A posteriori learning of closures for geophysical turbulence using ensemble Kalman inversion*

In the era of big data, one major question is whether data-driven methods can shed new light on traditional physics-based turbulence subgrid-scale models (SGM). To answer this question, we develop data-optimized SGMs for large eddy simulation (LES) of canonical geophysical turbulent flow, i.e., forced 2D (beta-plane) turbulence, using ensemble Kalman inversion (EKI). The EKI method has the merit of being derivative-free, thus enabling a posteriori learning: to learn SGM parameters from chosen direct numerical simulation (DNS) statistics, such as the turbulent kinetic energy spectra. To further quantify the uncertainty of the optimized coefficients, we use the calibrate, emulate, and sample (CES) algorithm. We apply EKI as the “calibration” part of CES to optimize traditional LES SGMs, e.g., the Smagorinsky, Leith, and backscattering models, for a wide range of systems. We have found that the optimized Smagorinsky and Leith coefficients are universal across different systems, i.e., independent of LES grid resolution, forcing wavenumber, Reynolds number, and beta-plane waves. The universality here is verified from large support of the intersections of posterior distribution across all the systems. While the optimized eddy-viscosity SGMs (Smagorinsky and Leith)  perform well in cases where backscattering is not significant, they cannot capture some of the flow statistics for those cases where the inverse cascade/ backscattering dominates. In such cases, backscattering SGMs are required for better performance. Specifically, we find that the Jansen-Held model consisting of a biharmonic diffusion and an anti-diffusion term, with optimized coefficients by EKI, performs the best in matching the flow statistics, particularly in capturing the extreme events and a priori metrics such as the interscale energy/enstrophy transfers. Our findings further show the power of the CES method in addressing the parametric uncertainty of physics-based SGMs when only a small amount of data is available from the original system, where invariant statistics can be used as the EKI target.