Sea ice surface patterns encode more information than can be represented solely by the ice fraction. The aim of this paper is thus to establish the importance of using a broader set of surface characterization metrics and to identify a minimal set of such metrics that may be useful for representing sea ice in Earth system models. Large-eddy simulations of the atmospheric boundary layer over various idealized sea ice patterns, with equivalent ice fractions and average floe areas, demonstrate that the spatial organization of ice and water can play a crucial role in determining boundary layer structures. Thus, various methods used to quantify heterogeneity in categorical lattice-based spatial data, such as those used in landscape ecology and Geographic Information System (GIS) studies, are employed here on a set of recently declassified high-resolution sea ice surface images. It is found that, in conjunction with ice fraction, patch density (representing the fragmentation of the surface), the splitting index (representing variability in patch size), and the perimeter–area fractal dimension (representing the tortuosity of the interface) are all required to describe the two-dimensional pattern exhibited by a sea ice surface. For surfaces with anisotropic patterns, the orientation of the surface relative to the mean wind is also needed. Finally, scaling laws are derived for these relevant landscape metrics, allowing for their estimation using aggregated spatial sea ice surface data at any resolution. The methods used in and the results gained from this study represent a first step toward developing further methods for quantifying variability in polar sea ice surfaces and for parameterizing mixed ice–water surfaces in coarse geophysical models.

The polar sea ice surface, a sensitive indicator of global climate change, exhibits persistent biases in sea ice fraction and extent in coarse-resolution Earth system models (ESMs)

The fringe zone that separates densely consolidated sea ice from the open ocean is known as the marginal ice zone (MIZ) – see

To this end, the complex geometric patterns formed by sea ice floes need to be analyzed. Larger floes will have a proportionally greater effect on surface–atmosphere fluxes, whereas smaller floes, with their more frequent transitions, will exacerbate the nonlinearity of the exchange processes. These surface–atmosphere fluxes have a large effect on the atmospheric boundary layer (ABL) overlaying the marginal ice zone (MIZ-ABL). As a thought experiment, we can consider an ice–water surface with a very fine checkerboard pattern and a sea ice fraction (

Given the importance of the topic and the challenges outlined above, previous work has attempted to quantify the heterogeneity of sea ice surfaces

Studies in landscape ecology have previously sought an optimal independent group of metrics for understanding the heterogeneity of lattice surfaces.

The questions that will be answered in this study are as follows:

Is the sea ice fraction of a MIZ surface, combined with some measure of average floe area, sufficient to predict the behavior of the overlying MIZ-ABL?

If not, what other surface information in a two-dimensional, lattice-based spatial pattern is needed to describe air–sea interactions?

How can this surface information be applied to sea ice surfaces in weather models and ESMs, considering factors such as availability of information, resolution-resampling invariance, and ease of understanding?

Large-eddy simulations (LESs) of the MIZ-ABL were conducted for different idealized configurations of sea ice. LESs are widely used to model heterogeneous high-Reynolds-number flows

Numerical details of the large-eddy simulation. Time is represented in terms of inertial periods (

LESs are thus used to model MIZ-ABL flow over

The bottom-boundary condition for each simulation can be thought of in terms of categorical lattice-based spatial data, where each node represents either the ice class or the water class. An ice node is prescribed a surface temperature (

Schematic of the domain setup for the large-eddy simulation. The ice (in gray) has a surface temperature (

All five patterns (Pattern1 to Pattern5), displayed in Fig.

Bird's-eye view of the five idealized

In addition to the Rossby number and the roughness ratio discussed earlier, an important dimensionless input parameter in these simulations is the heterogeneity Richardson number, defined as

While the LES utilizes idealized surfaces to examine the influence of patterns on the MIZ-ABL, examining what other landscape metrics might be important for surface characterization necessitates the use of real-world sea ice maps. The lattice spatial data used in the statistical analysis (see Sect.

