Prognostic Equations for the Mean Motions of Simple Fluid Systems and Their Relation to the Theory of Large Scale Atmospheric Turbulence

Abstract

By making use of the mathematical properties of certain special types of weighted means – taken either areawise or timewise – it has been shown that a variable's deviation from its mean can be expressed in terms of the mean itself. This result has been used to derive a closed system of equations for the mean motion of an incompressible autobarotropic fluid. Comparison of the equations for area-averaged and time-averaged motion reveals that the area mean has certain advantages, inherent in the form of the equation for the unaveraged motion. The eddy transport for the case of ≥strong averaging≤ is contrasted with that derived from the classical ≥mixing-length≤ theory of eddy diffusion. The usual restrictions on the statistics of eddy fluctuations are found to be inconsistent with the unrestricted form of the eddy transport. Finally, it has been shown that these methods can be extended to the derivation of closed systems of equations for the mean motions of simple baroclinic fluids.