufort, Chukchi, and East Siberian Seas from August 1992. They used the caliper-diameter method. The distributions of cumulative number as a function of floe diameter were well fit by a single power function with scaling exponents ranging from − 1.9 to -2.6. The floe diameters ranged from 1,000 to 20,000 m.

Inoue et al. (2004) analyzed two aerial images taken of the Sea of Okhotsk in February 18, 2000. They used the caliper-diameter method. They found the distributions of cumulative number as a function of floe diameter were well fit by a single power function with scaling exponents ranging from − 1.5 to -2.1. The floe diameters ranged from 10 to 100 m.

Toyota et al. (2006) analyzed four helicopter, icebreaker, and satellite images taken of the Sea of Okhotsk in February 2003. They used the caliper-diameter method. They found the distributions of cumulative number as a function of floe diameter were well fit by two scaling exponents. A scaling exponent of -1.15 was found for floes smaller than 40 m and − 1.87 for floes larger than 40 m. The floe diameters ranged from 1 to 1500 m.

Perovich and Jones (2014) analyzed nine aerial and satellite images taken of the Beaufort Sea between June and September 1998. They used the caliper-diameter method. They found the distributions of cumulative number as a function of floe diameter were well fit by a single power function with scaling exponents ranging from − 2.0 to -2.2. The floe diameters ranged from 10 to 10,000 m.

Gherardi and Lagomarsino (2015) analyzed three satellite images taken of the Kara Sea, Svalbard area, and Barents Sea from the Spring seasons of 2000, 2001, and 2009, respectively. They used the caliper-diameter method. They found the distributions of non-cumulative log-binned histograms of caliper-diameter were well fit by a single power function with a scaling exponent of -2.0 (Gherardi and Lagomarsino, 2015). The floe areas ranged from 2 to 5,000 m.

Wang et al. (2016) analyzed eighteen satellite images taken of the Beaufort and Chukchi Seas from summer through fall 2014. They used the caliper-diameter method. They found the distributions of cumulative number as a function of floe diameter were well fit by a single power function with scaling exponents ranging from − 2.77 to -4.12. The floe diameters ranged from 1 to 40,000 m.

Geise et al. (2017) analyzed six satellite images taken of the East Siberian Sea from June-August 2000–2002. They used pixel grouping to measure the floe surface area. The distribution of cumulative number as a function of floe area were well fit by two power functions with scaling exponents for the smaller floes ranged from − 0.3 to -0.6 and scaling exponent for the larger floes ranged from − 0.6 to -1.0. The size at the inflection point ranged from 280 x 10³ to 485 x 10³ m². The floe areas ranged from 30 m² to 28.4 x 10⁶ m².

Hwang et al. (2017) analyzed multiple satellite images taken of the Beaufort Sea from July 19, August 15, and August 23, 2014. They used the caliper-diameter method. They found the distributions of non-cumulative log-binned histograms of caliper-diameter were well fit by a single power function with scaling exponents ranging from − 2.7 to -3.0 (Hwang et al, 2017, Fig. 1). The floe diameters ranged from 245 to ~ 4000 m.

Stern et al. (2018) analyzed 273 satellite images taken of the Beaufort and Chukchi Seas from March to October 2013 and 2014. They used the caliper-diameter method. They found the distributions of non-cumulative log-binned histograms of caliper-diameter were well fit by a single power function with scaling exponents ranging from − 2 to -2.9 (Stern et al., 2018, Fig. 5). The floe diameters ranged from 10 to 30,000 m.

Table 1 provides a list of the results from our study and previous studies of ice floe scaling in the Antarctic and Arctic regions. To directly compare scaling exponents, we convert scaling exponents reported in the literature to equivalent scaling exponents of cumulative number as a function of floe area distributions. For each study, the uppermost value reported in Table 1 is the scaling exponent provided in the original research paper.

