The Application of Deterministic Chaos to Atmospheric
Physics
A. Mary Selvam* and Suvarna Fadnavis
Indian Institute of Tropical Meteorology, Pune-411
008, India
Abstract
A detailed theory of deterministic chaos is developed
for atmospheric turbulence and applied to mathematically model the formation
of global weather systems, climatological weather cycles and the electrical,
dynamical and microphysical characteristics of clouds.
I. Introduction
Satellite cloud pictures indicate organisation of cloud
patterns in rows, streets and Mesoscale (20-200 km) Cloud Clusters (MCC)
as a common feature of global weather systems1 and therefore
provide evidence for the existence of organised helical vortex roll circulations
or large eddies in the planetary atmospheric boundary layer (ABL)
extending from the earth's surface to the top of the troposphere (10-15
kms). The turbulent troposphere also sustains regular long term climatological
cycles e.g., the Quasi-Biennial Oscillation (QBO) and the sunspot
related 20 year cycle in weather patterns2. It is not clear
how such long-lived organised circulations are maintained in the dissipative
turbulent environment of the ABL3. In this paper it is
shown that the universal period doubling route to chaos is the growth
mechanism for organised weather systems in the global planetary atmosphere.
II. The Universal Period Doubling
Route to Chaos or Deterministic Chaos
Experimentally, many diverse systems have been found
to exhibit the characteristic behavior associated with deterministic
chaos, examples including chemical and biological systems as well those
from physics4. Mitchell Feigenbaum5 discovered that
a few universal ratios independent of any dynamical details characterised
all systems where periods doubled as they approached turbulence. At the
point of infinite period doubling the orbits of Feigenbaum's system
showed a complex behavior in which one could discern a scale invariant
or fractal structure6. Phenomenological observation of
fractal
structure in nature represents the two fundamental symmetries of nature,
namely, dilation (r®
r+c where r is the length scale and c is a constant) and
correspond respectively to change in unit of length or in the origin of
the co-ordinate system7. A self-similar object is identified
by its fractal dimension D that is defined as dlnM(R)
/ dlnR where M(R) is the mass contained within a distance R
from a typical point in the object. Self-similar growth process in nature
lead to the observed universal fractal geometry of macroscopic structures
in natural phenomena. However, the basic physical mechanism of the self-organised
fractal
geometry in nature is not yet identified7.
A striking example of self-similar
fractal
geometry in nature is exhibited by the global cloud cover pattern. Macroscopically
different shaped clouds are self-similar fractals over a number
of orders of magnitude of length scales8. Also, the structure
of the ABL as revealed by advances in remote sensing and insitu
measurement techniques indicate scale invariant energy structure for the
full continuum of atmospheric motions which follow a power law spectrum
of the form f -n where
f
is the frequency and n the exponent8. The physics
of the self-similar fractal cloud growth structures associated with
deterministic
chaos in the atmosphere is not yet identified.
III. Physics of Deterministic Chaos
in the Planetary Atmospheric Boundary Layer
Period doubling implies growth of self-similar large
eddy structures from turbulent eddies by successive incremental length
step growth equal to the turbulent eddy length. A representative example
is the organised growth of large eddy or vortex roll circulations from
the turbulence scale in the ABL and manifested as cloud rows and
streets in satellite pictures of global cloud cover9 pattern.
The Rayleigh-Benard instability observed in laboratory experiments10
simulates analogous vortex roll circulations and is an example of a self-organised
system. In summary, turbulent eddies of frictional origin at the planetary
surface posses inherent upward momentum flux which is progressively amplified
by buoyant energy supply from Microscale Fractional Condensation (MFC)
of water vapour on hygroscopic nuclei by deliquescence even in an unsaturated
environment11. Microscale Fractional Condensation (MFC)
occurs on hygroscopic nuclei when water vapour mixing ratio increases beyond
the threshold for condensation by deliquescence due to adiabatic cooling
of upward moving turbulent air parcels. The exponential decrease of atmospheric
density with height further accelerates the turbulence scale upward momentum
flux. The mean turbulence scale buoyant energy production gives rise to
the formation of progressively larger helical vortex roll circulations
in continuous succession, thereby generating a hierarchical eddy continuum.
The large eddy growth in the ABL from the turbulence scale energy
supply (energy pump) is analogous to the triggering of lower frequency
laser emission by a higher frequency laser pump in a non-linear optical
medium12. The larger eddies carry the turbulent eddies as internal
circulations which contribute to their (large eddies) further growth. A
conceptual model of the large and turbulent eddies in the ABL (
Atmospheric Boundary Layer) is given as Fig.1.
