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 . 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 [30-31]. The above visualization of the unified network of atmospheric flows as a quantum system is consistent with Grossing's  concept of quantum systems as order out of chaos phenomena. Order and chaos have been reported in strong fields in quantum systems .
Equation (3) 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 . Newton developed the idea of an inverse square law for gravitation in order to explain Kepler's laws, in particular, the third law. Kepler's laws were formulated on the basis of observational data and therefore are of empirical nature. A basic physical theory for the inverse square law of gravitation applicable to all objects, from macroscale astronomical objects to microscopic scale quantum systems is still lacking. The model concepts are analogous to a string theory  where, superposition of different modes of vibration in stringlike energy flow patterns result in material phenomena with intrinsic quantumlike mechanical laws which incorporate inverse square law for inertial forces, the equivalent of gravitational forces, on all scales of eddy fluctuations from macro- to microscopic scales. The cumulative sum of centripetal forces in a hierarchy of vortex circulations may result in the observed inverse square law form for gravitational attraction between inertial masses (of the eddies). Uzer et. al  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.
The apparent paradox of wave-particle duality in microscopic scale quantum systems  is however physically consistent in the context of macroscale atmospheric flows since the bi-directional energy flow structure of a complete atmospheric eddy results in the formation of clouds in updraft regions and dissipation of clouds in downdraft regions. The commonplace occurrence of clouds in a row is a manifestation of wave-particle duality in the macroscale quantum system of atmospheric flows (Fig. 1).
Figure 1. Wave-particle duality in atmospheric flows.
The above-described analogy of
quantumlike mechanics for atmospheric flows is similar to the concept of
a subquantum level of fluctuations whose space-time organization gives
rise to the observed manifestation of subatomic phenomena, i.e., quantum
systems as order out of chaos phenomena.
Puthoff  has shown that the observed stability of ground-state electronic orbits in atoms is a result of energy exchange with the sea of electromagnetic energy available in the vacuum zero point fluctuations. Historically, quantum mechanics had imposed arbitrary stability criterion for the ground state of electron orbits. Stable ground state is not possible in classical physics since attractive forces between the negative electron and positive nucleus will result in spiraling of orbital electrons into the nucleus accompanied by loss of energy due to emission of radiation by the accelerating electron, since all accelerating charges radiate energy. Puthoff  has also put forth the concept of "gravity as a zero-point fluctuation force". The vacuum zero-point fluctuation (electromagnetic) energy is manifested in the Casimir effect , namely a force between two closely spaced metal plates. Casimir effect is interpreted as due to imbalances in the zero-point energy caused by the presence of the plates and is analogous to the turbulent scale fluctuations whose spatial integration results in coherent large eddy structures. Recent studies show that background noise enhances weak signals in electronic circuits . El Naschie has proposed in a series of papers [15-27] that Cantorian-fractal conception of spacetime may effect reconciliation between quantum mechanics and gravity.
The spiral flow structure can
be visualized as an eddy continuum generated by successive length step
growths OR0 , OR1 , OR2 , OR3,....respectively
equal to R1 , R2 , R3
,....which follow Fibonacci mathematical series such that Rn+1
= Rn + Rn-1 and Rn+1
/ Rn = t
where t is the golden
mean equal to (1+Ö
5)/2 ( » 1.618).
Considering a normalized length step equal to 1 for the last stage
of eddy growth, the successively decreasing radial length steps can be
expressed as 1, 1/t
, 1/t 2,
,......The normalized eddy continuum comprises of fluctuation length scales
The probability of occurrence is equal to 1/t
and 1/t 2
for eddy length scale 1/t
in any one or both rotational (clockwise and anti-clockwise) directions.
Eddy fluctuation length of amplitude 1/t
, has a probability of occurrence equal to 1/t2
in both rotational directions, i.e., the square of eddy amplitude represents
the probability of occurrence in the eddy continuum. Similar result is
observed in the subatomic dynamics of quantum systems which are visualized
to consist of the superimposition of eddy fluctuations in wave trains (eddy
Nonlocal connections are intrinsic to quasiperiodic Penrose titling pattern. The phenomenon known as nonlocality or "action at a distance" characterize quantum systems. Experiments in quantum optics show that two distant events can influence each other instantaneously. Nonlocal connections in quantum systems apparently violate the fundamental theoretical law in modern physics that signal transmission cannot exceed the speed of light. The distinction between locality and nonlocality is related to the concept of a trajectory  of a single point object. The instantaneous nonlocal connection in the string-like energy flow patterns which represent extended objects can be visualised as shown in Fig. 3
Figure 3. Instantaneous non-local connection in atmospheric eddy circulations.
(b) Conventional continuous periodogram power spectral analyses of such spiral trajectories will reveal a continuum of periodicities with progressive increase in phase.
(c) The broadband power spectrum will have embedded dominant wave-bands the bandwidth increasing with period length. The peak periods En in the dominant wavebands will be given by the relation
En=TS(2+t )t n
where t is the golden mean equal to (1+Ö 5)/2 [@ 1.618] and Ts is the primary perturbation time period, for example, the solar powered annual cycle (summer to winter) of solar heating in a study of interannual climate variability. Ghil  reports that the most striking feature in climate variability on all time scales is the presence of sharp peaks superimposed on a continuous background. The model predicted periodicities are 2.2, 3.6, 5.8, 9.5, 15.3, 24.8, 40.1 and 64.9 years for values of n ranging from -1 to 6. Periodicities close to model predicted have been reported .
