A Scaling Theory for Horizontally Homogeneous, Baroclinically Unstable Flow on a Beta Plane

Abstract The scaling argument developed by the authors in a previous work for eddy amplitudes and fluxes in a horizontally homogeneous, two-layer model on an f plane is extended to a β plane. In terms of the nondimensional number ξ=U/(βλ2), where λ is the deformation radius and U is the mean thermal wind, the result for the rms eddy velocity V, the characteristic wavenumber of the energy-containing eddies and of the eddy-driven jets kj, and the magnitude of the eddy diffusivity for potential vorticity D, in the limit ξ ≫ 1, are as follows: V/U ≈ ξ, kjλ ≈ ξ−1, D/(Uλ) ≈ ξ2.Numerical simulations provide qualitative support for this scaling but suggest that it underestimates the sensitivity of these eddy statistics to the value of ξ. A generalization that is applicable to continuous stratification is suggested that leads to the estimates V ≈ (βT2)−1, kj ≈ βT, D ≈ (β2T3)−1,where T is a timescale determined by the environment; in particular, it equals λU−1 in the two-layer model and N(f∂zU)−1 in a continuous fl...