A. MARY SELVAM
*Indian Institute of Tropical Meteorology,
Pune, 411008, India*

(Retired) email: amselvam@gmail.com

web site: http://www.geocities.com/amselvam

*Canadian J. Phys. 68, 831 - 841 (1990)*

Abstract

The complex spaciotemporal
patterns of atmospheric flows that result from the cooperative existence
of fluctuations ranging in size from millimetres to thousands of kilometres
are found to exhibit long-range spacial and temporal correlations. These
correlations are manifested as the self-similar fractal geometry of the
global cloud cover pattern and the inverse power-law form for the atmospheric
eddy energy spectrum. Such long-range spaciotemporal correlations are ubiquitous
in extended natural dynamical systems and are signatures of deterministic
chaos or self-organized criticality. In this paper, a cell dynamical system
model for atmospheric flows is developed by consideration of microscopic
domain eddy dynamical processes. This nondeterministic model enables formulation
of a simple closed set of governing equations for the prediction and description
of observed atmospheric flow structure characteristics as follows. The
strange-attractor design of the field of deterministic chaos in atmospheric
flows consists of a nested continuum of logarithmic spiral circulations
that trace out the quasi-periodic Penrose tiling pattern, identified as
the quasi-crystalline structure in condensed matter physics. The atmospheric
eddy energy structure follows laws similar to quantum mechanical laws.
The apparent wave-particle duality that characterize quantum mechanical
laws is attributed to the bimodal phenomenological form of energy display
in the bidirectional energy flow that is intrinsic to eddy circulations,
e.g., formation of clouds in updrafts and dissipation of clouds in downdrafts
that result in the observed discrete cellular geometry of cloud structure.

**1. Introduction**

Traditional mathematical
models of dynamical systems, i.e., systems which evolve with time are based
on Newtonian continuum dynamics and consist of nonlinear partial differential
equations. The nonlinear partial differential equations do not have analytical
solutions, and numerical solutions obtained using digital computers are
found to exhibit sensitive dependence on initial conditions. This results
in chaotic solutions, thereby giving rise to deterministic chaos (1-4).
Such deterministic chaos was first identified in the computer realization
of a simple mathematical model of atmospheric flows (5). Even simple nonlinear
mathematical models are found to exhibit deterministic chaos, thereby imposing
limits on the long-term predictability of the dynamical system, e.g., long-range
weather prediction (6-11). Ruelle and Takens (12) were the first to identify
an analogy between deterministic chaos and turbulence, which is unpredictable,
as modelled by the Navier-Stokes (NS) equations for fluid flows (13). There
is now a growing conviction that traditional concepts of natural laws and
their mathematical formulations in model continuum dynamical systems are
inherently unstable for calculus-based long-term numerical computation
schemes that use digital computers, which have inherent roundoff errors
(14-16). There now exists a need for alternative concepts of natural laws
that can be used to formulate simple analytical equations to model space-time
continuum evolution of dynamical systems (17).

Long-range spaciotemporal
correlations, recently identified as self-organized criticality (18), are
signatures of deterministic chaos (19) in real world dynamical systems,
and they indicate sensitive dependence on initial conditions, i.e., microscopic
scale dynamical laws govern the spaciotemporal evolution of the macroscale
pattern. It has not yet been possible to idenify the laws governing the
dynamics of evolution for the microscopic scale internal structure of the
macroscale dynamical system that is characterized by a self-similar spatial
pattern concomitant with long-range temporal corelations or 1/n
noise. Such a self-organized robust spaciotemporal structure of the strange
attractor with fractal pattern formation is a collective phenomenon resulting
from the interaction of a large number of subsystems (19, 20). The mathematical
concept of fractals characterizes objects on various scales, large as well
as small, and thus reflects a hierarchical principle of organization. In
this paper, a cell dynamical system model (21) for deterministic chaos
in atmospheric flows is developed by consideration of microscopic domain
eddy dynamical processes. The model enables formulation of scale-invariant
governing equations for the observed atmospheric flow stucture characteristics
(22-26). To begin with, a brief summary of the latest developments in the
modelling of dynamical systems, in particular the concept of deterministic
chaos, is presented in Sects. 2-4. In Sects. 5-7, the signature of deterministic
chaos in the observed structure of atmospheric flows are identified, the
limitations of existing numerical weather prediction models are discussed,
and a cell dynamical system model for atmospheric flows is described. Finally,
in Sect. 8 it is shown that the laws governing atmospheric flows are similar
to quantum mechanical laws for subatomic dynamics. The model enables us
to predict the following. (*i*) The strange-attractor design of fluid
flows consists of a nested continuum of helical vortex-roll circulations
with ordered two-way energy feedback between the larger and smaller scales.
(*ii*) The microscopic scale internal structure of the overall logarithmic
spiral vortex-roll circulations consist of the quasiperiodic Penrose tiling
pattern identified as the quasi-crystalline structure in condensed matter
physics (27). (*iii*) The atmospheric eddy energy structure follows
laws similar to quantum mechanical laws.

