Long-range spatiotemporal correlations, recently identified as

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.

(

(

(

(

(

(

Atmospheric weather systems exist as coherent structures consisting of discrete cloud cells forming patterns of rows and (or) streets,

Lovejoy and Schertzer (35) have provided conclusive evidence for the signature of

(

(

(

(

(

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

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

[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.

*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

*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

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 '

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

(

(

(

(

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 (

*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

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.

(

(

(

(

(

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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