Free counter and web 
stats

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

email: amselvam@eth.net

web sites: http://www.geocities.com/~amselvam

http://amselvam.tripod.com/index.html

Proceedings of the IEEE National Aerospace and Electronics Conference - NAECON 1989, Dayton, OH, May 22-26, 1989, 1145-1152.

Abstract

    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.

Introduction

    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 systems.

    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 Flows

    Self-organized criticality 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)
(1)
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.
(2)
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 et 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 in Eq.(2).
    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
(3)
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
(4)
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 given by
(5)
From Eqs.(1)-(4)
(6)
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 flows.
    The self-organized 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 equation gives
(7)
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 Number

    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
from Eq.(1)
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, dq , 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

R
Wn
dR
dq
Wn+1
q
1
2
3.254
5.239
8.425
13.546
21.780
35.019
56.305
90.530
1
1.254
1.985
3.186
5.121
8.234
13.239
21.286
34.225
55.029
1
1.254
1.985
3.186
5.121
8.234
13.239
21.286
34.225
55.029
1
0.627
0.610
0.608
0.608
0.608
0.608
0.608
0.608
0.608
1.254
1.985
3.186
5.121
8.234
13.239
21.286
34.225
55.029
88.479
1
1.627
2.237
2.845
3.453
4.061
4.669
5.277
5.885
6.493

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 Flows

    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 RoR1R2R3R4R5 of the large eddy on either side of the primary perturbation traces out the logarithmic spiral

here
where t is the golden mean  winding number.
    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 10 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 Smalley, 1988).
    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 as
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.
(8)
where e 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 Distribution

    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, 2s, 3s, etc. since W2/W1 =lnZ2/lnZ1, where Z2 =Z12 and W1 =s . The corresponding probabilities of occurrence are P, P2, P4etc. 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 1, 1.618, 1.6182, ..... 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 W1 and W2 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 Chaos

    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 (Procaccia, 1988).
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 (Cvitanovic, 1988).
    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.

Conclusion

    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 Kelso, 1988).

References

Allison, M. 1988 Nature 336, 312-313.
Bak, P., C. Tang and K. Wiesenfeld 1988 Phys. Rev. A 38(1), 364-374.
Beck, C. and G. Roepstorff 1987 Physica Scripta 25D, 173-180.
Cherinkov, A. A., R. Z. Sagadeev and G. M. Zaslavsky 1988 Physics Today November, 27-37.
Curl, R. F. and R. E. Smalley 1988 Science 242, 1017-1022.
Cvitanovic, P. 1988 Phys. Rev. Lett. 61(24), 2729-2732.
Davies, P. 1988 New Scientist 15 Oct., 50-60.
Fairbairn, W. 1986 Phys. Bull. 37, 300-302.
Feigenbaum, M. J. 1980 Los Alamos Sci. 1, 4.
Fraedrich, K. 1988 Mon. Wea. Rev. 116(5), 1001-1012.
Grebogi, C., E. Ott and J. A. Yorke 1988 Phys. Rev. A 38(7), 3688-3692.
Harrison, R. G. and D. J. Biswas 1986 Nature 321, 394-401.
Holton, J. R. 1979 An Introduction to Dynamic Meteorology. Academic Press, N.Y., 39pp.
Janssen, T. 1988 Physics Reports 168(2).
Kraichnan, R. H. 1988 J. Stat. Phys. 51(5/6) 949-964.
Krinsky V. I. 1984 Selforganization, Autowaves and Structures far from Equilibrium. Springer-Verlag, New York, 263pp.
Lau, Ka-M, and P. H. Chan 1988 J. Atmos Sci. 45(3), 506-521.
Lebowitz, J. L., E. Presutti and H. Spohn 1988 J. Stat. Phys. 51(5/6), 841-862.
Levich, E. 1987 Physics Reports 151 (3/4), 129-238.
Lovejoy, S. and D. Schertzer 1986 Bull. Amer. Meteorol. Soc. 67, 21-32.
Mary Selvam, A. 1987 Proc. NAECON '87.
Mary Selvam, A. 1988 Proc. 8th Num. Wea. Pred. Conf. Amer. Meteorol. Soc., USA.
McCauley, J. L. 1988 Phys. Scrip. T20, 56pp.
Mort La Breque 1987 Mosaic 18(2), 22-41.
Oona, Y. and S. Puri 1988 Phys. Rev. A 38(1), 434-453.
Ottino, J. M., C. W. Leong, H. Rising and P. D. Swanson 1988 Nature 333, 419-425.
Procaccia, I. 1988 Nature 333, 618-623.
Pruppachar, H. R. and J. D. Klett 1978 Microphys. Clouds and Precipitation, D. Reidel.
Schoner, G. and J. A. S. Kelso 1988 Science 239, 1513-1520.
Townsend, A. A. 1956 The Structure of Turbulent Shear Flow. Cambridge Univ. Press.
Trenberth, K. E., G. W. Branslater and P. A. Arkin 1988 Science 242 1640-1644.
Warner, J. 1970 J. Atmos. Sci. 27, 682-688.
Vastano, J. A. and H. L. Swinney 1988 Phys. Rev. Lett. 60(18), 1773-1776.