Variational optimization-gradient descent for one-dimensional heat flux estimation in water with an adaptive line search algorithm

Vertical mixing of heat across thermal stratification in water has significant implications for ecology in lakes and for larger scale climate dynamics through effects on oceanic circulation. Current measurements of vertical heat fluxes often rely on sparse acoustic measurements. To address this, we created a data-driven tool to estimate heat fluxes from observed temperature profile data using variational optimization. This method converges to root mean square errors (RMSE) on the order of 10−4 °C for synthetic data sets with varying fluxes. In the process of testing, a novel line search algorithm FABLS (fully adaptive backtracking line search) was developed. This line search algorithm was developed with a goal of minimizing the number of function evaluations to optimize computational efficiency, motivated specifically by expensive fluid modeling problems. FABLS was compared with golden section search and a constant step size (CSS) approach and generally outperformed both on the basis of number of function evaluations to convergence. Comparisons suggest that optimizing for step size is generally more efficient, but a well-selected constant step size magnitude may be advantageous when the profile being assimilated is exceptionally smooth. Following validation, the solver was also applied to microstructure field data observations of shear instabilities in Lake Erie and converged to RMSE on the order of 10−2 °C. Solver-extracted fluxes from the field data set are used to approximate diffusivities using a flux-gradient approach. The method presented in this study is accurate, efficient, and increases our ability to analyze scalar mixing and transport in lakes and oceans.