Figure 1. Conceptual model of large and turbulent
eddies in the ABL
Townsend13 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 the turbulent motion as illustrated in the following example. Consider
a large eddy of radius R that forms in the field of isotropic
turbulence with turbulence length and velocity scales
2r
and w respectively. The dominant turbulent eddy radius is
therefore equal to r . The mean square circulation around
a circular path of large eddy radius R is given by.
where w, w1
are the tangential velocity components at the positions of the path elements
ds
and ds1. If the velocity product falls to zero
while the separation ds and ds1
is still small compared with the large eddy radius R, i.e.,
the motion in sufficiently separated parts of the flow is statistically
independent
W 2 = 4p
R 2r w*2
where w*
is the root mean square (r.m.s) circulation speed of the small (turbulent)
eddy.
The root mean square (r.m.s.) velocity of circulations
W
in the large eddy of radius R is
(1)
The above equation can be applied directly to
derive the r.m.s. circulation speed of the large eddy of radius R
generated by turbulence scale energy pump. The scale ratio z
is equal to the ratio of the radii of the large and turbulent eddies. Warming
and associated buoyant energy production occur by latent heat release during
condensation in the environment of the turbulent eddies and therefore results
in anomalous positive temperature lapse rates or Microscale Capping Inversion
(MCI) layer on the large eddy envelope (Fig.1). The MCI rises
up with the daytime convective growth of large eddy and may be identified
with the rising inversion of the daytime ABL seen in echosonde records14.
An incremental growth dR of large eddy radius equal to the
turbulent eddy radius r occurs in association with an increase
dW
in large eddy circulation speed as a direct consequence of the buoyant
velocity (w*
) production by MFC. The MCI is thus a region of wind shear
and temperature inversion in the ABL. The growth of large eddies
from the turbulence scale at incremental length steps equal to r
(turbulence length scale doubling) is therefore identified as the universal
period doubling route to chaos and MCI is therefore a region
of chaos. The growth of large eddy from the turbulence scale is shown at
Fig.2.
Figure 2. Growth of large eddy from the turbulent
eddies originating at the planetary surface
IV. Dilution of Large Eddy Volume
by Mass Exchange with Environment
The turbulent fluctuations mix overlying environmental
air into the growing large eddy volume and the non-dimensional steady state
fractional volume dilution k of the total large eddy volume
across unit cross section on its envelope is equal to
(2)
where w*
is the unidirectional turbulent eddy acceleration and dW
the corresponding acceleration of the large eddy circulation during the
large eddy incremental length step growth dR equal to r.
k > 0.5 for z <
10. Therefore organised large eddy growth can occur for scale ratio
z
³10
only since dilution by environmental mixing is more than half by volume
and erases the signature of large eddies for scale ratio z < 10.
Therefore a hierarchical, scale invariant, self-similar eddy continuum
with semi-permanent dominant eddies at successive decadic scale range intervals
is generated by the self-organised period doubling route to chaos
growth process. The large eddy circulation speed is obtained by integrating
Eq (2) for large eddy growth from the turbulence scale energy pump at the
planetary surface and is given as
(3)
k = 0.4 for z =
10. This is the well-known logarithmic wind profile relationship
in the ABL14 and k is designated as the
Von
Karman's constant and its value determined from observations is equal
to about 0.415. The period doubling route to chaos
growth process therefore generates a scale invariant eddy continuum where
energy flow structure is in the form of nested logarithmic spiral vortex
roll circulations, a complete circulation consisting of the outward and
inherent compensating inward flow. The region of chaos is the dynamic
growth region of large eddy by turbulence scale energy pumping and the
nested vortex hierarchical continuum energy structure is manifested as
the strange attractor design, a signature of organised chaos, first
identified by Lorenz16 during mathematical simulation
of atmospheric convection and later, widely investigated by mathematicians17.
A trajectory on the strange attractor exhibits widely different paths for
small differences in initial conditions thereby making long-term prediction
difficult. The strange attractor design of the atmospheric continuum vortex
roll circulations is manifested as the following atmospheric phenomena
(1) the spiral cloud formation in hurricanes18, (2) aerosol
concentration with layered fine structure observed in the stratospheric
Junge aerosol layer11, the arctic haze19 and more
spectacularly in the planetary rings of the major planets Jupiter, Saturn
& Uranus20 and (3) the layered structure of ozone concentration
in the Antarctic stratosphere21. Particles in the planetary
rings may be shown to follow the relation R3 / T2
= constant from Eqs. (1) & (2) and is therefore consistent
with Kepler's third law of planetary motions22.
Vertical mixing due to turbulent
eddy fluctuations progressively dilutes the rising large eddy and a fraction
f
of surface air reaches the normalised height z given by
Therefore W = w*
f z
From Eqs (1), (2) & (3)
(4)
The steady state fractional air mass flux f
from the surface is dependent only on the dominant turbulent eddy radius.
