Nonlinear Dynamics and Chaos:
Applications for Prediction of Weather and Climate
J.S.Pethkar and A.M.Selvam
Indian Institue of Tropical
Meteorology, Pune 411008, India
Proc. TROPMET 97, Bangalore,
India, 10-14 Feb. 1997.
Turbulence, namely, irregular
fluctuations in space and time characterize fluid flows in general
and atmospheric flows in particular. The irregular, i.e. nonlinear
space-time fluctuations on all scales contribute to the unpredictable
nature of both short-term weather and long-term climate. Quantification of
atmospheric flow patterns as recorded by meteorological parameters such as
temperature, wind speed, pressure, etc. will help exact prediction of weather
and climate and also provide a model for turbulent fluid flows in general.
Meteorologists have documented in detail the nonlinear variability of atmospheric
flows, in particular the interannual variability, i.e., the year-to-year
fluctuations in weather patterns. A brief summary of observational documentation
of interannual variability of atmospheric flows is given in the following.
The interannual variability of atmospheric flows is nonlinear and exhibits
fluctuations on all scales ranging up to the length of data period (time)
investigated. The broadband spectrum of atmospheric interannual variability
has embedded dominant quasiperiodicities such as the quasibiennial oscillation
(QBO ) and the ENSO (El Nino/Southern Oscillation) cycle of
3 to 7 years 1 which are identified as major contributors to local
climate variability, in particular, the monsoons which influence agriculture
dependent world economies. ENSO is an irregular (3 - 7 years), self
- sustaining cycle of alternating warm and cool water episodes in the Pacific
Ocean. Also called El Nino - La Nina, La Nina refers to the
cool part of the weather cycle while El Nino is associated with a
reversal of global climatic regimes resulting in anomalous floods and droughts
throughout the globe. It is of importance to quantify the total pattern of
fluctuations for predictability studies. Observations show that atmospheric
flows exhibit fluctuations on all scales (space-time) ranging from turbulence
(mm-sec) to planetary scale (thousand of kilometers-year). The power
spectra of temporal fluctuations are broadband and exhibit inverse
power law form 1/fB where f is frequency and
B, the exponent, is different for different scale ranges.
Inverse power-law form for power spectra implies scaling (self similarity)
for the scale range over which B is constant. Atmospheric flows
therefore exhibit multiple scaling or multifractal structure.
The fractal and multifractal nature of fluid turbulence in
general and also in atmospheric flows has been discussed in detail
by Sreenivasan2. The word fractal was first coined by Mandelbrot
3 to describe the selfsimilar fluctuations that are generic
to dynamical evolution of systems in nature. Fractals signify
non-Euclidean or fractional Euclidean geometrical structure. Traditional
statistical theory does not provide for a satisfactory description and quantification
of such nonlinear variability with multiple scaling. The apparently chaotic
nonlinear variability (intermittency) of atmospheric flows therefore exhibit
implicit order in the form of multiple scaling or multifractal structure
of temporal fluctuations implying long-range temporal correlations, i.e.
the amplitudes of long-term and short-term fluctuations are related by a
multiplication factor proportional to the scale ratio and therefore independent
of exact details of dynamical evolution of fluctuations 4-5. Recent
studies (since 1988) in all branches of science reveal that selfsimilar
multifractal spatial pattern formation by selfsimilar fluctuations
on all space-time scales is generic to dynamical systems in nature and is
identified as signature of self-organized criticality 6
. Such multifractal temporal fluctuations in atmospheric flows are
associated with selfsimilar multifractal spatial patterns for cloud
and rain areas documented and discussed in great detail by Lovejoy and his
group7-9. Standard meteorological theory cannot explain satisfactorily
the observed multifractal structure of atmospheric flows9
. Selfsimilar spatial pattern implies long-range spatial correlations.