To obtain equivalent scaling exponents, the order of operations is important. The conversion from caliper-diameter to area was determined for a cumulative distribution by Stern et al. (2018). For studies that reported the scaling exponent for a non-cumulative histogram distribution, we first obtain the equivalent dimension for the cumulative distribution. Second, for studies that reported a dimension for caliper-diameter, we convert to the equivalent dimension for floe area.

Conversion of scaling exponents from non-cumulative histogram distributions to cumulative number versus size distributions

Non-cumulative linearly-binned histogram distributions. The scaling exponent for a non-cumulative linearly binned histogram distribution that follows a power function has a slope that is steeper by an integer value of one (1) as compared to the same data plotted as a cumulative number versus size distribution (e.g., Fig. 1 of Burroughs and Tebbens, 2001). To obtain the equivalent scaling exponent for a cumulative number versus size distribution, when starting with the scaling exponent for a non-cumulative linearly binned histogram distribution, we add an integer value of one (1). As a specific example, Paget et al. (2017) reported scaling exponents of -1.9 to -3.5 for a non-cumulative linearly binned distribution, and we provide the equivalent scaling exponent for the cumulative number versus size distribution of -0.9 to -2.5 in square brackets [ ] (Table 1).

Non-cumulative logarithmically-binned histogram distribution. The scaling exponent for a cumulative number versus size distribution that follows a power function has a slope that is equal to the slope of the same data plotted as a non-cumulative linearly binned histogram distribution (e.g., Fig. 2 of Burroughs and Tebbens, 2001). In Table 1, for studies that reported the non-cumulative linearly binned histogram distributions (e.g., Gherardi & Lagomarsino, 2015 and Stern et al, 2018), the scaling exponents are the same as for the cumulative number vs. size distributions. Therefore, there is no conversion of the value of the scaling exponent.

Conversion from cumulative number versus caliper-diameter to cumulative number versus area

The scaling exponents derived from a 1-dimensional representation of the surface area (caliper-diameter) of a floe are not equivalent to those calculated from 2-dimensional measures of floe area. The scaling exponents calculated by the caliper-diameter method can be converted to the equivalent surface area scaling exponents by dividing the caliper-diameter scaling exponent by two (Stern et al., 2018). In Table 1, we show the equivalent surface area scaling exponent in parentheses ( ).

This study is based on four mostly or completely cloud free satellite images of four floe packs at different stages of fragmentation during the Antarctic summer months (November 2016, December 2015, January 2015, and February 2015) in the Weddell Sea (Fig. 2) obtained from the USGS Earth Explorer website (United States Geological Survey, 2016).

The Weddell Sea was selected because of the image clarity, quality, and cloud coverage less than twenty percent. The Weddell Sea is a preferred research location for many ice floe studies because it is protected by the Antarctic peninsula.

The images used in this study were collected by the Landsat 8 satellite. To measure the upper surface areas of individual floes, ideally the portions of the images analyzed must be cloud free because the bands used to view the ice floes do not penetrate through clouds. Melting ponds on the surface of the floes were not removed from the images. Satellite images were greyscale at 30 m per pixel resolution. We use floe area as the measure of floe size in contrast to the caliper-diameter method used in many previous studies. A visual inspection of the satellite images reveals no characteristic floes size (e.g., Figs. 3, 4, 5, 6, and 7), suggesting qualitatively that the distribution of floe sizes is scale invariant and might, therefore, be described by a power function.

The satellite images obtained from Earth Explorer were imported into ESRI’s geographic information system (GIS) program, ArcPro (2.2). Tiff pixels from these images were classified into two classes, ice and water, using ArcPro’s Supervised Classification tool. The high brightness pixels represent the ice, while the low brightness pixels represent water. Individual floes larger than 3 x 10⁶ m² were identified by grouping of ice class pixels and are surrounded by water class pixels. The Raster to Polygon tool was used to automatically identify the edges of each floe. The resulting floe polygons form the floes seen in the raster image (Fig. 7). The area and perimeter length of each floe in the raster image were automatically tabulated in the attribute table by ArcPro. Floes cut off by the edge of the image, regions obscured by cloud cover, and slush/finely crushed ice were not included in the analysis.