V. Deterministic Chaos and Richardson
Number for Atmospheric Turbulence
The Richardson number Ri14that
is used as an index of shear produced turbulence is defined as follows
Ri = NB2 / (wind shear)
2
where NB is the Brunt-Vaisala
frequency14. For the atmosphere, observations show that Ri
£
0.25 in regions of turbulence in the
atmosphere. In the following it is shown that Ri £
0.25 in the Microscale Capping Inversion
(MCI) for scale ratio z ³
10 23. The Brunt-Vaisala
frequency for the turbulent air parcels in the MCI is given in terms
of the virtual temperature (qv) lapse rate (dqv
/dR)
in the MCI where an incremental growth dR per second
of the large eddy is associated with a warming of
dqv
and an increase in wind speed dW as given below
where g is the acceleration due to
gravity.
wind shear in the MCI =
The buoyant vertical velocity
w*
production is a direct consequence of the temperature
perturbation dq*14
Therefore
dR being
the incremental large eddy growth per second is equal to dW
Therefore
Ri = 1/4 for scale ratio
z
= 10. It was shown in an earlier section that organised growth
of large eddy occurs for scale ratios z ³
10 and in the Microscale Capping Inversion
(MCI) at the crest of the large eddy the Richardson numberRi
£
0.25 and it is consistent with deterministic
chaos model predictions shown above. Richardson number may therefore
be alternatively considered to represent the ratio of the vertical velocity
in the large eddy to that in the turbulent eddy in the region of chaos,
the MCI.
In summary, the ABL consists
of a semi-permanent hierarchical system of eddies consisting of the convective,
meso-, synoptic and planetary scales which evolve basically from the dominant
turbulence scale at successive decadic scale range intervals and is manifested
as Mesoscale Cloud Clusters (MCC) and cloud rows in global synoptic
weather systems. Enhanced condensation inside clouds amplifies the myriads
of turbulent eddies and give rise to 'cloud top gravity oscillations'
(Fig.3).
Figure 3. Cloud formation in the updraft regions
of vortex roll (large eddy) circulations. The turbulent eddies get amplified
in the vertical by the latent heat released by condensation of water vapour
in the cloud and generate 'cloud-top gravity oscillations'. Electrical
charge separation occurs inside the cloud by transport upward (downward)
of positive (negative) space charges by the ascending (descending) flow
of the cloud top gravity oscillations.
Cloud water condensation in the innumerable
turbulent eddies is responsible for the observed cauliflower like surface
granularity of the cumulus clouds. The physical mechanism of growth of
the atmospheric buoyancy (gravity) waves from turbulent buoyant energy
production is analogous to the Condtional Instability of the Second Kind
(CISK) mechanism14 where hurricane systems are postulated
to derive their energy from convective scale cloud water condensation.
Also, there is an inherent two way energy feedback mechanism in the hierarchical
atmospheric eddy system discussed in this paper and given by Eq (1) which
is a statement of the law of conservation of energy, self-similarity and
self-consistency in atmospheric processes. The full continuum of atmospheric
eddies exist as a unified whole in time and space and contribute to the
manifested atmospheric phenomena in the global planetary atmosphere and
such a concept is similar to the 'Bootstrap' theory of Chew24
and the theory of implicate order envisaged by Bohm25.
The mechanism of evolution of the large eddies depends only on the turbulent
eddy size and is therefore universal and applicable to the global planetary
atmosphere and for all planetary atmospheres independent of their macroscopic
size and chemical composition.
The relationship between the size (R),
time period (T), circulation velocity (W) and
energy (E) scales of the convective (c), meso- (m),
synoptic (s) and planetary (p) scale atmospheric eddy systems
to the primary turbulence scale (t) is derived from Eq (1)
and is given below26, 27.
R : Rt = r : 10r : 10 2r
: 10 3r : 10 4r
T : tt = t : 40t : 40 2t
: 40 3t : 40 4t
W : Wt = w : 0.25w : 0.25 2w
: 0.25 3w : 0.25 4w
E : Et = e
: 62.5 e : 62.5 2e
: 62.5 3e
: 62.5 4e
The globally observed Quasi-Biennial
Oscillation (QBO) and the 20 -year cycle in weather patterns
may possibly result respectively from the fundamental semi-diurnal atmospheric
pressure oscillation (QBO ~ 12 hrs x 402) and the 5
minutes oscillation of the sun's atmosphere28 (20 years ~
5 min x 404) (Eq.5). Such a process is analogous to antistokes
laser emission triggered by a laser pump12.
VI. Atmospheric Eddy Energy Spectrum
The atmospheric eddy energy spectra obtained by observations
of turbulence spectra of wind in the ABL show the existence of a
continuous spectrum of eddies with universal characteristics of scale invariant
spectral slope29,30 implying the existence of self similarity
in atmospheric dynamical processes8. The universal period doubling
route to chaos eddy growth mechanism is shown to be responsible
for the observed scale invariant eddy energy spectrum in the atmosphere
as follows:
The eddy energy power spectrum is conventionally
plotted as lnE versus lnn
where E is the eddy energy and n
its frequency
The spectral slope S of the scale invariant
eddy energy spectrum is given as
S = D
lnE / D lnn
= ln (R 3W 2 / r
3w*2)
/ ln(R / r)
= -2 for
large z from Eq (1).
