General Systems Theory Concepts in Atmospheric Flows - Rain Formation in Warm Clouds: General Systems Theory - page 8

Fig. 1.4 Internal structure of
large eddy circulations
Based on Townsend's concept, the observed eddy continuum eddy growth in the
ABL is visualized to occur in the following two stages:
1. Growth of primary dominant turbulent eddy circulation from successive equal
fluctuation length step increment d z equal to one. Identifiable organized whole
turbulent eddy circulation forms at length step z = 10 associated with fractional
volume dilution by eddy mixing less than half so that its (eddy) identity is not
erased.
2. Large eddies are then visualized to form as envelopes enclosing these dominant
turbulent eddies starting from unit primary eddy as zero level. Spatial domain
of large eddy is obtained by integration from initial radius r to large eddy radius
R (Eq. 1.1). The growing large eddy traces logarithmic spiral circulation with
quasiperiodic Penrose tiling pattern (Fig. 1.4 ) for the internal structure such that
successive large eddy radii follow the Fibonacci number series and ratio of suc-
cessive radii R n + 1 / R n is equal to the golden mean τ (≈ 1.618).
3. Further it is shown that the variance (square of amplitude) spectrum and ampli-
tude probability distribution of fractal fluctuations are represented by the same
function, i.e., W and W 2 have the same probability distribution. Such a result that
the additive amplitude of eddies when squared represent the probability distribu-
tion of amplitudes is exhibited by microscopic scale quantum systems such as
the electron or photon. Fractal fluctuations therefore exhibit quantum-like chaos.
4. At each level, the integrated mean of eddy circulation speed w * gives the