Learning Stable and Generalizable Closed-form Equations for Geophysical Turbulence*

In atmospheric and oceanic processes where turbulent flows are prevalent, smaller-scale (subgrid-scale) processes that cannot be resolved in the model are parameterized. Here, we employ the 'equation-discovery' approach, a machine learning strategy streamlined to derive the governing equations of a system, to learn closures from filtered direct numerical simulations of 2D forced turbulence. This approach has shown that these closures rely on nonlinear combinations of gradients of filtered velocity, with constants that are not dependent on the fluid or flow properties but are contingent on filter type and size. We demonstrate that these closures are closely aligned with the nonlinear gradient model (GM), which is derivable analytically using Taylor-series expansions. We find that large-eddy simulations with higher-order terms from the Taylor series expansions are stable and they perform as well as or better than the traditional physics-based parameterizations. The proposed model has significant similarities between the true and GM-predicted fluxes and can capture the energy and enstrophy transfer between the resolved and the subgrid-scales, predicting both diffusion and backscattering. This interpretable closure is generalizable for closure modeling of any geophysical system.