Ocean–atmosphere turbulent flux algorithms in Earth system models do not always converge to unique and physical solutions: analysis and potential remedy in E3SMv2

Abstract. The development of physics parameterizations in Earth system models typically emphasizes whether the intended physics is reasonably represented, while mathematical aspects such as solvability of the governing equations and convergence of the numerical algorithms used to approximate their solutions receive far less attention. In this paper, we examine these mathematical issues for a widely used ocean–atmosphere turbulent flux parameterization and its implementation in the Energy Exascale Earth System Model version 2 (E3SMv2). We show that, under simulated meteorological conditions, the parameterization can yield no solution or multiple (including unintended) solutions. These problems arise primarily from (1) a discontinuity in the formulation of the neutral exchange coefficients and (2) the use of an ad hoc limiter on the Monin–Obukhov length to address a singularity in its definition. Compounding these problems is the fact that interventions of calculations such as limiters are often thought to have only a "minor" effect on numerical algorithms and are not documented in technical model descriptions. To address these solvability issues, we propose (1) a regularization that enforces continuity in the neutral exchange coefficients and (2) an adaptive procedure for selecting limiting values of the Monin–Obukhov length based on mathematical analysis of solution uniqueness. Implementing these revisions in E3SMv2 leads to statistically significant changes in the simulated latent heat fluxes over the mid-latitude oceans in the winter hemisphere as well as over the subtropical and tropical oceans. Overall, this work improves the well-posedness and numerical accuracy of ocean–atmosphere turbulent flux calculations in E3SMv2. Moreover, because discontinuities and ad hoc limiters are frequently encountered in physics parameterizations, this work serves as an example of how non-existence and non-uniqueness issues in parameterizations can be identified, analyzed, and resolved.