Data-Driven Diffusion Coefficient Estimation in Marine Lakes

Marine lakes, located near the sea, allow limited exchanges between the lake and the nearby sea. We provide an efficient method to estimate, as a function of depth, the diffusion coefficient, D, within the lake using easily available measurements for quantities such as temperature, salinity, or both. Our method solves for the spatially varying D by minimizing the misfit to data subject to a regularization condition. We model the system with a Helmholtz-type partial differential equation (PDE) that encapsulates specific characteristics of marine lakes, including the exchanges between the lake and the nearby sea and the absorption of solar heat by algae sheets beneath the lake's surface. Our empirical investigation unveils mathematical requirements of the model that underscore the previously unsuspected importance of the role played by algae sheets as a heat source in this ecosystem. By offering a versatile and insightful tool, our method not only contributes to a better understanding of marine ecosystems but also holds promise for broader applications.