Universal
Spectrum for Natural Variability of Climate :
J. S. Pethkar and A. M. Selvam^{*}
Indian
Institute of Tropical Meteorology, Pune 411008, India
Amer.
Meteorol. Soc.10th Conf. Applied Climatology, Fall 1997, Western USA.
^{*}^{(}Retired) email: selvam@ip.eth.net
web site: http:\\www.geocities.com\amselvam
1.
INTRODUCTION
The apparently irregular ( unpredictable ) spacetime fluctuations in
atmospheric flows ranging from climate ( thousands of kilometers  years ) to
turbulence ( millimeters  seconds ) exhibit the
universal symmetry of selfsimilarity. Selfsimilarity or scale invariance
implies longrange spatiotemporal correlations and is manifested in atmospheric
flows as the fractal geometry to spatial pattern concomitant with inverse
powerlaw form for power spectra of temporal fluctuations documented and
discussed in detail by Lovejoy and his group (Tessier et. al. 1996). Longrange
spatiotemporal correlations are ubiquitous to dynamical systems in nature and
are identified as signatures of selforganized criticality (Bak et. al.1988).
Standard meteorological theory cannot explain satisfactorily the observed
selforganized criticality. Numerical models for simulation and prediction of
atmospheric flows are subject to deterministic chaos and give unrealistic
solutions. Deterministic chaos is a direct consequence of roundoff error growth
in iterative computations. Roundoff error of finite precision computations
doubles on an average at each step of iterative computations ( Mary Selvam
1993a). Roundoff error will propagate to the mainstream computation and give
unrealistic solutions in numerical weather prediction (NWP) and climate models
which incorporate thousands of iterative computations in longterm numerical
integration schemes. A recently developed nondeterministic cell dynamical
system model for atmospheric flows (Mary Selvam 1990) predicts the observed
selforganized criticality as intrinsic to quantumlike mechanics governing flow
dynamics.
2.
MODEL CONCEPTS
In summary, (Mary Selvam 1990,1993a,b ,1997; Mary Selvam et. al. 1992,1996; Mary Selvam, Pethkar and Kulkarni 1995; Mary Selvam and Radhamani 1994,1995; Mary Selvam and Joshi 1995), based on Townsend's ( Townsend 1956) concept that large eddies are the envelopes of enclosed turbulent eddy circulations, the relationship between the large and turbulent eddy circulation speeds (W and w_{*} ) and radii ( R and r ) respectively is given as
(1)
Since the large eddy is the integrated mean of enclosed turbulent eddy
circulations, the eddy energy (kinetic) spectrum follows statistical normal
distribution. Therefore, square of the eddy amplitude or the variance represents
the probability. Such a result that the additive amplitudes of eddies, when
squared, represent the probability densities is obtained for the subatomic
dynamics of quantum systems such as the electron or photon (Maddox 1988a).
Atmospheric flows, therefore, follow quantumlike mechanical laws. Incidentally,
one of the strangest things about physics is that we seem to
need two different kinds of
mechanics, quantum mechanics for microscopic dynamics of quantum systems and
classical mechanics for macroscale phenomena (Rae 1988).The above visualization
of the unified network of atmospheric flows as a quantum system is consistent
with Grossing’s (Grossing 1989) concept of quantum systems as order out of chaos
phenomena. Order and chaos have been reported in strong fields in quantum
systems (Brown 1996). Writing Equation 1 in terms of the periodicities
T and
t of large and small eddies
respectively, where
and
we obtain
(2)
Equation 2 is analogous to Kepler’s third law of planetary motion, namely, the square of the
planet’s year (period) to the cube of the planet’s mean distance from the
Sun is the same for all planets (Narlikar 1982,1996; Weinberg 1993). Newton
developed the idea of an inverse square law for gravitation in order to explain
Kepler’s laws, in particular, the third law. Kepler’s laws were formulated on
the basis of observational data and therefore were of empirical nature. A basic
physical theory for the inverse square law of gravitation applicable to all
objects, from macroscale astronomical objects to microscopic scale quantum
systems is still lacking. The model concepts (Eq.2) are analogous to a string
theory (Kaku 1997) where, superposition of different modes of vibration result in
material phenomena with intrinsic quantumlike mechanical laws which incorporate
inverse square law for inertial forces, the equivalent of
gravitational forces, on all scales of eddy fluctuations from macro to
microscopic scales.
