Deterministic Chaos Model for Self-Organized
Adaptive Networks in Atmospheric Flows
A. Mary Selvam
Deputy Director (Retired)
Indian Institute of Tropical Meteorology,
Pune 411 008, India
Proceedings of the IEEE National Aerospace
and Electronics Conference - NAECON 1989, Dayton, OH, May 22-26,
The complex spatiotemporal
patterns of atmospheric flows resulting from the cooperative existence
of fluctuations ranging in size from millimeters to thousands of kilometers
are found to exhibit long-range spatial and temporal correlations manifested
as the selfsimilar fractal geometry to the global cloud cover pattern and
the inverse power law form for the atmospheric eddy energy spectrum. Such
long-range spatial and temporal correlations are ubiquitous to extended
natural dynamical systems and is a signature of the strange attractor design
characterizing deterministic chaos or self-organized criticality. The unified
network of global atmospheric circulations is analogous to the neural networks
of the human brain.
Long-range spatial and
temporal correlations in dynamical systems has been identified as a signature
of self-organized criticality or deterministic chaos (Bak, Tang and Wiesenfeld,
1988) and indicates self-organized information transport and long-term
memory in spatially extended selfsimilar geometrical networks (Vastano
and Swinney, 1988). Such spatially extended selfsimilar fractal geometrical
design associated with long-term memory or 1/n
noise, where n
is the frequency, is ubiquitous to natural dynamical systems. Deterministic
chaos in atmospheric flows is manifested as the fractal geometry to the
global cloud cover pattern and the inverse power law form, namely n-B
where B is the exponent for the atmospheric eddy energy spectrum (Lovejoy
and schertzer, 1986). Atmospheric teleconnections such as the ElNino/Southern
Oscillation (ENSO) cycle in the weather patterns which are responsible
for devastating changes in normal global weather regimes (Trenberth et
al., 1988) are also manifestations of long-range correlations in regional
weather activity. The physical mechanism responsible for the observed robust
spatiotemporal structure of strange attractor of dynamical systems is not
yet identified. In this paper a cell dynamical system model for atmospheric
flows is developed by consideration of microscopic domain eddy dynamical
processes. The non-deterministic model enables formulation of simple closed
set of governing equations for prediction of the observed long-range spatial
and temporal correlations in global weather systems (Mary Selvam, 1988).
The model concepts may find applications in design of artificial intelligence
systems with pattern recognition capabilities.
The mathematical models of natural dynamical
Mathematical models of
dynamical systems by tradition are based on Newtonian continuum dynamics
where it is assumed that all change is continuous and that evolution of
a system may be represented by equations with continuous rates of change.
The evolution equations of dynamical systems in general consist of nonlinear
partial differential equations which do not have analytical solutions and
therefore require digital computers for their numerical solutions. Digital
computer solutions involving long-term integration schemes for the continuum
dynamics incorporated in the nonlinear partial differential equations lead
to the following uncertainties in the model predictions. (1) The nonlinear
partial differential equations are sensitive to initial conditions and
give chaotic solutions characteristic of deterministic chaos. (2) Computer
capacity related truncations in model equations lead to errors of approximations.
(3) Computer precision related roundoff errors magnify exponentially with
time the above mentioned uncertainties and give unrealistic model predictions
(Beck and Roepstorff, 1987; McCauley, 1988; Grebogi and Yorke, 1988). The
abstract mathematical strange attractor traced by the evolution trajectory
in the six-dimensional phase space, i.e. three (x, y, z) position coordinates
and the corresponding three momenta coordinates is a computational artifact
and has no direct relationship with the physics of the dynamical evolution.
In recent years there
is growing conviction that conventional formulations of natural laws, their
mathematical models and computer realizations are inherently unrealistic
(Davies, 1988). An urgent need is now felt for alternative conceptual models
with exact solutions and robust computational techniques for the prediction
of the evolution of dynamical systems. Fluid flows in particular with their
enormous degrees of freedom cannot be modelled with conventional techniques
(Ottino et al., 1988). Several attempts have been made with limited
success to model fluid flows with molecular dynamics simulation using Navier-Stokes
equations (Kraichnan, 1988) over microscopic length scales. A non-deterministic
approach is the cell dynamical system model to which belongs the cellular
automata computational technique (Oona and Puri, 1988). Cell dynamical
system envisages an ensemble of identical unit cells evolving according
to predetermined arbitrary local laws of interaction between adjacent cells.
