Online learning of eddy-viscosity and backscattering closures for geophysical turbulence using ensemble Kalman inversion

This article is part of a Physical Review Collection on the Physics of Changing Climate.

Different approaches to using data-driven methods for subgrid-scale closure modeling of geophysical turbulence have emerged recently. Most of these approaches are data hungry and lack interpretability and out-of-distribution generalizability. Here, we use a hybrid approach that combines turbulence theory, physics-based modeling, and data-driven methods to overcome these challenges. Specifically, we address the parametric uncertainty of well-known physics-based large-eddy simulation (LES) closures: the Smagorinsky (Smag) and Leith eddy-viscosity models (one free parameter) and the Jansen-Held (JH) backscattering model (two free parameters). For various cases of two-dimensional turbulence, optimal parameters are first learned online from data via ensemble Kalman inversion (EKI), such that for each case, the LES energy spectrum matches that of direct numerical simulation (DNS). We quantify the uncertainties on these parameters using a modern machine-learning-accelerated Bayesian workflow, “Calibrate, Emulate, Sample.” Only a small training dataset is needed (to calculate the DNS spectra); i.e., the approach is data-efficient. We find the optimized parameter(s) and their associated uncertainty for each closure to be constant across broad flow regimes that differ in dominant length scales, eddy/jet structures, and dynamics, suggesting that these closures are generalizable. Next, we show that the online learned constants agree with the predictions of a recent semianalytical derivation, providing further interpretability. In both a priori and a posteriori tests that include examining the extreme events, LES with optimized closures, especially with JH, outperforms the baselines (LES with standard Smag, dynamic Smag, or Leith). This work shows the promise of combining advances in theory, physics-based modeling (e.g., JH), and data-driven modeling (e.g., online learning with EKI) to develop data-efficient frameworks for accurate, interpretable, and generalizable closures for geophysical turbulence, with ultimate applications in weather and climate prediction.