A theory of eddy diffusion in the atmosphere

Abstract

The theory of eddy diffusion in the atmosphere put forward almost simultaneously by G. I. Taylor and L. F. Richardson in England and by W. Schmidt in Austria is a direct generalisation of the classical theory of molecular diffusion. It is assumed that the mass effect of the eddies is entirely similar, except for a scale difference, to that of the molecules ; thus we find an eddy-diffusivity of the order of 10 2 to 10 11 cm. 2 /sec. replacing a molecular diffusivity of the order of 10 -1 cm. 2 /sec. in entirely similar differential equations. Recent researches,§ however, have shown that the difference between the eddy structure of a turbulent fluid and the molecular structure of a fluid at rest is more than one of scale, and it is now clear that there is need of an extended theory to express this difference. The failure of the earlier theory to account for the phenomenon of atmospheric diffusion has been made evident by the enormous variations found in K, the eddy conductivity. Richardson has evaluated K for the diffusion of smoke over short distances, for the distribution of volcanic ash, and for the scatter of small balloons, and has found K ’s varying from 10 4 to 10 8 cm. 2 /sec. Other writers have given estimates varying from 10 2 to 10 11 cm. 2 /sec., and in general it has been found that K increases rapidly with the scale of the phenomenon. The present paper is concerned with an attempt to define a new diffusion coefficient which is constant over a field of a few hundred metres to hundreds of kilometres. The basic idea, that the rate of diffusion is governed by what is termed the “ effective eddy ” is not entirely new ; it is inherent in much of Richardson’s work. In the earlier theory it was assumed that the size of the effective eddy remained constant ; here it is assumed that it varies with the distance travelled by the diffusing cluster.