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

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 two-dimensional (2-D) geophysical turbulence by adapting a Lilly-type scaling argument. Specifically, we 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 isotropic cases (

), the diagnosed amplitude constant is consistent with previous theoretical estimates based on closure, renormalisation-group and mode-coupling methods. The resulting semi-analytical parameters closely match the online-learned values obtained using ensemble Kalman inversion across several 2-D geophysical turbulence set-ups (Guan et al. Phys. Rev. Res. vol. 8, 2026, p. 023215). LES using these parameters reproduces key DNS statistics, including the tails of the vorticity distribution, and robustly outperforms dynamic Leith and Smagorinsky baselines.