Cellular automata is therefore basically the evolution of the system at
successive unit length intervals during unit time intervals. The cell dynamical
system evolution rules are arbitrary and does not incorporate the physics
of the dynamical process. Realistic simulation of the dynamical system
requires the identification of the physics of the self-organized criticality.
Physics of Self-Organized Criticality in Atmospheric
indicates the steady state existence of ordered structures amidst apparent
disordered background fluctuations, in particular with reference to the
critical phenomena of phase transitions. In the case of atmospheric flows
self-organized criticality relates to the existence of global coherent
structures, e.g. cloud rows/streets, meso-scale cloud clusters, hurricane
spiral cloud bands in an apparently dissipative turbulent background. The
physical mechanism responsible for the observed self-organized criticality
is described in the following.
The mean flow at the
planetary atmospheric boundary layer (ABL) possesses an inherent upward
momentum flux of surface frictional origin. The turbulence scale upward
momentum flux is amplified progressively by the exponential decrease of
atmospheric density with height coupled with buoyant energy generation
by microscale fractional condensation (MFC) on hygroscopic nuclei even
in an unsaturated environment (Pruppacher and Klett, 1978). The mean upward
momentum flux generates progressively larger vortex roll or large eddy
circulations which are manifested as cloud rows/ streets in global cloud
cover pattern. The space-time integrated mean of the non-trivial internal
turbulence scale buoyant energy circulations generates the observed large
eddies. The root mean square (r.m.s) circulation speed W of the
large eddy of radius R obtained by such integration of the internal
dominant turbulent scale energy circulations of length r and r.m.s
circulation speed w is given as (Townsend, 1956)
The above equation expresses the relationship
between domain size and buoyant energy content in terms of the microscopic
scale dynamical processes and is therefore analogous to (1) the renormalization
theory in statistical physics where the idea is to relate scale transformations
to changes of the variable, e.g. buoyant energy and (2) the Ising model
for phase transformations (Lebowitz et al., 1988) which solves the
puzzle of how it is that nearest-neighbour interactions propagate their
effect cooperatively to give rise to correlation length of macroscopic
length scale near the critical point.
Large eddy growth occurs
by turbulent buoyant acceleration w* generated during
turbulent eddy fluctuations of length r and therefore the incremental
length step growth dR of the large eddy is equal to r. The
corresponding increase in r.m.s. circulation speed dW of large eddy
is given by Eq. (1) as follows.
Turbulence scale yardsticks for length and
time for large eddy growth implied by Eq.(1) is intrinsic to the concept
of large eddy growth from turbulence scale buoyant energy generation. Such
a concept of large eddy growth by successive length scales doubling from
turbulence scale energy pumping with two-way energy feedback between the
larger and smaller scales is analogous to the following phenomena in other
fields of physics. (1) The universal period doubling route to chaos or
deterministic chaos in disparate dynamical systems (Fairbairn, 1986; Chernikov
al., 1988) (2) Stokes and anti-Stokes laser emission during chaos in
optical emissions in a nonlinear optical medium triggered by a laser pump
(Harrison and Biswas, 1986). (3) The growth of autowaves in a reactive
medium, i.e. wave growth occurs by energy supply from the medium during
stretching or dilation. (4) Wave cybernetics, i.e. cooperative existence
of a unified eddy continuum. (5) Wave synergetics which again means two-way
energy feedback in a unified eddy network (Krinsky, 1984).
The above method of
determination of large eddy evolution using turbulence scale yardsticks
for length and time is analogous to the concept of 'cellular automata'
computational technique which belongs to the non-deterministic cell dynamical
system method of determination of the evolution of dynamical systems (Oona
and Puri, 1988). Further, the macro-scale eddy evolution occurs as a natural
consequence of inherent microscopic scale dynamical processes as given
The growing large eddy
carries the turbulent eddies as internal circulations. The turbulent eddy
fluctuations mix environmental air into the large eddy volume. The steady
state fractional volume dilution k of large eddy by turbulent eddy
fluctuations is given as
Using Eq.(1) it may be computed and shown
that k>0.5 for scale ratio Z (=R/r) less than 10.