**2. Mathematical models
of dynamical systems and deterministic chaos**

Mathematical models
of dynamical systems, i.e., systems that evolve with time are traditionally
formulated using Newtonian continuum dynamics where it is assumed that
all change is continuous and the evolution equations of dynamical systems
are given by a system of partial diferential equations representing continuous
rate of change. The partial differential equations in general do not have
analytical solutions and therefore numerical solutions are obtained using
digital computers having finite precision. Such digital computer realizations
of continuum mathematical models for dynamical systems are inherently unrealistic
and result in deterministic chaos as explained earlier (Sect. 1). Mathematical
studies by scientists in diverse disciplines have revealed the existence
of deterministic chaos in disparate dynamical systems (1, 19, 28). The
computed trajectory of the dynamical system in the phase space comprising
the position and momenta coordinates traces out the self-similar fractal
geometrical shape of the strange attractor, so named because of its strange
convoluted shape being the final destination (attractor) of the trajectories.
Any two initially close points in the strange attractor rapidly diverge
with time and follow totaly different paths, though still within the strange-attractor
domain. Therefore, the future trajectories of initially close points are
unpredictable or random. The exact physical reason for the sensitive dependence
on initial conditions of deterministic nonlinear partial differential,
which are used for modelling dynamical systems as well as the self-similar
fractal geometry of the strange-attractor design that characterizes
the evolution trajectory in the phase space of the dynamical system, is
not yet identified (3, 4). Self-similarity implies scale invariance and
is a manifestation of dilation symmetry, whereby the shape of an object
is preserved during stretching. A self-similar object possesses the same
internal structure on all scales. Such self-similar objects are non-Euclidean
in shape and therefore possess a fractional or fractal dimension (29-31).
The fractal dimension ** D** is given by the relation

**3. Strange-attractor
design of real and model dynamical systems**

The computed strange
attractor of dynamical systems is a mathematical artifact (15) as explained
earlier and bears no relationship to the actual evolution trajectory. Further,
even for a realistic mathematical model, computer roundoff errors introduce
nontrivial uncertainities in the space differentials, namely dx, dy and
dz that result in artificial curvature for the trajectory, which eventually
ends up as limit cycles or periodicities for sufficiently long integration
time periods. Computer precision therefore, plays the role of a yardstick
in numerical model realizations and generates self-similar structures for
the continuum phase space trajectory, namely the strange attractor.

Recent studies show
that numerical model results scale with computer precision, and periodicities
in numerical model results are also a function of computer precision (14,
16). Computer model realizations that require long integration times are
therefore subject to computer precision uncertainities that result in the
loss of the predictability of the future state of the system.

However, such sensitive
dependence on initial conditions is actually exhibited by disparate real-world
dynamical systems and may be associated with information transport from
the microscale to the macroscale, which is indicated by the long-range
spacial and temporal correlations intrinsic to such systems. Therefore,
microscopic scale differences in initial conditions may contribute to appreciably
different large-scale space-time structures. It is important to identify
the exact microscopic scale mechanisms that contribute to the macroscale
space-time evolution of the robust self-similar strange-attractor design.
It should be possible to identify a simple conceptual model that is scale
invariant for the dynamical evolution of the system, i.e., a microscopic
scale scale unit-cell model that is directly applicable to the macroscale
multicellular model. Such a model for atmospheric flows is described in
Sect.7 and enables formulation of the dynamical processes of evolution
in simple mathematical formulations with analytical (algebraic) solutions
or where the numerical solutions does not require long-term integration
using digital computers.

