Adjoint-based Inversion for the Diffusion Coefficient in Marine Lake Models

Abstract

Marine lakes offer a unique opportunity to study how physical processes, such as turbulent mixing, biomixing, and tidal exchanges, regulate the vertical transport of heat, oxygen, and nutrients in relatively isolated ecosystems. The effective diffusion coefficient characterizes the overall mixing and is critical in models of marine lakes, but it is difficult to measure directly. This paper addresses the problem of inferring the depth-dependent diffusion coefficient from synthetic and real measurements of vertical temperature and salinity profiles. To do so, an inverse problem governed by a marine lake model described by the screened-Poisson equation (similar to Helmholtz) is formulated. The inverse problem is formulated as a nonlinear least-squares optimization problem, where the cost functional quantifies the misfit between the observed and recovered profiles. A Tikhonov regularization term is added to improve the conditioning of the discretized problem, with a regularization parameter chosen using the L-curve method. We solve this problem using an adjoint-based inexact Gauss-Newton method. To leverage the complementary information in these profiles, the formulation optimizes the fit across both datasets simultaneously. The accuracy of the reconstructed diffusion coefficient and the method's robustness to noise are investigated through comprehensive synthetic studies. Additionally, we contrast this framework with physics-informed neural networks, highlighting the advantages and limitations of each. When applied to real marine lake data, the adjoint method reconstructed a smaller diffusion coefficient in regions of strong density stratification, in line with physical expectations. The results indicate that the proposed approach provides a foundation for accurate modeling of marine lake dynamics.