Identifiable large eddy growth can occur for scale ratio Z>=10 only
since for smaller scale ratios the large eddy identity is erased by turbulent
eddy fluctuations. The r.m.s. circulation speed W of large eddy
which grows from the turbulence scale buoyant energy generation and originating
from the planetary surface may therefore be obtained by integrating Eq.(3)
and is given as
The above equation is the well known logarithmic
wind profile relationship for surface boundary layer obtained by conventional
eddy diffusion theories (Holton, 1979) where k is a constant of
integration named Von Karman's constant and its value obtained by observation
is equal to 0.4. Using Eq.(1) k is computed as equal to 0.38
for scale ratio equal to 11.09 which corresponds to golden mean
winding number for organized large eddy growth in the ABL as shown in a
later section. The cell dynamical system model or deterministic chaos model
for atmospheric flows enables to predict the logarithmic wind profile relationship
for the total planetary atmospheric boundary layer and also predicts the
observed value of 0.4 for the Von Karman's constant. Von Karman's
constant is a quantitative non-dimensional measure of the fractional volume
dilution of large eddy structure by steady state mixing with environment
intrinsic to an eddy ensemble and in particular is responsible for the
ubiquitous broken cloud surface geometry. Similar concepts of environmental
mixing apply to all open systems in diverse other fields and therefore
Von Karman's constant is more universal than the Feigenbaum's constants
(Feigenbaum, 1980) characterising deterministic chaos in disparate nonlinear
systems. The Von Karman's constant scales with the turbulence scale yardstick
for length, i.e. the precision for length measurement. Smaller and smaller
selfsimilar structures can be identified using progressively smaller yardsticks.
This concept can be applied to mathematical models of nonlinear systems
where computer precision may play a role in the generation of structures
in numerical models. Numerical studies indicate that computer results of
nonlinear systems scales with computer precision (Beck and Roepstorff,
1987) and periodicities in numerical models may be related to computer
precision (Grebogi et al., 1988).
The rising large eddy
gets progressively diluted by vertical mixing due to turbulent eddy fluctuations
and a fraction f of surface air reaches the normalized height Z
The steady state fractional air mass flux
from the surface is dependent only on the dominant turbulent eddy radius.
The vertical profile of the ratio of the actual liquid water content (q)
to the adiabatic liquid water content (qa) will therefore
follow the f distribution since the fraction f of the air
of surface origin which reaches the normalized height Z after dilution
by vertical mixing caused by the turbulent eddy fluctuations is given by
Eq.(6). The model predicted profile of q/qa is in close
agreement with observed profile as reported by Warner (1970), in particular
the cloud base value being equal to 0.6 corresponding to a scale
ratio of Z = 11.09 for dominant eddy growth. The cloud base vale
of 0.6 for q/qa is almost equal to fc
, the critical concentration for percolation threshold for critical phenomena,
i.e. when the liquid-gas mixture separates into gas and liquid phases (Mort
La Brecque, 1987). This percolation threshold for cloud growth by condensation
is consistent with the observed fractal geometry of individual cloud shapes
and also cloud ensembles. The percolation threshold for critical phenomena
may also be interpreted as the fractional probability of occurrence of
the initial perrturbation in the dominant eddy length scales, e.g. global
scale weather phenomena originating from the solar insolation driven buoyant
energy flux, namely the La Nina and El Nino relating respectively to normal
and abnormal global weather phenomena are found to have probabilities of
occurrences of 0.59 and 0.41 respectively (Fraedrich, 1988).
The critical concentration for percolation threshold therefore signifies
organized growth of a persistent perturbation into large-scale structures,
e.g. spread of diseases in the environment.