These maps, which have a horizontal extent of up to 10 km by 10 km, comprise the dataset used to calculate the landscape metrics. Some of the images did not fully cover this full extent; thus, in order to retain the real-world sea ice geometry, we reflected the images onto areas with no data. All metric calculations and analyses were conducted on these modified surfaces. The advantage of this high-resolution, large-extent dataset is that we can analyze how these metrics change with grain size. These maps are thus aggregated from a 1 m resolution to 2 m, 10 m, 50 m, 100 m, 200 m, 500 m, 1 km, and 2 km resolutions; resampling was done using the nearest-neighbor method from the Python Imaging Library (PIL). These resolutions cover a range of common resolutions used in fine large-eddy simulations and numerical weather prediction (NWP) models. Due to the excessive computational-processing time required for the original 1 m resolution data, the highest resolution at which landscape metrics were calculated was 2 m.

One important caveat of this dataset is that the histogram illustrating sea ice fraction (

The distribution of

FRAGSTATS, a spatial-pattern analysis program, was used to calculate landscape metrics

This resulted in 22 landscape metrics that focus on the global patterns of the surface. Many of these landscape metrics, however, are correlated with one another; for example, patch density and mean patch size are proportional to each other. This is due to the fact that there are limited observations one can make about a surface (such as the number of patches, the area of a patch, and the proportion of edge in a patch) but an infinite number of operations that can be performed on a surface. Therefore, many of these metrics (especially at the landscape level) simply represent different methods of aggregating or statistically analyzing these observations.

While collinearity between two metrics can be easily detected through a correlation matrix, multicollinearity (where one indicator is a linear combination of two or more other indicators) is more likely in these types of datasets. It is thus possible for two or more landscape metrics to jointly define another metric. An objective and statistical way of reducing these parameters is thus needed. Here, we chose the variance inflation factor (VIF),

Each of these metrics, listed in Table

The last two metric groups, Contrast and Diversity, are less important for the present application to sea ice. Contrast metrics refer to the magnitude of difference between adjacent patch types with respect to a given attribute – in the case of sea ice surfaces, which only correspond to two classes (ice and ocean), there is only one contrast between two categories. Thus, metrics in this group are simply represented by the contrast of surface temperature and roughness. Diversity metrics are influenced by the number of patch types present and the area-weighted distribution of these patch types. In this case, we only have two types of patches (ice and water), so the diversity is the same across all maps, and the weighted distributions of the patches are related to the ice fraction. Further information on all of these metric groups can be found in the FRAGSTATS manual

The large-eddy simulation technique detailed in Sect.

Normalized vertical profiles of

In all five simulations, the largest difference occurs between Pattern4 and Pattern5 (red and purple lines, respectively), which is expected since the geostrophic wind, and thus the near-surface wind, flows parallel (rather than perpendicular) to the strips of ice

Each simulation showed an increase in potential temperature of 2.1 %–2.2 %; this warming is consistent with the large ocean fraction in each of the patterns. Despite initializing the air temperature to produce zero fluxes according to the Monin–Obukhov flux models, which assume equilibrium between the air and water above each surface grid cell, the upward fluxes over warm water are larger than the downward fluxes over cooler ice. This is due to the effect of advection, which perturbs the equilibrium.

Major differences are also observed in the total streamwise and cross-stream stresses, displayed in Fig.

To further explain the differences seen in these patterns, we consider the decomposition of the total flux into its dispersive and turbulent contributions. Vertical turbulent heat flux is denoted as

Figure

The total heat flux in all these simulations linearly decreases with height, as dictated by the LES setup. As seen in Fig.

Lastly, analyzing the ratios between dispersive and total atmospheric vertical fluxes (Table

Overall, these LES results indisputably indicate that ice–water patterns hold key information on how the MIZ-ABL interacts with the underlying surface; thus, the rest of the paper is dedicated to characterizing these patterns.

Ratios between dispersive and total atmospheric vertical fluxes.