**4. Cell dynamical
system model: current concepts and limitations**

In this nondeterministic
computational technique, the dynamical system is assumed to consist of
an assembly of identical unit cells. Starting with arbitrary initial conditions,
the evolution of the dynamical system proceeds at successive unit length
steps during unit intervals of time following arbitrary laws of interaction
between adjacent cells. The 'cellular automata' belong to the cell dynamical
system described above and do not require calculus-based long-term integration
schemes (21). However, the cellular-automata rules for evolution are arbitrary
and do not have any physical basis. The relevant physical processes must
therefore be incorporated in the cellular-automata schemes. A cellular-automata
computational scheme that incorporates the physics of atmospheric flows
is described in Sect. 7.

**5. Observed structure
of atmospheric flows and signatures of deterministic chaos**

Recent advances in remote
sensing and *in situ* measurement techniques have enabled us to document
the following new observational characteristics of turbulent shear flows
in the planetary atmospheric boundary layer (ABL) where weather activity
occurs. The ABL extends to about 10 km above the surface of the earth.

(*i*) The atmospheric
flow consists of a full continuum of fluctuations ranging in size from
the turbulence scale of a few millimetres to the planetary scale of thousands
of kilometres.

(*ii*) The atmospheric
eddy energy spectrum follows an inverse power law of form n
^{-B}
, where n
is the frequency and **B** the exponent. The exponential power-law form
for the eddy energy spectrum indicates self-similarity and scale invariance.
The exponent **B** is found to be equal to 1.8 for both meteorological
(time period in days) and climatological (time period in years) scales,
which indicates a close coupling between the two scales (33-39).

(*iii*) Satellite
cloud-cover photographs give evidence for the existence of helical vortex-roll
circulations (or large eddies) in the ABL as indicated by the organization
of clouds in rows and (or) streets, mesoscale (up to 100km) cloud clusters
(MCC), and spiral bands in synoptic scale weather systems (40).

(*iv*) The structure
of atmospheric flows is invariably helical (curved) as manifested in the
visible cloud patterns of weather systems, e.g., all basic mesoscale structures
such as medium scale tornado generating storms, squall lines, hurricanes,
etc. (41), and in particular the supercell storm (42).

(*v*) Atmospheric
flows give an implicit indication of the upscale transfer of a certain
amount of energy inserted at much smaller scales, thereby generating the
observed helical fluctuations (41, 43).

(*vi*) The global
cloud-cover pattern exhibits self-similar fractal geometrical structure
and is consistent with the observed scale invariance of the atmospheric
eddy energy spectrum (35, 44) (see characteristic (*ii*) above).

Atmospheric weather
systems exist as coherent structures consisting of discrete cloud cells
forming patterns of rows and (or) streets, MCC, and spiral bands. These
patterns maintain their identity for the duration of their appreciable
lifetimes in the apparently dissipative turbulent shear flows of the ABL
(45). The existence of coherent structures (seemingly systematic motion)
in turbulent flows, in general, has been well established during the last
20 years of research into turbulence. However, it is still debated whether
these structures are the consequences of some kind of instabilities (such
as shear or centrifugal instabilities), or whether they are manifestations
of some intrinsic universal properties of any turbulent flow (41).

Lovejoy and Schertzer
(35) have provided conclusive evidence for the signature of deterministic
chaos in atmospheric flows, namely the fractal geometry of global cloud-cover
pattern and the inverse power-law form **n
^{-B}**
where

**6. Limitations of
conventional ABL models**

** **Presently available
models for ABL turbulent flows are incapable of identifying the coherent
helical structural form intrinsic to turbulence. Also, the models do not
give realistic simulations of the space-time averages for the thermodynamic
parameters and the fluxes of buoyant energy, mass, and momentum because
of the following inherent limitations.

