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 (Maddox 1988a).
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 (Rae 1988).The
above visualization of the unified network of atmospheric flows as a quantum
system is consistent with Grossing’s (Grossing 1989) concept of quantum
systems as order out of chaos phenomena. Order and chaos have been reported
in strong fields in quantum systems (Brown 1996). Writing Equation 1 in
terms of the periodicities T and t
of large and small eddies respectively, where
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 (Narlikar 1982,1996; Weinberg
1993). 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 were
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 (Equation 2) are analogous to a string theory (Kaku 1997) where,
superposition of different modes of vibration 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.
Uzer et al.
(1991) 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.
(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+Ö5)/2
[@1.618]
and T_{s} , the solar powered primary perturbation
time period is the annual cycle (summer to winter) of solar heating in
the present study of interannual variability. Ghil (1994) reports that
the most striking feature in climate variability on all time scales is
the presence of sharp peaks superimposed on a continuum 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.
Peridicities close to model predicted have been reported (Burroughs 1992;
Kane 1996).
(d) The ratio r/R also represents the increment dq in phase angle q (Equation 1) and therefore the phase angle q represents the variance. Hence, when the logarithmic spiral is resolved as an eddy continuum in conventional spectral analysis, the increment in wavelength is concomitant with increase in phase. Such a result that increments in wavelength and phase angle are related is observed in quantum systems and has been named 'Berry's phase' (Berry 1988; Maddox 1988b). The relationship of angular turning of the spiral to intensity of fluctuations is seen in the tight coiling of the hurricane spiral cloud systems.
(e) 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
The model predicted
logarithmic wind profile relationship such as Equation 4 is a longestablished
(observational) feature of atmospheric flows in the boundary layer, the
constant k, called the Von Karman ’s constant has
the value equal to 0.38 as determined from observations.
Historically, Equation 4 is basically an empirical law known as the universal
logarithmic
law of the wall, first proposed in the early 1930s by pioneering aerodynamicists
Theodor
von Karman and Ludwig Prandtl, describes shear forces exerted
by turbulent flows at boundaries such as wings or fan blades or the interior
wall of a pipe. The law of the wall has been used for decades by engineers
in the design of aircraft, pipelines and other structures (Cipra, 1996).
In Equation 4, W
represents the standard deviation of eddy fluctuations, since W
is computed as the instantaneous r.m.s. (root mean square) eddy perturbation
amplitude with reference to the earlier step of eddy growth. For two successive
stages of eddy growth starting from primary perturbation w_{*}
, the ratio of the standard deviations W_{n+1}
and W_{n} is given from Equation 4 as (n+1)/n.
Denoting by s
the standard deviation of eddy fluctuations at the reference level (n=1)
the standard deviations of eddy fluctuations for successive stages of eddy
growth are given as integer multiple of s,
i.e., s,
2s,
3s,
etc. and correspond respectively to
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 (Equation 4). 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. The variable
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
should follow the model predicted universal spectrum. The literature shows
many examples of pressure, wind and temperature whose shapes display a
remarkable degree of universality (Canavero and Einaudi,1987).
(f) Mary Selvam (1993a) 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 (Feigenbaum 1980) constants a and d as follows.
2a^{2} = pd
where pd
, the relative volume intermittency of occurrence contributes to the total
variance 2a^{2} of fractal structures.
The
Feigenbaum’s constant a represents the steady
state emergence of fractal structures. Therefore the total variance
of fractal structures for either clockwise or anticlockwise rotation
is equal to 2a^{2} . It was shown at Equation 5 above
that the steady state emergence of fractal structures in fluid flows
is equal to 1/k( = t^{2})
and therefore the Feigenbaum’s constant a is
equal to
a = t^{2} = 1/k = 2.62
(g) 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
(Equation 5 ). 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 ^{ 4n}
P = t ^{ 4t}
where t is the normalized
standard deviation (Equation 6) and are in agreement with statistical
normal distribution as shown in Table 1.
Model predicted and statistical normal probability density distributions






P =t^{4t} 













The periodicities T_{50}
and T_{95} up to which the cumulative percentage
contribution to total variances are respectively equal to 50
and 95 are computed from model concepts as follows.