(d) The overall logarithmic spiral flow structure is given by the relation
where the constant k is the steady state fractional volume dilution of large eddy by inherent turbulent eddy fluctuations . The constant k is equal to 1/t2 (@ 0.382) and is identified as the universal constant for deterministic chaos in fluid flows. The steady state emergence of fractal structures is therefore equal to
1/k @ 2.62
statistical normalized standard deviation t=0,1,2,3, etc.
The conventional power spectrum plotted as the variance versus the frequency in log-log scale will now represent the eddy probability density on logarithmic scale versus the standard deviation of the eddy fluctuations on linear scale since the logarithm of the eddy wavelength represents the standard deviation, i.e., the r.m.s. value of eddy fluctuations (5). The r.m.s. value of eddy fluctuations can be represented in terms of statistical normal distribution as follows. A normalized standard deviation t=0 corresponds to cumulative percentage probability density equal to 50 for the mean value of the distribution. Since the logarithm of the wavelength represents the r.m.s. value of eddy fluctuations the normalized standard deviation t is defined for the eddy energy as
where L is the period in years and T50 is the period up to which the cumulative percentage contribution to total variance is equal to 50 and t = 0. LogT50 also represents the mean value for the r.m.s. eddy fluctuations and is consistent with the concept of the mean level represented by r.m.s. eddy fluctuations. Spectra of time series of meteorological parameters when plotted as cumulative percentage contribution to total variance versus t have been shown to follow the model predicted universal spectrum [6-14].
(e) Mary Selvam  has shown that equation (1) represents the universal algorithm for deterministic chaos in dynamical systems and is expressed in terms of the universal Feigenbaum's  constants a and d as follows. The successive length step growths generating the eddy continuum OR0R1R2R3R4R5 analogous to the period doubling route to chaos (growth) is initiated and sustained by the turbulent (fine scale) eddy acceleration w* which then propagates by the inherent property of inertia of the medium of propagation. Therefore, the statistical parameters mean , variance , skewness and kurtosis of the perturbation field in the medium of propagation are given by w*, w*2 ,w*3 ,and w*4 respectively. The associated dynamics of the perturbation field can be described by the following parameters. The perturbation speed w* (motion) per second (unit time) sustained by its inertia represents the mass, w*2 the acceleration or force, w*3 the angular momentum or potential energy, and w*4 the spin angular momentum, since an eddy motion has an inherent curvature to its trajectory.
It is shown that Feigenbaum's constant a
is equal to 
where the subscripts 1 and 2 refer to two successive stages of eddy growth. Feigenbaum's constant a as defined above represents the steady state emergence of fractional Euclidean structures. Considering dynamical eddy growth processes, Feigenbaum's constant a also represents the steady state fractional outward mass dispersion rate and a2 represents the energy flux into the environment generated by the persistent primary perturbation w* . Considering both clockwise and counterclockwise rotations, the total energy flux into the environment is equal to 2a2 . In statistical terminology, 2a2 represents the variance of fractal structures for both clockwise and counterclockwise rotation directions.
The Feigenbaum's constant d is shown to be equal to 
and represents the fractional volume intermittency of occurrence of fractal structures for each length step growth. Feigenbaum's constant d also represents the relative spin angular momentum of the growing large eddy structures as explained earlier.
Equation (1) may now be written as
where dR equal to r represents the incremental growth in radius for each length step growth, i.e., r relates to the earlier stage of eddy growth.
Substituting the Feigenbaum's constants a and d defined above (9 and 10) equation (11) can be written as
2a2 = p d
a = t2 = 1/k = 2.62
(f) The relationship between Feigenbaum's constant a and statistical normal distribution for power spectra is
derived in the following.
The steady state emergence of fractal structures is equal to the Feigenbaum's constant a (6). The relative variance of fractal structure for each length step growth is then equal to a2. The normalized variance 1/a2n will now represent the statistical normal probability density for the nth step growth according to model predicted quantumlike mechanics for fluid flows . Model predicted probability density values P are computed as
P = t- 4 n
P = t-4 t
where t is the normalized standard deviation (7) and are in agreement with statistical normal distribution as shown in Table 1.
Model predicted and statistical
normal probability density distributions
since the corresponding value for both direction
is equal to a (6 ).
The emerging fractal space-time structures have moment coefficient of kurtosis given by the fourth moment equal to
The moment coefficient of skewness
for the fractal space-time structures is equal to zero for the symmetric
eddy circulations. Moment coefficient of kurtosis equal to 3
and moment coefficient of skewness equal to zero characterise
the statistical normal distribution underlying the fractal space-time
eddy continuum structure.
Normal distribution characteristics for the eddy continuum fluctuation field can also be derived from model concept as follows.