Now that we have established the need for surface characteristics beyond sea ice fraction, we aim to examine the indicators that can be used for this purpose. For each of the nine resolutions considered, 44 observed sea ice images were analyzed. When conducting this analysis at the 2 m resolution, four metrics (including sea ice fraction) remained after the VIF elimination process and did not exhibit multicollinearity with one another: sea ice fraction (

The first aggregation metric, PD

The second metric, SPLIT, with a VIF of 1.9, was first described in

The only Shape metric, PAFRAC, with a VIF of 2.1, is obtained by regressing each patch's perimeter (

Thus, in addition to sea ice fraction (

Landscape metrics of the simulations conducted in Sect.

For the real ice maps obtained from Arctic images, the ice fraction varies from 0.19 to 0.99, PD varies from 5.4 to 42.5, SPLIT varies from 1.23 to 4.07, and PAFRAC varies from 1.338 to 1.726 at a 2 m map resolution. In many cases, however, numerical simulations also require the resampling of high-resolution surfaces by increasing the grain (pixel) size. For example, sea ice maps from reconnaissance satellites may have a resolution of up to 1 m, but this is computationally impractical for numerical weather models. Large-eddy simulations of the ABL can have resolutions down to 50 m, while NWP models have resolutions of 2 to 10 km. Therefore, even with high-resolution data, the aggregation and resampling of these surface patterns are inevitable in modeling. Furthermore, considering the operational use of these metrics, regularly updating these values would likely involve multiple satellite products with differing resolutions; thus, metrics that can be extrapolated/interpolated across different grid cell sizes would allow for consistent computation when standardized to a single weather model grid cell.

Landscape metrics plotted against resolution with respect to

Therefore, it is useful to examine how these chosen metrics vary as an image is aggregated to a resolution applicable to numerical weather models (or other numerical models, such as LESs); an appealing metric would be one that is invariant to resolution changes. Sea ice fraction (

Some metrics, such as SPLIT and PAFRAC, seem to exhibit near-invariant behavior after a certain jump in the resolution. For example, when starting from the 10 m resolution, SPLIT stays fairly constant as the resolution decreases. There is also variation in PAFRAC as the resolution decreases from 10 m. This is consistent with results from previous studies as some landscape metrics exhibit large errors when surfaces are aggregated to lower resolutions

Thus far, we have identified four surface pattern indicators that characterize the MIZ surface: sea ice versus water concentration (

We observe that in Pattern4 and Pattern5, the difference in geostrophic direction is related to the directionality of sea ice organization. In Pattern4, the wind blows consistently over an infinitely repeating pattern of sea ice and water at regular intervals. In Pattern5, the wind blows over much longer strips of ice and water, even though some sea-ice–water transitions are present. Any other oblique flow is thus in between these two parallel and perpendicular regimes. We characterize the differences between these regimes by examining the variance in the surface exposed to the wind. In other words, Pattern4 exhibits high variance since the surface wind flows over a maximum of eight ice–water transitions within one domain length, whereas Pattern5 exhibits low variance since the wind flows over a maximum of two sea-ice–water transitions. This then raises the question of how to determine a principal direction for a more complex surface.

We therefore attempted to characterize this anisotropy by computing the direction of the eigenvector (the eigendirection) for the surface with the least amount of variance, resulting in the fewest ice–water transitions possible. This was done using the

Eight maps from the sea ice dataset, each overlaid with their principal eigendirection (long arrow) and secondary eigendirection (short arrow), computed via principal component analysis.

Eigenvector (

It is hypothesized that, for a fixed sea ice fraction (

However, some of these maps exhibit a higher degree of anisotropy than others, such as Fig.

By definition,

Although stability over an ice- or water-dominated surface depends on many factors, such as wind direction and potential air temperature, in the cases where potential air temperature falls between the surface temperatures of ice and water, the ice fraction of a sea ice surface can be a fair indicator of the behavior of the MIZ-ABL. However, this is only the case when the ice fraction approaches

While Fig.