Coherent Helicity in Atmospheric Flows
The existence of coherent
structures (seemingly systematic motion) in turbulent flows has been well
established during the last 20 years of research in turbulence. It is still
however debated whether these structures are the consequence 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 (Levich, 1987). The existence of coherent helical structures in weather
systems has been documented in observational studies of cloud cover pattern,
in particular the hurricane spiral, the tornado funnel cloud and also wind
flow trajectories in supercell storms. In the following the cell dynamical
system model enables to show that coherent helicity is intrinsic to atmospheric
eddy growth in the ABL involves successive radial growth steps equal to
turbulence length scale along with a corresponding eddy angular rotation
from the origin. The eddy growth
originating from O (Figure 1) follows the spiral curve OAB because of the
inherent logarithmic eddy circulation trajectory at Eq.(4).
Figure 1: The logarithmic spiral geometric
design of circulation trajectories in atmospheric flows
The angular rotation from the origin at
location A is measured with respect to axis OX. Let OA and OB denote the
locations of the large eddy radii
R and R+dR for a growth
period of one second. The angular rotation dq
is given by
where rR is the turbulent
eddy radius corresponding to large eddy radius R. AB is the tangent
at A to the circle drawn with center O and radius R so that
AB will also represent the tangent of the
spiral at A for limited range. The angle BAC between the logarithmic spiral
and its tangent is called the crossing angle a
of the spiral.
Substituting b=tan a
and integrating for eddy growth from r to R, the above
This is the equation for an equiangular logarithmic
spiral when the crossing angle a
is a constant. At any location A the wind flow into the eddy continuum
system traces out a logarithmic spiral geometrical pattern.
Deterministic Chaos and Golden Mean Winding
In the following it is
shown that the universal period doubling route to chaos growth process
generates periodicities which follow the Fibonacci mathematical number
series such that the ratio of successive period lengths is equal to the
golden mean, namely
(McCauley, 1988). Considering eddy growth per second, the instantaneous
incremental growth dR of the large eddy of length R is equal
to Wn , the large eddy circulation speed at the nth
incremental time step. The corresponding angular turning rate dq
per second is equal to Wn . The large eddy circulation
speed Wn+1 at the (n+1)th time
step is equal to
The radius R and the corresponding
angular rotation q
are obtained respectively as the cumulative sums of dR and dq
. The instantaneous values of R, Wn ,dR,
, Wn+1 , and q
are tabulated in Table 1 for eddy growth starting from unit length onwards
applying the above concept of period doubling large eddy growth process.
Table 1 shows that the period doubling
growth sequence generates as a natural consequence successive large eddy
lengths R which follow the Fibonacci series, i.e.
with the golden mean winding number
Quasicrystalline Structure for Atmospheric
The cell dynamical system
model for atmospheric flows enables to show that the growth of large eddies
from turbulence scale gives rise to the geometric pattern of the quasiperiodic
Penrose tiling pattern identified as the quasicrystalline structure in
condensed matter physics (Janssen, 1988). Earlier it was shown that large
eddy growth occurs in length step increments following the Fibonacci number
series. Therefore any primary perturbation RoO (Figure 2) generates
compensating return circulations on either side along isosceles triangles
with 108 degrees vertex angles.
Figure 2: The internal structure of
dominant large eddy circulation. The small-scale internal circulation structure
forms the quasiperiodic Penrose tiling pattern with adjacent fat (unshaded)
and thin (shaded) rhombi.
A complete large eddy circulation is therefore
completed in 5 radial length step increments and associated angular rotation
of 36 degrees on either side of the primary perturbation. The envelopes
of the large eddy on either side of the primary perturbation traces out
the logarithmic spiral
is the golden mean
One complete large
eddy circulation is traced out in five length steps and therefore the radius
of the dominant large eddy =OR5 =rt5
=11.09r. The dominant large
eddy radius R being equal to 11.09r is consistent with earlier
intuitive deduction of large eddy growth for scale ratios greater than
alone. The internal structure of one complete large eddy circulation consists
of adjacent balanced counter rotating circulations tracing out the Penrose
tiling pattern (Figure 2) identified as the quasicrystalline structure
in condensed matter physics (Janssen, 1988). The short range circulation
balance requirements impose long-range orientational order in the quasicrystalline
structure for large eddies in atmospheric flows and is consistent with
the observed long-range correlations in global weather phenomena, e.g.