(6)
The above model prediction is consistent with
observations of a universal spectral slope approaching -2 29.
VII. Quantum Mechanical Nature
of Atmospheric Eddy Energy Structure
The Kinetic energy
KE
per unit mass of an eddy of frequency n
in the hierarchical eddy continuum is shown to be equal to Hn
where H is the spin angular momentum of unit mass of the
largest eddy in the hierarchy. The circulation speed of the largest eddy
in the continuum is equal to the integrated mean of all the inherent turbulent
eddy circulations. Let Wp be this mean circulation
speed or the zero level about which all the smaller frequency fluctuations
occur.
Therefore 2 R = wp /
n
from Eq.(1)
= Hn
H is
equal to the product of the momentum of unit mass of planetary scale eddy
and its radius and therefore represents the spin angular momentum of unit
mass of planetary scale eddy about the eddy center. Therefore the Kinetic
energy of unit mass of any component eddy of frequency nof
the scale invariant eddy continuum is equal to
Hn
. Further, since the largest eddy is but the sum total of all the inherent
smaller scales, the large eddy energy content is equal to the sum of all
its individual component eddy energies and therefore the kinetic energy
KE
distribution is normal and the kinetic energy KE of any eddy of
radius R in the eddy continuum expressed as a fraction of
the energy content of the largest eddy in the hierarchy will represent
the cumulative normal probability density distribution. The eddy continuum
energy spectrum is therefore the same as the cumulative normal probability
density distribution plotted on a log-log scale, i.e., the eddy
energy probability density distribution is equal to the square of the eddy
amplitude. Therefore the eddy continuum energy structure follows quantum
mechanical laws31. The energy manifestation of radiation and
other subatomic phenomena appear to possess the dual nature of wave and
particles since one complete eddy energy circulation structure is inherently
bi-directional and associated 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. Also from Eq. (7)
DE. DT
= H = constant
The above statement is
analogous to Heisenberg's uncertainty principle for subatomic dynamics31.
In the context of the atmospheric eddy continuum the above equation implies
that large changes in eddy energy can occur only during short intervals
of time and vice-versa, illustrative examples being the hurricane systems
on the one hand and climate changes on the other.
VIII. Statistical Distribution
Characteristics of the Atmospheric Eddy Continuum
Fundamental classical statistical distribution
functions commonly occurring in natural phenomena are shown in the following
to be inherent characteristics of the universal period doubling growth
phenomena. The distribution of means for sample size n has
a variance W22 that is related
to the population variance W12 as
follows32 .
W22
= W12 / n
The above statistical relation may be
derived from Eq.(1) in the context of the variance of the eddy parameters
for two different ratios z1 and z2
where n = z2 / z1 as follows
Therefore 
For eddy growth from smaller scale to
the larger scale the ratio of eddy energy for unit mass of large eddy to
that of turbulent eddy is equal to 1/n where n
is the scale ratio. The eddy continuum generated by the successive integration
of smaller scale circulations into large-scale circulation patterns will
therefore have eddy energy spectral slope S given as
= - 2
S = - 2 is
in agreement with earlier derivation (Eq.6) for large-scale ratios. Since
large eddy energy is the integrated mean of all inherent small scale eddies,
in general, coarse mesh observations give a spectral slope - 2 for
the eddy energy spectrum, e.g., climatological data23.
IX. Physical Meaning of Normal
Distribution Parameters
In the following it is shown that the universal period
doubling route to chaos growth phenomena in the atmosphere gives
rise to the observed statistical normal distribution characteristics
for atmospheric phenomena as a natural consequence. The period doubling
growth is initiated and sustained by the turbulent (fine scale) eddy acceleration
w*
across unit cross section that then propagates by the inherent property
of inertia of the medium.
In the context of atmospheric turbulence,
the statistical parameters, mean, variance, skewness
and kurtosis represent respectively the net vertical velocity, intensity
of turbulence, vertical momentum flux and intermittency of turbulence and
are given respectively by w*,w*2,w*3,
w*4
. The momentum coefficient of skewness
equal to zero and the moment coefficient of kurtosis equal
to 3 are the characteristics of the normal distribution32
. In the following it is shown that the observed normal distribution characteristics
of atmospheric phenomena is a signature of deterministic chaos.
By analogy, the perturbation speed
w*
(motion) per second of the medium sustained by its inertia represents the
mass, w*2
the acceleration or force, w*3
the momentum (or potential energy) and w*4
the spin angular momentum since an eddy motion has an inherent curvature
to its trajectory. The eddy motion is inherently symmetric with bi-directional
energy flow and therefore the skewness factor w*3is
equal to zero for one complete eddy circulation thereby satisfying the
law of conservation of momentum. The moment coefficient of kurtosis is
given by (dw) 4 / w*4
and represents the relative magnitude of the spin angular momentum of large
eddy generated by the period doubling growth from the turbulent eddy. The
statistical moment coefficient of kurtosis (dW)4/ w*4
represents the intermittency of turbulence and is shown in the following
to be equal to 3 as a natural consequence of the growth phenomenon
by the period doubling route to chaos.
