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**THE APPLICATION OF DETERMINISTIC CHAOS TO ATMOSPHERIC SCIENCES** **A. MARY SELVAM ^{*} and SUVARNA FADNAVIS^{+}**

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 systems^{1} 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 patterns^{2}. It is not clear how such long-lived organised circulations are maintained in the dissipative turbulent environment of the *ABL*^{3}. 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 physics^{4}. Mitchell Feigenbaum^{5} 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 structure^{6}. 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 system^{7}. 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 identified^{7}.

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 scales^{8}. 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

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

Figure 1. Conceptual model of large and turbulent eddies in the ABL

Townsend^{13} 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

where ** w**,

*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

(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

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

** k > 0.5** for

(3)

** k = 0.4** for

Vertical mixing due to turbulent eddy fluctuations progressively dilutes the rising large eddy and a fraction ** f **of surface air reaches the normalised height

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.

**VI. DETERMINISTIC CHAOS AND RICHARDSON NUMBER FOR ATMOSPHERIC TURBULENCE** ** **

** **The** ***Richardson number*** Ri **

*Ri = N _{B}^{2} / (wind shear) ^{2}*

** **

where ** N_{B}** is the

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

Therefore

** dR **being the incremental large eddy growth per second is equal to

Therefore

** Ri = 1/4** for scale ratio

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*) mechanism^{14} 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 *Chew* ^{24} and the *theory of implicate order* envisaged by *Bohm*^{25}. 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 (

*R : R _{t} = r : 10r : 10 ^{2}r : 10 ^{3}r : 10 ^{4}r*

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 40^{2}) and the *5* minutes oscillation of the sun's atmosphere^{28} (*20 years ~* 5 min x 40^{4}) (Eq.5). Such a process is analogous to *antistokes* laser emission triggered by a laser pump^{12}.

**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 slope^{29,30} implying the existence of self similarity in atmospheric dynamical processes^{8}. 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 *ln E* versus

The spectral slope ** S** of the scale invariant eddy energy spectrum is given as

*S = D lnE / D lnn*

*= ln (R ^{3}W ^{2} / r ^{3}w*

** **

** **

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

Therefore *2 R = w _{p} / *

from Eq.(1) = ** H**n

* 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

*DE. DT = H = constant*

** ** The above statement is analogous to *Heisenberg's uncertainty principle* for subatomic dynamics^{31}. 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

** W_{2}^{2}** =

** **

The above statistical relation may be derived from Eq.(1) in the context of the variance of the eddy parameters for two different ratios ** z_{1}** and

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

**= - 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

**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*

By analogy, the perturbation speed *w***_{*}** (motion) per second of the medium sustained by its inertia represents the mass,

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**) =

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

**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

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

The eddy energy spectral slope becomes steeper than *-2* with high resolution (fine scale) observations^{30} 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

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**,

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 *2***s** and *3***s** are respectively equal to *100 ^{2}* and

**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* model^{17} given below.

*X _{n+1} = F(X_{n}) = LX_{n}(1-X_{n})*

** **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*'s^{5} research showed that the following two universal constants ** a** and

a and d therefore denote the successive spacing ratios of** X**

The universal constants ** a** and

The physical concept of large eddy growth by the period doubling process enables to derive the universal constants ** a** and

From Eq. (1) the function ** a** may be defined as

(8)

** a** is therefore equal to

(9)

Therefore ** 2a^{2} = 3d** from Eqs.(8) & (9). The variable

The universal *Feigenbaum*'s constants ** a** and

**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 predicted^{9} 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 vortices^{34}. 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 radius* R *so that

* AB *will 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 angle* a *of the spiral.

Substituting *b = tan *** a **; and integrating for eddy growth from

*R = re ^{b}^{q}*

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

*dR = r and R =* *S r*** ** Cloud bandwidth =

The dominant turbulent eddy radius determines the angular turning *d***q** 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 radius

*Dvorak*^{35} 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 *T _{1}(a)* to

Figure 5 Deterministic chaos model prediction of the hurricane spiral cloud bands (second row) and Dvorak cloud diagrams (first row) for storm intensities T_{1} (a) to T_{4} (a)

Growth time of the eddy system

The eddy growth time ** T** for an eddy radius

*T = dR /W*

** **

where ** li** is the logarithm integral or the

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

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 *T _{1}(a)* to

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 Holland^{36}.

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 *T _{1}(a)* to

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 workers^{18,36,37}.

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 workers^{38}.

**XII. DETERMINISTIC CHAOS AND CLOUD PHYISCAL 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 shown^{39} (i) the vertical profile of the ratio of the actual cloud liquid water content (** q**) to the adiabatic liquid water content (

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 (*q*_{a}) 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

**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 distribution^{41}. The above theory also helps to explain the observed^{42} 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

**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

**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 Chakravorthy^{44} 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 convection^{45}. 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 areas^{8} and thus may account for the radiation hot spot fall out pattern reported following the *Chernobyl* nuclear reactor accident^{46}. 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 systems^{47}. 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

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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**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 |

**LEGEND**

Figure 1. Conceptual model of large and turbulent eddies in the ABL

Figure 2. Growth of large eddy from the turbulent eddies originating at the planetary surface.

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.

Figure 4. The spiral air flow track in hurricanes.

Figure 5. Deterministic chaos model prediction of the hurricane spiral cloud bands (second row) and *Dvorak* cloud diagrams (first row) for storm intensities *T _{1}(a)* to

Figure 6. Deterministic chaos model prediction of horizontal surface pressure field pattern for hurricanes.

Figure 7. Deterministic chaos model prediction of the horizontal wind field pattern for hurricanes.

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 (*q*_{a}) and comparison with observations (J. Warner*, J. Atmos. Sci.*, **27**, 682-688, 1970)

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