Semi-analytical eddy-viscosity and backscattering closures for 2D geophysical turbulence

Abstract

Physics-based eddy-viscosity and backscattering closures are widely used for large-eddy simulation (LES) of geophysical turbulence, but their key parameters are often chosen empirically. Here, we develop a semi-analytical framework for estimating these parameters in 2D geophysical turbulence. Specifically, we extend a Lilly-type scaling argument, previously used for 3D turbulence, to 2D geophysical turbulence and obtain closed-form estimates, up to an amplitude constant, for the coefficients of the Leith and Smagorinsky eddy-viscosity closures, a biharmonic eddy-viscosity closure, and the Jansen--Held backscattering closure with a prescribed backscattering fraction. The amplitude constant appears in the turbulent kinetic energy direct-cascade spectrum and can be diagnosed from a few direct numerical simulation (DNS) or eddy-resolving snapshots. For the $\beta$-free cases, the diagnosed amplitude constant is consistent with previous theoretical estimates based on closure, renormalization-group, and mode-coupling methods. The resulting semi-analytical parameters closely match the online-learned values obtained using ensemble Kalman inversion across several 2D geophysical turbulence setups. LES using these parameters reproduces key DNS statistics, including the tails of the vorticity distribution, and robustly outperforms dynamic Leith and Smagorinsky baselines.