From Eq (3)
Therefore
Organised eddy growth occurs for
scale ratio equal to 10 and identifies the large eddy on
whose envelope period doubling growth process occurs. Therefore for a dominant
eddy
(dz/z) = 1/2 for
one length step growth by period doubling process since z = dz +
dz .
Therefore moment coefficient of kurtosis is equal
to
In other words, period doubling growth
phenomena result in a threefold, increase in the spin angular momentum
of the large eddy for each period doubling sequence. This result is consistent
since period doubling at constant pump frequency involves eddy length step
growth dR an either side of the turbulent eddy length dR
. The intermittency of turbulence, i.e., episodes of turbulent fluctuations
on relative time scale is also equal to 3.
X. Physics of the Generalised Scale
Invariance for the Atmospheric Eddy Continuum Energy Spectra
The scale invariant atmospheric
eddy continuum of the form f- -n,
where f is the frequency and n the exponent
observed in the ABL30 is shown to be the result of eddy
growth by the universal period doubling route to chaos, the exponent
n
being a function of the scale ratio.
The eddy energy spectrum is shown (see
Section
VII) to be the same as the cumulative normal probability curve plotted
on a log-log scale. The eddy energy spectral slope is derived from
the cumulative normal probability distribution curve as follows. The period
doubling sequence generates a large eddy of radius
R equal
to 2r, so that the cumulative probability of occurrence P1
of turbulent eddy fluctuations of either sign is given by (0.5r /(R+r))
and is equal to
0.167 (=1/6) in close agreement with the
cumulative normal probability value corresponding to one standard deviation
s
. The cumulative normal probabilities and corresponding slopes on log-log
scale for standard deviation 1s
, 2s
and 3s
are given in Table 1.
Table 1
Normal Distribution
| Deviation
(s
= standard
deviation) |
Cumulative
occurrence
frequency |
Slope of cumulative
occurrence frequency
curve (log-log scale) |
|
1s
|
.1587
|
- 1.8
|
|
2s
|
.0228
|
- 5.0
|
|
3s
|
.0013
|
- 10.0
|
The slope of the log-log plot
of cumulative normal probability distribution curve at one standard deviation
is equal to -1.8 in agreement with computed (see Section
VIII) and observed33 values and relates to coarse mesh observations
e.g.,
climatological data. The eddy energy spectral slope becomes steeper than
-2
with high resolution (fine scale) observations30 which include
perturbations due to more than one period doubling sequence as shown below.
Considering period doubling of large
eddy R giving rise to larger eddy of length R2
= 2R considerations as above give a cumulative probability of occurrence
P2
of large eddy length R equal to (0.5R / (R+R2))
= 1/6 »
0.167. Therefore the cumulative probability
of occurrence of turbulent length scale r in the large eddy
length section R2 is equal to P1
x P2 = 1/36 = 0.028 in close agreement with cumulative
normal probability value corresponding to two standard deviation (2s
.) (Table 1). The slope of the eddy energy spectrum as derived from
the log-log plot of the cumulative normal probability curve is -5
at 2s
(Table 1). Steep spectral slopes for localised eddy energy spectra
are associated with regions of energy concentration such as severe weather
systems. Eddy energy spectral slopes steeper, than -2 have been
reported in the ABL30. Similarly period doubling associated
with R2 corresponds to cumulative probability
of occurrence equal to (1/36)»
0.0008 of the primary turbulence scale
eddy at three standard deviations and the corresponding eddy energy spectral
slope is -10 (Table 1).
In summary, period doubling at one
standard deviation generates a semi-permanent dominant large eddy with
scale ratio equal to 10 with respect to the fine scale turbulent
eddy and a corresponding eddy energy spectral slope equal to -1.8.
It may also be inferred that the primary period doubling in successive
cumulative radial length steps r, 2r, 3r
, etc., generated perturbations of increasing magnitude w*sinq
for q =0
to 90o so that the eddy energy
spectrum has positive slope for the primary eddy circulation updraft half
cycle. At the completion of one complete primary eddy circulation the spectral
slope becomes equal to zero and with the generation of large eddies by
space-time integration of complete cycles of small-scale fluctuations,
the spectral slope becomes negative.
The scale ratio for the period doubling
at one standard deviation is equal to 10 with respect to
the turbulence scale. If the turbulence scale itself is assumed to consist
of 10 successive sections, then the primary scale ratio at
one standard deviation is equal to 100 and by similar reasoning
the scale ratios at 2s
and 3s
are respectively equal to 1002 and 1004.
XI. Physical Meaning of the Universal
Feigenbaum's Constants of the Period Doubling Route to Chaos
The universal period doubling route to chaos
has been studied extensively by mathematicians. The basic example with
the potential to display the main features of the erratic behaviour is
the Julia model17 given below.
