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

Abstract

The scaling argument developed by Larichev and Held (1995) for eddy amplitudes and fluxes in a horizontally homogeneous, two-layer model on an f-plane is extended to a beta-plane. In terms of the non-dimensional number x = U/(beta*lambda^2), where lambda 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, and the magnitude of the eddy diffusivity for potential vorticity D, in the limit x>>1, are as follows: V/U ~ x ; k*lambda ~ 1/x ; D/(U*lambda) ~ x^2. Numerical simulations provide qualitative support for this scaling, but suggest that it underestimates the sensitivity of these eddy statistics to the value of x. A generalization that is applicable to continuous stratification is suggested which leads to the estimates: V ~ 1/(beta T^2); k ~ beta*T; D ~ 1/{[beta^2][T^3]} where T is a time-scale determined by the environment; in particular, it equals lambda/U in the two-layer model and N/(fS) in a continuous flow with uniform shear S and stratification N. This same scaling has also been suggested as relevant to a continuously stratified fluid in the opposite limit, x<<1 (Held, 1980). Therefore, we suggest that it may be of general relevance in planetary atmospheres and in the oceans.