Atmospheric flows therefore exhibit long-range spatiotemporal correlations,
namely, self-organized criticality, signifying order underlying
apparent chaos. Prediction may therefore be possible. Statistical prediction
models are based on observed correlations, which, however, change with time,
thereby introducing uncertainties in the predictions. Traditionally, prediction
of atmospheric flow patterns has been attempted using mathematical models
of turbulent fluid flows based on Newtonian continuum dynamics. Such models
are nonlinear and finite precision computer realizations give chaotic solutions
because of sensitive dependence on initial conditions, now identified as
deterministic chaos, an area of intensive research in all branches of science
since 1980 10. Sensitive dependence on initial conditions in computed
solutions implies long-range spatiotemporal correlations, namely self-organized
criticality, similar to that observed in real world dynamical systems.
Deterministic chaos in computed solutions precludes long-term prediction.
The fidelity of computed solutions is questionable in the absence of analytical
(true) solutions11. Deterministic chaos is a direct consequence
of round-off error growth in finite precision computer solutions of error
sensitive dynamical systems such as X n+1 = F(Xn
) , where Xn+1, the (n+1)th value of the variable
X at the (n+1)th instant is a function F of Xn. Mary
Selvam12 has shown that round-off error approximately doubles
on an average for each iteration in iterative computations and give
unrealistic solutions in numerical weather prediction (NWP)
and climate models which incorporate thousands of iterations in long-term
numerical integration schemes. Computed model solutions are therefore mere
mathematical artifacts of the universal process of round-off error growth
in iterative computations. Mary Selvam12 has shown that the computed
domain is the successive cumulative integration of round-off error
domains analogous to the formation of large eddy domains as envelopes
enclosing turbulent eddy fluctuation domains such as in atmospheric
flows13-16. Computed solutions, therefore qualitatively
resemble real world dynamical systems such as atmospheric flows with
manifestation of self-organized criticality. Self-organized criticality
, i.e., long-range spatiotemporal correlations, originates with the
primary perturbation domains corresponding respectively to round-off
error and dominant turbulent eddy fluctuations in model and real world
dynamical systems. Computed solutions, therefore, are not true solutions.
The vast body of literature investigating chaotic trajectories in recent
years (since 1980) document, only the round-off error structure in
finite precision computations. The physical mechanism underlying self-organized
criticality in model and real world dynamical systems is not yet identified.
A recently developed non-deterministic cell dynamical system model for atmospheric
flows13-16 predicts the observed self-organized criticality as
intrinsic to quantumlike mechanics governing flow dynamics. El Naschie (1997:
Chaos, Solitons and Fractals8(11), 1873 - 1886) has shown
mathematically fractal spacetime fluctuation characteristics for quantum
systems. The model provides for a universal quantification for self-organized
criticality by predicting the universal inverse power-law form of the statistical
normal distribution for the power spectrum of temporal fluctuations. The
model predictions are in agreement15-16 with continuous periodogram
spectral analysis of meteorological data sets. A complete review
of literature relating to studies on Nonlinear Dynamics and Chaos
and applications for prediction for weather and climate is given in Selvam
and Fadnavis 17 .
The authors are grateful to Dr.A.S.R.Murty for his keen
interest and encouragement during the course of the study.
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Oscillation. Academic Press, NY, International Geophysical Series
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14 Mary Selvam, A., Pethkar, J. S., and Kulkarni, M. K.
Signatures of a universal spectrum for atmospheric interannual variability
in rainfall time series over the Indian region. Int'l. J. Climatol
. 1992, 12, 137-152.
15 Selvam, A. M. and Joshi, R. R., Universal spectrum
for interannual variability in COADS global air and sea surface temperatures.
Int'l.J.Climatol. 1995, 15, 613 - 623.
16 Selvam, A. M., Pethkar, J. S., Kulkarni, M. K., and
Vijayakumar, R. Signatures of a universal spectrum for atmospheric interannual
variability in COADS surface pressure time series. Int'l. J. Climatol
. 1996, 16, 393-404.
17 Selvam, A. M., and Fadnavis, S. Signatures
of a universal spectrum for atmospheric interannual variability in
some disparate climatic regimes. Meteorology and Atmospheric Physics
1998, 66, 87-112.(Springer-Verlag, Austria)