Xn+1 = F(Xn) = LXn(1-Xn)
The above non-linear model represents
the population values of the parameter X at different time
periods
n, and L parameterises the rate of
growth of X for small X.
The Eq.(1) representing large eddy
growth as integrated space time mean of turbulent eddy fluctuation given
as 
is analogous
to the Julia model since large eddy growth is dependent on the energy
input from the turbulence scale with ordered two way energy feedback between
the larger and the smaller scales. Therefore the well-established abstract
mathematical results for the Julia model can be interpreted in terms
of physical processes occurring in nature as follows. Feigenbaum's5
research showed that the following two universal constants a
and d are independent of the details of the non-linear equation
for the period doubling sequence:
a and d therefore
denote the successive spacing ratios of X*and
L
respectively for adjoining period doublings.
The universal constants a
and d assume different numerical values for period tripling,
quadrupling, etc., and the appropriate values are computed by Delbourgo17
and shown to follow the relation 3d = 2a 2 over
a wide domain.
The physical concept of large eddy
growth by the period doubling process enables to derive the universal constants
a
and d and their mutual relationship as functions inherent
to the scale invariant eddy energy structure as follows.
From Eq. (1) the function a may
be defined as
(8)
a is therefore equal to
1/k
from Eq.(2) where k is the Von Karman's constant representing
the non dimensional steady state fractional volume dilution rate of large
eddy by turbulent eddy fluctuations across unit cross section on the large
eddy envelope. Therefore a represents the non-dimensional
total fractional mass disperson rate and is inherently negative. The variable
a2
represents the fractional energy flux into the large eddy environment.
Let d represent the ratio of the spin angular moments for
the total mass of the large and turbulent eddies
(9)
Therefore 2a2 = 3d
from Eqs.(8) & (9). The variable 2a2 represents
the total eddy energy flux into the medium. The spin angular momentum of
the resulting large eddy accounts for the observed value of three
for the moment coefficient of kurtosis of the normal distribution (see
Section
IX). Therefore the above equation relating the universal
Feigenbaum's
constants is a statement of the law of conservation of energy, i.e., the
period doubling growth process generates a threefold increase in the spin
angular momentum of the resulting large eddy and propagates outward as
the total large eddy energy flux in the medium. The property of inertia
enables propagation of turbulence scale perturbation in the medium by release
of the latent energy potential of the medium. An illustrative example is
the buoyant energy generation by water vapour condensation in the updraft
regions in the atmospheric boundary layer.
The universal Feigenbaum's constants
a
and d are respectively to -2.52 and 4.05
as computed from Eqs.(8) & (9) since the scale ratio z
is equal to 10 for the self-organised eddy growth mechanism
in the atmospheric boundary layer.
XII. Deterministic Chaos and Organised
Weather Systems in the ABL
The global weather systems are the patterns of eddy
energy manifestation in time and space of the rhythm of the unified whole
of the planetary atmospheric eddy continuum. There is inherent coupling
and continuity of global weather systems in time and space with universal
characteristics for the thermodynamic anomaly patterns with respect to
the normalised length scale. In the following, the model predicted9
unique thermodynamic anomaly patterns for the most intense weather system,
the hurricane is compared with precise well-established observational results.
It was shown earlier (Section IV)
that the wind profile in the ABL follows the logarithmic law. Since
large eddy growth involves increase in radius simultaneous with angular
displacement from origin, the trajectory of airflow associated with the
large eddies will follow a logarithmic spiral pattern both in the horizontal
and vertical. The complete eddy circulation consisting of the ascent and
the return descent airflow therefore occurs in the form of logarithmic
spiral vortices34. The full continuum of atmospheric eddies
exist as a unified whole in the form of vortices within vortices as displayed
in the extreme cases of the tornado funnel and the dust devil.
Spiral cloud bands of cyclone systems
The spiral airflow track for a synoptic scale
large eddy is shown at Fig.4.
Figure 4. The spiral airflow track in hurricanes
The eddy growth originating from O
follows the spiral curve OAB. The angular rotation from the origin
at location A is measured with respect to the axis OX.
Let OA and OB denote
the locations of the large eddy radii* R and 
for a growth period of one second.
The angular rotation 
is given by
AB is the tangent at A to
the circle drawn with center O and radiusR so that
BC = d R = w*f
rR / rR+dR
ABwill
also represent the tangent to the spiral at A for a limited range.
The angle BAC between the logarithmic spiral and its tangent is
called the crossing angleaof
the spiral.
Substituting b = tan a;
and integrating for eddy growth from r to R
the above equation gives
R = rebq
This is the equation for an equiangular logarithmic
spiral when the crossing angle is a constant.
At any location A the horizontal airflow
path into a synoptic scale cyclone system follows a logarithmic spiral
track.
