# Mathematical Models for Global Mean Sea Level Rise

by Stephen Kaczkowski

Year of Award: 2018

Award: Pólya

Publication Information: The College Mathematics Journal, Volume 48, Number 3, May 2017, pages 162-169.

Summary:  It is not every day that one can address a topic of urgent interest in the calculus classroom. We often say that mathematics training develops modeling and problem solving skills, but the examples we present are often poor, with the result that our students lack confidence when it is time to address a real problem. The paper fills a gap, by illustrating to an undergraduate audience how mathematical modeling can be used to explain climate phenomena including, here, the important phenomenon of sea level rise caused by global climate change. The paper first presents the main causes for the rise in ocean level, namely (1) thermal expansion due to rising temperatures of the ocean water, and (2) melting of the ice caps over Greenland and Antarctica, which releases huge amounts of fresh water into the oceans. The mathematics of the paper then focuses on (1) for the period 1971 to 2010. Under a simplifying hypothesis of constant salinity, the paper first presents the relationship between seawater density on one side, and temperature and pressure on the other side. These relationships do not exist as “formulas”, and have to be determined empirically from curves or data appearing in the literature. It is the decrease in the seawater density that results in a rise of the global mean sea level. This occurs in the upper layers of the oceans, down to a depth of 700 meters. Since the phenomenon depends on the depth, the global mean sea level rise h is obtained by integrating the expansion of the volume of the ocean in thin layers of constant depth, and this expansion of the volume comes from the change of density Δρ of the seawater due to the temperature increase. Evaluating this Δρ as a function of depth from 1971 to 2010 requires using data in the literature at four different depths and extrapolating from these. The results obtained are within the range of the estimations of the 2013 Intergovernmental Panel on Climate Change (IPCC) report, namely an average of 0.6 mm/yr of sea level rise since 1971, inside a confidence interval [0:4; 0:8]. The paper then briefly addresses the effect of the melting of the ice sheets and glaciers on land.

The paper is very well written. The hypotheses and simplifications are clearly stated, both in the modeling part and in the analysis of the equations. Although the author cautions that the model is not developed enough to use for future predictions, the paper ends with further questions and an invitation to more exploration. Climate studies are very complex and good modeling is an art: one must capture in the model the essential features and neglect the others. But mathematics alone cannot tell what are the essential features! Reading this paper, inexperienced mathematicians can gain confidence in their ability to contribute to the modeling of a complex system; yet the paper is honest about the limitations of a simple model. It is hoped that it will inspire students of calculus—and their teachers—to pursue these ideas further and learn more about the subject!

Response from the Author:

I am so grateful that you have chosen to recognize my work by honoring me with this award. I thank the editors and referees of The College Mathematics Journal for their role in encouraging teaching and research in mathematics. This project was inspired by my wife’s coastal engineering work along various U.S. East Coast beaches; sea level rise is frequently discussed by the engineers and scientists who work in shoreline management, and it is good to see that an environmental topic like this one has struck a chord within the mathematical community.