Uzer et al.(1991) have discussed new developments within the last two
decades which have spurred a remarkable revival of interest in the application
of classical mechanical laws to quantum systems. The atom was originally
visualized as a miniature solar system based on the assumption that the laws of
classical mechanics apply equally to electrons and planets. However within a
short interval of time the new quantum mechanics of Schrodinger and Heisenberg became
established (from the late 1920s) and the analogy between the structure of the
atom and that of the solar system seemed invalid and classical mechanics became
the domain of the astronomers. There is now a revival of interest in classical
and semiclassical methods which are found to be unrivaled in providing
an intuitive and computationally tractable approach to the study of
atomic, molecular and nuclear dynamics.
2.1 Model
Predictions
(a) Atmospheric
flows trace an overall logarithmic spiral trajectory with the quasiperiodic Penrose
tiling pattern for the internal structure
(b) Conventional
continuous periodogram power spectral analyses of such spiral trajectories will
reveal a continuum of periodicities with progressive increase in phase.
(c) The
broadband power spectrum will have embedded dominant wavebands, the bandwidth
increasing with period length. The peak periods
E_{n} in the dominant wavebands will be given by the relation
E_{n} = T_{s}(2+t)t^{n}
(3)
where
t is the golden mean equal to (1+Ö5)/2 [@1.618] and
T_{s}
, the solar
powered primary perturbation time period is the annual cycle (summer to winter)
of solar heating in the present study of interannual variability. Ghil (1994)
reports that the most striking feature in climate variability on all time scales
is the presence of sharp peaks superimposed on a continuous background.
The model predicted periodicities are
2.2, 3.6, 5.8, 9.5, 15.3, 24.8, 40.1,
and
64.9 years for values of n
ranging
from 1 to 6. Periodicities close to model predicted have been reported (Burroughs
1992; Kane 1996).
(d) The
ratio r/R
also represents the increment dq
in phase angle q
(Equation 1 )and therefore the phase angle q represents the variance. Hence, when the logarithmic spiral
is resolved as an eddy continuum in conventional spectral analysis, the
increment in wavelength is concomitant with increase in phase. Such a result
that increments in wavelength and phase angle are related is observed in quantum
systems and has been named 'Berry's
phase' (Berry 1988; Maddox 1988b). The relationship of angular turning
of the spiral to intensity of fluctuations is seen in the tight coiling of the
hurricane spiral cloud systems.
(e) The overall logarithmic spiral flow structure is given by the following relation containing the length scale ratio z equal to R/r
(4)
where the constant k
is the steady state fractional volume dilution of large eddy by inherent
turbulent eddy fluctuations . The constant
k is equal to 1/t^{2} (@
0.382) and is identified as the universal constant for deterministic chaos in
fluid flows. The steady state emergence of fractal structures is therefore equal
to
1/k @ 2.62
(5)
The model predicted logarithmic wind profile relationship such as
Equation 4 is a longestablished (observational) feature of atmospheric flows in
the boundary layer, the constant k, called the Von
Karman ’s constant has the value equal to 0.38
as determined from observations. Historically, Equation 4 is basically an
empirical law known as the universal
logarithmic law of the wall , first proposed in the early 1930s by pioneering
aerodynamicists Theodor von Karman and Ludwig Prandtl, describes shear forces
exerted by turbulent flows at boundaries such as wings or fan blades or the
interior wall of a pipe.
The law of the wall has been used for decades by engineers in the design of
aircraft, pipelines and other structures (Cipra, 1996).
In Equation 4, W represents the
standard deviation of eddy fluctuations, since W
is computed as the instantaneous r.m.s. (root mean square) eddy perturbation
amplitude with reference to the earlier step of eddy growth. For two successive
stages of eddy growth starting from primary perturbation w_{*}
the ratio of the standard deviations W_{n+1}_{ } and
W_{n} is given from
Equation 4
as (n+1)/n. Denoting by s
the standard deviation of eddy
fluctuations at the reference level (n=1)
the standard deviations of eddy fluctuations for successive stages of eddy
growth are given as integer multiple of s, i.e., s, 2s, 3s, etc. and correspond respectively to
statistical normalized standard deviation t
The conventional power spectrum plotted as the variance versus the
frequency in loglog scale will now represent the eddy probability density on
logarithmic scale versus the standard deviation of the eddy fluctuations on
linear scale, since the logarithm of the eddy wavelength represents the standard
deviation i.e., the r.m.s. value of eddy fluctuations (Equation
4). The r.m.s. value of eddy fluctuations can be represented in terms of
statistical normal distribution as follows. A normalized standard deviation
t=0 corresponds to cumulative
percentage probability density equal to 50
for the mean value of the distribution. Since the logarithm of the wavelength
represents the r.m.s. value of eddy fluctuations the normalized standard
deviation t is defined for the
eddy energy as
(7)
where L is the period in
years and T_{50}_{
} is the period up to which the
cumulative percentage contribution to total variance is equal to 50
and t
= 0. The variable logT_{50}_{ } also represents the mean value for the r.m.s. eddy
fluctuations and is consistent with the concept of the mean level represented by
r.m.s. eddy fluctuations. Spectra of time series of meteorological parameters
when plotted as cumulative percentage contribution to total variance versus t
should follow the model predicted universal spectrum. The literature shows many
examples of pressure, wind and temperature whose shapes display a remarkable
degree of universality (Canavero and Einaudi,1987).