Storm intensity and cloud band configuration
The cloud bands identify the circulation
path of the synoptic cyclonic eddy whose radial growth dR
is equal to the dominant turbulent eddy radius r and dq
is the corresponding angular rotation.
dR = r and R
= S r
Cloud bandwidth =
The dominant turbulent eddy radius determines
the angular turning dq
and incremental large eddy radius growth dR and therefore
the synoptic scale spiral cloud band has different crossing angles and
band widths at different locations with respect to the storm center. Observations
show that increased condensation results in decrease in dominant turbulent
eddy radius23. There is heavy condensation close to the storm
center in association with tighter coiling of the spiral with overlapping
cloud bands.
Dvorak35 has classified
cyclonic storms according to the appearance of cloud bands as related to
observed storm intensities. The cloud band pattern relating to the categories
T1(a)
to T4(a) of the Dvorak classification are simulated
by suitably altering r along the radial distance R
and computing R and cloud band width from equations given
above. The model simulated cloud bands and the corresponding Dvorak
cloud diagrams are given in Fig.5 and there is good agreement between the
two.
Figure 5 Deterministic chaos
model prediction of the hurricane spiral cloud bands (second row) and Dvorak
cloud diagrams (first row) for storm intensities T1 (a) to T4
(a)
Growth time of the eddy system
The eddy growth time T for
an eddy radius R is computed as follows.
T = dR /W
where li is the logarithm integral
or the Soldner's integral.
Horizontal profile of cyclone pressure field
The low-pressure field of the cyclone system
is created by the upward ascent of surface air. At any location distance
R
from the storm center O there is an upward mass flux of air equal
to w*rr
per second across unit area where r
is the air density and w*
is the production of vertical velocity per second by MFC at surface
layers. A synoptic scale weather system which has been in existence for
a time period TN and extending to a radial distance
RN
developes a central pressure departure equal to w*rrTN
with respect to the ambient pressure field at the periphery X. At
the intermediate location B the corresponding pressure departure
is equal to w*rrTR
where TR is the time period for the eddy to grow
from B to X. The normalised pressure departure NPD
at the intermediate location with respect to the extreme pressure departure
at the storm center is computed as
The horizontal profile of the hurricane
pressure field normalised to the ambient pressure given by NPD in
the above equation is computed for the 6 categories T1(a)
to T6(a) of the Dvorak storm intensity categories
and shown at Fig.6.
Figure 6 Deterministic
chaos model prediction of the horizontal surface field pattern for
hurricanes.
The computed horizontal profile of NPD
closely resembles the corresponding log / linear pressure profiles for
the nine Florida hurricanes by Holland36.
Horizontal profile of wind
The horizontal profile of wind in a cyclone
system follows the logarithmic law and depends only on the turbulent eddy
radius. The horizontal wind profile for the 6 categories of storm
intensities T1(a) to T6(a) of the Dvorak35
classification are computed and shown at Fig.7.
Figure 7. Deterministic
chaos model prediction of the horizontal wind field pattern for hurricanes.
The model predicted wind variation with
distance from storm center resembles the observed wind field around storms
reported by several workers18,36,37.
The airflow speed is due mainly to
the dynamic buoyant energy production by MFC and thus is not influenced
by the rotation of the earth. Therefore the Coriolis force does
not influence the airflow into the synoptic scale eddy in agreement with
theoretical studies by other workers38.
XIII. Deterministic Chaos and Cloud
Physical Processes
Cloud growth occurs in the updraft regions of vortex
roll circulations in the low-pressure field of synoptic scale systems.
From the theory of atmospheric eddy dynamics it is derived and shown39
(i) the vertical profile of the ratio of the actual cloud liquid water
content (q) to the adiabatic liquid water content (qa
) follows the f distribution (Fig.8)
Figure 8. Deterministic chaos model prediction
of the vertical profile of the ratio of cloud liquid water content (q)
to the adiabatic liquid water content (qa)
and comparison with observations (J. Warner, J. Atmos. Sci., 27,
682-688, 1970)
(2) the vertical profiles of the vertical velocity
W
and the total cloud liquid water content qt
are respectively given by W = w*f
z and qt
= q*f
z where t represents
the total values and *
represents cloud base values (3) the cloud growth time
where li is the logarithm integral (4) the cloud drop size
spectrum follows the naturally occurring Junge aerosol size spectrum11
and (5) the computed raindrop size spectrum closely resembles the observed
Marshall-Palmer
raindrop size distribution20 at the surface.
XIV. Deterministic Chaos and Atmospheric
Electrification
Fair Weather Electric Field and Geomagnetic Field
The atmospheric eddy continuum circulations give rise
to vertical mass exchange in the ABL such that a net positive space
charge current flows upward with a simultaneous downward transport of negative
space charges from ionospheric levels and this dynamical two way charge
transport is shown to be of the right order of magnitude and direction
to sustain the fair weather atmospheric electric field and also explain
the horizontal component of the geomagnetic field distribution41.
The above theory also helps to explain the observed42 close
similarity between the geomagnetic field lines and atmospheric circulation
patterns. Therefore changes in atmospheric circulation patterns preceding
climatic changes can be detected in geomagnetic field pattern variations.
