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 ? plane. In terms of the nondimensional number ?=U/(??2), where ? 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 ? ? 1, are as follows: V/U ≈ ?, kj? ≈ ??1, D/(U?) ≈ ?2.Numerical simulations provide qualitative support for this scaling but suggest that it underestimates the sensitivity of these eddy statistics to the value of ?. A generalization that is applicable to continuous stratification is suggested that leads to the estimates V ≈ (?T2)?1, kj ≈ ?T, D ≈ (?2T3)?1,where T is a timescale determined by the environment; in particular, it equals ?U?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, ? ? 1. Therefore, the authors suggest that it may be of general relevance in planetary atmosphere and in the oceans.