The power spectrum, when plotted
as normalised standard deviation t versus cumulative
percentage contribution to total variance represents the statistical normal
distribution (Equation 7), i.e., the variance represents the probability
density. The normalised standard deviation values corresponding to cumulative
percentage probability densities P equal to 50
and 95 respectively are equal to 0 and 2
from statistical normal distribution characteristics. Since t
represents the eddy growth step n (Equation 6) the dominant
periodicities T_{50} and T_{95}
upto which the cumulative percentage contribution to total variance are
respectively equal to 50 and 95 are obtained
from Equation 3 for corresponding values of n , i.e., 0
and 2. In the present study of interannual variability, the
primary perturbation time period T_{s} is equal
to the annual (summer to winter) cycle of solar heating and T_{50}
and T_{95} are obtained as
T_{50} = (2+t)t^{0}@ 3.6 years
(h) 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 11)
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
The power spectrum therefore follows
inverse power law form, the slope decreasing with increase in t.
Increase in t corresponds to large eddies ( low frequencies)
and is consistent with observed decrease in slope at low frequencies in
dynamical systems.
(I) 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 (Equations
6 & 9) 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
where M is the mass contained within a distance R from a point in the fractal object.Considering growth from n^{th} to (n+m)^{th} step
The fractal dimension increases
with the number of growth steps.The dominant wavebands increase in length
with successive growth steps. The fractal dimension D
indicates the number of periodicities incorporated. Larger fractal
dimension indicates more number of periodicities and complex patterns.
(j) 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 comprising of five growth steps each for clockwise and counterclockwise rotation, the total variance is equal to
2a^{2} x 10 = 137.07
For each complete cycle ( comprising
of five growth steps each ) in simultaneous clockwise and counterclockwise
rotations, the relative energy increase is equal to 137.07
and represents the fine structure constant for eddy energy structure.
Incidentally, the fine
structure constant in atomic physics (Davies 1986; Gross 1985; Omnes
1994) designated as a^{
1} , a dimensionless number
equal to 137.03604 is very close to that derived above
for atmospheric eddy energy structure. This fundamental constant has attracted
much attention and it is felt that quantum mechanics cannot be interpreted
properly until such time as we can derive this physical constant from a
more basic theory.
(k) 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 19,
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 primary
circulation structure inside this large eddy
= (2a^{2} x 10)^{2}/10
@
1879
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
(Omnes 1994).
(a) In a majority of spectra, periodicities
up to 4 years contribute up to 50% of total variance (see
references of Mary Selvam et al.) and is in agreement with
model prediction (Equation 12). The model also predicts that, periodicities
upto 9.5 years contribute upto 95% of total variance (Equation
13). Dominant periodicities, such as the widely documented QBO,
ENSO
and decadal scale fluctuations may be used for predictability studies.
(b) Model predicted universal spectrum
(Equation 7) has been identified in the interannual variability of rainfall
(Mary Selvam et al. 1992; Mary Selvam 1993b; Mary Selvam
et
al. 1995), temperature (Mary Selvam and Joshi 1995) and surface pressure
(Mary Selvam et al. 1996) and imply laws analogous to Kepler’s
laws (Equation 2) for eddy circulation dynamics. Universal spectrum for
atmospheric interannual variability provides precise quantification for
the apparently irregular natural variability. The concept of universal
spectrum for fluctuations rules out linear secular trends in meteorological
parameters with regard to climate change. Global warming, either natural
or man  made (industrialization related) will result in enhancement of
fluctuations of all scales (Equation 1). The following studies indicate
intensification of spacetime fluctuations in atmospheric flows in recent
years (since 1970s). The report of IPCC (Intergovernmental
Panel on Climate Change) shows that recent increases have been found
in the intensity of the winter atmospheric circulation over the extratropical
Pacific and Atlantic (Houghton et al. 1996). There have been
relatively more frequent
El Nino episodes since 1976/77 with only
rare excursions into the other extreme (La Nina episodes). An assessment
of ENSO  scale secular vartiability shows that ENSO  scale
variance is relatively large in recent decades (Wang and Ropelewski 1995).
Hurrel and Van Loon (1994) have reported a delayed breakdown of the polar
vortex in the troposphere and lower stratosphere after the late 1970s coincident
with the beginning of the ozone deficit in the Antarctic spring. It is
possible that enhanced vertical mixing (Equations 6 and 7) inside the polar
vortex may contribute to the ozone loss. Regions of enhanced convective
activity in the monsoon regime are found to be associated with lower levels
of atmospheric columnar total ozone content (Hingane and Patil 1996). Incidentally,
it was found that enhancement of background noise, i.e. energy input into
the eddy continuum results in amplification of faint signals in electrical
circuits (Brown 1996).