(*i*) The physics
of the observed coherent helical geometric structure inherent in turbulent
flows is not yet identified, and therefore the structural form of turbulent
flows cannot be modelled.

(*ii*) By convention,
the Newtonian continuum dynamics of the atmospheric flows are simulated
by the NS equations which are inherently nonlinear, and being sensitive
to initial conditions, give chaotic solutions characteristic of deterministic
chaos.

(*iii*) The governing
equations do not incorporate the mutual coexistence and interaction of
the full spectrum of atmospheric fluctuations that form an integral part
of atmospheric flows (10, 36, 49, 50).

(*iv*) The limitations
of available computer capacity necessitate severe truncations of the governing
equations, thereby generating errors of approximations.

(*v*) The above-mentioned
uncertainties are further magnified exponentially with time by computer
roundoff errors and result in unrealistic solutions (14, 15). Recent exhaustive
studies by Weil (51) and others also indicate that existing numerical models
of atmospheric boundary layer flows require major revisions to incorporate
an understanding of turbulence and diffusion in boundary layer flows. Recently,
there has been growing conviction that curent numerical weather prediction
models are inadequate for accurate forecasts (16, 52-55). Numerical modelling
of atmospheric flows, diffusion, and cloud growth therefore require alternative
concepts and computational techniques.

*6.1 Deterministic
chaos and weather prediction: current status*

At present, the signatures
of deterministic chaos, namely the fractal geometrical structure concomitant
with **1/****n**
noise, have been conclusively identified in model and real atmospheric
flows, and the fractal dimension of the strange attractor traced by atmospheric
flows has been estimated with recently developed numerical algorithms (55),
which use the time series data of meteorological parameters, e.g., rainfall,
temperature, windspeed, etc. However, such estimations of the fractal dimension
have not helped resolve the problem of the formulation of a simple closed
set of governing equations for atmospheric flows (57-60) mainly because
the basic physics of deterministic chaos is not yet identified.

**7. Cell dynamical
system model for atmospheric flows**

The nondeterministic model described below incorporates the physics of the growth of macroscale coherent structures from microscopic domain fluctuations in atmospheric flows. In summary, the mean flow at the planetary ABL posesses an inherent upward momentum flux of frictional origin at the planetary surface. This turbulence-scale upward momentum flux is progressively amplified by the exponential decrease of the atmospheric density with height coupled with the buoyant energy supply by microscale fractional condensation on hygroscopic nuclei, even in an unsaturated environment (61). The mean large-scale upward momentum flux generates helical vortex-roll (or large eddy) circulations in the planetary atmospheric boundary layer and is manifested as cloud rows and (or) streets, and MCC in the global cloud cover pattern. A conceptual model of large and turbulent eddies is shown in Fig. 1.

FIG
1. Conceptual model of large and turbulent eddies in the planetary ABL.
The mean air flow at the planetary surface carries the signature of the
fine scale features of the planetary surface topography as turbulent fluctuations
with a net upward momentum flux. This persistent upward momentum flux of
surface frictional origin generates large-eddy (or vortex-roll) circulations,
which carry upward the turbulent eddies as internal circulations. Progressive
upward growth of a large eddy occurs because of buoyant energy generation
in turbulent fluctuations as a result of the latent heat of condensation
of atmospheric water vapour on suspended hygroscopic nuclei such as common
salt particles. The latent heat of condensation generated by the turbulent
eddies forms a distinct warm envelope or a *microscale capping inversion*
layer at the crest of the large-eddy circulations as shown in the upper
part of the figure. The lower part of the figure shows the progressive
upward growth of the large eddy from the turbulence scale at the planetary
surface to a height ** R** and is seen as the rising inversion
of the daytime atmospheric boundary layer. The turbulent fluctuations at
the crest of the growing large-eddy mix overlying environmental air into
the large-eddy volume, i.e., there is a two-stream flow of warm air upward
and cold air downward analogous to superfluid turbulence in liquid helium
(see ref. 79). The convective growth of a large eddy in the atmospheric
boundary layer therefore occurs by vigorous counter flow of air in turbulent
fluctuations (see also Fig. 4), which releases stored buoyant energy in
the medium of propagation, e.g., latent heat of condensation of atmospheric
water vapour. Such a picture of atmospheric convection is different from
the traditional (see ref. 78) concept of atmospheric eddy growth by diffusion,
i.e., analogous to the molecular level momentum transfer by collision.