A plot of the cumulative number versus floe area is shown in Fig. 8. The data for each image are well fit by two power functions (Eq. 1), with an inflection point between them.

The choice of an algorithm for fitting a distribution that appears to be a power function has been debated in recent years (Deluca & Corral, 2013; Holme, 2019). One method is that of Clauset et al. (2009). Corral & Gonzalez (2019) enumerated three issues with the Clauset et al. (2019) method: the method cannot be extended to truncated power distributions; there is no justification why the minimization of the Kolmogorov-Smirnov distance should work to find a meaningful scaling exponent; and the method was found to fail when applied to simulated data with power function tails, despite the data being a known power function. For this paper, the specific scaling exponent is not as important as the result that the data are well represented by a power function. Further, we are not extrapolating the results beyond the range of the data, which would amplify any errors in the reported values. Given these issues, we have used the same method used in our previous paper studying ice floe distributions in the Artic (Geise et al., 2017), to facilitate direct comparison of results.

For each of the data sets we analyze, we find the parameters C and β using the Levenberg-Marquardt algorithm to minimize Chi squared (Press et al., 2001) using the Wavemetrics software IgorPro version 6.37. This method is appropriate for fitting the power functions to data and calculating the associated error; the method assumes that the errors in fitting the functions to the data are normally distributed (Press et al., 2001). In all analyses for this study, the data were fit using IgorPro by the power function y₀ + Cx^-β, where y₀ was set to zero and the fit was found to converge. Errors are reported to plus/minus one standard deviation in Table 1.

Each of the four data sets analyzed and plotted on Fig. 8 exhibit two power scaling regimes where larger floes have a distribution whose scaling exponent is greater than that of the small floes. The range of floe areas on either side of the inflection point is greater than one order of magnitude for the smaller floes and nearly one order of magnitude for the larger floes, where one order of magnitude is considered to be the minimum range for uniquely fitting a power function to a dataset.

The scaling exponents -β exhibit a range of values (Table 1). The scaling region for smaller floes has scaling exponents -β ranging from − 0.8 to -1.0. The larger floes have scaling exponents -β ranging from − 1.5 to -1.8. The inflection point between the two regions ranges from 86 x 10⁶ to 174 x 10⁶ m². The number of floes analyzed in each image varies from 303 (February 2015) to 1,078 (December 2015).

Inflection point sizes decrease roughly sequentially from November to February as sea ice evolves from larger to smaller floes throughout the season (Fig. 8). Note that cushioning of larger floes by smaller floes is present for almost all floe sizes. This suggests that once the floe size distribution is established at the time of initial breakup, further size reduction is primarily by grinding and melting of the sides of floes, not by fracturing across the floes (Fig. 3b).

Summing the areas of the larger floes, those above the inflection point, and dividing by the total floe area for each image indicates that the area of the large floes is 21–44% of the total floe area (Table 1). Note that this differs from the percentage of total floe area above and below the inflection points found in the Arctic by Geise et al. (2017) where the larger floes, those greater than the inflection points, accounted for 72–95% of the total floe area.

Geise et al. (2017) proposed a model that produces a power distribution of fragment sizes, which may explain the power distribution of floe areas but not the existence of an inflection point between the larger and smaller floe size distributions. Figure 9 illustrates their probabilistic model for fragmentation of a cubic volume for any material where fracture is the method of fragmentation. At iteration n = 1 the volume is broken into eight smaller cubes of equal size. At iteration n = 2 some of the eight cubes are randomly selected and broken into eight smaller cubes. The probability that a cube will be fragmented is an adjustable variable. In Fig. 9, the probability of fragmentation is p = 0.25. At iteration n = 3 the model bifurcates based on whether (A) only the smallest of the cubes or (B) a cube of any size, can be randomly selected and broken. With a large number of iterations, the rule for A or B determines the cumulative frequency versus size distribution of the fragments. Rule A leads to a power fragment size distribution and Rule B leads a lognormal fragment size distribution. As we observe power distributions in the size (area) of floes, we propose that a process following Rule A is appropriate for ice floes. The choice of probability of fragmentation controls the scaling exponent of the power law distribution (e.g., p = 0.25 yields a b of -0.5). We note that this fragmentation model generates a single power distribution without an inflection point. This model could be appropriate for ice floe distributions reporting a single power distribution of floe areas (Table 1, studies that report “b with no inflection”).

