General Systems Theory Concepts in Atmospheric Flows - Rain Formation in Warm Clouds: General Systems Theory - page 9

respectively, the mean and corresponding standard deviation of eddy circulation
speed at that level and the ratio W / w * is the normalized standard deviation σ. The
primary turbulent eddy with circulation speed w * and radius r is the reference
level and normalized standard deviation σ values from 0 to 1 refer to the primary
eddy growth region.
5. Primary eddy growth begins with unit length step perturbation followed by suc-
cessive 10 unit length growth steps (Selvam 1990 , 2012a , b , 2013 ; see Sect. 1.5
below).

1.5 Primary Dominant Eddy Growth Mechanism

in the ABL

1.5.1 Steady-State Fractional Volume Dilution of Large

Eddy by Turbulent Fluctuations

As seen from Figs. 1.2 and 1.3 and from the concept of large eddy growth, vigorous
counter flow (mixing) in turbulent eddy fluctuations characterizes the large-eddy
volume. The total fractional volume dilution rate of the large eddy by turbulent
(eddy) vertical mixing of environmental air across unit cross-section of the large
eddy surface is derived from Eq. (1.1) and is given as follows.
The ratio of the upward mass flux of air in the turbulent eddy to that in the large
eddy across unit cross-section (of the large eddy) per second is equal to w ∗ /d W ,
where w ∗ is the increase in vertical velocity per second of the turbulent eddy due to
the MFC process, and d W is the corresponding increase in vertical velocity of large
eddy. This fractional volume dilution of the large eddy occurs in the environment of
the turbulent eddy. The fractional volume of the large eddy which is in the environ-
ment of the turbulent eddy where dilution occurs is equal to r / R .
Therefore, the total fractional volume dilution k of the large eddy per second
across unit cross-section can be expressed as follows:

w
dW

r
R

*

(1.2)

k = .

The value of k ≈ 0.4 when the length scale ratio R / r is equal to 10 since d W ≈ 0.25
w ∗ (Eq. 1.1). The growing large eddy cannot exist as a recognizable entity for length
scale ratio values less than 10.
Identifiable large eddies can grow only for scale ratios z > 10. The convective
scale eddy of radius R c evolves from the turbulent eddy of radius r for the size ratio
( z ). R c / r = 10. This type of decadic scale range eddy mixing can be visualized to oc-
cur in successive decadic scale ranges generating the convective, meso-, synoptic,
and planetary scale eddies of radii R c , R m , R s , and R p where c, m, s, and p represent,
respectively, the convective, meso-, synoptic, and planetary scales.