A. M. Selvam and Suvarna Fadnavis

Indian Institute of Tropical Meteorology, Pune 411 008, India

Atmospheric flows exhibit long-range spatiotemporal correlations manifested as the fractal geometry to the global cloud cover pattern concomitant with inverse powerlaw form for power spectra of temporal fluctuations of all scales ranging from turbulence (millimeters-seconds) to climate (thousands of kilometers-years) [1]. Long-range spatiotemporal correlations are ubiquitous to dynamical systems in nature and are identified as signatures of self-organized criticality [2,3]. Standard models for turbulent fluid flows in meteorological theory cannot explain satisfactorily the observed multifractal (space-time) structures in atmospheric flows. Numerical models for simulation and prediction of atmospheric flows are subject to deterministic chaos [4] and give unrealistic solutions. Deterministic chaos is a direct consequence of round-off error growth in iterative computations. Round-off error of finite precision computations doubles on an average at each step of iterative computations [5]. Round-off error will propagate to the mainstream computation and give unrealistic solutions in numerical weather prediction (NWP) and climate models which incorporate thousands of iterative computations in long-term numerical integration schemes. A recently developed non-deterministic cell dynamical system model for atmospheric flows [6-14] predicts the observed self-organized criticality as intrinsic to quantumlike mechanics governing flow dynamics. Further, the fractal space-time structure to the stringlike atmospheric flow trajectory is resolved into a continuum of eddies. The eddy circulations obey Kepler's third law of planetary motion and therefore eddy inertial masses obey Newton's inverse square law of gravitation on all scales from microscopic to macroscale. El Naschie [15-27] has discussed, in a series of publications, the fractal structure to space-time and also states that fractalisation of microspace is the origin of gravity.

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 [30-31]. 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].

The square of the eddy amplitude W ² represents the kinetic energy E given as (from equation .1)

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 [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 wave-particle duality in microscopic scale quantum systems [31] 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).

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 [38] 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 [39] 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 [40], 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 [41]. 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.

2.1. Model Predictions

(a) Atmospheric flows trace an overall logarithmic spiral trajectory OR₀R₁R₂R₃R₄R₅ simultaneously in clockwise and anti-clockwise directions with the quasi-periodic Penrose tiling pattern [42] for the internal structure Fig. 2.

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

In summary, energy pumping at a fundamental frequency generates a broad-band eddy continuum and has been observed as chaos in laser and non-linear optical systems which are basically governed by quantum mechanical laws [44].

(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 E_n in the dominant wavebands will be given by the relation

(d) The overall logarithmic spiral flow structure is given by the relation

The model predicted logarithmic wind profile relationship such as equation (5) 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. Historically, equation (5), 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 [47].

statistical normalized standard deviation t = 0,1,2,3, etc. (7)

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].

It is shown that Feigenbaum's constant a is equal to [5]

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

(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 n^th step growth according to model predicted quantumlike mechanics for fluid flows . Model predicted probability density values P are computed as

or

where t is the normalized standard deviation (7) and are in agreement with statistical normal distribution as shown in Table 1.

The statistical normal distribution characteristics underlie the fractal space-time continuum fluctuations as shown in the following

The steady state emergence of fractal structures for each length step growth for any one direction of rotation ( either clockwise or anticlockwise ) is equal to

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

For any two successive stages of eddy growth (1)

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} (17)

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.

(h) The fractal dimension D can be expressed as a function of the universal Feigenbaum's constant a as follows.

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

similarly

Therefore the fractal dimension D is given as

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 which superimpose to give the observed four dimensional space-time structure to the flow pattern. The above concept of dimension for real world spacetime patterns is consistent with El Naschie's[25,26] interpretation of dimensions for superstring theories in particle physics, namely a string rotates in ordinary space and only uses the extra dimensions for vibrations which simulate particle masses. El Naschie has also derived mathematically and shown that the golden mean is intrinsic to the geometry of fractal structures [49-54]. The correlated fluctuations of the fractal spacetime eddy continuum is analogous to the Bose-Einstein condensation phenomena observed in liquid helium [55-58].

(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

2a² x 10 = 137.07 (22)

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.

(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 = (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

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.

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_ec² 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

Summarising [64]

The radius of the electron is about one hundred-thousandths 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].

A no scale energy M can be defined for the large eddy energy with respect to the primary eddy energy m_p [60-61] as follows

Since phenomenological manifestation of energy occurs only during one half cycle of eddy perturbation

from equation (1)

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[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] .

The authors express their gratitude to Dr. A. S. R. Murty for his keen interest and encouragement during the course of this study.

Figures

Figure 1. Wave-particle duality in atmospheric flows.

Figure 2. The quasiperiodic Penrose tiling pattern.

Figure 3. Instantaneous non-local connection in atmospheric eddy circulations.

Table 1

Model predicted and statistical

normal probability density distributions