The generation of turbulent buoyant energy
by the microscale fractional condensation is maximum at the crest of the
large eddies and results in the warming of the large-eddy volume. The turbulent
eddies at the crest of the large eddies are identifiable by a *microscale
capping inversion* that rises upward with the convective growth of the
large eddy during the course of the day. This is seen as the rising inversion
of the daytime planetary boundary layer in echosonde and radiosonde records
and has been identified as the entrainment zone (62) where mixing with
the environment occurs.

Townsend (63) has investigated
the structure and dynamics of large-eddy formations in turbulent shear
flows and has shown that large eddies of appreciable intensity form as
a chance configuration of turbulent motion as illustrated in the following
example. Consider a large eddy of radius ** R** that forms in
a field of isotropic turbulence with turbulence length and velocity scales

**= 2(2p R)w_{*}(2r
w_{*})**

where *w***_{*}**
is tangential to the path elements

The above equation enables us to compute
the instantaneous acceleration **d W** for a large-eddy of radius

Equation [1] signifies a two-way ordered
energy (kinetic energy) flow between the smaller and larger scales and
[1] is therefore identified as the statement of the *law of conservation
of energy *for the universal period doubling route for chaos eddy growth
processes in atmospheric flows. Figure 2 shows the concept of the universal
period doubling route for chaotic eddy growth process by the self-sustaining
process of ordered energy feedback between the larger and smaller scales,
the smaller scales forming the internal circulations of the larger scales.

FIG
2. Physical concept of the universal period doubling route
to chaotic eddy growth process by the self-sustaining process of ordered
energy feedback between the larger and smaller scales, the smaller scales
forming the internal circulations of the larger scales. The figure shows
a uniform distribution of dominant turbulent scale eddies of length scale
2** r** . Large-eddy circulations such as ABCD form as coherent
structures sustained by the enclosed turbulent eddies. The r.m.s. circulation
speed of the large eddy is equal to the spatially integrated mean of the
r.m.s. circulation speeds of the enclosed turbulent eddies. Such a concept
envisages large-eddy growth in unit length step increments during unit
intervals of time with turbulence-scale yardsticks for length and time,
and is therefore analogous to the cellular automata computational technique.
The growth of the large-eddy by successive period doubling, namely, discrete
length step increments equal to the turbulence length scale is identified
as the physics of the universal period doubling route to chaos eddy growth
process.

Atmospheric boundary layer flows, therefore,
generate, as a natural consequence of surface friction, persistent microscopic
domain turbulent fluctuations that amplify and propagate upward and outward
spontaneously as a result of the buoyant energy supply from the latent
heat of condensation of atmospheric water vapour on suspended hygroscopic
nuclei in the upward fluctuations of air parcels. The evolution of the
macroscale atmospheric eddy continuum structure occurs in successive microscopic
fluctuation length steps in the ABL and therefore has a self-similar scale-invariant
fractal geometrical structure by concept and also according to [1]. Equation
[1] is therefore identified as the universal algorithm that defines the
space-time continuum evolution of the atmospheric eddy energy structure
(strange attractor). Such a concept of the autonomous growth of the atmospheric
eddy continuum with ordered energy flow between the scales is analogous
to the 'bootstrap' theory of Chew (64), the theory of implicate order envisaged
by Bohm (65), and Prigogine's concept of the spontaneous emergence of order
through a process of self-organization (65).

The turbulent eddy
circulation speed and radius increase with the progressive growth of the
large eddy as given in [1]. The successively larger turbulent fluctuations,
which form the internal structure of the growing large eddy, may be computed
from [1] as

During each length step growth **d R**
, the small-scale energizing perturbation

[4]

The angular turning **d****q**
inherent to eddy circulation for each length step growth is equal to **d R/R**
. The perturbation

Table 1. The computed spatial growth of the strange-attractor design traced by the macroscale dynamical system of atmospheric flows as shown in Fig. 3.

