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.

Submitted to Journal of the Atmospheric Sciences. Latex source file. Figures are included in an uuencoded file. Revision only involves submission format, not content of paper. User may wish to obtain postscript version of full paper from http://www.gfdl.gov/~ih

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