A scaling theory for the macroturbulence of weakly supercritical planetary atmospheres

Abstract

A theory for the general circulation of the atmosphere must be based on a theory of its macroturbulence. A central component is the near-surface eddy heat flux, which is closely tied to the mass transport of the surface branch of the circulation. Existing scaling theories are largely formulated within the two-layer quasi-geostrophic framework, which assumes supercritical states and an inverse cascade of kinetic energy---conditions that are frequently not satisfied in planetary atmospheres, including Earth's. Nevertheless, such theories have shown some empirical success, including that the Rhines scale is associated with the mixing length even in the absence of any inverse energy cascade. Here, we analyze hundreds of idealized, dry general circulation model simulations spanning wide ranges of rotation rates, meridional temperature gradients, vertical stratification, and seasonality. Most simulations reside in the marginally critical regime where strongly nonlinear scaling theories fail. We propose a new theory that captures both domain-averaged and local eddy heat flux behavior across all simulations. The theory uses scaling arguments for the zonal momentum balance and accounts for kinematic effects on eddy mixing. The scaling depends on a non-dimensional thermal Rossby number and implicitly accounts for the role of nonlinear eddy momentum fluxes across jets. We demonstrate the skill of the proposed scaling in explaining global near-surface temperature distributions across a wide range of climate states in a simple energy balance model. Implications for general circulation theory and extending the scalings to moist atmospheres are discussed.