(f) Mary
Selvam (1993a) has shown that Equation 1 represents the universal algorithm for
deterministic chaos in dynamical systems and is expressed in terms of the
universal Feigenbaum's
(Feigenbaum
1980) constants a
and d as follows.
2a^{2} =
p
d
(8)
where
p
d , the relative volume intermittency of
occurrence contributes to the total variance 2a^{2} of fractal structures. The Feigenbaum’s constant a
represents the steady state emergence of fractal structures. Therefore, the total
variance of fractal structures for either clockwise or anticlockwise rotation is
equal to 2a^{2}. It was shown at Equation 5 above that the steady
state emergence of fractal structures in fluid flows is equal to
1/k(
= t^{2}) and therefore the Feigenbaum’s
constant a is equal to
a = t^{2}
= 1/k = 2.62
(9)
(g) The
relationship between Feigenbaum’s
constant a and statistical normal distribution for power spectra is
derived in the following.
The steady state emergence of fractal structures is equal to the Feigenbaum’s
constant a (Equation 5 ). The
relative variance of fractal structure for each length step growth is then equal
to a^{2}. The normalized variance
1/a^{2n}^{
} will now represent the
statistical normal probability density for the n^{th} step growth according to
model predicted quantumlike mechanics for fluid flows . Model predicted
probability density values P are
computed as
P = t
^{ 4n}
(10)
or
P = t
^{}^{ }^{4t}
(11)
where t is the normalized
standard deviation (Equation 6) and
are in agreement with statistical normal distribution as shown in Table 1.
Table
1
Model
predicted and statistical normal probability density distributions
Growth
step 
Normalized
std dev 
probability
densities 

n 
t 
model
predicted P = t
^{4t} 
statistical
normal distribution 
1 
1 
.1459 
.1587 
2 
2 
.0213 
.0228 
3 
3 
.0031 
.0013 
The
periodicities T_{50}
and T_{95} up to which
the cumulative percentage contribution to total variances are respectively
equal to 50 and 95 are computed from model concepts as follows.
The power spectrum, when plotted as
normalised standard deviation t
versus cumulative percentage contribution to total variance represents the
statistical normal distribution (Equation 7), i.e., the variance represents the
probability density. The normalised standard deviation values corresponding to
cumulative percentage probability densities P
equal to 50 and 95 respectively are equal to 0
and 2 from statistical normal
distribution characteristics. Since t
represents the eddy
growth step n
(Equation 6) the
dominant periodicities T_{50}
and T_{95} up to which the
cumulative percentage contribution to total variance are respectively equal to 50
and 95 are obtained from Equation 3 for corresponding values of n
, i.e., 0 and 2. In the present
study of interannual variability, the primary perturbation time period T_{s}
is equal to the annual (summer to winter) cycle of solar heating and T_{50} and T_{95}
are obtained as
T_{50} = (2+t)t^{0} @
3.6
years
(12)
T_{95} = (2+t)t^{2} @
9.5 years
(13)
(h) The
power spectra of fluctuations in fluid flows can now be quantified in terms of
universal Feigenbaum’s constant a
as follows.
The normalized variance and therefore the statistical normal
distribution is represented by (from Equation 11)
P = a ^{
2t}
(14)
where P is the probability
density corresponding to normalized standard deviation
t. The graph of P
versus t will represent the power
spectrum. The slope S of the power spectrum
is equal to
(15)
The power spectrum therefore follows inverse power law form, the slope
decreasing with increase in t.
Increase in t corresponds to large
eddies ( low frequencies) and is consistent with observed decrease in slope at
low frequencies in dynamical systems.
(I) The
fractal dimension D can be expressed as a
function of the universal Feigenbaum’s constant a
as follows.
The steady state emergence of fractal structures is equal to a
for each length step growth (Equations 6 & 9) and therefore the fractal
structure domain is equal to a^{m}
at m^{th}^{
} growth step starting from
unit perturbation. Starting from unit perturbation, the fractal object occupies
spatial (two dimensional) domain a^{m}
associated with radial extent t^{m}
since successive radii follow Fibonacci
number series. The fractal dimension D is defined as
where M is
the mass contained within a distance R
from a point in the fractal object. Considering growth from
n^{th}
to (n+m)^{th}^{
} step
(16)
similarly
(17)
Therefore from
Equation 9
(18)
The fractal dimension increases with the number of
growth steps. The dominant wavebands increase in length with successive growth
steps. The fractal dimension D
indicates the number of periodicities
incorporated. Larger fractal dimension indicates more number of
periodicities and complex patterns.