the ElNino/Southern Oscillation (ENSO) cycle. The large eddy internal structure
therefore has five-fold symmetry of the dodecahedron which is referred
to as the icosahedral symmetry, e.g. the geodesic dome conceived by Buckminster
Fuller. Recently Carbon macromolecules C60 formed by condensation
from a carbon vapour jet are found to have such icosahedral symmetry of
the closed soccer ball and has been named buckminsterfullerene (Curl and
The time period of
large eddy circulation made up of internal circulations with 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 Fibonacci winding number on either
side of the initial perturbation. Therefore large eddy time period T
is directly proportional to the total circulation path traversed on any
one side and is given in terms of the turbulence scale time period t
Therefore large eddy circulation time period
is also related to the geometrical structure of the flow pattern.
Dominant Weather Cycles (Limit Cycles)
It was shown above that
identifiable large eddy growth occurs for successive scale ratio ranges
11.09. Therefore from Eq.(1) the following relations are derived for length,
time and energy scales of limit cycles in atmospheric flows.
the turbulent eddy energy is equal to (4/3)pr3w2
and E the large eddy energy content similarly can be shown to be
equal to C where C =(7/11)Z2 . The limit
cycles in atmospheric flows originating from solar insolation powered primary
oscillations are given in the following. (1) The 40-50 day oscillation
in the atmospheric general circulation and the quasi-5 yearly ENSO phenomena
(Lau and Chan, 1988) may possibly arise from diurnal surface heating. (2)
The 40-year cycle in climate may be a direct consequence of the annual
solar cycle (summer and winter). (3) The QBO (quasi-biennial oscillation)
in the tropical stratospheric wind flows may arise as a result of the semi-diurnal
pressure oscillation. (4) The 20-year cycle in weather patterns associated
with the solar sunspot cycle may be related to the newly identified 5-minute
oscillation of the sun's atmosphere. The growth of large eddies by energy
pumping at smaller scales, namely the diurnal surface heating, the semi-diurnal
pressure oscillation and the annual summer-winter cycle as cited above
is analogous to the generation of chaos in optical emissions triggered
by a laser pump (Harrison and Biswas, 1986). Continuous periodogram analysis
of long-term high resolution surface pressure data will give the amplitude
and phase of the limit cycles in the regional atmospheric flow pattern.
Recent barometer data on planet Mars reveal oscillations with periods very
close to one (Martian) day and half a day preceding episodes of global
dust storms (Allison, 1988).
Deterministic Chaos and Statistical Normal
The statistical distribution
characteristics of natural phenomena commonly follow normal distribution
associated conventionally with random chance. The normal distribution is
characterized by (1) the moment coefficient of skewness equal to zero,
signifying symmetry and (2) the moment coefficient of kurtosis equal
to three representing intermittency of fluctuations on relative time scales.
The large eddy grows from the space-time integration of inherent turbulent
eddies and therefore the eddy energy spectrum follows the cumulative normal
probability density distribution. The probability P of occurrence
of the turbulent fluctuations of energy e
in the dominant eddy fluctuations of energy E is given as P=(e/E)
multiplied by the relative frequency of occurrence. The relative frequency
of occurrence of the primary perturbation in the first stage of dominant
eddy growth with a total of ten length step growth is equal to 1/10
and therefore P =0.1571 from Eq.(8). The successive dominant
eddy growths have standard deviations of inherent primary perturbations
equal to s,
etc. since W2/W1 =lnZ2/lnZ1,
where Z2 =Z12 and W1
=s . The
corresponding probabilities of occurrence are P, P2,
where P=0.1571 and agree more or less with the normal distribution
values. Since an eddy motion is inherently symmetric with bidirectional
energy flow, the skewness factor is equal to zero for one complete eddy
circulation thereby satisfying the law of conservation of momentum. The
moment coefficient of kurtosis which represents the intermittency of turbulence
is shown in the following to be equal to three. For dominant eddy growth
vigorous counter flow occurs during the successive oppositely directed
radial length step growths which follow the Fibonacci number series
..... The critical concentration
f for percolation threshold in
critical phenomena, i.e. the steady state fractional outward mass flux
resulting from unit primary perturbation = 1/1.618 =0.618
for each radial length step growth and the corresponding fractional mass
dilution, namely the Von Karman's constant as defined earlier is equal
to 1-0.618=0.382 in agreement with earlier value calculated for
the dominant large eddy growth (Eq.3). The ratio of momentum flux corresponding
to the primary perturbation propagation in the successive radial length
step growths with respective perturbation speeds
is given by W1/kW2 =lnZ1/klnZ2
=1/2k since Z2 =Z12 . The
moment coefficient of kurtosis for each period doubling length step growth
is (W1/kW2)4 and is equal
to 1/(2x0.38)4 ~3 and corresponds to that for normal
distribution. Incidentally it follows that the percolation threshold for
critical phenomena of the ubiquitous period doubling growth process in
nature is equal to 1-k = 0.618 and is in agreement with reported
values (Mort La Brecque, 1987).