R |
W_{n} |
dR |
dq |
W_{n+1} |
q |

2 3.254 5.239 8.425 13.546 21.780 35.019 56.305 90.530 |
1.254 1.985 3.186 5.121 8.234 13.239 21.286 34.225 55.029 |
1.254 1.985 3.186 5.121 8.234 13.239 21.286 34.225 55.029 |
0.627 0.610 0.608 0.608 0.608 0.608 0.608 0.608 0.608 |
1.985 3.186 5.121 8.234 13.239 21.286 34.225 55.029 88.479 |
1.627 2.237 2.845 3.453 4.061 4.669 5.277 5.885 6.493 |

It is seen that the succesive values of
the circulation speed ** W** and radius

(c)

FIG. 3.
The internal structure of large-eddy circulations. (*a*) Turbulent
eddy growth from primary perturbation **OR _{o}** starting from
the origin

Turbulent eddy growth from primary perturbation
**OR _{o}**
starting from the origin

The time period of large-eddy circulation made up of internal circulations with the Fibonacci winding number is arrived at as follows. Assuming turbulence-scale yardsticks for length and time, the primary turbulence-scale perturbation generates successively larger perturbations with the Fibonacci winding number on either side of the initial perturbation. Therefore, the large-eddy time period

[5] *T
= t *[ 2 (1 + **t +t^{2}
+ t^{3}+
t^{4} ) + t^{5}
] = 43.74 t**

Therefore, the large-eddy circulation time period is also related to the geometrical structure of the flow pattern.

*7.1 Dominant weather
cycles *(*limit cycles*)

It was shown above that
dominant large-eddy growth occurs from turbulence-scale energy pumping
for successive scale ratio ranges **t ^{5}
= 11.09** .Therefore, from [1] the
following relations are derived for the length and time scales of limit
cycles in atmospheric flows.

*r *:* R = r *:* *t^{5}*r** ***
: ****t ^{10}
r : t^{15}
r : t^{20}
r**

The limit cycles or
dominant periodicities in atmospheric flows (71), possibly originating
from solar-powered primary oscillations, are given in the following. (*i*)
The 40- to 50-day oscillation in the atmospheric general circulation and
the quasi-five yearly ENSO phenomena (49) may possibly arise from diurnal
surface heating. (*ii*) The 40- to 50-year cycle in climate may be
a direct consequence of the annual solar cycle (summer and winter oscillation).
(*iii*) The quasi-biennial oscillation (QBO) in the tropical stratospheric
wind flows may arise as a result of the semidiurnal pressure oscillation.
(*iv*) The 22-year cycle in weather patterns associated with the solar
sunspot cycle may be related to the newly identified 5-min oscillations
of the sun's atmosphere (72). The growth of large eddies by energy pumping
at smaller scales, namely the diurnal surface heating, the semidiurnal
pressure oscillation, and the annual summer-winter cycles as cited above
is analogous to the generation of chaos in optical emissions triggered
by a laser pump (73). Recent barometer data on the planet Mars, whose tenuous
atmosphere magnifies atmospheric oscillations, reveal oscillations with
periods very close to 1.5 Martian days preceding episodes of global dust
storms (74), which indicates a possible cause and effect mechanism as given
in [6]. The identification of limit cycles in atmospheric flows is possible
by means of the continuous periodogram analysis of long-term high-resolution
surface pressure data and this will help long-term prediction of regional
atmospheric flow pattern (75).

As seen from Fig. 3
and from the concept of eddy growth, vigorous counter flow (mixing) characterizes
the large-eddy volume. the steady-state fractional volume dilution ** k**
of the large-eddy volume by environmental mixing is given by

Earlier it was shown that the successive
eddy length step growths generate the angular turning **d****q**
of the large-eddy radius ** R** given by

[8]
*k = *1/**t^{2}=
0.382**

Since the steady-state fractional volume
dilution of the large eddy by inherent turbulent eddy fluctuations during
successive length step increments is equal to **0.382**, i.e., less
than half, the overall Euclidean geometrical shape of the large eddy is
retained as manifested in the cloud billows, which resemble spheres.

