In this paper we advance physical background of the energy- and flux-budget turbulence closure based on the budget equations for the turbulent kinetic and potential energies and turbulent fluxes of momentum and buoyancy, and a new relaxation equation for the turbulent dissipation time-scale. The closure is designed for stratified geophysical flows from neutral to very stable and accounts for the Earth rotation. In accordance to modern experimental evidence, the closure implies maintaining of turbulence by the velocity shear at any gradient Richardson number Ri, and distinguishes between the two principally different regimes: strong turbulence at Ri << 1 typical of boundary-layer flows and characterised by the practically constant turbulent Prandtl number; and weak turbulence at Ri > 1 typical of the free atmosphere or deep ocean, where the turbulent Prandtl number asymptotically linearly increases with increasing Ri (which implies very strong suppression of the heat transfer compared to the momentum transfer). For use in different applications, the closure is formulated at different levels of complexity, from the local algebraic model relevant to the steady-state regime of turbulence to a hierarchy of non-local closures including simpler down-gradient models, presented in terms of the eddy-viscosity and eddy-conductivity, and general non-gradient model based on prognostic equations for all basic parameters of turbulence including turbulent fluxes.
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