4.2** **Model Predictions

(a) Atmospheric flows trace an overall logarithmic spiral trajectory R_{o}R_{1}R_{2}R_{3}R_{4}R_{5 }with the quasiperiodic *Penrose tiling pattern* for the internal structure (Figure 6 2.5 Fivefold and Spiral Symmetry Associated with Fibonacci Sequence).

(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(R_{o}OR_{1}, R_{1}OR_{2}, R_{2}OR_{3}, R_{3}OR_{4}, R_{4}OR_{5}, etc.) the bandwidth increasing with period length(Figure 6). The peak periods *E _{n}* in the dominant wavebands will be given by the relation

* E_{n} = T_{S}(2+t )t ^{n}* (5)

where t is the golden mean equal to *(1+**Ö 5)/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. The most striking feature in climate variability on all time scales is the presence of sharp peaks superimposed on a continuous background(Ghil,1994 References ).

(d) The ratio *r/R* also represents the increment *d**q *in phase angle q (Equation 3 and Figure 5) 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. The angular turning, in turn, is directly proportional to the variance( Equation 3) 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;Simon et al.,1988;Maddox,1988b,1991; Samuel and Bhandari,1988; Kepler *et al*. 1991;Kepler,1992;Anandan,1992 References ). 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

(6)

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* (7)

Logarithmic wind profile relationship such as Equation 6 is a long-established (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 (Wallace and Hobbs , 1977 References ) .Equation 6 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 References ).

In Equation 6, *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_{*}*

* statistical normalized standard deviation t = 0,1,2,3, etc*. (8)

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 6). 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

(9)

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

The theoretical basis for formulation of the universal spectrum is based on the *Central Limit Theorem in Statistics*, namely, sample averages from almost any population encountered in practice tend to become normally distributed as the sample size increases. Therefore, when the spectra are plotted in the above fashion, they tend to closely (not exactly) follow cumulative normal distribution (see Section 6)

Though there is more information retained in the original spectra than conveyed in this form, universal spectrum for climate variability, if found to exist, will unambiguously rule out linear trends and predict changes in intensity of spectral components in response to changes (increase or decrease) in energy input into the atmospheric system.

(f) Mary Selvam (1993a References ) has shown that Equation 3 represents the universal algorithm for deterministic chaos in dynamical systems and is expressed in terms of the universal *Feigenbaum*'s (Feigenbaum , 1980 References ) constants ** a** and

* 2a^{2} = p d* (10)

where, p ** d** ,the relative volume intermittency of occurrence contributes to the total variance

The *Feigenbaum*'s constants ** a** and

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

* a = t ^{2} = 1/k = 2.62* (11)

(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 7 ). The relative variance of fractal structure for each length step growth is then equal to

* P = t ^{- 4n}* (12)

or

* P = t ^{- 4t}* (13)

where *t* is the normalized standard deviation(Equation 8) and are in agreement with statistical normal distribution as shown in Table 1.

The periodicities *T _{50}* and

The power spectrum, when plotted as normalized standard deviation *t* versus cumulative percentage contribution to total variance represents the statistical normal distribution(Equation 9),i.e, the variance represents the probability density. The normalized 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 8) the dominant periodicities *T _{50}* and

* T_{50} = (2+t )t ^{0} @ 3.6 years* (14)

* T_{95} = (2+t )t ^{2} @ 9.5 years* (15)

(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 13)

* P = a ^{- 2t}* (16)

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 8 & 11) and therefore the fractal structure domain is equal to

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

(18)

similarly

(19)

Therefore

(20)

The fractal dimension increases with the number of growth steps. The dominant wavebands increase in length with successive growth step (Figure 6). 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.

*2 a^{2}* = relative variance of fractal structure (both clockwise and anticlockwise rotation) for each growth step.

For one dominant large eddy (Figures 5 & 6) comprising of five growth steps each for clockwise and counterclockwise rotation, the total variance is equal to

** 2a^{2} x 10 = 137.07** (21)

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 References ) designated as a **^{-1}** ,a dimensionless number equal to

(k) The ratio of proton mass *M* to electron mass *m _{e}* , i.e. ,

From Equation 21,

The energy ratio for two successive dominant eddy growth *= ( 2a^{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}*

@ 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 References ).