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 [29]. 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 [3031]. The above visualization of
the unified network of atmospheric flows as a quantum system is consistent
with Grossing's [32] concept of quantum systems as order out of chaos
phenomena. Order and chaos have been reported in strong fields in quantum
systems [33].
E=Hn
and
we obtain
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 [35]. 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 [36] 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 [37] 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 waveparticle
duality in microscopic scale quantum systems [31] is however physically
consistent in the context of macroscale atmospheric flows since the bidirectional
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 waveparticle
duality in the macroscale quantum system of atmospheric flows (Fig.
1).
Figure 1. Waveparticle duality in atmospheric flows.
The abovedescribed analogy of
quantumlike mechanics for atmospheric flows is similar to the concept of
a subquantum level of fluctuations whose spacetime organization gives
rise to the observed manifestation of subatomic phenomena, i.e., quantum
systems as order out of chaos phenomena.
Puthoff [38] has shown that the
observed stability of groundstate 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 [39] has also put
forth the concept of "gravity as a zeropoint fluctuation force". The vacuum
zeropoint fluctuation (electromagnetic) energy is manifested in the Casimir
effect [40], namely a force between two closely spaced metal plates.
Casimir
effect is interpreted as due to imbalances in the zeropoint 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 [41]. El Naschie has proposed in a series
of papers [1527] that Cantorianfractal 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 OR_{0} , OR_{1} , OR_{2} , OR_{3},....respectively
equal to R_{1} , R_{2} , R_{3}
,....which follow Fibonacci mathematical series such that R_{n+1}
= R_{n} + R_{n1} and R_{n+1}
/ R_{n} = 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},
1/t^{3}
,......The normalized eddy continuum comprises of fluctuation length scales
1,
1/t
, 1/t^{2},........
The probability of occurrence is equal to 1/t
and 1/t ^{2
}respectively
for eddy length scale 1/t
in any one or both rotational (clockwise and anticlockwise) directions.
Eddy fluctuation length of amplitude 1/t
, has a probability of occurrence equal to 1/t^{2}
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
continuum).
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 [43] of a single
point object. The instantaneous nonlocal connection in the stringlike
energy flow patterns which represent extended objects can be visualised
as shown in Fig. 3
Figure 3. Instantaneous nonlocal 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 wavebands the bandwidth increasing with period length. The peak periods E_{n} in the dominant wavebands will be given by the relation
E_{n}=T_{S}(2+t )t ^{n}
where t
is the golden mean equal to (1+Ö
5)/2 [@ 1.618] and
T_{s
}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 [45] 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 [46].
(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/t^{2
}(@
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 loglog
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 T_{50}
is the period up to which the cumulative percentage contribution to total
variance is equal to 50 and t = 0. LogT_{50}
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 [614].
(e) Mary Selvam [5] 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 [48] constants a and d as follows. The successive length step growths generating the eddy continuum OR_{0}R_{1}R_{2}R_{3}R_{4}R_{5} 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 [5]
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 a^{2} 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 2a^{2} . In
statistical terminology, 2a^{2} represents the variance
of fractal structures for both clockwise and counterclockwise rotation
directions.
The Feigenbaum's constant d is shown to be equal to [5]
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
2a^{2} = p d
a = t^{2} = 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 a^{2}. The normalized variance 1/a^{2n}
will now represent the statistical normal probability density for the
n^{th}
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 spacetime
structures have moment coefficient of kurtosis given by the fourth
moment equal to
The moment coefficient of skewness
for the fractal spacetime 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 spacetime
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
therefore
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
as follows.
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
as follows.
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 a^{m}
at
m^{th
}growth step starting from unit perturbation. Starting
from unit perturbation, the fractal object occupies spatial (two
dimensional) domain a^{m} associated with radial extent
t^{m}
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.
2a^{2} = relative variance of fractal structure (both clockwise and anticlockwise rotation) for each growth step.
For one dominant large eddy (Fig. 2) OR_{0}R_{1}R_{2}R_{3}R_{4}R_{5} comprising of five growth steps each for clockwise and counterclockwise rotation, the total variance is equal to
2a^{2}x10=137.07
(j) The ratio of proton mass M to electron mass m_{e} , i.e. M/m_{e} is another fundamental dimensionless number which also awaits derivation from a physically consistent theory. M/m_{e} 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/m_{e}.
From Equation (22),
The energy ratio for two successive dominant eddy growth = (2a^{2} 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
= (2a^{2} x 10)^{2}/10
@ 1879
The primary circulation corresponds
to OR_{0}R_{1} (Fig.2) with length scale OR_{0}
equal to t^{5}
and the dominant large eddy length scale OR_{5} is then
equal to (t^{5})^{6}
. The length scale ratio OR_{5 }/ OR_{0} is equal
to ( t^{5}
)^{6} /t^{5}
= t ^{25}@10^{5.22}
. The ratio of the radii of atom and electron is also approximately equal
to 10^{5} [64].
Quantum mechanical concepts relating
to fundamental particles and universal constants are summarised in the
following [62]. 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 m_{e}c^{2}
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}» 10^{5}
Summarising [64]
radius of electron »2.82´ 10^{13 }cms = r
radii of most atoms »2´ 10^{8} cms = R
The scale ratio z =R/r »10^{5}
The radius of the electron is about one hundredthousandths
of the radius of an average atom [64].
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 [63].
and
from equation (1)
The above concept is analogous
to the no scale super gravity model of Lahanas and Nanapoulous [65] where
M,
the superPlanck mass is given in terms of the Planck scale
m_{p}
(»10^{19}
Gev ) which corresponds to the first excited
state of these strings [66]. 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 m_{p} and acceptable
low energy phenomenology. The no scale structure is super symmetric since
it fuses together the nontrivial 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
in nature.
The string theory visualises
particles as extended objects and thereby avoids singularities, a major
problem in the application of pointlike concept for particles in traditional
physics [67].
The string theory for quantumlike
mechanics in atmospheric flows is analogues to Bohm's concept of implicate
order for subatomic dynamics of quantum systems [68] .
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