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 quantum-like 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 Grossingís (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
Equation 3 is analogous to Keplerís third law of planetary motion, namely, the square of the planetís year (period) to the cube of the planetís 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 Keplerís laws, in particular, the third law. Keplerís 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 quantum-like 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 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)tn
where t is the golden mean equal to (1+Ö5)/2 [@1.618] and Ts , 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/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
The model predicted logarithmic wind profile relationship such as Equation 4 is a long-established (observational) feature of atmospheric flows in the boundary layer, the constant k, called the Von Karman ís 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 Wn+1 and Wn 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 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 (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 T50 is the period up to which the cumulative percentage contribution to total variance is equal to 50 and t = 0. The variable 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 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 Feigenbaumís (Feigenbaum 1980) constants a and d as follows.
2a2 = pd
where pd , the relative volume intermittency of occurrence contributes to the total variance 2a2 of fractal structures. The Feigenbaumís 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 2a2 . It was shown at Equation 5 above that the steady state emergence of fractal structures in fluid flows is equal to 1/k( = t2) and therefore the Feigenbaumís constant a is equal to
a = t2 = 1/k = 2.62
(g) The relationship between Feigenbaumís 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 Feigenbaumís constant a (Equation 5 ). 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 - 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
The periodicities T50
and T95 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 T50 and T95 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 Ts is equal to the annual (summer to winter) cycle of solar heating and T50 and T95 are obtained as
T50 = (2+t)t0@ 3.6 years
(h) The power spectra of fluctuations in fluid flows can now be quantified in terms of universal Feigenbaumís 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 Feigenbaumís
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 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
where M is the mass contained within a distance R from a point in the fractal object.Considering growth from nth 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 Feigenbaumís 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 comprising of five growth steps each for clockwise and counterclockwise rotation, the total variance is equal to
2a2 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 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 19,
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 primary
circulation structure inside this large eddy
= (2a2 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
(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
al. 1995), temperature (Mary Selvam and Joshi 1995) and surface pressure
(Mary Selvam et al. 1996) and imply laws analogous to Keplerís
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 space-time 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).