The fractional outward
mass flux of air across a unit cross section for any two successive steps
of eddy growth is given by

*f _{c} = *1/t
= 0.618

** f_{c}**
is therefore equal to the percolation threshold for critical phenomena,
i.e., where the liquid-gas mixture separates into the liquid and gas phases
with the formation of self-similar fractal structures (76) and in the case
of atmosphereric flows this is associated with the manifestation of coherent
vortex-roll structures. The ratio of the actual (observed) cloud liquid
water content

The vigorous counterflow of air (mainly vertically) in turbulent eddy fluctuations characterizes the internal structure of the growing large eddy. The turbulent eddies carried upward by the growing large eddy are amplified to form 'cloud-top gravity oscillations' and are manifested as the distictive cauliflower-like surface granularity of the cumulus cloud growing in the large-eddy updraft regions under favourable conditions of moisture supply (Fig. 4). Therefore, atmospheric convection and the associated mass, heat, and momentum transport in the ABL occur by the vigorous counterflow of air in intrinsic fractal structures and not by eddy diffusion processes postulated by the conventional theories of atmospheric convection (78). Such a concept of atmospheric convection is analogous to superfluid turbulence in liquid helium (79).

FIG. 4. Cloud structure in the ABL. The turbulent eddies carried upward by the growing large eddy (see Fig. 1) are amplified to form cloud-top gravity (buoyancy) oscillations and are manifested as the distinctive cauliflower-like surface granularity of the cumulus cloud growing in the large-eddy updraft regions under favourable conditions of moisture supply in the environment. The fractal or broken cloud structure is a direct result of cloud water condensation and evaporation, respectively, in updrafts and downdrafts of the innumerable microscale turbulent eddy fluctuations in the cloud volume. Therefore, atmospheric convection and the associated mass, heat, and momentum transport in the ABL occur by the vigorous counterflow of air in intrinsic fractal structures and not by eddy diffusion processes postulated by the conventional theories of atmospheric convection (see Fig. 1).

The r.m.s circulation speed

The above equation is the well-known logarithmic
spiral relationship for wind profile in the surface ABL derived from conventional
eddy diffusion theory (78) where ** k** is a constant of integration
and its magnitude is obtained from observations as

**8. Deterministic chaos
and quantumlike mechanics in atmospheric flows**

Historically, macroscale
physical phenomena are described by classical dynamical laws (Newton's
laws) for all practical purposes, while subatomic phenomena, e.g., electromagnetic
radiation require quantum mechanical laws to explain their physical manifestation.
It has not yet been possible to identify a universal theory of everything
(TOE) for the totality of manifested phenomena from the macro- to the subatomic
scales. So far classical dynamical laws have failed to explain deterministic
chaos in macroscale dynamical systems. On the other hand, the standard
interpretation of quantum mechanics, chiefly the ad hoc assumption of the
wave-particle duality for the quantum system, e.g., electron or photon,
does not provide a complete description of a quantum system. Though quantum
mechanical laws are successful in describing subatomic phenomena, the following
inconsistencies are yet to be resolved.

(*i*) The interpretation
of Shrodinger's wave function as quantities whose squared amplitudes give
the probability density that a particle will be at a particular place (if
the arguments of the wave function are in space coordinates). Such a declaration
that algebraically additive amplitudes must be squared to obtain probability
densities is unsatisfactory in the absence of physically consistent mathematically
rigorous proof (81).

(*ii*) The unresolved
issue of nonlocality in quantum mechanics, namely, the Einstein-Podolsky-Rosen
(EPR) 'paradox', whereby the spatially separated parts of a quantum system
(photon, electron, etc.) respond as a unified whole to local perturbations
(84, 85).