(j)
The relationship between fine
structure constant, i.e. the eddy energy ratio between successive
dominant eddies and Feigenbaum’s constant a is derived as follows.
2a^{2} = relative
variance of fractal structure (both clockwise and anticlockwise rotation) for
each growth step.
For one dominant large eddy comprising of five growth steps each for
clockwise and counterclockwise rotation, the
total variance is equal to
2a^{2} x 10 = 137.07
(19)
For each complete cycle ( comprising of five growth steps each ) in
simultaneous clockwise and counterclockwise rotations, the relative energy
increase is equal to 137.07
and represents the fine structure constant for eddy energy structure.
Incidentally ,the fine structure constant in atomic
physics (Davies 1986; Gross 1985; Omnes 1994) designated as a
^{ 1}
, a dimensionless number equal to 137.03604
is very close to that derived above for atmospheric eddy energy structure. This
fundamental constant has attracted much attention and it is felt that quantum
mechanics cannot be interpreted properly until such time as we can derive this
physical constant from a more basic theory.
(k) The
ratio of proton mass M
to electron mass m_{e} , i.e., M/m_{e}
is another fundamental dimensionless number which also awaits derivation from a
physically consistent theory.
M/m_{e} determined by observation is equal to about 2000.
In the following it is shown that ratio of energy content of large to small
eddies for specific length scale ratios is equivalent to M/m_{e}.
From
Equation 19,
The energy ratio for two successive dominant eddy growth =
(2a^{2}
x10)^{2}
Since each large eddy consists of five growth steps each for clockwise
and anticlockwise rotation,
The relative energy content of primary circulation structure inside this
large eddy
= (2a^{2}
x 10)^{2}/10 @ 1879
The cell dynamical system model concepts therefore enable physically
consistent derivation of fundamental constants
which define the basic structure of quantum systems. These two fundamental
constants could not be derived so far from a basic theory in traditional quantum
mechanics for subatomic dynamics (Omnes 1994).
3. APPLICATIONS FOR PREDICTION
(a)
In a majority of spectra, periodicities up to 4
years contribute up to 50% of total
variance (see references of Mary Selvam et. al.) and is in agreement with model
prediction (Equation 12) The model also predicts that periodicities
up to 9.5
years contribute up to 95% of total variance( Equation 13). Dominant periodicities,
such as the widely documented QBO, ENSO and decadal scale fluctuations may be
used for predictability studies.
(b)
Model predicted universal
spectrum (Equation 7) has been identified in the interannual variability of rainfall
(Mary Selvam et al. 1992; Mary Selvam 1993b; Mary Selvam et al. 1995); temperature
(Mary Selvam and Joshi 1995) and surface pressure (Mary Selvam et al. 1996) and
imply laws analogous to Kepler’s laws (Equation 2) for eddy circulation dynamics.
Universal spectrum for atmospheric interannual variability provides precise
quantification for the apparently irregular natural variability. The concept of
universal spectrum for fluctuations rules out linear secular trends in
meteorological parameters with regard to climate change. Global warming, either
natural or man  made (industrialization related) will result in enhancement of
fluctuations of all scales (Equation 1).The following studies indicate
intensification of spacetime fluctuations in atmospheric flows in recent years
(since 1970s). Intergovernmental Panel on Climate Change (IPCC) report shows
that recent increases have been found in the intensity of the winter atmospheric
circulation over the extratropical Pacific and Atlantic (Houghton et
al 1996).There have been relatively more frequent El Nino episodes since
1976/77 with only rare excursions into the other extreme (La Nina episodes) An
assessment of ENSO  scale secular variability shows that ENSO  scale variance
is relatively large in recent decades (Wang and Ropelewski 1995). Hurrel and Van
Loon (1994) have reported a delayed breakdown of the polar vortex in the
troposphere and lower stratosphere after the late 1970s coincident with the
beginning of the ozone deficit in the Antarctic spring. It is possible that
enhanced vertical mixing (Equations 6 and 7) inside the polar vortex may
contribute to the ozone loss. Regions of enhanced convective activity in the
monsoon regime are found to be associated with lower levels of atmospheric
columnar total ozone content (Hingane and Patil 1996). Incidentally, it was found
that enhancement of background noise, i.e., energy input into the eddy continuum
results in amplification of faint signals in electrical circuits (Brown 1996).
4. ACKNOWLEDGEMENTS
The authors express their gratitude to Dr. A.S.R. Murty for his keen
interest and encouragement during the course of this study
5.
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