4.1 Basic Equations

Based on Townsend's ( Townsend ,1956 References) concept that large eddies are the envelopes of enclosed turbulent eddy circulations, the relationship between the large and turbulent eddy circulation speeds (*W* and *w** _{*} *) and radii (

(3)

Since the large eddy is the integrated mean of enclosed turbulent eddy circulations, the eddy energy (kinetic) spectrum follows statistical normal distribution. Therefore, square of the eddy amplitude or the variance represents the probability. Such a result that the additive amplitudes of eddies, when squared, represent the probability densities is obtained for the subatomic dynamics of quantum systems such as the electron or photon(Maddox,1988a,1993 References). Atmospheric flows, therefore, follow quantumlike mechanical laws. Incidentally, one of the strangest things about physics is that we seem to need two different kinds of mechanics, quantum mechanics for microscopic dynamics of quantum systems and classical mechanics for macroscale phenomena (Rae,1986,1988;clough,1988 References). Visualization of atmospheric flows as a quantum system is consistent with the well known fact that spectral analyses of meteorological data (spatial or temporal) show that superposition of periodicities of all scales contribute to the observed fluctuations at any location or instant of time.

The above visualization of the unified network of atmospheric flows as a quantum system is consistent with Grossing's (Grossing, 1989 References) concept of quantum systems *as order out of* *chaos* phenomena. Macroscale quantum systems have been reported(Bardeen,1990 References). Classical physical concepts such as wave-trains help visualize quantum systems (Nauenberg et al., 1994 References).Order and chaos have been reported in strong fields in quantum systems(Richards,1988;Brown,1996 References).

Writing Equation 3 in terms of the periodicities *T* and *t* of large and small eddies respectively, where

and

we obtain

(4)

Equation 4 is analogous to *Kepler*'s third law of planetary motion, namely, the square of the planet's year (period) to the cube of the planet's mean distance from the Sun is the same for all planets (Uvarov et al. 1979;Narlikar ,1982;1996; Weinberg ,1993 References). Uzer et al.(1991) have discussed new developments within the last two decades which have spurred a remarkable revival of interest in the application of classical mechanical laws to quantum systems. The atom was originally visualized as a miniature solar system based on the assumption that the laws of classical mechanics apply equally to electrons and planets. However within a short interval of time the new quantum mechanics of *Schrodinger* and *Heisenberg *became established (from the late 1920s) and the analogy between the structure of the atom and that of the solar system seemed invalid and classical mechanics became the domain of the astronomers. There is now a revival of interest in classical and semiclassical methods which are found to be unrivaled in providing an intuitive and computationally tractable approach to the study of atomic, molecular and nuclear dynamics.

Observations show (Section 6) that atmospheric eddy flow structure follows *Kepler*'s third law of planetary motion consistent with model prediction (Equation 4).