(*iii*) Energy
propagation and interchange in quantum systems occur in discrete quanta
or packets of energy content *h***n**
where ** h** is a universal constant of nature (Planck's
constant) and

(

In the following it is shown that atmospheric flow structure follows laws similar to quantum mechanical laws for subatomic dynamics. The apparent inconsistencies of quantum mechanical laws described above are explained in terms of the physically consistent characteristics inherent in eddy circulation patterns in atmospheric flows.

In summary: the kinetic energy (KE) of an eddy of rotation frequency

*W _{p} = *2pn

From [1],

Furthermore,

**KE = p Hn
= (1/2)Hw**

** H** is equal to the product
of the momentum of the planetary scale eddy and its radius and therefore
represents the angular momentum of the planetary scale eddy about the eddy
centre. Therefore, the

FIG. 5. Quantum mechanical analogy with macroscale phenomena of atmospheric flows. The upper part of the figure illustrates the concept of wave-particle duality as physically consistent in the common place observed phenomena of the formation of clouds in a row as a natural consequence of cloud formation and dissipation, respectively, in the updrafts and downdrafts of vortex roll circulations in the ABL. The lower part of the figure illustrates the concept of non-locality by analogy with instantaneous transfer of energy from effort to load in a pulley and as also inferred by the physically consistent phenomena of instantaneous circulation balance in the atmospheric vortex-roll circulations with alternating balanced high- and low-pressure areas.

(

(

(

The continuously evolving atmospheric eddy continuum traces out the quasi-periodic Penrose tiling pattern as shown in Fig. 3, where, as a natural consequence the eddy growth is associated with an increase in phase angle and is analogous to Berry's phase in quantum mechanics (84). Long-range correlations in regional weather activity as displayed markedly in the ENSO phenomena may be a manifestation of non-local connection associated with Berry's phase in quantum systems.

The macroscale atmospheric flow structure may therefore provide physically consistent interpretations for the apparent inconsistencies of quantum mechanical laws thereby unifying the laws of natural phenomena.

**9. Conclusions**

The nondeterministic
cell dynamical system model for atmospheric flows described in this paper
enables us to formulate simple analytical (algebraic) equations for the
atmospheric flow structure pattern, i.e., the strange attractor, and to
predict the following new results.

(*i*) The strange-attractor
design of atmospheric flow structure consists of a nested continuum of
vortex-roll circulations with ordered energy flow between the scales.

(*ii*) Large-eddy
circulations grow from space-time integration of internal nontrivial small-scale
energy pumping, e.g., solar-powered turbulent buoyant energy of frictional
origin in atmospheric flows. Therefore, small-scale circulation balance
requirements impose long-range orientational order and are manifested as
the quasi-periodic Penrose tiling design for the internal structure of
large-eddy circulations with overall logarithmic spiral flow pattern. The
growth of large eddies by successive unit length step increments equal
to the turbulence-scale length is identified as the universal period doubling
route to a chaotic eddy growth process in atmospheric flows.

(*iii*) The universal
constant for deterministic chaos is identified as Von Karman's constant
and is equal to 0.382, which quantifies the steady state fractional volume
dilution of large eddies by turbulent fluctuations.

(*iv*) Convective
growth of large eddies in atmospheric flows occurs by vigorous counterflow
in inherent tubulent eddy fluctuations and not according to the conventional
concept of eddy diffusion, i.e., momentum transfer by collision. Such a
concept is analogous to superfluid turbulence in liquid helium.

(*v*) Atmospheric
flows follow laws similar to quantum mechanical laws. The quantum mechanical
analogy with macroscale atmospheric flows is seen in commonplace
events such as the formation of clouds at the crests (updrafts) of eddy
(wave) circulations, e.g., clouds in a row, thereby resolving the apparent
paradox of wave-particle duality.

Since the strange attractor
design of atmospheric flow structure consists of periodicities with fine
structure (continuum) a continuous periodogram analysis of time series
data will enable a complete description of the strange attractor and such
a concept has recently been put forth by Cvitanovic (85). Further, identification
of dominant periodicities, i.e., limit cycles in atmospheric flows by continuous
periodogram analyses of multistation high-resolution surface pressure data
may help long-range (months to years) forecasts of global weather patterns.

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