General Systems Theory Concepts in Atmospheric Flows - Rain Formation in Warm Clouds: General Systems Theory - page 27

Earlier studies on the application of the concept of maximum entropy in atmo-
spheric physics are given below. A systems theory approach based on maximum
entropy principle has been applied in cloud physics to obtain useful information on
droplet size distributions without regard to the details of individual droplets (Liu
et al. 1995 ; Liu 1995 ; Liu and Hallett 1997 , 1998 ; Liu and Daum 2001 ; Liu et al.
2002 , 2002 ). Liu et al. ( 2002 ) conclude that a combination of the systems idea
with multiscale approaches seems to be a promising avenue. Checa and Tapiador
( 2011 )have presented a maximum entropy approach to rain drop size distribution
(RDSD) modeling. Liu et al. ( 2011 ) have given a review of the concept of entropy
and its relevant principles, on the organization of atmospheric systems and the prin-
ciple of the second law of thermodynamics, as well as their applications to atmo-
spheric sciences. The maximum entropy production principle (MEPP), at least as
used in climate science, was first hypothesized by Paltridge ( 1978 ).
In the following it is shown that the eddy continuum energy distribution P
(Eqs. 1.16 and 1.20) is the same as the Boltzmann distribution for molecular ener-
gies. From Eq. (1.40)

r
R

R
r

z == e

(1.43)

or

r
R

r
R

−

= e .

The ratio r/R represents the fractional probability P (Eqs. 1.16 and 1.20) of occur-
rence of small-scale fluctuations ( r ) in the large eddy ( R ) environment. Consider-
ing two large eddies of radii R 1 and R 2 ( R 2 greater than R 1 ) and corresponding r.m.s
circulation speeds W 1 and W 2 which grow from the same primary small-scale eddy
of radius r and r.m.s circulation speed w * we have from Eq. (1.1)

2

R
R

W
W

1 2

= .

2

2

1

From Eq. (1.43)

2

W
W

R
R

2

1
2

−

R
R

−

2

(1.44) 1

2

= e = e .

1

The square of r.m.s circulation speed W 2 represents eddy kinetic energy. Following
classical physical concepts (Kikoin and Kikoin 1978 ), the primary (small-scale)
eddy energy may be written in terms of the eddy environment temperature T and
Boltzmann's constant K B as