Nonlinear Dynamics and Chaos: Applications for Prediction of Weather and Climate

  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/f^B where f is frequency and B, the exponent, is different  for different scale ranges.  Inverse power-law form for power spectra  implies scaling (selfsimilarity) 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 group^7-9. 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(X_n ) , where X_n+1, the (n+1)^th value of the variable  X at the (n+1)^th instant is a function F of X_n. 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 flows^13-16. 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 flows^13-16 predicts the observed self-organized criticality as intrinsic to quantumlike mechanics governing flow dynamics. El Naschie (1997: Chaos, Solitons and Fractals  8(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 agreement^15-16 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 .