Elsevier

Continental Shelf Research

Volume 210, 26 December 2020, 104278
Continental Shelf Research

Research papers
Mathematical modeling of Arctic sea ice freezing and melting based on nonlinear growth theory

https://doi.org/10.1016/j.csr.2020.104278Get rights and content

Highlights

  • Nonlinear growth theory is proposed as the freezing and melting model of Arctic sea ice.

  • Mathematical expressions are obtained to simulate the periodic cycle of sea ice coverage.

  • Net sea surface heat flux primarily contributes to the dynamic freezing and melting processes.

Abstract

The spatiotemporal distribution of sea ice in the cold Arctic region is of climatic and engineering significance. However, in terms of statistics, there remains a lack of quantitative mathematical descriptions of the annual periodic sea ice cycle for the entire Arctic and its regional seas. Using the semi-enclosed and seasonally ice-covered Kara Sea as a systematic case study, this quantitative study focuses on the theoretical mathematical expressions of the nonlinear freezing and melting processes of sea ice. The periodic cycle of sea ice coverage was divided into freezing, frozen, and melting stages, with the continuous ice-coverage curves clearly highlighting temporal nonlinearity during both freezing and melting periods. Thus, nonlinear growth theory was used to quantitatively simulate ice-coverage curves. Results from a comparison of the various functions showed the Logistic function to most accurately quantify the annual periodic cycle, followed by the Gompertz and von Bertalanffy functions, with the Negative exponential and Brody functions having the least suitable results. Further, our results indicate that net sea surface heat flux primarily contributes to the nonlinear temporal freezing and melting of sea ice in the periodic cycle. The nonlinear growth theory was also applied to other Arctic sea areas, including the Laptev Sea, East Siberian Sea, Chukchi Sea, Beaufort Sea, and Baffin Bay. By using the developed mathematical equations, we are able to quantify sea ice conditions at any given time for different Arctic sea areas.

Introduction

The extensive sea ice cover in the cold Arctic region forms one of the most remarkable geophysical settings. On the one hand, sea ice plays an important role in polar and global climate systems, with changes having a profound influence on the marine environment, atmospheric circulation, and climate change (e.g., Smetacek and Nicol, 2005; Screen and Simmonds, 2010; Collow et al., 2019). On the other hand, the occurrence of ice can be hazardous and potentially threaten navigation passages and marine structures (e.g., Timco and Weeks, 2010; Kujala et al., 2019; Li et al., 2020). Thus, the spatiotemporal variations of Arctic sea ice are an important research topic.

Since the beginning of satellite monitoring in 1979, the majority of Arctic sea ice research has focused primarily on long-term and inter-annual variability of sea ice at a global (e.g., Laxon et al., 2003; Comiso et al., 2008; Stroeve et al., 2012), or regional and local scale (e.g., Belchansky et al., 2004; Meier et al., 2007; Onarheim et al., 2018). Results from these studies indicate that sea ice extent reaches a seasonal maximum in March, and retreats to a minimum in September. Notably, a decline in sea ice extent is accelerating over the summer months. In general, satellite-observed spatiotemporal variations of sea ice can be simulated using a number of thermal-dynamic coupling models, such as CICE, LIM, MITgcm, GFDL, and neXtSIM (Losch et al., 2010; Stroeve et al., 2012; Hunke, 2014; Bouillon and Rampal, 2015). However, in terms of statistics, quantitative mathematical descriptions of the annually periodic sea ice cycle have not yet been investigated or resolved for the entire Arctic and its regional seas. The use of suitable mathematical expressions allows for the theoretical investigation of the continuous and dynamic sea ice processes, thereby, improving our understanding of time-varying sea ice conditions, such as dynamic freezing and melting.

In view of the complicated Arctic land and sea distributions, the Kara Sea, owing to its semi-enclosed geographical nature (Divine et al., 2004; Duan et al., 2019), can be represented as a suitable case study for the investigation of periodic sea ice cycles using quantitative mathematical descriptions. The Kara Sea is a seasonally ice-covered Arctic sea shelf, enclosed by various continents and islands, including southern and eastern Eurasia, northeastern Severnaya Zemlya, northwestern Franz Josef Land, and western Novaya Zemlya (Fig. 1). Unlike the western Barents Sea, which receives large amounts of inflowing heat from the North Atlantic warm waters, the Kara Sea remains relatively colder due to the blocking effect of Novaya Zemlya.

Using the semi-enclosed Kara Sea as a systematic case study, we developed theoretical mathematical expressions and used nonlinear growth theory to model and quantitatively describe the periodic cycle of sea ice in the Arctic. The remainder of this article consists of four sections. Firstly, we define and describe sea ice coverage and nonlinear growth theory. Secondly, we present the results from our analyses of sea ice spatiotemporal characteristics, sea ice development stages, and simulated periodic cycles of nonlinear sea ice coverage. Thirdly, we use the thermodynamic conditions of sea ice to interpret the results from established nonlinear models. For a broader perspective, we apply the nonlinear growth theory to other Arctic areas, with further sea ice freezing and melting simulations. Finally, we summarize the relevant conclusions from our study.

Section snippets

Sea ice coverage

Long-term daily satellite sea ice concentration (SIC) data from the Nimbus-7 SMMR and DMSP SSM/I-SSMIS Passive Microwave Data from the National Snow and Ice Data Center (NSIDC) (Cavalieri et al., 1996) were used in this study. The temporal coverage ranged from January 1979 to December 2018, with a spatial resolution of 25 × 25 km. Sea ice coverage was selected to quantitatively identify temporal variations in sea ice extent. Notably, sea ice coverage, especially in marginal ice zones, can

Spatiotemporal variability of sea ice

It is a basic requirement to overview the spatiotemporal distributions of sea ice cover in the Kara Sea. Hence, we used NSIDC daily satellite SIC data from 1979 to 2018 to investigate mean monthly SIC spatial patterns.

Our results indicate that the distribution of sea ice shows a regular periodic freezing and melting cycle for the Kara Sea during the 40-year period from 1979 to 2018 (Fig. 2). Both first-year and multi-year sea ice occur in the Kara Sea. In general, from January to April, sea ice

Comparison of nonlinear growth theory and polynomial method

The sea ice curves modeled in this study are characterized by nonlinear temporal freezing and melting processes, with the Logistic growth function showing the best results in the simulation of periodic sea ice coverage cycles. This finding is in good agreement with the previous attempt (Duan et al., 2019), and is more scientific on the basis of the added frozen stage. Specifically, the developed obtained mathematical expressions showed better results for the freezing stage than the melting

Conclusion

The results of our study allow for a better understanding of the dynamic freezing and melting processes of Arctic sea ice. Using nonlinear growth theory, we quantitatively obtained a theoretical mathematical description of the annual sea ice periodic cycle.

The semi-enclosed and seasonally ice-covered Kara Sea was used as a case study to conduct a systematic modeling analysis. The periodic cycle of sea ice coverage could be divided into freezing, frozen, and melting stages. The nonlinear growth

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors are very grateful to the two excellent anonymous reviewers and Editor Andrea S. Ogston for the valuable comments and suggestions. The authors are also grateful for the public satellite sea ice data from NSIDC (https://nsidc.org/data/NSIDC-0051/) and the metocean reanalysis data from ECMWF (http://apps.ecmwf.int/datasets/). This work was financially supported by the NSFC-Shandong Joint Foundation (No. U1706226), the National Natural Science Foundation of China (No. 51779236), and the

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