Eddy Diffusivity in Barotropic Turbulence on a β-plane with Quadratic Bottom Drag

Geostrophic turbulent eddies are crucial in the oceans for transporting and mixing properties, but they cannot be fully resolved by IPCC-class models, which therefore rely on adequate eddy parameterization schemes. A useful reduced model for geostrophic turbulence is barotropic (2D) turbulence, which can be used to develop a fundamental understanding of meso-scale turbulent mixing. The focus of this study is on 2D β-plane turbulence with quadratic bottom drag, which, although is arguably the most realistic 2D model of ocean turbulence, has remained unexplored thus far. Both β and the magnitude of the quadratic drag affect the halting scale of the inverse energy cascade. While quadratic drag can halt the cascade by removing eddy kinetic energy (EKE), the β effect introduces a wave-turbulence crossover and causes a channeling of EKE into zonal jets. Their relative importance are governed by a single non-dimensional parameter, ξ, which is defined as the ratio of the frictional halting wavenumber to the wave-turbulence crossover wavenumber. For large ξ the eddy diffusivity becomes independent of β, while for small ξ it becomes approximately independent of friction. In between is a "transition regime", which is likely to be most relevant for Earth's ocean. In the transition regime, the eddy mixing length is significantly suppressed by the β-effect and is well approximated by the Rhines scale, but the EKE level remains primarily controlled by the bottom drag. By parameterizing the non-linear eddy-eddy interaction as a stochastic white noise forcing and linear damping process, we derive a generalized analytical solution for the eddy diffusivity across all 3 regimes. The result connects recent arguments about the suppression of eddy mixing by flow-relative phase propagation, to traditional scaling theories for β-plane turbulence. The theory is tested using fully nonlinear numerical simulations at eddy-resolving resolution. The numerical results confirm the predictions of the analytical theory.