# A Mathematical Perspective on Marine Biogeochemistry and the Oceanic Carbon Cycle

Oceans contain approximately 96.5 percent of the water on Earth, covering about 70 percent of the Earth’s surface. “The ocean is quite a unique feature of Earth,” Francisco de Melo Viríssimo of the London School of Economics and Political Science said. “Everything you think about the ocean is so grand.” During a minisymposium presentation at the 2022 SIAM Conference on Mathematics of Planet Earth, which took place this week concurrently with the 2022 SIAM Annual Meeting, de Melo Viríssimo provided an overview of marine biogeochemistry and explored the ways in which mathematics contributes to modeling in this complex area.

The ocean is not solely made up of water — it also contains dissolved salts such as sodium and calcium, as well as dissolved gasses, dissolved organic material, and a wide variety of particles in suspension. “That’s very much how a biogeochemist sees the ocean: as a big pool of chemical elements that are being transformed though biological, chemical, and geological processes,” de Melo Viríssimo said. This point of view leads to several fundamental questions: What controls the mean abundance of elements in the ocean? How about their variation in space? And what controls their change over time?

**Figure 1.**The carbon cycle. Public domain image courtesy of the National Oceanic and Atmospheric Administration.

The carbon cycle became an area of study relatively recently, around the 1980s. Tyler Volk and Martin Hoffert wrote one of the first studies that began to elucidate the carbon cycle through the idea of carbon pumps. They proposed one physical pump (a solubility pump), as well as two biological pumps: a carbonate pump and a soft-tissue pump. The solubility pump is driven by physics and causes carbon to sink in cold regions of the ocean. The biological pumps, however, are not as well understood. They involve taking carbon from the water into organic matter, which will eventually sink in the ocean after the organism’s death. The carbon in the organic matter may be consumed by microorganisms; but if it sinks past the bottom of the mesopelagic zone, where light no longer penetrates through the water and photosynthesis cannot occur, then the carbon may remain there for much longer.

The first dedicated expeditions for understanding the carbon cycle in the ocean occurred during the 1980s, including the Vertical Transport and Exchange program which made a massive impact in marine biogeochemistry. This period also saw the first modeling results in the field, which primarily involved fitting the very few data points that were available to develop equations and make predictions.

Three-dimensional biogeochemical modeling had its beginnings in the 1990s. Figure 2 illustrates the evolution that led to this point and occurred afterwards as researchers added new components to modeling frameworks over the decades. “It evolved pretty quickly, from the 1970s when your general circulation models were just the atmosphere, to the 1990s when you had three components and the ocean was really introduced,” de Melo Viríssimo said. “Today you have a lot of things — even detailing dynamical ice sheets.” The first marine biogeochemistry model of which de Melo Viríssimo was aware is the HAMburg Ocean Carbon Cycle Model, which was introduced in 1990. Though at the time it consisted of a single equation for phytoplankton, the model has since been updated and is still in use today.

Scientific fieldwork to investigate the oceanic carbon cycle is quite difficult, so the advent of modeling made it much easier to test hypotheses and allowed the pace of research to grow exponentially. In the 21st century, this has led to new concepts and theories based on modeling—including new types of carbon pumps—and many dedicated research projects and programs. However, still relatively little is known about this incredibly complex system. Researchers do not know the amount of carbon that is exported from the ocean’s surface to the deep ocean, for example; while there have been many papers on this subject, nothing is yet conclusive.

**Figure 2.**Progress over the decades in three-dimensional biogeochemical modeling. Figure from [1].

Marine biogeochemistry models involve a complex set of partial differential equations, and there are several areas in which mathematics and mathematicians can contribute significantly. For instance, mathematical techniques can help significantly speed up simulations. de Melo Viríssimo provided an example of pre-storing ocean circulation, which was the most computationally expensive component of a complicated partial differential equation model — doing so allowed him to perform an extensive study with 50 or more simulations.

“The last bit where mathematics can help a lot is with new, out-of-the-box ideas,” de Melo Viríssimo continued. “As mathematicians, we are trained to think in a very specific way that is quite different from how other sciences are trained. It’s complementary: by the way that you approach these problems, you sometimes can see things that others can’t.”

The talks that followed in the minisymposium session provided a sampling of current work in marine biogeochemistry that mathematicians and other researchers are accomplishing. Enrico Ser-Giacomi (Massachusetts Institute of Technology) presented a talk about a network theory perspective on ocean dynamics. Ser-Giacomi and his collaborators are working to develop a framework for studying physical transport in the ocean—describing fluid flow from the point of view of networks—with several examples of possible applications to marine science.

Dan Lu (Oak Ridge National Laboratory) also presented a talk on approaches to advance biogeochemical modeling and further the knowledge of the carbon cycle through machine learning. She investigated different questions related to data-driven Earth system modeling, including how to create interpretable machine learning models to simulate the terrestrial carbon cycle as well as uncertainty quantification methods for machine learning predictions of a changing climate.

The minisymposium continued with a second session in the afternoon that also showcased aspects of marine biogeochemistry and its modeling. Emily Zakem (Carnegie Institution for Science) presented an unified theory for organic matter accumulation, followed by Camila Serra-Pompei's (Massachusetts Institute of Technology) talk on the links between phytoplankton size spectra, community composition, and carbon export efficiency. Christopher Follett (Massachusetts Institute of Technology) presented exciting results on the predictive power of niche models to understand plankton dynamics in a warming world, and James Watson (Oregon State University) closed out the session with a description of the cross-scale oceanographic drivers of human activity on the oceans.

Overall, marine biogeochemistry is a very exciting field of study that, despite its growing appeal, still has many open questions. de Melo Viríssimo hoped that this minisymposium session would draw more attention from the mathematics community for the many interesting and important problems remaining in this area.

**Acknowledgements:** Francisco de Melo Viríssimo acknowledged Adrian Martin, Steph Henson, and Andrew Yool (National Oceanography Centre), Jamie Wilson (University of Bristol), and Raffa Bernadello (Barcelona Supercomputing Center). de Melo Viríssimo also thanked the Institute of Mathematics and its Applications and the London Mathematical Society for their generous early-career grants that partially funded this activity.

**References**

[1] Verronen, P.T., & Schmidt, H. (2016). Numerical models of atmosphere and ocean. In K. Matthes, T.D. de Wit, & J. Lilensten (Eds.), *Earth’s climate response to a changing Sun*. Les Ulis, France: EDP Sciences.

Jillian Kunze is the associate editor of SIAM News. |