In recent years there is growing conviction that conventional formulations of natural laws, their mathematical models and computer realizations are inherently unrealistic (Davies, 1988). An urgent need is now felt for alternative conceptual models with exact solutions and robust computational techniques for the prediction of the evolution of dynamical systems. Fluid flows in particular with their enormous degrees of freedom cannot be modelled with conventional techniques (Ottino

The mean flow at the planetary atmospheric boundary layer (ABL) possesses an inherent upward momentum flux of surface frictional origin. The turbulence scale upward momentum flux is amplified progressively by the exponential decrease of atmospheric density with height coupled with buoyant energy generation by microscale fractional condensation (MFC) on hygroscopic nuclei even in an unsaturated environment (Pruppacher and Klett, 1978). The mean upward momentum flux generates progressively larger vortex roll or large eddy circulations which are manifested as cloud rows/ streets in global cloud cover pattern. The space-time integrated mean of the non-trivial internal turbulence scale buoyant energy circulations generates the observed large eddies. The root mean square (r.m.s) circulation speed

(1)

The above equation expresses the relationship
between domain size and buoyant energy content in terms of the microscopic
scale dynamical processes and is therefore analogous to (1) the renormalization
theory in statistical physics where the idea is to relate scale transformations
to changes of the variable, e.g. buoyant energy and (2) the Ising model
for phase transformations (Lebowitz Large eddy growth occurs by turbulent buoyant acceleration

(2)

Turbulence scale yardsticks for length and
time for large eddy growth implied by Eq.(1) is intrinsic to the concept
of large eddy growth from turbulence scale buoyant energy generation. Such
a concept of large eddy growth by successive length scales doubling from
turbulence scale energy pumping with two-way energy feedback between the
larger and smaller scales is analogous to the following phenomena in other
fields of physics. (1) The universal period doubling route to chaos or
deterministic chaos in disparate dynamical systems (Fairbairn, 1986; Chernikov
The above method of determination of large eddy evolution using turbulence scale yardsticks for length and time is analogous to the concept of 'cellular automata' computational technique which belongs to the non-deterministic cell dynamical system method of determination of the evolution of dynamical systems (Oona and Puri, 1988). Further, the macro-scale eddy evolution occurs as a natural consequence of inherent microscopic scale dynamical processes as given in Eq.(2).

The growing large eddy carries the turbulent eddies as internal circulations. The turbulent eddy fluctuations mix environmental air into the large eddy volume. The steady state fractional volume dilution

(3)

Using Eq.(1) it may be computed and shown
that (4)

The above equation is the well known logarithmic
wind profile relationship for surface boundary layer obtained by conventional
eddy diffusion theories (Holton, 1979) where The rising large eddy gets progressively diluted by vertical mixing due to turbulent eddy fluctuations and a fraction

(5)

From Eqs.(1)-(4)
(6)

The steady state fractional air mass flux
from the surface is dependent only on the dominant turbulent eddy radius.
The vertical profile of the ratio of the actual liquid water content (The self-organized eddy growth in the ABL involves successive radial growth steps equal to turbulence length scale along with a corresponding eddy angular rotation

Figure 1: The logarithmic spiral geometric
design of circulation trajectories in atmospheric flows

The angular rotation from the origin at
location A is measured with respect to axis OX. Let OA and OB denote the
locations of the large eddy radii
*R *and *R+dR* for a growth
period of one second. The angular rotation *d**q
i*s given by

(7)

This is the equation for an equiangular logarithmic
spiral when the crossing angle a
is a constant. At any location A the wind flow into the eddy continuum
system traces out a logarithmic spiral geometrical pattern.
Table 1

R |
W_{n} |
dR |
dq |
W_{n+1} |
q |

2 3.254 5.239 8.425 13.546 21.780 35.019 56.305 90.530 |
1.254 1.985 3.186 5.121 8.234 13.239 21.286 34.225 55.029 |
1.254 1.985 3.186 5.121 8.234 13.239 21.286 34.225 55.029 |
0.627 0.610 0.608 0.608 0.608 0.608 0.608 0.608 0.608 |
1.985 3.186 5.121 8.234 13.239 21.286 34.225 55.029 88.479 |
1.627 2.237 2.845 3.453 4.061 4.669 5.277 5.885 6.493 |

Table 1 shows that the period doubling
growth sequence generates as a natural consequence successive large eddy
lengths *R* which follow the Fibonacci series, i.e.

with the golden mean winding number

here

One complete large eddy circulation is traced out in five length steps and therefore the radius of the dominant large eddy =OR

The time period of large eddy circulation made up of internal circulations with Fibonacci winding number is arrived at as follows. Assuming turbulence scale yardsticks for length and time, the primary turbulence scale perturbation generates successively larger perturbations with Fibonacci winding number on either side of the initial perturbation. Therefore large eddy time period

(8)

where The atmospheric eddy continuum energy structure follows quantum mechanical laws (Mary Selvam, 1987). The energy manifestation of radiation and other subatomic phenomena appear to posses the dual nature of wave and particles since one complete eddy energy circulation is inherently bidirectional with corresponding bimodal form of manifested phenomena, e.g. formation of clouds in the updraft regions and dissipation of clouds in the downdraft regions giving rise to discrete cellular structure to cloud geometry. The geometric phase difference between successive eddies, i.e. the crossing angle is related to the periodicities as shown earlier (Eq.7) and therefore large eddy growth is inherently associated with a geometric phase change and this result is consistent with the recently identified relation between geometric phase and frequency in laser propagation (Simon

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