To understand what other information can be obtained from a two-dimensional, binary lattice surface, we examined 44 spatial metrics traditionally used in the field of landscape ecology since knowing the cover fraction (ice fraction) and the number and median area of the floes is not enough to able to fully describe the ice–water–atmosphere physics. These 44 additional spatial metrics were applied to LIDPs of real-world satellite sea ice imagery to determine which metrics were important, and the variance inflation factor was used to detect and remove any multicollinearity in this dataset. The remaining metric set included the following metrics: ice fraction, patch density (representing the number of sea ice floes and, thus, their mean size in a given area), the splitting index (representing variance in floe sizes), and the perimeter–area fractal dimension (representing edge tortuosity). We also proposed using the surface eigendirection relative to the mean wind direction to characterize the influence of surface anisotropy and its interaction with the wind direction.

The resulting set of five metrics, including eigendirection, is useful for describing a two-dimensional surface. However, based on the VIF analysis, it also represents a minimal set of indicators needed to describe such a surface since these metrics contain distinct and important information. Nevertheless, the development of practical parameterizations for sea ice and the MIZ-ABL will ultimately need to include additional considerations, including the ease of obtaining these parameters for modeling applications, the computing time needed to calculate these surface metrics dynamically versus that needed to resolve surface features when running an ESM, and the availability of easier-to-compute surrogate metrics.

The first step in answering this broad question is to investigate to what degree these other metrics affect the MIZ-ABL compared to the first-order effect of ice fraction on the MIZ-ABL. In other words, given a specific ice fraction, how will changing any of the metrics in the resulting set affect the overlying MIZ-ABL? While this study addressed this question for idealized surfaces and established the relevance of these parameters under certain conditions, determining which parameters will be critical for real ice maps – and understanding their frequency and impact – requires additional simulations, and a follow-up to this study is underway (see

Another crucial step in answering this question involves figuring out how one would go about creating an accurate parameterization based on available external grid cell variables; the resources needed to answer this question may be extensive, especially considering that, in this age of machine learning, high-resolution synthetic satellite imagery is generated more frequently (see

In this study, incompressible filtered Navier–Stokes equations (along with a Boussinesq approximation for the mean state) and a heat budget are solved for a horizontally periodic flow, where a variable with a tilde represents a quantity filtered via the numerical grid spacing (

An overbar denotes averaging over time, which is used as a surrogate for ensemble Reynolds averaging, while spatial averaging over the heterogeneous domain (in both

The LES employs boundary conditions that are periodic in the horizontal direction, with zero vertical velocity at the top and bottom of the domain, as well as a stress-free top lid (

The initial potential temperature was chosen so that the mean heat flux over the entire domain was zero; in other words, the heat flux going into the ice was equal in magnitude to the heat flux coming from the water, based on the area fraction of the domain (in this case, the ice fraction). Thus, the initial potential air temperature of the large-eddy simulation was chosen to ensure that the ice-fraction-weighted heat flux over the ice (

Table

All landscape metrics used in the VIF analysis conducted in Sect.

A dataset containing the simulation results for the five patterns and the FRAGSTATS output for the sea ice maps are publicly available at

The supplement related to this article is available online at:

JF: conceptualization, methodology, formal analysis, investigation, writing (original draft), and visualization. EBZ: methodology, investigation, formal analysis, writing (review and editing), supervision, and funding acquisition. MB: writing (review and editing) and validation. LB: writing (review and editing) and validation. All authors reviewed the results and approved the final version of the paper.

The contact author has declared that none of the authors has any competing interests.

The statements, findings, conclusions, and recommendations are those of the authors and do not necessarily reflect the views of the National Oceanic and Atmospheric Administration or the US Department of Commerce. Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

We would like to acknowledge the support provided by NCAR's Computational and Information Systems Laboratory for high-performance computing on Cheyenne

This research has been supported by the National Science Foundation (grant no. AGS 2128345) and the US Department of Commerce (grant no. NA18OAR4320123).

This paper was edited by Michel Tsamados and reviewed by Ian Brooks and Christof Lüpkes.