The wandering of the geomagnetic North Pole is therefore closely related
to global climatic variation and incidentally also is reflected in the
subatomic dynamics of ferromagnetic substances, which naturally align themselves
along geomagnetic N-S direction.
Cloud Electrification
It is shown that cloud top gravity oscillations
mix overlying environmental air into the cloud such that there is a downward
transport of negative space charges from above cloud top regions and a
simultaneous upward transport of positive space charges from below cloud
base levels to the cloud top region (Fig.3). Positive dipole cloud charging
occurs by the vertical mixing. The electric field at the surface due to
the cloud dipole charge, the strength of the cloud dipole, the cloud electrical
conductivity, the point discharge current are expressed in terms of the
basic non-dimensional parameters f and z43.
XV. Deterministic Chaos and Atmospheric
Urban Effects
The thermal energy input from industrial / urban sites
in combination with hygroscopic nuclei and moisture lead to enhanced cloud
growth process with taller clouds and heavier rainfalls particularly in
the downwind region. A fraction f of the surface nuclei form
cloud / raindrops and therefore the same fraction f of atmospheric
pollution content will also get washed down in the rain. Even in clear
weather conditions the pollution content of atmospheric air in the form
of aerosols will be equal to a fraction f of the value at
the source location. The steady state flux of pollution transport is therefore
given by the f distribution both in the horizontal and vertical.
Though f is small at large values of the normalised distance
z,
yet long term accumulation of pollution will be appreciable resulting in
irreversible environmental degradation. Also enhanced dynamics associated
with thermal energy supply from urban industrial sites leads to a faster
transport of pollutants in all directions.
XVI. Deterministic Chaos and Stratospheric
Dynamics
Thermal energy sources are regions of enhanced eddy
dynamics and vertical mixing extending to the stratosphere and above. Enhanced
downward flux of stratospheric ozone occurs above regions of industrial
/ urban activity. Beig and Chakravorthy44 have reported a sharp
decrease in stratospheric ozone in association with a major fire in an
offshore oil well in India. Downward transport of stratospheric ozone occurs
in regions of deep convection45. Stratospheric aerosol and radioactive
debris from volcanic eruptions and nuclear experiments / accidents are
transported downwards to surface levels in regions of deep convection where
intense vertical mixing occurs. Such regions of stratospheric contamination
deposition on surface, even in fair weather will occur in discrete areas
of fractal nature analogous to rainfall areas8 and thus
may account for the radiation hot spot fall out pattern reported following
the Chernobyl nuclear reactor accident46. Also, the recently
reported ozone hole in the Antarctic stratosphere may possibly
be caused by enhanced vertical mass exchange due to increased international
exploration activities in Antarctica during the spring / summer season
in recent years.
XVII. Deterministic Chaos and Ionospheric
Dynamics
It is known that solar flares perturb the ionosphere
and cause intensification of weather systems47. Therefore ionospheric
heating experiments and the numerous earth orbiting satellites may possibly
create fine scale magnetosphere / ionospheric perturbations and lead to
inadvertent modification of climate.
XVIII. Conclusions
Deterministic chaos in the planetary atmospheric
boundary layer is identified as the growth of large eddy (helical vortex
roll circulation) from turbulence scale buoyant energy generation with
implicit ordered two way energy feed back mechanism between the larger
and smaller scales. It is shown that such a process generates a scale invariant,
hierarchical self-similar atmospheric eddy continuum energy structure with
dominant eddies (limit cycles) at decadic scale range intervals as manifested
in Meso-scale Cloud Clusters (MCC) and the fractal geometry
of cloud cover pattern. Further, the eddy continuum energy structure obeys
quantum mechanical laws and the apparent wave-particle duality is attributed
to the inherent bi-directional eddy energy flow associated with bimodal
phenomenological manifestation, e.g., formation of clouds in updrafts and
dissipation of clouds in downdrafts of the eddy circulation. The pressure
and wind anomaly patterns of global weather systems and the cloud electrical,
microphysical and dynamical characteristics could result from the simple
universal unique functions of the turbulence scale energy generation. The
universal Feigenbaum's constants a and d
defining the period doubling growth process in the atmosphere are respectively
equal to -2.52 and 4.05 in agreement with mathematical
computations. The universal relation 2a2=3d is
shown to be a statement of the law of conservation of energy for the period
doubling growth process, since period doubling growth on either side of
the primary turbulent eddy generates a threefold increase in the spin angular
momentum (3d) of the resulting large eddy and is equal to
the total energy flux (2a2) into the environment
of the large eddy.
The bi-directional energy flow in the planetary
atmospheric eddy continuum is manifested as various tropospheric, ionospheric
and magnetospheric phenomena in a continuous chain of individual perturbation
events in a space-time continuum which is super symmetric in the macroscale
ABL,
being the fusion of the individual component eddy symmetries.
Acknowledgements
The authors expresse their deep gratitude to Dr. A.
S. R. Murty for his keen interest and encouragement during the course of
this study.
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