Let P represent the probability of occurrence in the medium of bidirectional eddy energy flux with characteristics of a particular large eddy of radius R. Since W originates from w*
substituting from equation (1)
substituting from equation (5)
Substituting for k , namely,
The probability P is obtained as
Linearising equation (16) for two successive stages of eddy growth
Therefore statistical normal distribution characteristics are followed by the probability P of occurrence of eddy fluctuation W originating from earlier stage perturbation w*.
(g) The power spectra of fluctuations in fluid flows
can now be quantified in terms of universal Feigenbaum's constant a
The normalized variance and therefore the statistical normal distribution is represented by (from equation 14)
P = a - 2t
where P is the probability density corresponding to normalized standard deviation t. The graph of P versus t will represent the power spectrum. The slope S of the power spectrum is equal to
(h) The fractal dimension D can be
expressed as a function of the universal Feigenbaum's constant a
The steady state emergence of fractal structures is equal to a for each length step growth (7 & 13) and therefore the fractal structure domain is equal to am at mth growth step starting from unit perturbation. Starting from unit perturbation, the fractal object occupies spatial (two dimensional) domain am associated with radial extent tm since successive radii follow Fibonacci number series. The fractal dimension D is defined as
(i) The relationship between fine structure constant, i.e. the eddy energy ratio between successive dominant eddies and Feigenbaum's constant a is derived as follows.
2a2 = relative variance of fractal structure (both clockwise and anticlockwise rotation) for each growth step.
For one dominant large eddy (Fig. 2) OR0R1R2R3R4R5 comprising of five growth steps each for clockwise and counterclockwise rotation, the total variance is equal to
(j) The ratio of proton mass M to electron mass me , i.e. M/me is another fundamental dimensionless number which also awaits derivation from a physically consistent theory. M/me determined by observation is equal to about 2000. In the following it is shown that ratio of energy content of large to small eddies for specific length scale ratios is equivalent to M/me.
From Equation (22),
The energy ratio for two successive dominant eddy growth = (2a2 x10)2
Since each large eddy consists of five growth steps each for clockwise and anticlockwise rotation,
The relative energy content of large eddy with respect to primary circulation structure inside this large eddy
= (2a2 x 10)2/10
The primary circulation corresponds
to OR0R1 (Fig.2) with length scale OR0
equal to t5
and the dominant large eddy length scale OR5 is then
equal to (t5)6
. The length scale ratio OR5 / OR0 is equal
to ( t5
= t firstname.lastname@example.org
. The ratio of the radii of atom and electron is also approximately equal
to 105 .
Quantum mechanical concepts relating to fundamental particles and universal constants are summarised in the following . The only objects that appear to be exactly the same every where are the atoms and their constituent particles. A natural unit of mass is the nucleon mass, equal approximately to that of the hydrogen atom. Nucleons (i. e. , protons and neutrons) have a mass 1836 times the mass of the electron.
The constants of nature can be arranged to form natural numbers (often referred to as dimensionless numbers) that are independent of our units of measurement. The ratio of the nucleon and electron masses equal to approximately 1836 is one such number. Another example is the Sommerfeld's fine structure constant defined by a
where e is the charge on electron, h,
the Plank's constant and c, the velocity of light . The fine
structure constant appears whenever radiation interacts with particles,
and the combination of c, h and e indicates a wave
like (h) interaction between particles (e) and light (c).
The classical electron radius is the size of an electron as calculated prior to the introduction of quantum mechanics. It is obtained by assuming that all the energy mec2 of the electron is in the form of electrical energy equal to
Thus giving a radius r expressed by
A characteristic size of atoms is the radius R of the hydrogen atom
known as the Bohr orbit radius. The absence of the gravitational constant G and c indicates that gravity and relativity are not of primary importance in the structure of atoms.
The electron radius
Where R is the radius of the hydrogen atom. Therefore ratio of radii of atom to electron is equal to a-2» 105
radius of electron »2.82´ 10-13 cms = r
radii of most atoms »2´ 10-8 cms = R
The scale ratio z =R/r »105
The radius of the electron is about one hundred-thousandths
of the radius of an average atom .
The cell dynamical system model concepts therefore enable physically consistent derivation of fundamental constants which define the basic structure of quantum systems. These two fundamental constants could not be derived so far from a basic theory in traditional quantum mechanics for subatomic dynamics .
from equation (1)
The above concept is analogous
to the no scale super gravity model of Lahanas and Nanapoulous  where
the super-Planck mass is given in terms of the Planck scale
Gev ) which corresponds to the first excited
state of these strings . The virtues of the no scale super gravity
model are automatically vanishing cosmological constant (at least
at the classical level), dynamical determination of all mass scales in
terms of fundamental Planck scale mp and acceptable
low energy phenomenology. The no scale structure is super symmetric since
it fuses together the non-trivial internal symmetries of the internal small
scale eddies with the spacetime (Poincare) symmetries of the eddy
continuum structure and accounts for the observed fractal geometry
The string theory visualises particles as extended objects and thereby avoids singularities, a major problem in the application of point-like concept for particles in traditional physics .
The string theory for quantumlike mechanics in atmospheric flows is analogues to Bohm's concept of implicate order for subatomic dynamics of quantum systems  .
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