A scaling theory for horizontally homogeneous, baroclinically unstable flow on a beta plane

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 k_j, and the magnitude of the eddy diffusivity for potential vorticity D, in the limit ξ ≫ 1, are as follows: V/U ≈ξ, kⁱλ ≈ ^-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, k_j ≈ 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 flow with uniform shear and stratification. This same scaling has also been suggested as relevant to a continuously stratified fluid in the opposite limit, ξ ≪ 1. Therefore, the authors suggest that it may be of general relevance in planetary atmospheres and in the oceans.