Sammis and King (2007) propose a physical process that results in power scaling of fragmentation sizes of geologic materials in a fault gauge. Adjacent fragments must be approximately the same size to break each other by fracture. They observe that in the process of fragmentation by grinding and crushing, larger fragments are surrounded by smaller fragments that cushion them and protected them thus greatly reducing the probability of further fragmentation. They propose that such a process leads to a power distribution in fragmentation size. Visual inspection of Figs. 3a, 4a, 5, and 6 reveal that for ice floes the large fragments are surrounded by smaller fragments which may similarly have cushioned them and reduced the probability of further fragmentation.

The distribution of floe area as a function of cumulative number versus floe area in each image were well fit by two power functions that meet at an inflection point between the larger floes and the smaller floes (Fig. 8, Table 1). The floe areas range from 3 x 10⁶ to 550 x 10⁶ m². Floe areas larger than the inflection points are well fit by a power function with scaling exponents of -1.5 to -1.8 while the floes areas smaller than the inflection points are well fit by a power function with a scaling exponents of -0.8 to -1.0. The inflection points range from 58 x 10⁶ to 155 x 10⁶ m².

This study supports some of the previous studies that found that the cumulative number versus floe area distribution of seasonal floes are distributed in two power scaling regimes in the Antarctic region (Steer et al., 2008; Toyota et al., 2011; and Toyota et al., 2015) and in the Arctic (Toyota et al., 2006 and Geise et al., 2017). We suggest that the two power scaling regimes and the inflection between the large and small floes are established during the initial breakup of the sea ice solely by the process of fracture. The distributions of floe size regimes retain the same scaling exponent (-b are within 2 standard deviations for three out of four of the images; see Fig. 8 and Table 1) as the sea ice evolves from larger to smaller floes throughout the season due to grinding and melting as seen in Fig. 8.

When the initial ice breaks up by fracture the floe size distribution and the scaling exponents are established. The scaling exponents established during the initial ice break-up are maintained through the summer season and change minimally (by not more than 0.1) during the reduction of floe sizes at all scales through the processes of grinding and melting. While the scaling exponents b do not appreciably change during the season in our study, the activity level C tends to decrease (Fig. 8, Eq. 1). As C decreases, the number of floes of any and all sizes present is smaller. If there is not change is slope, the decrease in size is equal in log space at all sizes. indicating the floe sizes have systematically decreased. This can be seen in Fig. 8 where the data points for Feb 2015 are below those of Jan 2015, and the largest sized floe is also smaller.

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

The authors have no relevant financial or non-financial interests to disclose.

Author Contribution

Coffey obtained the satellite data, analyzed the images for floe area, and created many of the figures.Barton conceived and coordinated the project.Tebbens lead the statistical analysis of the data sets.We all worked on writing the manuscript.

Acknowledgements

The authors thank Greg Geise and Dan Khoel for their help in processing the GIS data. We also thank two anonymous reviewers for constructive criticism that greatly improved an early version of this manuscript.

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Table 1 is available in the Supplementary Files section.

No competing interests reported.

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Power Scaling of Ice Floe Areas in the Weddell Sea, Southern Ocean With a Summary of Previous Ice Floe Scaling Studies

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Abstract

Figures

1. Introduction

2. Previous Studies

2.1 Southern Ocean

2.2 Arctic Ocean

3. Calculation of Equivalent Scaling Dimensions

4. Data

5. Analysis

6. Fragmentation Model

7. Discussion and Conclusions

Declarations

Author Contribution

Acknowledgements

References

Table

Additional Declarations

Supplementary Files

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