The Mathematical Language for Deterministic
Intensive numerical studies
of mathematical models of diverse nonlinear systems has enabled formulations
of deterministic chaos in terms of the following mathematical functions
(McCauley, 1988). (1) The F-A spectrum of multifractal structure is defined
as follows. The field of chaos is characterized by a selfsimilar fractal
geometrical structure with fractal dimension D which is defined
as dlnM(R)/dlnR where M(R) is the mass contained within a
distance R from a typical point in the object. In the context of
atmospheric eddy energy structure the eddy energy (potential) spectrum
is directly proportional to the mass distribution for various lengths R.
Therefore the slope of the eddy energy spectrum plotted on a log-log scale
will give the fractal dimension for the atmospheric eddy energy structure.
Since the atmospheric eddy energy spectrum is the same as the cumulative
normal probability density distribution the fractal dimension is equal
to the slope of the cumulative normal probability density distribution
plotted on a log-log scale. The eddy energy spectrum therefore has multifractal
dimension and is consistent with observations in other nonlinear systems.
A convenient way of characterizing a multifractal is by the function F
which measures how many times N(A)dA
one finds the scaling A falling in an interval of size dA
The F-A spectrum can be computed from
the cumulative normal distribution curve. (2) The Kolmogorov-Sinai entropy
and the Lyapunov exponent (McCauley, 1988) both are a measure of the exponential
growth of initial uncertainties in the field of chaos and therefore are
represented by b equal to 0.618 in Eq.(7) which represents
the steady state period doubling growth of atmospheric eddies with golden
mean winding number t
. (3) The Von Karman's constant is shown (Mary Selvam, 1987) to be more
universal than the Feigenbaum's constants (Feigenbaum, 1980) characterizing
chaos in disparate nonlinear systems. The above concept of a scale invariant
continuum of eddies for the field of chaos is analogous to the Sine Circle
map technique (McCauley, 1988) and enables quantification of universal
characteristics of nonlinear systems and is consistent with recent investigations
The atmospheric eddy
continuum energy structure follows quantum mechanical laws (Mary Selvam,
1987). The energy manifestation of radiation and other subatomic phenomena
appear to posses the dual nature of wave and particles since one complete
eddy energy circulation is inherently bidirectional with corresponding
bimodal form of manifested phenomena, e.g. formation of clouds in the updraft
regions and dissipation of clouds in the downdraft regions giving rise
to discrete cellular structure to cloud geometry. The geometric phase difference
between successive eddies, i.e. the crossing angle is related to the periodicities
as shown earlier (Eq.7) and therefore large eddy growth is inherently associated
with a geometric phase change and this result is consistent with the recently
identified relation between geometric phase and frequency in laser propagation
(Simon et al., 1988) and is analogous to manifestation of Berry's
phase in subatomic phenomena. The inherent continuity of the eddy circulations
in the unified network give rise to non-local connections manifested as
the long-range spatiotemporal correlations in the robust architecture of
the strange attractor in dynamical systems.
The unified network of
the atmospheric eddy continuum circulations with inherent ordered two-way
energy cascade between the component eddies provides for dynamic information
storage and global response to individual eddy circulation perturbations
and is analogous to the neural network of the human brain (Schoner and
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